Quantum Magnetism, Spin Waves, and Light
Silvia Viola Kusminskiy

 


Quantum Magnetism,





 





Spin Waves, 





 





and Optical Cavities





 






Silvia Viola Kusminskiy






Max Planck Institute for the Science of Light





 





Staudtstrae 2, 91058 Erlangen, Germany





 and






Friedrich-Alexander University Erlangen-Nuremberg






Staudtstrae 7, 91058 Erlangen, Germany








CHAPTER: ABSTRACT


Both magnetic materials and light have always played a predominant
role in information technologies, and continue to do so as we move
into the realm of quantum technologies. In this course we review the
basics of magnetism and quantum mechanics, before going into more
advanced subjects. Magnetism is intrinsically quantum mechanical in
nature, and magnetic ordering can only be explained by use of quantum
theory. We will go over the interactions and the resulting Hamiltonian
that governs magnetic phenomena, and discuss its elementary excitations,
denominated magnons. After that we will study magneto-optical effects
and derive the classical Faraday effect. We will then move on to the
quantization of the electric field and the basics of optical cavities.
This will allow us to understand a topic of current research denominated
Cavity Optomagnonics.

These notes were written as the accompanying material to the course
I taught in the Summer Semesters of 2018 and 2019 at the Friedrich-Alexander Universitt
in Erlangen. The course is intended for Master or advanced Bachelor
students. Basic knowledge of quantum mechanics, electromagnetism,
and solid state at the bachelor level is assumed. Each section is
followed by a couple of simple exercises which should serve as to
" fill in the blanks" of what has been
derived and a couple of check-points for the main concepts developed.





CHAPTER: ELECTROMAGNETISM


The history of magnetism is ancient: just to give an example, the
magnetic compass was invented in China more that 2000 years ago. The
fact that magnetism is intrinsically connected to moving electric
charges (and not to " magnetic charges"
) however was not discovered until much later. In the year 1820, Oersted
experimentally demonstrated that a current-carrying wire had an effect
on the orientation of a magnetic compass needle placed in its proximity.
In the following few years, Ampere realized that a small current loop
generates a magnetic field which is equivalent to that of a small
magnet, and speculated that all magnetic fields are caused by charges
in motion. In the next few sections we will review these concepts
and the basics of magnetostatics.



§ BASIC MAGNETOSTATICS


As the name indicates, magnetostatics deals with magnetic fields that
are constant in time. The condition for that is a steady-state current,
in which both the charge density ρ and the current density j=I/A_ s
(A_ s cross-sectional area) are independent of time 

    ∂ρ/∂ t    =0
    ∂𝐣/∂ t    =0 .

From the continuity equation 

    ∇·𝐣+∂ρ/∂ t=0

we moreover obtain 

    ∇·𝐣=0 .


In these notes we will call magnetic induction to 𝐁
and magnetic field to 𝐇.[Some authors call instead 𝐁 the magnetic field and 𝐇
the auxiliary field. ] In free space, these two fields are related by 

    𝐁=μ_0𝐇

being μ_0=4π×10^-7 NA^-2 the permeability of
free space. We will use the SI units system throughout these notes,
therefore 𝐁 is measured in Teslas (T= V. s. m^-2)
and 𝐇 in Amperes per meter (A. m^-1).

The magnetic induction at point 𝐫 due to a current loop
can be calculated using the Biot-Savart law 

    d𝐁=μ_0I/4π r^2 dℓ×𝐫/r

where dℓ points in the direction of the current
I, see Fig. <ref>. Equivalent to the Biot-Savart
law is Ampere's law, which reads 

    ∮_𝒞𝐁· d𝐬=μ_0I

where I is the current enclosed by the closed loop 𝒞,
see Fig. <ref>. Ampere's law is general, but
it is useful to calculate magnetic fields only in cases of
high symmetry, for example the magnetic field generated by an infinite
straight wire. Using Stoke's theorem, we can put Ampere's law in differential
form 

    ∇×𝐁=μ_0𝐣 .

Ampere's law together with the absence of magnetic monopoles condition

    ∇·𝐁=0

constitute the Maxwell equations for magnetostatics. 


These equations give us indeed time independent magnetic fields, and
if we compare the magnetostatic equations with the full microscopic
Maxwell equations 

    ∇·𝐁    =     0
    ∇·𝐄    =    ρ/ε_0
    ∇×𝐄    =     -∂𝐁/∂ t
    ∇×𝐁    =    μ_0(𝐣+ε_0∂𝐄/∂ t)

(with ε_0=8.85×10^-12 Fm^-1
the vacuum permittivity) we see that we have moreover decoupled the
magnetic and electric fields.



 §.§ Check points



  * What is the magnetostatic condition? 

  * Write the magnetostatic Maxwell equations 




§ MAGNETIC MOMENT


The magnetic moment of a current loop is defined as 

    𝐦=IA𝐧̂ ,

where A is the area enclosed by the loop and 𝐧̂
is the normal to the surface, with its direction defined from the
circulating current by the right-hand rule, see Fig. <ref>.
𝐦 defines a magnetic dipole in the limit of A→0
but finite moment.



Using Eq. <ref> we can calculate the magnetic induction
generated by a small current loop of radius R 

    𝐁(𝐫)     =μ_0I/4π∫ dℓ'×Δ𝐫/Δ r^3
        =-μ_0I/4π∫ dℓ'×∇(1/Δ𝐫)

with Δ𝐫=𝐫-𝐫' (see Fig. <ref>).
From Eq. <ref> we know we can define a vector potential
𝐀(𝐫) such that 𝐁(𝐫)=∇×𝐀(𝐫).
By a simple manipulation of Eq. <ref>, one can
show that in the far field limit (Δ𝐫≫ R ), 

    𝐀(𝐫)     =μ_0/4π𝐦×𝐫̂/r^2
    𝐁(𝐫)     =μ_0/4π3(𝐦·𝐫)𝐫-r^2𝐦/r^5

which is the magnetic induction generated by a magnetic dipole. 


More generally, for an arbitrary current density distribution 𝐣(𝐫')
one can define <cit.>

    𝐦=1/2∫ d^3𝐫'[𝐫'×𝐣(𝐫')]

and Eq. <ref> is the lowest non-vanishing term
in a multipole expansion of the vector potential (in the Coulomb gauge,
∇·𝐀=0) 

    𝐀(𝐫)=μ_0/4π∫ d^3𝐫'𝐣(𝐫')/|𝐫-𝐫'| .


The energy of a magnetic dipole in a magnetic field is given by 

    E_ Z=-𝐦·𝐁

and therefore is minimized for 𝐦∥𝐁. This
is called the Zeeman Energy. 


  * Exercise: derive Eqs.<ref> and
<ref> (tip: use the " chain rule"
and a multipole expansion). 

  * Exercise: show that Eq. <ref> follows
from <ref> (tip: 1-D Delta-function distributions have units
of 1/length). 




 §.§ Check points



  * How do you show the equivalence between the magnetic field of a small
current loop and that of a small magnet? A conceptual explanation
suffices. 




§ ORBITAL ANGULAR MOMENTUM


The magnetic moment 𝐦 can be related to angular momentum.
In order to do this, we consider the limit of one electron e (with
negative charge -e) orbiting around a fixed nucleus, see Fig. <ref>.
Note that here we get the first indication that magnetism is a purely
quantum effect: stable orbits like that are not allowed classically,
and we need quantum mechanics to justify the stability of atoms. Our
argument here is therefore a semiclassical one. 


The average current due to this single electron is 

    I=-e/T=-eω/2π

where T is one period of revolution. The electron also possesses
orbital angular momentum 𝐋=m_e𝐫×𝐯.
Measured from the center of the orbit, 

    𝐋=m_eR^2ω⃗

and using Eq. <ref> we obtain 

    𝐦=-e/2m_e𝐋 .

Therefore we have linked the magnetic moment of a moving charge to
its orbital angular momentum. The coefficient of proportionality is
called the gyromagnetic ratio 

    γ_ L=-e/2m_e ,

which is negative due to the negative charge of the electron. Hence
in this case the magnetic moment and angular momentum are antiparallel.
In solids, electrons are the primary source of magnetism due to their
small mass compared to that of the nucleus. Since m_p≈10^3m_e,
the gyromagnetic ratio for the nucleus is strongly suppressed with
respect to the electronic one.



 §.§ Check points



  * What is the gyromagnetic ratio? 

  * Why is the gyromagnetic ratio of the nucleus suppressed with respect
to the electronic one? 




§ SPIN ANGULAR MOMENTUM


Although we performed a classical calculation, the result obtained
for the gyromagnetic ratio in Eq. <ref> is consistent
with the quantum mechanical result. We know however that the electron
posses an intrinsic angular momentum, that is, the spin 𝐒.
The total angular momentum of the electron is therefore given
by 

    𝐉=𝐋+𝐒 .

The spin has no classical analog and the coefficient of proportionality
γ_ S between magnetic moment and spin 

    𝐦_ S=γ_ S𝐒

needs to be calculated quantum mechanically via the Dirac equation
(see e.g. Chapter 2 of Ref. <cit.>).
The results is 

    γ_ S≈-e/m=2γ_ L ,

where the approximate symbol indicates that there are relativistic
corrections (also contained in the Dirac equation!) to this expression.
The γ_ S value agrees with experimental observations.

The total magnetic moment of the electron is therefore given by 

    𝐦_ TOT≈γ_ L(𝐋+2𝐒)

and hence is not simply proportional to the total angular momentum!
To understand the relation between 𝐦_ TOT and 𝐉,
given by the Land factor, we need to resort to quantum mechanics
and the operator representation of angular momentum. We will do that
in the next chapter, where we discuss the atomic origins of magnetism.



 §.§ Check points



  * What is the relation between the magnetic moment of an electron and
the angular momentum operators? 




§ MAGNETIC MOMENT IN A MAGNETIC FIELD


A magnetic moment in a magnetic field experiences a torque 

    𝒯=𝐦×𝐁 .

Therefore, the classical equation of motion for the magnetic dipole
(considering for the moment only the orbital angular momentum) is

    d𝐋/ dt=𝐦×𝐁=γ_ L𝐋×𝐁 .



Using the geometry depicted in Fig. <ref> we obtain 

    𝐋̇    =Lsinθφ̇𝐞_ϕ
    γ_ L𝐋×𝐁    =|γ_ L|LBsinθ𝐞_ϕ .

Hence the magnetic moment precesses around 𝐁 at a frequency

    ω_ L=φ̇=|γ_ L𝐁|

which is denominated the Larmor frequency. Therefore the angular
momentum will precess around the 𝐁 field at a fixed angle
θ and with constant angular frequency ω_ L.



This is consistent with the energy expression defined in Eq. <ref>.
The work per unit time performed by the torque is given by the usual
expression for the power 

    dW/ dt=𝒯·ω⃗

where the angular velocity vector is perpendicular to the plane of
rotation and its direction is given by the right hand rule. We see
therefore that there is no power transfer in the Larmor precession,
since ω_ L𝐞̂_z·𝒯=0. There
is however an energy cost if we want to change the angle θ
of precession, since the resultant angular velocity θ̇𝐞_φ
is collinear with the torque. Using (for simplicity we defined now
θ as the angle between 𝐦 and 𝐁, see
Fig. <ref>). 

    𝒯    =-mBsinθ𝐞_φ
    ω⃗_θ    =θ̇𝐞_φ

we find 

    𝒯θ̇=-mBsinθ dθ/ dt

and therefore the work exerted to rotate 𝐦 up to an angle
θ is (up to a constant) 

    W=-∫ mBsinθ dθ=𝐦·𝐁=-E_ Z .

We can therefore take the Zeeman energy E_ Z as the potential
energy associated with the necessary work required to rotate the dipole
𝐦 with respect to an external 𝐁 field. 


We will see equations of motion in the form of Eq. <ref>
reappearing throughout this course, even as we treat the angular momenta
as quantum operators. The reason is that, even though the total magnetic
moment 𝐦_ TOT is not proportional to the total
angular momentum 𝐉 (see Eq. <ref>), their quantum
mechanical expectation values are proportional to each other through
the Land factor g. We will see this more formally when we start
dealing with the quantum mechanical representation of the angular
momenta. For now, we assume that 𝐦_ TOT and 𝐉
are related by 

    𝐦_ TOT·𝐉=gγ_ L𝐉·𝐉

from which we can obtain a classical expression for g, by replacing
Eqs.<ref> and <ref> into <ref>
and noting that 

    𝐋·𝐒=1/2(J^2-L^2-S^2)

from J^2=(𝐋+𝐒)^2. One obtains

    g_ cl=3/2+S^2-L^2/2J^2 ,

where the superscript indicates this is a classical approximation
for g, which coincides with the quantum mechanical result in the
limit J^2, S^2, and L^2 large <cit.>.



 §.§ Check points



  * Write the equation of motion for an angular momentum in the presence
of a magnetic field 

  * What is the dynamics of an angular momentum in the presence of a magnetic
field? 

  * What is the Larmor frequency? 




§ MAGNETIZATION


Inside a material, the magnetic induction 𝐁 indicates
the response of the material to the applied magnetic field 𝐇.
Both vector fields are related through the magnetization in
the sample 

    𝐁=μ_0(𝐇+𝐌) ,

where the magnetization is defined as the average magnetic moment
per unit volume, 

    𝐌(𝐫)=N⟨𝐦⟩_V/V ,

and where the average indicates that we average over all atomic magnetic
moments in a small volume V around position 𝐫 [We average over a " microscopically large but macroscopically
small" volume V.] containing N magnetic ions. In this way, a smooth vectorial function
of position is obtained. From Eq. <ref>, we see that
the magnetization has the same units as the magnetic field 𝐇
(A. m^-1). In Eq. <ref>, both 𝐁
and 𝐇 indicate the fields inside the material,
and hence 𝐇 contains also the demagnetizing fields
(that is, it is not just the external applied field). We will see
more on demagnetization fields in the next section.

The response to the magnetic field of the magnetization and field
induction are characterized by the magnetic susceptibility
χ and the permeability μ, respectively 

    𝐌    =χ𝐇
    𝐁    =μ𝐇 ,

where we have written the simplest expressions for the case in which
all fields are collinear, static (that is, independent of time), and
homogeneous in space (𝐪=ω=0). In general however
the response functions are tensorial quantities, e.g. M_i=∑_jχ_ijH_j,
and depend on frequency ω and momentum 𝐪. Note
that from Eq. <ref> we obtain 

    μ_ r=μ/μ_0=1+χ

again in the simple collinear case. μ_ r is the relative
permeability, is dimensionless and equals to unity in free space.

The quantities defined in Eqs. <ref> and <ref> are
still allowed to depend on temperature T and magnetic field 𝐇.
We will now consider qualitatively the dependence on 𝐇.
For linear materials, χ and μ are independent of 𝐇.
A linear material with negative constant susceptibility is diamagnetic,
whereas a positive susceptibility indicates either paramagnetism
(no magnetic order) or antiferromagnetism (magnetic order with
magnetic moments anti-aligned and zero total magnetization). In these
cases, the magnetization is finite only in the presence of a magnetic
field. On the other hand, if χ and μ depend on 𝐇,
the relations Eqs.<ref> and <ref> are non linear.
This is the case for magnetically ordered states with net magnetization,
namely ferromagnets (magnetic moments aligned and pointing
in the same direction) and ferrimagnets (magnetic moments anti-aligned
but of different magnitude, so that there is a net magnetization).
In these materials, the magnetization increases non-linearly with
the applied field and saturates when all the magnetic moments are
aligned. When decreasing the magnetic field, there is a remanent,
finite magnetization at zero field. This process is called hysteresis
and it is used to magnetize materials. As we learned in the previous
section, the magnetic moment, and hence the magnetic characteristics
of a material, are related to the total angular momentum of the electrons,
and therefore on the atomic structure. We will learn more about this
in the next chapter.



 §.§ Check points



  * What is the relation between magnetic moment and magnetization? 




§ MAGNETOSTATIC MAXWELL
EQUATIONS IN MATTER


To calculate the magnetic dipole moment 𝐦 from Eq. <ref>
we have to know the microscopic current density. In general however
we are not interested in microscopic, fast fluctuations. We already
saw an example in which we considered the average current I generated
by one orbiting electron, to obtain semiclassically the gyromagnetic
ratio γ_ L in Sec.<ref>.
We have also defined the magnetization 𝐌 as a macroscopic
quantity which entails the average density of the microscopic 𝐦.
In a material, in general we have access to the magnetization, which
is due to bound microscopic currents, and to the macroscopic current
density due to free charges, which we will denominate 𝐣_ F.
This motivates defining a macroscopic vector potential 𝐀
in terms of these two macroscopic quantities, and not the microscopic
currents as in Eq. <ref> 

    𝐀(𝐫)=μ_0/4π∫ d^3𝐫'[𝐣_ F(𝐫')/|𝐫-𝐫'|+𝐌(𝐫')×(𝐫-𝐫')/|𝐫-𝐫'|^3] .

Note that this is simply re-writing Eq. <ref>, separating
the bound- and free- currents contributions. The bound-current contribution,
the second term in Eq. <ref>, is written in terms
of the magnetization and is equivalent to an averaged Eq. <ref>.

Eq. <ref> allows us to define an effective current
density associated with the magnetization, by noting that <cit.>

    ∫_𝒱 d^3𝐫'𝐌(𝐫')×(𝐫-𝐫')/|𝐫-𝐫'|^3    =∫_𝒱 d^3𝐫'𝐌(𝐫')×∇'(1/|𝐫-𝐫'|)
        =∫_𝒱 d^3𝐫'∇'×𝐌(𝐫')(1/|𝐫-𝐫'|)+∮_𝒮𝐌(𝐫')× d𝐚'/|𝐫-𝐫'| .

We can therefore define an effective bound volume current density

    𝐣_ B=∇×𝐌

and an effective bound surface current density 

    𝐊_ B=𝐌×𝐧̂

where the surface element is defined as d𝐚= da𝐧̂.
In the bulk, for a well behaved magnetization function, the surface
integral vanishes and we obtain 

    𝐀(𝐫)=μ_0/4π∫ d^3𝐫'[𝐣_ F(𝐫')/|𝐫-𝐫'|+𝐣_ B(𝐫')/|𝐫-𝐫'|] .

The surface current 𝐊_ B enters usually through
boundary conditions at interfaces.

If we now go back to Ampere's Eq. <ref> and separate
the total current density into free and bound contributions 𝐣=𝐣_ F+𝐣_ B,
we obtain 

    ∇×𝐁=μ_0(𝐣_ F+∇×𝐌)

which defines the magnetic field 𝐇 

    𝐇=1/μ_0𝐁-𝐌

such that 

    ∇×𝐇=𝐣_ F .

Therefore the magnetic field 𝐇 takes into account in an
average way the bound currents, and has as its only source the free
currents. Eq. <ref> is equivalent to Eq. <ref>,
just re-written in a more convenient form for macroscopic magnetostatics
in matter. Note that 𝐇 is, on the contrary to 𝐁,
not divergence-free: 

    ∇·𝐇=-∇·𝐌 .


The magnetostatic Maxwell equations in matter (also known as " macroscopic" )
therefore read 

    ∇·𝐁    =0
    ∇×𝐇    =𝐣_ F

and have to be complemented by the constitutive equation 𝐁=μ𝐇
in linear media (taking μ as a constant, independent of 𝐇)
or 𝐁=F(𝐇) in non-linear (e.g., ferromagnetic)
media, where F is a characteristic function of the material.

We finish this section by stating the magnetostatic boundary conditions
at an interface between two different media 1 and 2 

    (𝐁_2-𝐁_1)·𝐧̂    =0
    𝐧̂×(𝐇_2-𝐇_1)     =𝐊_ F

where 𝐊_ F is a free surface current density (usually
0). 


  * Exercise: Prove Eqs <ref>. 




 §.§ Check points



  * What is the meaning of the magnetic field 𝐇? 

  * What are the magnetostatic Maxwell equations in matter? 




§ DEMAGNETIZING FIELDS


A crucial difference between magnetic and electric fields is the lack
of free magnetic charges, or monopoles.[Magnetic monopoles, if they exist, have evaded experimental detection
so far. They can however emerge as effective quasiparticles in condensed
matter systems, and have been detected in materials which behave magnetically
as a " spin ice" <cit.>.] They are however a useful mathematical construction in some cases,
for example, to calculate the so called demagnetization fields. In
finite systems, we can consider the magnetization as dropping to zero
abruptly at the boundary of the material, giving rise to an accumulated
" magnetic charge density" at the surface
which acts as an extra source of magnetic fields inside of the material.
These fields in general oppose to an externally applied magnetic field
and are therefore dubbed demagnetizing fields. A surface magnetic
charge density is energetically costly, and for finite magnetic systems
at the microscale it can determine the spatial dependence of the magnetically
ordered ground state, giving rise to magnetic textures.

If we consider the special case of no free currents, 𝐣_ F=0,
Eqs. <ref> imply that we can define a magnetic scalar
potential ϕ_ M such that 

    𝐇=-∇ϕ_ M

and using Eq. <ref> we obtain a Poisson equation 

    ∇^2ϕ_ M=-∇·𝐌

with solution 

    ϕ_ M(𝐫)=-1/4π∫_𝒱 d^3𝐫'∇·𝐌(𝐫')/|𝐫-𝐫'|+1/4π∮_𝒮 da'𝐧̂'̂·𝐌(𝐫')/|𝐫-𝐫'| .

Analogous to the case of the vector potential in Eq. <ref>,
this allows us to define an effective magnetic-charge density

    ρ_ M=-∇·𝐌

and an effective magnetic surface-charge density 

    σ_ M=𝐌·𝐧̂ .

We see that ρ_ M can only be finite for a non-homogeneous
magnetization 𝐌(𝐫), whereas a finite σ_ M
indicates a discontinuity of 𝐌 at the chosen surface 𝒮. 


  * Exercise: Uniformly magnetized sphere 

For a ferromagnet at saturation, the magnetization can be considered
as given, so we can in principle calculate the resulting magnetic
field for a given geometry using Eq. <ref>. We consider
here as an example the case of a uniformly magnetized sphere as depicted
in Fig. <ref>. 


a) Choosing 𝐞̂_z in the direction of 𝐌
we can write 𝐌=M_0𝐞̂_z. Calculate ρ_ M
and σ_ M and write the Poisson equation for ϕ_ M
.

b) Show that the scalar potential inside of the sphere is 

    ϕ_ M^ in=1/3M_0z

and find the magnetic field 𝐇^ in and magnetic
induction 𝐁^ in inside of the sphere.

c) The magnetic field 𝐇^ in inside of the sphere
opposes the magnetization and it is therefore called a demagnetizing
field. The proportionality coefficient between 𝐇^ in
and 𝐌 is called the demagnetizing factor N.
What is the value of N in this case? Demagnetization factors are
geometry dependent, and can moreover be defined only in very special
cases with simple geometries [The demagnetizing fields are always present, but it is only in very
simple geometries that one can describe them with simple numerical
factors.]. Besides the sphere, one can define demagnetization factors for an
infinite plane, an infinite cylinder, and a spheroid.

d) Let's assume that now the sphere is placed in an external magnetic
field 𝐇_0. Using linearity, write the solution for 𝐇^ in
and 𝐁^ in in this case.

e) Let's now consider the case that the sphere is not permanently
magnetized, but we now the material has a permeability μ. From
the constitutive equation 

    𝐁^ in=μ𝐇^ in ,

obtain the magnetization as a function of the external magnetic field
𝐌_μ(𝐇_0), where the notation 𝐌_μ
implies that in this case we consider the magnetization not as given,
but it depends on the permeability of the material. Show that 𝐌_μ(0)=0,
and therefore the obtained expression is not valid for materials with
permanent magnetization.



 §.§ Check points



  * Which is the origin of the demagnetization factors? 




CHAPTER: ATOMIC ORIGINS OF MAGNETISM


In the previous chapter we reviewed the basic concepts of magnetism
and magnetostatics using some semiclassical considerations. In particular,
we attributed the magnetic moment of atoms to " small
current loops" and to the angular momentum of electrons.
In this chapter, we will put these concepts into more solid footing
with the help of quantum mechanics.



§ BASICS OF QUANTUM MECHANICS


We first review some basic concepts of quantum mechanics. In quantum
mechanics, we describe a particle of mass m in a potential V
by a wavefunction ψ(𝐫,t) which satisfies the Schrdinger
equation 

    iħ∂ψ(𝐫,t)/∂ t=-ħ^2/2m∇^2ψ(𝐫,t)+V ψ(𝐫,t) .

The probability of finding the particle at a time t in a volume
element d^3r around position 𝐫 is given by
|ψ(𝐫,t)|^2 d^3r . If the potential V is
independent of time, ψ(𝐫,t)=ψ(𝐫)f(t) and
ψ(𝐫) is an eigenfunction of the time-independent
Schrdinger equation 

    -ħ^2/2m∇^2ψ(𝐫)+V(𝐫) ψ(𝐫)=Eψ(𝐫)

with energy E. Equivalently, we can write the eigenvalue equation
for the Hamiltonian in Dirac notation 

    Ĥψ=Eψ

where the quantum state of the particle is represented by the ket
ψ, the respective wavefunction is ψ(𝐫)=⟨𝐫|ψ⟩,
and the Hamiltonian operator is 

    Ĥ=p̂^2/2m+V(𝐫̂).

In the position representation, 𝐩̂→-iħ∇
so that, for example, ⟨𝐫|𝐩̂|ψ⟩=-iħ∇ψ(𝐫)
.

In general, for any operator Â we can write the eigenvalue
equation Âψ_α=αψ_α where
ψ_α is an eigenstate with eigenvalue α.
For a hermitian operator, Â^†=Â, α
is real and the eigenstates form a basis of the Hilbert space where
the operator acts. This is called an observable. The expectation
value for Â if the system is in the eigenstate ψ_α
then is simply ψ_αÂψ_α=α.
If we consider a second operator B̂ acting on the same Hilbert
space, it is only possible to find a common basis of eigenstates of
Â and B̂ if and only if the two operators commute:
[Â,B̂]=ÂB̂-B̂Â=0. In this case,
the two operators can be measured simultaneously to (in principle)
arbitrary precision. If the operators do not commute, then we run
into the Heisenberg uncertainty principle. The most well known
example is that of the momentum and position operators, which satisfy
[x̂,p̂]=iħ. How precise we measure one of the operators
will determine the precision up to which we can know the value of
the other: Δ xΔ p≥ħ/2. In general 

    Δ AΔ B≥1/2|⟨[Â,B̂]⟩|

where Δ A=√(⟨Â^2⟩-⟨Â⟩^2)
corresponds to the standard variation of Â and analogously
for operator B̂.



§ ORBITAL ANGULAR MOMENTUM IN
QUANTUM MECHANICS


The orbital angular momentum operator expression in quantum mechanics
is inherited from its classical expression, 𝐋̂=𝐫̂×𝐩̂.
In the position representation it is given by 

    𝐋̂=-iħ𝐫×∇ .

From this expression it is easy to verify that the different components
of 𝐋̂ do not commute with each other. Instead, one
obtains 

    [L̂_i,L̂_̂ĵ]=iħϵ_ijkL̂_k ,

where ϵ_ijk is the Levi-Civita tensor and the Einstein
convention for the implicit sum or repeated indices has been used.
Therefore, it is not possible to measure simultaneously with arbitrary
precision all components of the angular momentum. Let's assume we
choose to measure L̂_z . In this case, the Heisenberg uncertainity
principle reads 

    Δ L_xΔ L_y≥ħ/2|⟨ L_z⟩| .

It is however possible to find a common basis for L̂^2 and
one of the angular momentum components, since [L̂^2,L̂_i]=0.
Typically, L̂_z is taken and the respective eigenvalues
are labelled by l and m. These are called the quantum
numbers. The eigenstates satisfy 

    L̂^2ψ_lm_l    =ħ^2l(l+1)ψ_lm_l
    L̂_zψ_lm_l    =ħ mψ_lml .

For the orbital angular momentum, l is an integer and -l≤ m_l≤ l.
These conditions can be depicted pictorically as in Fig. <ref>.


The angular momentum vector has a magnitude ħ√(l(l+1))
and its projection on the z-axis is quantized and takes one
of the possible values ħ m_l. The maximum value of L_z
is ħ l, instead of ħ√(l(l+1)) as one would expect
classically. We recover the classical expectation in the limit l≫1
. The L_x and L_y components do not have a definite value
and are represented as a precession of 𝐋 around the z-axis.



 §.§ Check points



  * Explain graphically the properties of an angular momentum operator
and how it differs from a classical angular momentum. 




§ HYDROGEN ATOM


For a problem with rotational symmetry the angular momentum is conserved:
[Ĥ,𝐋̂]=0. Hence we can find a basis of eigenstates
that are also energy eigenstates, such that 

    Ĥψ_nlm=E_nψ_nlm .

The quantum numbers are denominated as principal, azimuthal,
and magnetic respectively for n, l, and m. If we ignore
spin, we have all the tools to solve the energy levels and orbitals
for the hydrogen atom, in which the electron is subject to the Coulomb
potential 

    V_C(r)=-e^2/4πε_0r

due to the nucleus. Due to the spherical symmetry of the problem,
is is convenient to write the wavefunction in spherical coordinates.
From Eq. <ref> it can be shown that 

    ψ_nlm(r,θ,ϕ)=R_nl(r)Y_lm(θ,ϕ)

The principal number n=1,2,3... gives the quantization of energy
E_n∝-1/n^2. The R_nl(r) are associated Laguerre
functions and determine the radial profile of the probability distribution
for the electron. Y_lm(θ,ϕ) are spherical harmonics,
which can also be written in terms of the associated Legendre
functions, Y_lm(θ,ϕ)=P_l^m(cosθ)e^imϕ.
For m=0, Y_l0(θ)=P_l(cosθ) are simply the Legendre
polynomials. The azimuthal number l=0,1,...,n-1 labels the usual
s, p, d, ... orbitals. The s orbitals are spherically symmetric, since
Y_00(θ,ϕ)=1/√(4π). The higher the azimuthal number,
the higher the probability to find the electron further away from
the nucleus, whereas n gives the number of nodes of the wavefunction
in the radial direction.

That the azimuthal quantum number l is quantized was demonstrated
experimentally in what we now know as the Zeeman effect. In
order to see why, we come back to the relation between the orbital
angular momentum and the magnetic moment. We can write 

    m_ L    =μ_ B√(l(l+1))
    𝐦_ L·𝐳    =-μ_ Bm_l

where μ_ B=ħγ_ L is the Bohr magneton
and the expressions correspond to expectation values. If the
atom is placed in an external magnetic field, there will be an extra
contribution to the energy due to the Zeeman term, see Eq. <ref>.
We can take the z-axis to coincide with the magnetic field,
hence 

    E_ Z=μ_ Bm_lB

and we say that the degeneracy of the l level, originally 2l+1,
is split for all l>0. This splitting can be measured in the absorption
spectrum of atoms with total spin angular momentum equal to zero.
To add the effect of the spin degree of freedom, we have however first
to understand how to combine angular momentum operators in quantum
mechanics.



 §.§ Check points



  * What are the quantum numbers for the Hydrogen atom and what do they
tell us? 




§ ADDITION OF ANGULAR MOMENTUM AND
MAGNETIC MOMENT


The orbital angular momentum is the generator of rotations in position
space. In general however, we can define an angular momentum simply
by its algebra, determined by the commutation relation <ref>.
The spin angular momentum generates rotations in spin space and satisfies

    [Ŝ_i,Ŝ_̂ĵ]     =iħϵ_ijkŜ_k
    Ŝ^2s    =ħ^2s(s+1)s
    Ŝ_zs    =ħ m_ss .

In contrast to the orbital angular momentum, the spin quantum number
s is not constrained to be an integer, and can take also half-integer
values. Fermions (e.g., the electron) have half-integer values
of spin and bosons integer values. For electrons, s=1/2 and m_s=±1/2
.

The spin operator commutes with the orbital angular momentum 

    [𝐒̂,𝐋̂]=0

since they act on different state spaces. One would be therefore tempted
to choose {𝐒̂^2,𝐋̂^2,L̂_z,Ŝ_z}
as a set of commuting observables. Spin and orbital angular momentum
however interact via the spin-orbit interaction, a relativistic
correction to the Hamiltonian <ref> which
reads 

    Ĥ_ SO=ħ^2/2m_e^2c^21/r∂ V/∂ r𝐒̂·𝐋̂ .

For an atomic system, V is the Coulomb potential. This correction
is usually small, but it increases with the atomic number. Since 𝐒̂
and 𝐋̂ are coupled by Eq. <ref>, they
are not conserved and they do not commute separately with the Hamiltonian.
The total angular momentum 𝐉̂=𝐒̂+𝐋̂
however is conserved. We choose therefore {𝐒̂^2,𝐋̂^2,𝐉̂^2,Ĵ_z}
as a set of commuting observables, and s, l , j and m_j
are our quantum numbers. The change of basis is achieved via the Clebsch-Gordan
coefficients 

    sljm_j=∑_m_sm_lsm_slm_l⟨ sm_slm_l|sljm_j⟩ .

We now have the task of relating the magnetic dipolar moment 𝐦_ TOT
to the total angular momentum operator Ĵ. From Eq. <ref>
we know that 𝐦_ TOT is not collinear with Ĵ,
but it is proportional to 𝐋̂+2𝐒̂. 


We note however that the projection 𝐦̂_ TOT·𝐉̂
is well defined, since 

    [𝐋̂·𝐉̂,𝐉̂^2]     =[𝐋̂·𝐉̂,𝐋̂^2]=[𝐋̂·𝐉̂,𝐒̂^2]=[𝐋̂·𝐉̂,𝐉̂_̂ẑ]=0
    
    [𝐒̂·𝐉̂,𝐉̂^2]     =[𝐒̂·𝐉̂,𝐋̂^2]=[𝐒̂·𝐉̂,𝐒̂^2]=[𝐒̂·𝐉̂,𝐉̂_̂ẑ]=0 .

The magnetic moment therefore precesses around Ĵ,
as depicted in Fig. <ref>.

The Wigner-Eckart theorem allows us to relate 𝐦̂_ TOT
with Ĵ in terms of expectation values by noting
that <cit.> 

    γ_ L⟨ sljm_j|𝐋̂+2𝐒̂|sljm_j'⟩=γ_ Lg(slj)⟨ sljm_j|𝐉̂|sljm_j'⟩ .

Therefore in the (2j+1) degenerate subspace with fixed quantum
numbers (s,l,j) we can think of 𝐦̂_ TOT
as being proportional to Ĵ. In the literature, this
is sometimes denoted as "𝐦̂_ TOT=γ_ LgĴ
" in an abuse of notation. The value of g(slj) we
can find by projecting 𝐦̂_ TOT·𝐉̂

    ⟨ sljm_j|(𝐋̂+2𝐒̂)·𝐉̂|sljm_j⟩=g(slj)⟨ sljm_j|𝐉̂^2|sljm_j⟩

and using Eqs.<ref> we obtain the Land factor

    g(s,l,j)=3/2+s(s+1)-l(l+1)/2j(j+1) .

We see that this expression coincides with the classical one obtained
in Eq. <ref> in the limit of s,l,j≫1.

Going back to the Zeeman splitting, we see that in general the Zeeman
correction to the atomic energy in the presence of a magnetic field
is given by 

    E_ Z=μ_ Bg(s,l,j)m_jB

and therefore the Zeeman splitting depends on all the atomic orbital
numbers, and it is not simply μ_ BB. This is denominated
the anomalous Zeeman effect since at the time of the experiments
the spin was still not known, and it was not possible to explain the
effect. Note however that the normal Zeeman effect can be observed
only for atoms with zero total spin, and therefore the anomalous one
is much more common. 


  * Exercise: Prove Eqs.<ref> 

  * Exercise: Derive Eq. <ref>. 




 §.§ Check points



  * How do you relate the magnetic moment of an electron to its total
angular momentum? Why? 

  * What is the Land factor? 




§ GENERALIZATION TO MANY ELECTRONS


Generalizing the previous concepts beyond the hydrogen atom / single
electron problem is impossible to do in an exact manner, since the
problem turns into a many-body problem: the many electrons interact
not only with the nucleus, but among themselves. We can however make
analytical progress by doing some reasonable approximations. The first
one is called the Hartree approximation, in which we consider
that each electron moves in an effective central potential V_ eff(r)
generated by the nucleus plus all the other electrons. The other electrons
are said to screen the potential of the nucleus, since their
charge is opposite.

The second approximation concerns the spin-orbit coupling Eq. <ref>.
For all except the heaviest atoms, the spin-orbit interaction is weak
and can be treated within perturbation theory. In this case, we can
first neglect this interaction, an consider that 𝐋̂_i
and 𝐒̂_i for each electron i are independent.
Hence, we can calculate the total orbital and spin angular momentum
simply by summing them separately: 𝐋̂_ TOT=∑_i𝐋̂_iand
𝐒̂_ TOT=∑_i𝐒̂_i , and
then proceed to calculate the total angular momentum 𝐉̂_ TOT=𝐋̂_ TOT+𝐒̂_ TOT
and the corresponding magnetic moment. 𝐒̂_ TOT^2
has eigenvalues ħ S(S+1) with S=∑_im_s,i and, respectively,
𝐋̂_ TOT^2 has eigenvalues ħ L(L+1)
with L=∑_im_l,i. The allowed values of 𝐉̂_ TOT
are given by the angular momentum summation rules: J:|L-S|,|L-S|+1,...L+S.
This approximation is denominated the Russell-Saunders coupling
and it gives rise to the well known Hund's rules. To calculate
𝐉̂_ TOT we need first a prescription to obtain
𝐒̂_ TOT and 𝐋̂_ TOT.

A closed shell means that we have occupied all 2(2l+1) levels in
it, where the factor of 2 comes for the spin s=±1. Therefore
both the total orbital and spin angular momentum of the shell are
zero, and hence also the total angular momentum 𝐉̂
in the shell. The total angular momentum of the atom, and therefore
its magnetic properties, will be determined by the last, partially
unoccupied shell. If all shells are closed (that is, full), then 𝐉̂_ TOT=0
and the atom is diamagnetic. The Hund rules tell us how to distribute
our n≤2(2l+1) " left over" electrons
in the last shell.The total spin 𝐒̂_ TOT we
obtain by applying the Pauli principle: since electrons are fermions,
their wavefunction is antisymmetric and two electrons can not have
the same quantum numbers. Each orbital characterized by l can then
be occupied by only two electrons, one with spin up, and one with
spin down. The first Hund rule tells us to maximize S, since
this will tend to put one electron with in each orbital until half
filling, (2l+1), and then continue with spin down. This minimizes
Coulomb repulsion by putting electrons, in average, as far apart as
possible. The second Hund rule tells us to maximize the orbital
angular momentum, once the spin is maximized. This also minimizes
Coulomb repulsion, by making the electrons orbit as far apart as possible.
The third Hund rule sounds more mysterious: if the shell is
more than half filled (n≥2l+1) then J=L+S, and for a less
than half filled shell (n≤2l+1), J=|L-S| . This is actually
due to the spin-orbit interaction λ𝐋̂_ TOT·𝐒̂_ TOT,
where it can be shown that λ's sign changes between these
two configurations. Therefore to minimize spin-orbit coupling requires
𝐒̂_ TOT and 𝐋̂_ TOT
parallel or antiparallel, depending on the filling.

To finish this section, we point out that for the heavier elements
the Russell-Saunders coupling prescription is not valid anymore, due
to the strong spin-orbit coupling. For these elements, a different
prescription, denominated jj coupling, is used. There 𝐉̂_̂î
for each electron is first calculated, and then the total 𝐉̂_ TOT.

As we mentioned, if all shells are closed the total angular momentum
is zero and the atom is diamagnetic. Diamagnetism can be understood
by the Faraday law: as a magnetic field is turned on, it induces a
change in the orbital motion of the electrons which opposes the change
in magnetic flux. Diamagnetism is usually a weak effect and it is
overshadowed by paramagnetism in atoms with partially unfilled shells.
Once the atoms are ordered in a lattice and form a solid, it can happen
that magnetic order develops as the temperature is lowered. This will
depend on the electronic interactions, as we will see in the next
chapter.



 §.§ Check points



  * What is the Russell-Saunders coupling scheme? 




CHAPTER: MAGNETISM IN SOLIDS


In the previous chapter we showed how to calculate the magnetic moment
of an atom. We saw that the problem is already quite involved even
for a single atom if we go beyond a hydrogen-like one. When atoms
come together to form a solid, to treat the magnetic problem atom
per atom is not only impossible but also not correct, since we have
to take into account the binding between the atoms that form the solid,
and what matters is the collective behavior of the material. In a
solid, the orbitals of the constituent atoms overlap to form bands
instead of discrete energy levels. Depending on the character of the
orbitals involved in the magnetic response of a material, we can divide
the problem into two big subsets: metals and insulators. In metals,
the orbitals are extended and have a good amount of overlap, so the
electrons are delocalized and free to move around the solid.
In insulators, the orbitals are narrow and we talk about localized
magnetic moments. Of course, this is an oversimplified view: there
are systems in which both localized and delocalized electrons participate
in magnetism, or in which magnetism and electric conduction are due
to different groups of orbitals. An example is that of heavy
fermion systems, which can be modeled as a Kondo Lattice:
a lattice of magnetic impurities embedded in a sea of free electrons.
Itinerant magnetism (that one due to delocalized electrons) and mixed
systems belong to what one calls strongly correlated systems,
highly complex many-body problems in which electron-electron interactions
have to be taken into account. In this course we will concentrate
mostly on magnetic insulators. Moreover, to understand magnetic
ordering we have to introduce electronic interactions and take into
account the Pauli principle.



§ THE CURIE-WEISS LAW


We consider first a system of localized, N identical non interacting
magnetic moments, and calculate their collective paramagnetic response.[As we saw in the last chapter, all atoms present a diamagnetic behavior,
but since it is very weak compared to the paramagnetic response, only
atoms with closed shells (e.g. noble gases) present an overall diamagnetic
response. An exception is of course superconducting materials, which
can have a perfect diamagnetic response. This is however a collective,
macroscopic response due to the superconducting currents which oppose
the change in magnetic flux.] We can obtain the magnetization from the Helmoltz free energy (see
e.g. Ref. <cit.>) 

    F     =-k_ BTln Z
    
    M     =-N/V∂ F/∂ B

where k_ B is the Boltzmann constant, T is the temperature,
and Z the canonical partition function for one magnetic moment

    Z=∑_ne^-E_n/k_ BT .

If we consider a magnetic moment with total momentum J, we have
2J+1 possible J_z values and 

    Z=∑_J_z=-J^Je^-1/k_ BTgμ_ BJ_zB .

The sum can be performed since it is a geometric one. Putting all
together, one finds 

    M=N/Vgμ_ BJℬ_J(μ_ BgJB/k_ BT)

where 

    ℬ_J(x)=2J+1/2J(2J+1/2Jx)-1/2J(1/2Jx)

is the Brillouin function. For μ_ BB≫ k_ BT this
function goes to 1 and the magnetization saturates: all momenta are
aligned with the B-field. For large temperatures instead, μ_ BB≪ k_ BT,
one obtains the inverse-temperature dependence of the magnetization
known as the Curie law, characterized by the susceptibility

    χ_ c=M/H=N/Vμ_0(gμ_ B)^2/3J(J+1)/k_ BT .



Experimentally, it is found that this result is good for describing
insulating crystals containing rare-earth ions (e.g. Yb, Er), whereas
for transition metal ions in an insulating solid agreement is found
only if one takes J=S. This is due to an effect denominated angular
momentum quenching, where effectively L=0. This quenching is due
to crystal fields: since now the atoms are located in a crystalline
environment, rotational symmetry is broken and each atom is located
in electric fields due to the other atoms in the crystal. In the case
of rare-earth ions, the magnetic moments come from f-shells, which
are located deep inside the atom and therefore better isolated form
crystal fields. For transition metals instead, the magnetic moments
come from the outermost d-shells, which are exposed to the symmetry
breaking fields. These however are spatial dependent fields which
do not affect directly the spin degree of freedom. As a consequence
of the breaking of spatial rotational invariance, the orbital angular
momentum is not conserved and precesses instead around the crystal
fields, averaging to zero.

The Curie law is followed by all materials with net magnetic moment
for large enough temperature. For low temperatures however, in certain
materials magnetic order develops and there is a deviation from the
Curie law in the susceptibility. Weiss postulated the existence of
molecular fields, which are proportional to the magnetization
in the material and are responsible for the magnetic ordering. The
total magnetic field acting on a magnetic moment within this picture
is therefore 𝐇_tot=𝐇_ext+λ𝐌
and from the Curie susceptibility 

    χ_ c=M/H_tot=M/H_ext+λ M=C/T

with 

    C=N/Vμ_0(gμ_ B)^2/3J(J+1)/k_ B

we obtain 

    M=CH_ext/T-λ C

and therefore the susceptibility in the external field follows the
Curie-Weiss law 

    χ_ W=C/T-λ C .

We see that for large temperatures this susceptibility follows the
Curie inverse law, but as the temperature approaches a critical temperature
T_C=λ C, χ_ W diverges indicating a phase
transition to the magnetically ordered phase.

A magnetically ordered phase implies M≠0 for an external field
B_ext=0. From Eq. <ref> we see that, if we do not
consider the molecular fields postulated by Weiss, M(0)=0. Let's
now consider B_ext=0 but the existence of a molecular field B_W=μ_0λ M.
Inserting this field in Eq. <ref> we obtain an implicit
equation for M: 

    M=M_0ℬ_J(μ_ Bμ_0gJλ M/k_ BT) ,

where we have defined the saturation magnetization 

    M_0=N/Vgμ_ BJ

since ℬ_J(x→∞)=1. Eq. <ref>
has still a solution M=0. However, if 

    [ d(M_0ℬ_J)/ dM]_M=0≥1

we see that a second solution to the transcendental equation is possible.
From 

    ℬ_J(x→0)≈J+1/3Jx

we obtain 

    M_sp=λ CM_sp/T_C

and therefore T_C=λ C, in agreement with Eq. <ref>.
The subscript sp indicates this is a spontaneous magnetization,
not induced by an external magnetic field. For T>T_C, Eq. <ref>
is not fulfilled and M=0 is the only solution. This can be seen
graphically in Fig. <ref>. 


At the time, the origin of these postulated molecular fields was not
known. Naively, one could expect the dipole-dipole interaction between
the magnetic moments to be the origin of the magnetic ordering. This
energy scale is however too small to explain magnetism at room temperature.
The potential energy of one magnetic dipole 𝐦_2 in the
magnetic field created by another dipole 𝐦_1, V_pot=-𝐦_2·𝐁_1(𝐫)
is 

    V_pot=-𝐦_2·μ_0/4π3(𝐦_1·𝐫)𝐫-r^2𝐦_1/r^5 .

A quick estimate corresponds to taking the distance between the dipoles
as the interatomic distance, and the dipolar moments simply as Bohr
magnetons. Equating this energy to k_ BT_ C results
in a critical temperature for magnetic ordering of T_ C≈1K.
Therefore dipolar interactions could be responsible magnetic ordering
only below this temperature, which is very low. We know however that
magnetic ordering at room temperature is possible. Instead, a rough
estimate of the repulsive Coulomb energy between two electrons gives

    U_c/k_ B=e^2/k_ B4πε_0a≈10^5K

which is very large! This could provide us with the necessary energy
scale. In the following section we will see that magnetic ordering
is due to a combination of the electrostatic energy and a very quantum
effect: the Pauli principle.


  * Exercise: Prove Eq. <ref>. 

  * Exercise: estimate T_ C from the dipole-dipole
interaction. 




 §.§ Check points



  * What is the Curie law and when is it valid? 

  * What is the Curie-Weiss law? 

  * Why the dipolar-dipolar interaction cannot in general explain magnetic
ordering? 




§ EXCHANGE INTERACTION


Let's consider first the case of two electrons subject to a Hamiltonian

    H=p_1^2/2m_e+p_2^2/2m_e+V(𝐫_1,𝐫_2) .

This Hamiltonian is independent of spin, however the Pauli exclusion
principle imposes a spatial symmetry on the wavefunction ψ(𝐫_1,𝐫_2)
solution of Hψ(𝐫_1,𝐫_2)=Eψ(𝐫_1,𝐫_2).
Since two electrons with the same quantum numbers are not allowed
at the same place, the total wavefunction ψ(𝐫_1,𝐫_2;s_1,s_2)
must be antisymmetric with respect to exchange of both spin and space:

    ψ(𝐫_1,𝐫_2;s_1,s_2)=-ψ(𝐫_2,𝐫_1;s_2,s_1) .

If there is no spin-orbit coupling in our Hamiltonian, we can separate
the wavefunction 

    ψ(𝐫_1,𝐫_2;s_1,s_2)=ψ(𝐫_1,𝐫_2)χ(s_1,s_2)

and therefore we see that the antisymmetry implies a correlation between
the spin and orbital parts of the wavefunction. This is a constraint
of the problem and solutions that do not fulfill this constraint are
infinite in energy. For two electrons, we have the total S=0,1
and correspondently S_z=0,-1,0,1. We can write the corresponding
states in explicit symmetric and antisymmetric linear combinations,
the antisymmetric singlet state 

    |0;0⟩1/√(2)(|↑↓⟩-|↓↑⟩)

and the symmetric triplet state with S=1, S_z=-1,0,1 :


    |1;1⟩    =|↑↑⟩
    
    |1;0⟩    =1/√(2)(|↑↓⟩+|↓↑⟩)
    
    |1;-1⟩    =|↓↓⟩

where the states are labeled as |S;S_z⟩. Since the Hamiltonian
does not depend on spin, we can work simply with the spatial component
of the wavefunction, but imposing the right symmetry. The spatial
part of the wavefunction corresponding to the singlet configuration,
ψ_singlet(𝐫_1,𝐫_2) will therefore be
symmetric in space coordinates, whereas ψ_triplet(𝐫_1,𝐫_2)
has to be antisymmetric 

    ψ_singlet(𝐫_1,𝐫_2)     =ψ_singlet(𝐫_2,𝐫_1)
    ψ_triplet(𝐫_2,𝐫_1)     =-ψ_triplet(𝐫_2,𝐫_1) ,

and we can write the full state as


    |Ψ_singlet⟩    =|ψ_singlet⟩|0;0⟩
    
    |Ψ_triplet(S_z)⟩    =|ψ_triplet⟩|1;S_z⟩

The Schrdinger equation therefore reads 

    H|ψ_singlet⟩    =E_s|ψ_singlet⟩
    
    H|ψ_triplet(S_z)⟩    =E_t|ψ_triplet⟩

and if E_s≠ E_t, the ground state is spin-dependent even
though H is not. If that is the case, since spatial and spin sectors
are correlated, we search for a Hamiltonian operating in spin space
H̃, coupling 𝐬̂_1 and 𝐬̂_2
and such that it is equivalent to H:


    H̃|0;0⟩    =E_s|0;0⟩
    H̃|1;S_z⟩    =E_t|1;S_z⟩ .

The Hamiltonian that does the trick is <cit.>

    H̃=1/4(E_s+3E_t)-1/ħ^2(E_s-E_t)(𝐬̂_1·𝐬̂_2)

since 

    𝐬̂_1·𝐬̂_2/ħ^2=1/2S(S+1)-3/4=
    -3/4    S=0
    1/4    S=1
     .

We have therefore constructed a Hamiltonian acting on spin space which
is in principle equivalent to the interacting Hamiltonian of Eq. <ref>,
which gives an effective interaction between the spins. This is called
the molecular Heisenberg model and can be written as

    H̃=J_0-J_12𝐬̂_1·𝐬̂_2

with 

    J_12=1/ħ^2(E_s-E_t) .

If J_12>0, this interaction favors a ferromagnetic alignment
of the spins, consistent with the fact that the singlet energy is
higher than the triplet one. We now show an example in which E_s≠ E_t
and calculate explicitly J_12. We will see that part of J_12
has no classical analog and comes from interchanging particles 1 and
2, since in quantum mechanics they are indistinguishable. 


  * Exercise: Prove Eq. <ref> and show that H̃
given in Eq. <ref> satisfies <ref>. 




 §.§ Check points



  * Why can you write a Hamiltonian in spin space which is equivalent
to the Hamiltonian defined in position space? 

  * How do you impose the equivalence for the two-electrons system? 




§ HYDROGEN MOLECULE


We consider now two hydrogen atoms that are brought close together
to form a hydrogen molecule. We consider the nuclei, a and b,
as fixed at positions 𝐑_a and 𝐑_b, whereas
the electrons, at positions 𝐫_1 and 𝐫_2
are subject to the nuclei Coulomb potential plus the repulsive Coulomb
interaction between them . This corresponds to taking the nuclei mass
m_a,m_b→∞ and it is an example of the denominated
Born Oppenheimer approximation. The Hamiltonian for each, separate
hydrogen atom a and b are given by 

    H_a    =-ħ^2∇_1^2/2m_e-e^2/4πε_01/|𝐑_a-𝐫_1|
    
    H_b    =-ħ^2∇_2^2/2m_e-e^2/4πε_01/|𝐑_b-𝐫_2|

and we know the respective eigenfunctions, ϕ_a,b with eigenenergies
E_a,b. If the distance between the two atoms |𝐑_a-𝐑_b|→∞,
these are the exact solutions. If however the atoms are brought close
together to form a molecule, there will be an interaction term 

    H_I=e^2/4πε_01/|𝐑_a-𝐑_b|-e^2/4πε_01/|𝐑_b-𝐫_1|-e^2/4πε_01/|𝐑_a-𝐫_2|+e^2/4πε_01/|𝐫_2-𝐫_1|

and the total Hamiltonian is given by 

    H_tot=H_a+H_b+H_I .

Even though this is a quite simple system, this problem cannot be
solved exactly and we have to resort to approximations. We treat H_I
as a perturbation and use the atomic wavefunctions ϕ_a,b as
a basis for a variational solution of the full wavefunction. This
implies we are assuming the electrons are quite localized at their
respective atom. This approach is denominated the Heitler-London
method.

If we consider the unperturbed Hamiltonian 

    H_0=H_a+H_b

the eigenfunctions will be simply linear combinations of the product
of the original orbitals, ϕ_aϕ_b, with eigenenergy E_a+E_b
since the two systems do not interact with each other. To preserve
the indistinguishability of the particles, we cannot simply write
a solution as ϕ_a(𝐫_1)ϕ_b(𝐫_2),
since the probability density |ϕ_a(𝐫_1)|^2|ϕ_b(𝐫_2)|^2
is not invariant under exchanging 𝐫_1⟷𝐫_2.
The symmetric and antisymmetric combinations fulfill however the indistinguishability
condition 

    ψ_s    (𝐫_1,𝐫_2)=1/√(2)(ϕ_a(𝐫_1)ϕ_b(𝐫_2)+ϕ_b(𝐫_1)ϕ_a(𝐫_2))
    ψ_t    (𝐫_1,𝐫_2)=1/√(2)(ϕ_a(𝐫_1)ϕ_b(𝐫_2)-ϕ_b(𝐫_1)ϕ_a(𝐫_2))

where with the notation ψ_s, ψ_t we have anticipated
that the symmetric (antisymmetric) solution in space corresponds to
the singlet (triplet) solution in spin space, with a full wavefunction
as given in Eq. <ref>. Note that both
ψ_s, ψ_t are eigenfunctions of H_0 with eigenvalue
E_a+E_b, and therefore without the interaction H_I, E_t=E_s=E_a+E_b
and the soulutions are 4-fold degenerate.

We now calculate E_t and E_s perturbatively in the presence
of the interaction H_I and using ψ_s, ψ_t as
variational wavefunctions 

    E_s/t=⟨ψ_s/t|H_tot|ψ_s/t⟩/⟨ψ_s/t|ψ_s/t⟩ .

We are interested in the ground state solutions, so that ϕ_a,b
are solutions of each hydrogen atom with E_a,b=E_0. The variational
principle tells us that the energies calculated by Eq. <ref>
are always greater or equal that the true groundstate.[Note that in general the variational procedure would be to write ψ_s/t(𝐫_1,𝐫_2)=c_1ϕ_a(𝐫_1)ϕ_b(𝐫_2)± c_2ϕ_b(𝐫_1)ϕ_a(𝐫_2)
and find the coefficients c_1,2 by minimizing Eq. <ref>.
One then finds c_1=c_2=1/√(2). In Eq. <ref>
we used our knowledge of the symmetry of the problem plus the normalization
of the wavefunctions ϕ_a,b to write the result immediately.]

Let's first analyze the simple overlap 

    ⟨ψ_s/t|ψ_s/t⟩=∫ d^3r_1 d^3r_2ψ_s/t^*(𝐫_1,𝐫_2)ψ_s/t(𝐫_1,𝐫_2) .

If the two atoms are infinitely apart |𝐑_a-𝐑_b|→∞,
then the orbitals corresponding to different atoms have zero overlap,
⟨ϕ_a|ϕ_b⟩=0, and 

    ⟨ψ_s/t|ψ_s/t⟩_0=∫ d^3r_1 d^3r_2|ϕ_a(𝐫_1)|^2|ϕ_b(𝐫_2)|^2=1 .

When the atoms are brought close together, their orbitals will overlap:
⟨ϕ_a|ϕ_b⟩≠0. We do not calculate this
overlap explicitly, but simply note that it will be finite and denote
it by O 

    O^2=∫ d^3r_1 d^3r_2ϕ_a^*(𝐫_1)ϕ_b(𝐫_1)ϕ_a(𝐫_2)ϕ_b^*(𝐫_2)

and hence 

    ⟨ψ_s/t|ψ_s/t⟩=1± O^2 .

We now turn to the numerator in Eq. <ref>.
We already know that the non-interacting contribution is 

    ⟨ψ_s/t|H_0|ψ_s/t⟩/⟨ψ_s/t|ψ_s/t⟩=2E_0 .

The correction to this non-interacting energy is given by the term
containing ⟨ψ_s/t|H_I|ψ_s/t⟩, which we
see contains two kind of terms. One of them is simply the Coulomb
electrostatic interaction between the two atoms, assuming that they
are close enough to interact but electron 1(2) still "
belongs" to atom a(b) 

    K=∫ d^3r_1 d^3r_2ϕ_a^*(𝐫_1)ϕ_b(𝐫_2)H_Iϕ_a(𝐫_1)ϕ_b^*(𝐫_2) .

The remaining term has no classical analog, and it measures the Coulomb
energy cost upon exchanging the two electrons 

    X=∫ d^3r_1 d^3r_2ϕ_a^*(𝐫_1)ϕ_b(𝐫_1)H_Iϕ_a(𝐫_2)ϕ_b^*(𝐫_2) .

and it is called the exchange integral, or exchange interaction.
Putting all together we obtain 

    E_s/t=2E_0+K± X/1± O^2

where the + (-) corresponds to the singlet (triplet) solution
ψ_s (ψ_t ). In general O≪1 and we can replace
the denominator by 1. We have therefore shown that E_s-E_t≠0
and hence for this problem 

    J_12≈2/ħ^2X ,

which justifies the name exchange parameter for J_12.



 §.§ Check points



  * What is the Heitler-London model? 

  * What is the exchange interaction and why does it not have a classical
analog? 




§ HEISENBERG, ISING, AND XY MODELS


In our variational solution for the Hydrogen molecule from the last
section, double occupation is forbidden, so two electrons cannot be
in the same atom at the same time. This indicates our treatment is
valid for insulators, where electrons are quite localized, but in
turn leads necessarily to small values of the exchange parameter,
since J_12 relies on the overlap of the single-atom orbitals.
The above example therefore must be taken as a toy model which reveals
the character of the ferromagnetic interaction. In general, the exchange
constant is generated by more complex interactions, e.g. superexchange
where the ferromagnetic exchange interaction between two spins is
mediated by an exchange interaction with an atom in between those
with a net angular momentum.

We further postulate that our model can be generalized to N multielectron
atoms 

    H=-1/2∑_ijJ_ij𝐒̂_i·𝐒̂_j

where the exchange coefficient J_ij is taken as a parameter of
the model, that has to be calculated for each particular material.
The Hamiltonian in Eq. <ref> is the Heisenberg
Hamiltonian. The factor of 1/2 accounts for the double-counting
in the sum. We write 𝐒̂ by convention, and we refer
to the magnetic moment as " spins" in
an abuse of language: in reality, unless the orbital angular momentum
is quenched, the total angular momentum of the ions is meant. The
spin operators follow the angular momentum algebra when located at
the same site, and commute with operators at different sites: 

    [Ŝ_i^α,Ŝ_̂ĵ^β]=iħδ_ijϵ_αβγŜ_i^γ

where α, β, γ indicate the spatial components
of the angular momentum x, y, z. These commutation relations
make the quantum Heisenberg model, despite its simple appearance,
quite a rich model and exactly solvable only for a few simple cases.
The input of the model is the lattice connectivity and dimensionality,
and the exchange parameter J_ij .

Besides insulators, the Heisenberg Hamiltonian is a valid model for
localized magnetic moments embedded in a metal. In that case, the
exchange interaction is mediated by the conduction electrons, which
gives rise to the RKKY interaction (Ruderman–Kittel–Kasuya–Yosida).
The calculated exchange function J_ij is an oscillating function
of position, alternating between positive and negative values, and
is longer ranged than in the insulating case.

If there are no local magnetic moments but the system still presents
magnetic order, the conduction electrons are also responsible for
the magnetic order. In this case the magnetism is denominated itinerant
and it is described by a different Hamiltonian: the Hubbard
Hamiltonian. This model takes into account the kinetic energy of
the electrons, who can " jump" from lattice
site to lattice site, and penalizes double occupation with a local
Coulomb repulsion term. The Hubbard model takes an effective Heisenberg
form in the particular case of a half-filled band and strong Coulomb
interaction.

The Heisenberg Hamiltonian is the " father"
Hamiltonian of other well known models in magnetism. In a crystal,
crystal fields can give rise to anisotropies in the exchange parameter
J_ij. If the anisotropy is along only one direction one can write

    H=-∑_ijJ̃_ij(Ŝ_i^xŜ_j^x+Ŝ_i^yŜ_j^y+Δ Ŝ_i^zŜ_j^z) .

If Δ>1, magnetic ordering occurs along the z axis, which
is denominated the easy axis. For Δ≫1 the Hamiltonian
turns effectively into the Ising Model 

    H_ Ising=-∑_ijJ_ij S_i^zS_j^z .

 Note that in this particular case, all operators in the Hamiltonian
commute, and therefore the model is in this sense classical. If Δ<1
then we have an easy plane ordering. For Δ≪1 the
system is effectively two-dimensional and isotropic, which is termed
the XY model


    H_ XY=-∑_ijJ_ij (Ŝ_i^xŜ_j^x+Ŝ_i^yŜ_j^y) .


If the system is placed in an external magnetic field, a Zeeman term
is added to the Hamiltonian 

    H     =-1/2∑_ijJ_ij𝐒̂_i·𝐒̂_j-gμ_ B𝐁·∑_i𝐒̂_i
        =-1/2∑_ijJ_ij𝐒̂_i·𝐒̂_j-gμ_ BB∑_iŜ_i^z .

In Eq. (<ref>) the spin operators are dimensionless
and ħ has been absorbed in μ_ B (correspondingly,
in the commutation relation Eq. (<ref>) ħ
should be set to 1 if this convention is used). Note that the Zeeman
energy is -𝐦·𝐁 (see Eq. (<ref>))
and tends to align the magnetic moment with the magnetic field, and
anti-align the angular momentum. Sometimes by convention the extra
minus sign is not used, since from now onwards one always works with
the angular momentum operators, and one takes gμ_ B>0.
This corresponds simply to transform 𝐁→-𝐁
if we want to translate into the magnetization or magnetic moments,
and does not affect the results.



 §.§ Check points



  * Write the Heisenberg Hamiltonian in a magnetic field 




§ MEAN FIELD THEORY 


Once we have a Hamiltonian that models our system, we want in principle
to (i) find the ground state, that is, the lowest energy eigenstate
of the system, which is the only state populated at zero temperature,
(ii) find the excitations on top of this ground state, which will
determine the behavior of the system at T≠0, and (iii) study
phase transitions, either at T=0 (denominated a quantum phase
transition) where by changing some other external parameter like
the magnetic field, the ground state of the system changes abruptly,
or at finite temperature, where an order parameter of the system (given
by a quantum-statistical average of some relevant quantity to describe
the system), goes to zero as a function of temperature or other external
parameters. An example of the latter is the magnetization M(B,T).
We go back now to the issue of magnetic ordering armed with the Heisenberg
Hamiltonian. As we pointed out above, this is a very rich model, and
there are very few general statements that can be made about the three
points mentioned above. We turn therefore first to a well know approximation
denominated mean field theory. This approximation is in general
good only for long-range interactions and high dimensions, it is however
widely used to get a first idea of, for example, what kind of phases
and phase transitions our model can present. For the Heisenberg model,
we will see that mean field theory will give us a microscopic justification
of the Weiss molecular fields we introduced at the beginning of this
chapter.

We start by assuming that ⟨𝐒̂_i⟩ is
finite. For example, for a ferromagnetic ground state, ⟨𝐒̂_i⟩
is uniform and such that 

    𝐌=N/Vgμ_ B⟨𝐒̂_i⟩ ,

where N is the number of lattice sites and V the total volume.
We write now 𝐒̂_i in the suggestive form 

    𝐒̂_i=⟨𝐒̂_i⟩+(𝐒̂_i-⟨𝐒̂_i⟩) ,

which corresponds to splitting the operator into its quantum statistical
average value and the fluctuations with respect to this average. In
mean field theory, these fluctuations are assumed to be small. The
mean field Hamiltonian is obtained from the Heisenberg Hamiltonian
by keeping only terms up to first order in the fluctuation, 

    H_ MF=1/2∑_ijJ_ij⟨𝐒̂_i⟩·⟨𝐒̂_j⟩-∑_ijJ_ij⟨𝐒̂_i⟩·𝐒̂_j-gμ_ B𝐁·∑_i𝐒̂_i ,

where we already included an external magnetic field 𝐁.
The first term in Eq. (<ref>) is simply a constant
shift in the energy. The second term corresponds to a spin a site
j in the presence of a magnetic field generated by all other spins.
We can therefore define an effective magnetic field 

    𝐁_ eff=𝐁+1/gμ_ B∑_iJ_ij⟨𝐒̂_i⟩

and our problem is reduced from an interacting problem (where spins
interact with each other via the exchange interaction), to that of
non-interacting spins in the presence of a magnetic field. If we go
back to Eq. (<ref>), we see that now we can give a
microscopic explanation to the Weiss molecular fields, which were
assumed to be proportional to to the magnetization. In particular
we find that we can write 

    𝐁_ eff=𝐁+μ_0λM

with 

    λ=1/μ_0V/N1/(gμ_ B)^2J_0

where we have used translational symmetry and defined J_0=∑_iJ_ij
independent of j. This system therefore presents a transition to
an ordered state as a function of temperature and magnetic field as
discussed for the Curie-Weiss law, but we now have a microscopic explanation
for the phenomenological model.



 §.§ Check points



  * What is the meaning of the mean field theory? 

  * How is it related to the Curie-Weiss law? 




§ GROUND STATE OF THE FERROMAGNETIC HEISENBERG
HAMILTONIAN


For the particular case in which J_ij≥0 ∀ i, j it
is possible to find the ground state of the Heisenberg Hamiltonian
without further specifications. This is called the ferromagnetic
Heisenberg model since the ground state is ferromagnetically ordered,
as we show in the following.

We consider hence the Hamiltonian 

    Ĥ=-1/2∑_ijJ_ij𝐒̂_i·𝐒̂_j  ,  with  J_ij=J_ji≥0 .

f the spins were classical vectors, the state of lowest energy would
be that one with all N spins are aligned. Hence a natural candidate
for the ground state of Ĥ is 

    |0⟩=|S,S⟩_1|S,S⟩_2...|S,S⟩_N

where 𝐒̂_i^2|S,S⟩_i=S(S+1)|S,S⟩_i
and Ŝ_i^z|S,S⟩_i=S|S,S⟩_i, that is all
spins take their maximum projection of Ŝ^z (we consider
all spins identical). The individual spin operators however do not
commute with the Hamiltonian <ref>, 

    [𝐒̂_i,Ĥ]≠0

and therefore a product states of the form 

    |ψ⟩=|S,m_1⟩|S,m_2⟩...|S,m_N⟩

(with Ŝ_i^z|S,m_i⟩=m_i|S,m_i⟩) span
a basis of the Hilbert space, but are not necessarily eigenstates
of Ĥ. The total spin operator 𝐒̂_ TOT
however does commute with Ĥ 

    [𝐒̂_ TOT,Ĥ]=0

and we can construct an eigenbasis for Ĥ, 𝐒̂_ TOT^2,
and 𝐒̂_ TOT^z. The state <ref>
is a state with maximum 𝐒̂_ TOT. We will now
prove that (i) <ref> is an eigenstate of <ref>,
and (ii) that there is no eigenstate with higher energy <cit.>.

To follow with the proof it is convenient to recast Ĥ in
term of ladder operators 

    Ŝ_i^±    =Ŝ_i^x± iŜ_i^y
    Ŝ_i^±|S_i,m_i⟩    =√((S_i∓ m_i)(S_i+1± m_i))|S_i,m_i±1⟩ ,

from which it is clear why they are call ladder operators: Ŝ^+
(Ŝ^-) increases (decreases) the projection of Ŝ^z
by one unit. In terms of <ref>, the Hamiltonian <ref>
reads 

    Ĥ=-1/2∑_ijJ_ij[1/2(Ŝ_i^+Ŝ_j^-+Ŝ_i^-Ŝ_j^+)+Ŝ_i^zŜ_j^z] .

With this expression it is now straightforward to show that |0⟩
is an eigenstate of Ĥ 

    Ĥ|0⟩=-1/2∑_ijJ_ijŜ_i^zŜ_j^z|0⟩=-S/2^2∑_ijJ_ij|0⟩

since Ŝ_i^+|0⟩=0 ∀ i (remember that spin
operators at different sites commute, and J_ii=0). Therefore

    Ĥ|0⟩=E_0|0⟩

with 

    E_0=-S/2^2∑_ijJ_ij .

We now need to prove that E_0 is the minimum possible energy.
For that we consider the expectation value of Ĥ with any
other arbitrary eigenstate |ψ⟩

    E_0'=⟨ψ|Ĥ|ψ⟩≥-1/2∑_ijJ_ijmax⟨ψ|𝐒̂_i·𝐒̂_j|ψ⟩≥-S^2/2∑_ijJ_ij

where we have used that J_ij≥0, J_ii=0, and that ⟨ψ|𝐒̂_i·𝐒̂_j|ψ⟩≤ S^2
for i≠ j<cit.>. Hence 

    E_0'≥ E_0 ,

which proves that the fully polarized state |0⟩ given in
Eq. <ref> is indeed the ground state. This ground
state is however not unique, but it is (2S_ TOT+1)
degenerate in spin space, with S_ TOT=NS. This can be easily
visualized in the 2-spins case, where for the triplet state, S_ TOT=1
and the state has 3 possible projections of Ŝ_z, see Eq.
<ref>. Note that each of these states is moreover infinitely
degenerate in position space, since we are able to choose the quantization
axis freely. This is an example of what is called spontaneous
symmetry breaking, which occurs when the ground state has a lower
symmetry than the Hamiltonian <cit.>. The
ground state of Ĥ is a specific realization of the (2S_ TOT+1)
possible ground states, and therefore has " picked"
a preferred direction in spin space, which is not determined by the
symmetry of Ĥ. We mention in passing that our results are
valid in all its generality strictly for T=0. For T>0, the Mermin-Wagner
theorem tells us that, in 1 and 2 dimensions and for short range
interactions continuous symmetries cannot be spontaneously broken.

If we add a magnetic field to Ĥ, our total Hamiltonian is
the one in Eq. <ref>. In this case, it
is favorable for the system also to maximize the projection of Ŝ_z,
and in this case |0⟩ given in Eq. <ref>
is the only ground state, with energy 

    E_0(B)=-S/2^2∑_ijJ_ij -gμ_ BBNS .

In this case the symmetry is not spontaneously broken, since is is
already broken at the Hamiltonian level by the applied magnetic field.



 §.§ Check points



  * What is the ground state of the ferromagnetic Heisenberg Hamiltonian? 

  * How do you prove it is the ground state? 

  * What happens in the presence of an external magnetic field to the
degeneracy of the state? 




§ GROUND STATE OF THE ANTIFERROMAGNETIC
HEISENBERG HAMILTONIAN


Aside from the ferromagnetic case treated in the previous section,
finding the ground state of the Heisenberg Hamiltonian is in general
difficult and has to be studied case by case. The ground state will
depend on the nature of the interactions (short or long range, sign,
anisotropy), from the lattice structure, and from the dimensionality
of the system. To illustrate this difficulty, we discuss here briefly
the antiferromagnetic Heisenberg model on a bipartite lattice <cit.>.
In this case J_ij≤0 ∀ i, j and the lattice can be
sub-divided in two sublattices A and B, such that J_ij
is finite only when i and j belong to two different sublattices.
The simplest example is that of a square lattice with nearest neighbor
interactions, where all nearest neighbors of A belong to the B
sublattice, and vice-versa, see Fig.



A guess of the ground state based on the classical model is the
so called Nel state: the two sublattices are fully polarized,
but in opposite directions 

    |0?⟩_ AF=∏_i∈ A|S,S⟩_i∏_j∈ B|S,-S⟩_j .

It is easy to see however that this state is not an eigenstate of
the Heisenberg Hamiltonian. Using the representation of Ĥ
in terms of ladder operators, see Eq. <ref>, we see
that 

    Ĥ|0?⟩_ AF=-1/2∑_i∈ A∑_j∈ BJ_ij[1/2(Ŝ_i^+Ŝ_j^-|0?⟩_ AF+Ŝ_i^-Ŝ_j^+|0?⟩_ AF)+Ŝ_i^zŜ_j^z|0?⟩_ AF] .

The first term is simply zero, since the sublattices A and B
are fully polarized in " the right way"
with respect to the operators. The last term is simply proportional
to |0?⟩_ AF. However for the second term


    Ŝ_i^-Ŝ_j^+|0?⟩_ AF∝|S,⟩...|S,-S+1⟩_j...|S,S-1⟩_i...|S,m_N⟩

which shows that |0?⟩_ AF cannot be an eigenstate.
In general, one can prove that the true ground state |0⟩_ AF
is non-degenerate and a singlet of total spin: Ŝ_ TOT|0⟩_ AF=0.
This is called the Marshall's Theorem. Which kind of singlet
is the actual ground state is however not determined.



 §.§ Check points



  * What is a Nel state? 

  * Why it is not the ground state of the antiferromagnetic Heisenberg
Hamiltonian? 




§ GROUND STATE OF THE CLASSICAL HEISENBERG
MODEL


The classical Heisenberg model is obtained by replacing the spin operators
𝐒̂_i by simple vectors 𝐒_i with fixed
length |𝐒_i|=S. As we saw in previous sections, this
kind of approximation is valid in the limit of large spin S. For
the classical model it is straightforward to find the ground state
for translational invariant systems, 

    J_ij=J_ji=J(𝐑_i-𝐑_j)

where 𝐑_i indicates the points on a Bravais lattice.
In Fourier space 

    𝐒_i    =1/√(N)∑_𝐤𝐒_𝐤e^i𝐤·𝐑_i
    𝐒_𝐤    =1/√(N)∑_i𝐒_ie^-i𝐤·𝐑_i

and hence 

    H_ cl=-1/2N∑_𝐑_i𝐑_jJ(𝐑_i-𝐑_j)∑_𝐤𝐤'(e^i𝐤·𝐑_i𝐒_𝐤·𝐒_𝐤'e^i𝐤'·𝐑_j)

Defining Δ𝐑=𝐑_i-𝐑_j and using

    ∑_𝐑_ie^i𝐤·𝐑_i=Nδ_𝐤,0

being δ_𝐤,𝐤' the Kronecker Delta, we obtain

    H_ cl    =-1/2N∑_𝐑_iΔ𝐑J(Δ𝐑)∑_𝐤𝐤'(e^i𝐤·𝐑_i𝐒_𝐤·𝐒_𝐤'e^i𝐤'·𝐑_ie^-i𝐤'·Δ𝐑)
        =-1/2N∑_Δ𝐑J(Δ𝐑)∑_𝐤𝐤'Nδ_𝐤+𝐤',0𝐒_𝐤·𝐒_𝐤'e^-i𝐤'·Δ𝐑
        =-1/2∑_Δ𝐑J(Δ𝐑)∑_𝐤𝐒_𝐤·𝐒_-𝐤e^i𝐤·Δ𝐑

and defining 

    J(𝐤)     =∑_iJ(Δ𝐑)e^i𝐤·Δ𝐑

we obtain 

    H_ cl=-1/2∑_𝐤J(𝐤)𝐒_𝐤·𝐒_-𝐤

Imposing the constraint 𝐒^2=S^2, one can show that
the minimum of the energy given by Eq. <ref> is given
by setting 𝐤=𝐐, where 𝐐 determines
the global maximum of J(𝐐) <cit.>.
One obtains the equilibrium configuration 

    𝐒_i=S(cos(𝐐.𝐑_i),sin(𝐐.𝐑_i),0)

which is in general a planar helical state. Since 𝐐 is
determined by the maximum of J(𝐐), the order is not necessarily
commensurate with the lattice, unless this maximum coincides with
a high symmetry point in the Brillouin zone. Special cases are 𝐐=0,
where we recover ferromagnetic order, or 𝐐 taking a value
at the edge of the Brillouin zone, in which case the order is antiferromagnetic.
Applying a magnetic field in the z direction tilts the magnetic
order out of plane.



 §.§ Check points



  * What is the classical Heisenberg model and how does one in general
find its ground state? 




§ DIPOLE-DIPOLE INTERACTIONS


In the Heisenberg Hamiltonian we have not included the dipole-dipole
interaction term 

    Ĥ_d-d=-μ_0/4π(μ_ Bg)^2/2∑_i≠ j1/|𝐫_i-𝐫_j|^3[3𝐒̂_i·(𝐫_i-𝐫_j)𝐒̂_j·(𝐫_i-𝐫_j)/|𝐫_i-𝐫_j|^2-𝐒̂_i·𝐒̂_j] .

We argued this term is too small to justify magnetic ordering at the
observed temperatures, and we found out that the exchange interaction
is orders of magnitude larger. However the exchange interaction depends
on the overlap of orbitals, and as such is generally short ranged:
it decays exponentially with distance, at least for insulators. From
Eq. <ref> we see that the dipole-dipole interaction
decays instead algebraically, as 1/r^3, and that means that it
will be dominant at large distances. This gives rise, in large samples,
to the formation of domains. In intermediate-size samples, usually
micrometer sized, the competition between dipole-dipole interactions
and exchange interactions leads to textured ground states. Being long-ranged,
the dipolar interactions are sensitive to the boundaries of the sample,
where the spins try to align with the boundary, so as to minimize
the " magnetic monopoles" at the surface
as introduced in Chapter <ref>. Dipole-dipole
interaction terms can be included in the Heisenberg Hamiltonian in
the long-wavelength limit as demagnetization fields.



CHAPTER: SPIN WAVES AND MAGNONS


In the last chapter we dealt with the ground state of the Heisenberg
model for a few solvable examples. The ground state is the only eigenstate
that is occupied at strictly zero temperature. The excitations on
top of this ground state will determine the behavior of the system
at low temperatures. These are collective excitations of the magnetic
system, and their quanta are denominated magnons <cit.>.
One can draw an analogy to another physical phenomenon which is perhaps
more intuitive: that of mechanical vibrations. There, all atoms participate
in the collective mechanical vibration, and we call phonons the respective
quanta. In the next sections we will study the magnetic elementary
excitations in more detail. We will focus on the simplest case of
the Heisenberg ferromagnet, but the concepts are general.



§ EXCITATIONS OF THE HEISENBERG FERROMAGNET


For the Heisenberg ferromagnet at T=0 we found that all spins are
aligned with and the magnetization is given by the saturation magnetization

    M_ s=gμ_ BN/VS .

At T≠0 some spins will " flip" or,
more generally, decrease by one unit, which in turn will decrease
the magnetization from it saturation value. An obvious candidate for
an excited state is therefore 

    |i⟩=|S,S⟩_1...|S,S-1⟩_i...|S,S⟩_N

which can be obtained from the ground state Eq. <ref>
as 

    |i⟩=1/√(2S)Ŝ_i^-|0⟩ .

One can easily see that |i⟩ is an eigenstate of the Ŝ^z
operator 

    Ŝ_i^z|i⟩    =(S-1)|i⟩
    Ŝ_j^z|i⟩    =S|i⟩  ∀   j≠ i .

However Ŝ_i^+|i⟩≠0, instead one obtains 

    Ŝ_j^-Ŝ_i^+|i⟩=2S|j⟩ .

Therefore the Heisenberg Hamiltonian <ref> shifts
the site of the " flipped" spin i→ j,
and |i⟩ as given in Eq. <ref> is not an eigenstate
of <ref>. It is also not a good approximation to an
excited eigenstate, since flipping a spin in such a manner has a very
high energy cost, of the order of the exchange interaction. For example,
for nearest neighbor interaction with exchange constant J, flipping
a spin has an energy cost of Δ E∼ zJS, where z is the
coordination number (that is, z is the number of nearest
neighbors, e.g. for a square lattice z=4). We have already seen
that the exchange constant is of the order of the critical temperature,
quite a high energy for regular ferromagnets.

One can prove that an actual eigenstate of <ref> is
given by 

    |𝐤⟩=1/√(N)∑_𝐑_ie^-i𝐑_i·𝐤|i⟩ ,

where 𝐑_i are the lattice sites. We see therefore that
an eigenstate is formed by distributing the " flipped"
spin over all sites, and therefore it is a collective excitation
of the system. This collective excitation has well defined momentum
ħ𝐤 (up to a reciprocal lattice vector) and energy
ħω(𝐤) and we call it a quasiparticle.
In particular for magnetic systems these quasiparticles are denominated
magnons. In the following we will show that |𝐤⟩
is an eigenstate of <ref> and will calculate its dispersion
relation ħω(𝐤).

In order to do this, we first perform a Fourier transform of the Hamiltonian
<ref> by defining the spin operators in momentum space
(in analogy to the classical case, see Eqs. <ref>) 

    Ŝ_𝐤^α    =1/√(N)∑_𝐑_ie^-i𝐤·𝐑_iŜ_i^α
    Ŝ_i^α    =1/√(N)∑_𝐤Ŝ_𝐤^αe^i𝐤·𝐑_i

where α=x, y, z, ±. Note that 

    (Ŝ_𝐤^+)^†    =Ŝ_-𝐤^-
    (Ŝ_𝐤^z)^†    =Ŝ_-𝐤^z

and one can verify that the commutators now read 

    [Ŝ_𝐤_1^+,Ŝ_𝐤_2^-]     =2/√(N)Ŝ_𝐤_1+𝐤_2^z
    [Ŝ_𝐤_1^z,Ŝ_𝐤_2^±]     =±1/√(N)Ŝ_𝐤_1+𝐤_2^± .

Using that J_ij=J_ji and that spin operators at different sites
commute, we re-write slightly the Heisenberg Hamiltonian <ref>
as 

    Ĥ=-1/2∑_ijJ_ij[Ŝ_i^+Ŝ_j^-+Ŝ_i^zŜ_j^z] .

From Eqs. <ref> and using again Eq. <ref>
we obtain 

    Ĥ    =-1/2N∑_𝐑_i𝐑_jJ(𝐑_i-𝐑_j)∑_𝐤𝐤'(Ŝ_𝐤^+e^i𝐤·𝐑_iŜ_𝐤'^-e^i𝐤'·𝐑_j+Ŝ_𝐤^ze^i𝐤·𝐑_iŜ_𝐤'^ze^i𝐤'·𝐑_j)
        -gμ_ B/√(N)B_0∑_𝐑_i∑_𝐤Ŝ_𝐤^ze^i𝐤·𝐑_i

where we added an external magnetic field. We use Δ𝐑=𝐑_i-𝐑_j
and Eq. <ref> to write 

    Ĥ=     -1/2N∑_Δ𝐑J(Δ𝐑)∑_𝐑_i∑_𝐤𝐤'e^i(𝐤+𝐤')·𝐑_i(Ŝ_𝐤^+Ŝ_𝐤'^-+Ŝ_𝐤^zŜ_𝐤'^z)e^-i𝐤'·Δ𝐑
        -gμ_ B√(N)B_0∑_𝐤Ŝ_𝐤^zδ_𝐤,0
    
    =     -1/2N∑_Δ𝐑J(Δ𝐑)N∑_𝐤𝐤'δ_𝐤+𝐤',0(Ŝ_𝐤^+Ŝ_𝐤'^-+Ŝ_𝐤^zŜ_𝐤'^z)e^-i𝐤'·Δ𝐑-gμ_ B√(N)B_0Ŝ_0^z
    
    =     -1/2∑_Δ𝐑J(Δ𝐑)∑_𝐤(Ŝ_𝐤^+Ŝ_-𝐤^-+Ŝ_𝐤^zŜ_-𝐤^z)e^i𝐤·Δ𝐑-gμ_ B√(N)B_0Ŝ_0^z .

Using the definition for J(𝐤) given in Eq. <ref>
we finally obtain an expression for the Heisenberg Hamiltonian in
momentum space 

    Ĥ=-1/2∑_𝐤J(𝐤)(Ŝ_𝐤^+Ŝ_-𝐤^-+Ŝ_𝐤^zŜ_-𝐤^z)-gμ_ B√(N)B_0Ŝ_0^z .


First we check the action of Ĥ given in Eq. <ref>
on the ground state |0⟩. For that we note 

    Ŝ_𝐤^z|0⟩=1/√(N)∑_𝐑_ie^-i𝐤·𝐑_iŜ_i^z|0⟩    =S/√(N)∑_𝐑_ie^-i𝐤·𝐑_i|0⟩=S√(N)δ_𝐤,0|0⟩
    Ŝ_i^+|0⟩    =0  ⇒  Ŝ_𝐤^+|0⟩=0

and examine the action of each term in Eq. <ref>
on |0⟩. Using the commutators in Eq. <ref>,
for the first term in the sum we obtain 

    -1/2∑_𝐤J(𝐤)Ŝ_𝐤^+Ŝ_-𝐤^-|0⟩    =-1/2∑_𝐤J(𝐤)(Ŝ_-𝐤^-Ŝ_𝐤^++2/√(N)Ŝ_0^z)|0⟩
        =-1/2√(N)∑_𝐤J(𝐤)(2Ŝ_0^z)|0⟩=-S∑_𝐤J(𝐤)|0⟩=0

where the last equality stems from 

    ∑_𝐤J(𝐤)=∑_𝐤∑_Δ𝐑J(Δ𝐑)e^i𝐤·Δ𝐑=∑_Δ𝐑J(Δ𝐑)Nδ_Δ𝐑,0=NJ(Δ𝐑=0)=0 .

For the second term in Eq. <ref> 

    -1/2∑_𝐤J(𝐤)Ŝ_𝐤^zŜ_-𝐤^z|0⟩    =-1/2√(N)S∑_𝐤J(𝐤)Ŝ_𝐤^zδ_-𝐤,0|0⟩
        =-1/2√(N)SJ(𝐤=0)Ŝ_0^z|0⟩
        =-1/2NS^2J(𝐤=0)|0⟩ .

We now note that 

    NJ(𝐤=0)=N∑_Δ𝐑J(Δ𝐑)=∑_𝐑_iΔ𝐑J(Δ𝐑)=∑_𝐑_i𝐑_jJ(𝐑_i-𝐑_j)=∑_ijJ_ij

and therefore 

    -1/2∑_𝐤J(𝐤)Ŝ_𝐤^zŜ_-𝐤^z|0⟩=-S^2/2∑_ijJ_ij|0⟩ .

For the third term in Eq. <ref> we obtain simply

    -gμ_ B√(N)B_0Ŝ_0^z|0⟩=-gμ_ BNB_0S|0⟩ .

Putting all together we obtain Ĥ|0⟩=E_0|0⟩
with 

    E_0(B_0)=-S/2^2∑_ijJ_ij -gμ_ BB_0NS

which coincides with our result in Eq. <ref> as expected.

We now want to show that |𝐤⟩ given in Eq. <ref>
is a eigenstate of <ref>. Using <ref>
we can write |𝐤⟩ in terms of the ladder operators
in momentum space, 

    |𝐤⟩=1/√(2S)Ŝ_𝐤^-|0⟩ ,

hence it is enough to show that Ŝ_𝐤^-|0⟩
is an eigenstate <cit.>. Writing 

    ĤŜ_𝐤^-|0⟩    =(Ŝ_𝐤^-Ĥ+[Ĥ,Ŝ_𝐤^-])|0⟩=(Ŝ_𝐤^-E_0+[Ĥ,Ŝ_𝐤^-])|0⟩
        =E_0Ŝ_𝐤^-|0⟩+[Ĥ,Ŝ_𝐤^-]|0⟩ ,

we see that that amounts to showing that [Ĥ,Ŝ_𝐤^-]|0⟩∝Ŝ_𝐤^-|0⟩.
Using similar manipulations as above, and taking into account that
J(𝐤)=J(-𝐤), one obtains 

    [Ĥ,Ŝ_𝐤^-]|0⟩=[gμ_ BB_0-S(J(𝐤)-J(𝐤=0))]Ŝ_𝐤^-|0⟩

and therefore |𝐤⟩ is an eigenstate of <ref>
with eigenvalue 

    E(𝐤)=E_0(B_0)+gμ_ BB_0-S(J(𝐤)-J(𝐤=0)) .


The energy on top of the ground state is the excitation energy

    ħω(𝐤)=gμ_ BB_0-S[J(𝐤)-J(𝐤=0)]

which is simply the energy of one magnon — the quasiparticle
energy is always defined with respect to the ground state energy,
in field theory it is common to call the ground state of a system
the " vacuum" . Eq. <ref> is
also called the dispersion relation, since it gives the dependence
of the energy with the wave vector. Note that for B_0=0, ħω(0)=0
and therefore any infinitesimal temperature will cause excitations
with 𝐤=0. This " zero mode"
is an example of a Goldstone mode, which is always present
when there is spontaneous symmetry breaking in the system. In general,
Goldstone modes are massless bosons quasiparticles which appear in
systems where there is a spontaneously broken continuous symmetry.

If we compare with the ground state energy E_0(B_0), in particular
the Zeeman term, we see that the magnetic moment of the system has
been modified by one unit. We can therefore conclude that a magnon
has spin 1, and therefore it is a bosonic quasiparticle. The
expectation value of a local spin operator Ŝ_i^z with
respect to the one-magnon state |𝐤⟩ can be shown
to be 

    ⟨𝐤|Ŝ_i^z|𝐤⟩=S-1/N=⟨0|Ŝ_i^z|0⟩-1/N  ∀   i, 𝐤

which shows that the spin reduction is indeed of one unit and it is
distributed uniformly over all sites 𝐑_i. Semiclassically,
one can picture each spin in the lattice precessing around the z
axis with a projection of ħ(S-1/N). However, except for 𝐤=0,
the spins do not precess in phase but instead they have a phase difference
of e^-i𝐤·(𝐑_i-𝐑_j), forming
a spin wave. This semiclassical picture can be better understood
if we look at the equations of motion for the spins, which we do in
the following section.

We finishing this section by stating that our intuition fails again
if we want to construct two-magnon states. The obvious choice 

    |𝐤,𝐤'⟩∝Ŝ_𝐤^-Ŝ_𝐤'^-|0⟩

is actually not an eigenstate of the Heisenberg Hamiltonian. This
is due to magnon-magnon interactions present in the Hamiltonian, which
are not taken into account in a simple product state as <ref>.
The same of course holds for multiple-magnon states. Therefore, when
having multiple magnons excited in a system, they can interact and
magnon states will decay due to magnon-magnon interactions. These
excited states therefore have a certain lifetime. Besides magnon-magnon
interactions, scattering with impurities or phonons in a material
will determine the lifetime of magnon states.



 §.§ Check points



  * What is a magnon? 

  * How do you obtain the energy dispersion of a magnon? 




§ EQUATION OF MOTION APPROACH


In the Heisenberg picture, the time dependence is included in the
operators. Instead of the Schrdinger equation for the state vectors,
we write the Heisenberg equation of motion for the spin operators

    ħd𝐒̂_i/dt=i[Ĥ,𝐒̂_i] .

Using the Heisenberg Hamiltonian Eq. <ref>
and the commutation relations for the spin operators, it is straightforward
to show 

    ħd𝐒̂_i/dt=-(∑_jJ_ij𝐒̂_j+gμ_ B𝐁_0)×𝐒̂_i .

This equation is exact. We can compare it however with the classical
equation of motion for the angular momentum given in Eq. <ref>,
and we see that, by recovering the units of the spin and absorbing
the Planck constant into the definition of the exchange constant,
Eq. <ref> can be directly translated into a classical
equation of motion by taking the expectation values of the spin operators,
and in particular 

    𝐁_eff=𝐁_0+1/gμ_ B⟨∑_jJ_ij𝐒̂_j⟩

is the effective magnetic field including the Weiss molecular fields.

We turn now to a different approximation, in which we retain for now
the operator character of 𝐒̂_i. We are, as in the
previous section, interested in the low energy excitations of the
system on top of the ground state. Since the ground state is fully
polarized, we expect the projection Ŝ_i^z to remain almost
constant and close to S. From Eq. <ref> we
see this is valid as long as NS≫1, that is, the total spin number
is much larger than the number of excitations in the system. From
Eq. <ref> we obtain 

    ħdŜ_i^x/dt   ≈-S∑_jJ_ij(Ŝ_j^y-Ŝ_i^y)+gμ_ BB_0Ŝ_i^y
    ħdŜ_i^y/dt   ≈-S∑_jJ_ij(Ŝ_i^x-Ŝ_j^x)-gμ_ BB_0Ŝ_i^x
    ħdŜ_i^z/dt   ≈0 .

These equations are decoupled for the ladder operators 

    ħdŜ_i^±/dt=∓ i[S∑_jJ_ij(Ŝ_i^±-Ŝ_j^±)+gμ_ BB_0Ŝ_i^±] ,

where we write the equal sign in the understanding that the equation
is valid in the limit established for Eqs. <ref>.

Eqs. <ref> still couple spin operators at different
sites. To decouple them, we go once more to the Fourier representation.
Using Eq. <ref>, for the lowering operator one obtains

    ħdŜ_𝐤^-/dt=iS[J(𝐤=0)-J(𝐤)]Ŝ_𝐤^-+igμ_ BB_0Ŝ_𝐤^- .

Hence we see that the equation of motion for each wave vector 𝐤
is decoupled from the rest, in the spirit of a normal modes decomposition.
Eq. <ref> is easily solved by 

    Ŝ_𝐤^-=M̂_𝐤e^iω(𝐤)t+iα_𝐤

with the time dependence given entirely by the exponential term, and
ω(𝐤) is given by Eq. <ref> (α_𝐤
is an arbitrary phase). We have therefore re-derived the dispersion
relation obtained in the previous section for the one-magnon state.

In the semiclassical picture and considering only one excited mode,
for the real-space components of the spin we obtain 

    S_i^x    =M_𝐤/√(N)cos(𝐤·𝐑_i+ω(𝐤)t)
    
    S_i^y    =M_𝐤/√(N)sin(𝐤·𝐑_i+ω(𝐤)t)
    
    S_i^z    =S

which are the components of a plane wave with frequency ω(𝐤).

Coming back to the Heisenberg equation of motion for Ŝ_𝐤^-

    ħdŜ_𝐤^-/dt=i[Ĥ,𝐒̂_𝐤^-]

from Eq. <ref> we can write 

    ħω(𝐤)Ŝ_𝐤^-|0⟩=[Ĥ,𝐒̂_𝐤^-]|0⟩=ĤŜ_𝐤^-|0⟩-Ŝ_𝐤^-E_0|0⟩

and therefore we recover 

    ĤŜ_𝐤^-|0⟩=[E_0-ħω(𝐤)]Ŝ_𝐤^-|0⟩

as in the exact result.

As an example, we give the dispersion relation for spins on a cubic
lattice with nearest-neighbor interactions. For this case, 

    J(𝐤)=2J(cos(k_xa)+cos(k_ya)+cos(k_za))

and therefore 

    ħω(𝐤)=2JS(3-cos(k_xa)+cos(k_ya)+cos(k_za))+gμ_ BB_0 .

For small 𝐤 we obtain a quadratic dispersion 

    ħω(𝐤)≈ JSk^2a^2+gμ_ BB_0 ,

which is gapped as long as B_0≠0. Note that this dispersion
is different from the usual acoustic phonon dispersion in solids,
which is linear in k. The dispersion <ref>
is actually similar to that encountered for flexural phonon
modes in materials with reduced dimensionality: for example, the out-of-plane
phonon modes of a graphene membrane. The k^2 dispersion is a
signature of the rotational symmetry of the problem.



 §.§ Check points



  * Derive the Heisenberg equation of motion for the spins from the Heisenberg
Hamiltonian 

  * Relate the concept of magnon from the previous section, with the semiclassical
picture given by the equations of motion. 




§ HOLSTEIN-PRIMAKOFF TRANSFORMATION


Due to their algebra (that is, their commutation relations), angular
momentum operators are difficult to treat in an interacting theory.
There are however transformations which write the angular momentum
operators in terms of second-quantization creation and annihilation
operators, either fermionic or bosonic. The idea of these transformations
is to simplify the commutation rules, so that one can use well known
methods of second quantization. The price to pay is that the transformations
are non linear. In this section we go over one of these transformations
which is widely used, the Holstein-Primakoff transformation. Within
this transformation, the angular momentum operators are written as
non-linear functions of bosonic creation and annihilation operators,
that is, a collection of harmonic oscillators.

Before writing the transformation explicitly we remind briefly the
properties of creation (â_i^†) and annihilation
(â_i) harmonic-oscillator operators. The subscript i
indicates the lattice site, that is, we have a harmonic oscillator
at every site on the lattice. The commutation relations for these
bosonic operators are 

    [â_i, â_j^†]     =δ_ij
    [â_i, â_j]     =[â_i^†, â_j^†]=0

The Hamiltonian of a single oscillator reads 

    H_ osc=ħω_i(â_i^†â_i+1/2) ,

where ω_i is the frequency of oscillator i. If the oscillators
are independent, the total Hamiltonian is simply the sum over the
respective Hamiltonians. Usually however the original operators are
not independent, but if the Hamiltonian is quadratic in these,
one can find a linear transformation which diagonalizes the Hamiltonian,
so that in the new basis the Hamiltonian is a sum of harmonic oscillators
Eq. <ref>. A state |n_i⟩ with n_i
" particles" at site i can be constructed
from the vacuum |0_i⟩ by applying â_i^†
(below we will identify n_i with the number of flipped spins
at site i). In general, we have 

    â_i|0_i⟩    =0
    â_i^†|n_i⟩    =√(n_i+1)|n_i+1⟩
    â_i|n_i⟩    =√(n_i)|n_i-1⟩
    â_i^†â_i|n_i⟩    =n_i|n_i⟩=n̂_i|n_i⟩ ,

where n̂_i=â_i^†â_i is the number
operator at site i. Therefore |n_i⟩ is an eigenstate
of H_ osc with energy E(n_i)=ħω_i(n_i+1/2),
with n_i=0, 1, 2, ....

Comparing Eqs. <ref> with Eq. <ref> and Ŝ_i^z

    Ŝ_i^±|S,m_i⟩    =√((S∓ m_i)(S+1± m_i))|S,m_i±1⟩
    Ŝ_i^z|S,m_i⟩    =m_i|S,m_i⟩

we see that the creation and annihilation bosonic operators act in
a similar way to the spin ladder operators, whereas the number operator
is diagonal in this basis just as Ŝ_i^z. However we cannot
replace simply Ŝ_i^± by â_i^†, â_i
since this would not satisfy the commutation relations for the spin.
This is accounted for by using a non-linear transformation, the Holstein-Primakoff
transformation 

    Ŝ_i^+    =√(2S)√(1-â_i^†â_i/2S)â_i
    Ŝ_i^-    =√(2S)â_i^†√(1-â_i^†â_i/2S)
    Ŝ_i^z    =(S-â_i^†â_i) .

As we anticipated, in this case the number operator counts the number
of flipped spins, which we can see from the last equality in Eqs.
<ref> 

    Ŝ_i^z|n_i⟩=(S-n̂_i)|n_i⟩=(S-n_i)|n_i⟩≡ m_i|n_i⟩ .

For n_i=0, the spin is fully polarized and m_i=S, hence
the Holstein-Primakoff vacuum |n_i=0⟩ corresponds to the
fully polarized spin state. We see however that the spectrum of n̂_i
is constrained, due to the square root in Eqs. <ref>
we must impose n_i=0, 1, ...2S. The maximum value of n_i
corresponds to the spin " fully flipped"
, m_i=-S. We observe that 

    Ŝ_i^-|m_i=-S⟩    =√(2S)â_i^†√(1-n̂_i/2S)|n_i=2S⟩=√(2S)â_i^†√(1-2S/2S)|n_i=2S⟩=0
    Ŝ_i^+|m_i=-S-1⟩    =√(2S)√(1-n̂_i/2S)â_i|n_i=2S+1⟩
        =√(2S)√(2S+1)√(1-n̂_i/2S)|n_i=2S⟩=0

and hence the ladder operators do not connect the physical subspace
n_i=0, 1, ...2S with the unphysical one n_i>2S.

Let's now define 

    ϕ(n̂_i)=√(2S)√(1-n̂_i/2S)

and write the Heisenberg Hamiltonian Eq. <ref>
in terms of the Holstein-Primakoff operators, Eqs. <ref>.
We find 

    Ĥ=-NS^2J_0/2+SJ_0∑_in̂_i-S∑_ijJ_ijϕ(n̂_i)â_iâ_j^†ϕ(n̂_j)-1/2∑_ijJ_ijn̂_in̂_j

with J_0=∑_iJ_ij. We see that, due to Eq. <ref>,
this Hamiltonian is not quadratic in the â_i^†,
â_i operators and therefore we cannot write it as a sum
of independent harmonic oscillators. We have hence transformed our
original Heisenberg Hamiltonian of interacting spins into a Hamiltonian
of interacting bosons.



 §.§ Check points



  * Write the Holstein-Primakoff transformation 

  * What is special about it? 




§ SPIN-WAVE APPROXIMATION


We will now proceed to reformulate our Hamiltonian into a non-interacting
term (namely, non-interacting magnons), plus interaction terms. If
we expand Eq. <ref> as a series 

    ϕ(n̂_i)/√(2S)=1-n̂_i/4S-n̂_i^2/32S^2- ...

we can write Eq. <ref> also as a series 

    Ĥ=-NS^2J_0/2+∑_n=1^∞:Ĥ_2n:

with :Ĥ_2n: containing n creation and n annihilation
operators in normal order (all â_i^† to the left,
all â_i to the right, e.g. for n=2, â_i^†â_j^†â_mâ_l).
The terms with n>1, that is, beyond quadratic, give rise to magnon-magnon
interactions as we will see below. We will however first study the
spin-wave approximation, where only the quadratic, non-interacting
terms are kept in the expansion Eq. <ref> 

    :Ĥ_2n:=SJ_0∑_in̂_i-S∑_ijJ_ijâ_i^†â_j .

This truncation of the series is justified at low temperatures, where
the number of excitations (total number of flipped spins) is small
compared with the total number of spins NS. For that to hold, the
average number of flipped spins per site has to be small n_i≪ S
and therefore we can approximate the square root in Eq. <ref>
to 1 and hence ϕ(n̂_i)≈√(2S). Within this approximation,
the spin ladder operators are indeed approximated by simple harmonic
oscillators, while the z component is kept at saturation 

    Ŝ_i^+   ≈√(2S)â_i
    Ŝ_i^-   ≈√(2S)â_i^†
    Ŝ_i^z   ≈ S ,

and it can be directly seen that the Heisenberg Hamiltonian Eq. <ref>
is quadratic in the â_i^†, â_i operators.
This approximation is completely analogous to the one we performed
when working with the equation of motion, Eq. <ref>.
Note that, although the Hamiltonian is quadratic, it is not diagonal
in i, j (see Eq. <ref>). Just as we did for the equations
of motion, we need to go to Fourier space to obtain a diagonal Hamiltonian
and therefore decoupled harmonic oscillators. In this case, we transform
simply the bosonic operators 

    â_𝐤    =1/√(N)∑_𝐑_ie^-i𝐤·𝐑_iâ_i
    â_𝐤^†    =1/√(N)∑_𝐑_ie^i𝐤·𝐑_iâ_i^†

and, within this approximation, we can show that the Heisenberg Hamiltonian
reduces to 

    Ĥ_ sw=E(B_0)+∑_𝐤ħω(𝐤)â_𝐤^†â_𝐤

where we have added an external magnetic field for completeness, and
E(B_0) and ω(𝐤) are given by Eqs. <ref>
and <ref> respectively.

The Hamiltonian <ref> describes a system of uncoupled harmonic
oscillators. Its eigenstates are simply products of one-magnon states,
that can be obtained from the vacuum by applying repeatedly â_𝐤^†

    |ψ_ SW⟩=∏_𝐤(â_𝐤^†)^n_𝐤|0⟩

where n_𝐤 is the number of magnons with wavevector 𝐤,
eigenvalue of the number operator in Fourier representation n̂_𝐤=â_𝐤^†â_𝐤.
We see that we can write the one-magnon state defined in Eq. <ref>
as |𝐤⟩=â_𝐤^†|0⟩.
This state is an eigenstate of both the full Heisenberg Hamiltonian
and of the non-interacting Hamiltonian <ref>. States with
n_𝐤>1 are however only eigenstates of <ref>.



 §.§ Check points



  * What is the meaning of the spin-wave approximation in terms of the
Holstein-Primakoff transformation? 

  * What is a magnon in this language? 

  * What is the meaning of the Heisenberg Hamiltonian in terms of the
bosonic operators? 




§ MAGNON-MAGNON INTERACTIONS


We now proceed to investigate the higher order terms (n>1) in Eq.
<ref>. We consider for simplicity a Heisenberg Hamiltonian
with nearest neighbors interactions 

    Ĥ_ n.n.=-J/2∑_⟨ ij⟩(Ŝ_i^+Ŝ_j^-+Ŝ_i^-Ŝ_j^+/2+Ŝ_i^zŜ_j^z) .

Inserting Eqs. <ref> generally we obtain 

    Ĥ_ n.n.    =-J/2∑_⟨ ij⟩[S√(1-â_i^†â_i/2S)â_iâ_j^†√(1-â_j^†â_j/2S)+Sâ_i^†√(1-â_i^†â_i/2S)√(1-â_j^†â_j/2S)â_j]
        -J/2∑_⟨ ij⟩(S-â_i^†â_i)(S-â_j^†â_j) .

We now keep the first two terms in the expansion of ϕ(n̂_i),
see Eq. <ref>. Therefore, simply inserting into Eq. <ref>

    Ĥ_ n.n.   ≈-J/2∑_⟨ ij⟩S(1-â_i^†â_i/4S)â_iâ_j^†(1-â_j^†â_j/4S)
        -J/2∑_⟨ ij⟩Sâ_i^†(1-â_i^†â_i/4S)(1-â_j^†â_j/2S)â_j
        -J/2∑_⟨ ij⟩(S-â_i^†â_i)(S-â_j^†â_j) .

To be consistent with the approximation we keep terms with up to four
creation/annihilation operators in Eq. <ref>.
One obtains 

    Ĥ_ n.n.   ≈-NzJS/2^2+JS∑_⟨ ij⟩(â_i^†â_i+â_j^†â_j-â_i^†â_j-â_j^†â_i)
        -J∑_⟨ ij⟩[â_i^†â_iâ_j^†â_j-1/4(â_i^†â_i^†â_iâ_j+â_i^†â_j^†â_jâ_j+â_j^†â_i^†â_iâ_i+â_j^†â_j^†â_jâ_i)]

with the following terms in the expansion being of order 1/S or
higher.

We already saw that the quadratic terms in Eq. <ref>
can be diagonalized by going to the Fourier representation of the
operators â_i, after which one obtains the Hamiltonian
<ref> — in this case with B_0=0 and ω(𝐤)
the corresponding one for nearest neighbors interaction. Here we pay
attention to the new terms, for simplicity we look at one of them,
e.g. the term containing â_j^†â_i^†â_iâ_i.
We denote with Δ the nearest-neighbor vector. Hence

    J/4∑_⟨ ij⟩â_j^†â_i^†â_iâ_i    =J/4∑_𝐑_j,Δâ_j^†â_j+Δ^†â_j+Δâ_j+Δ
        =J/4N^2∑_𝐑_j,Δ∑_𝐤_1,𝐤_2,𝐤_3,𝐤_4â_𝐤_1^†e^-i𝐤_1·𝐑_jâ_𝐤_2^†e^-i𝐤_2·(𝐑_j+Δ)×
       ×â_𝐤_3e^i𝐤_3·(𝐑_j+Δ)â_𝐤_4e^i𝐤_4·(𝐑_j+Δ)
        =J/4N^2∑_𝐑_j,Δ∑_𝐤_1,𝐤_2,𝐤_3,𝐤_4e^-i(𝐤_1+𝐤_2-𝐤_3-𝐤_4)·𝐑_j×
       ×â_𝐤_1^†â_𝐤_2^†â_𝐤_3â_𝐤_4e^-i(𝐤_2-𝐤_3-𝐤_4)·Δ
        =J/4N∑_𝐤_1,𝐤_2,𝐤_3,𝐤_4δ_𝐤_1+𝐤_2,𝐤_3+𝐤_4â_𝐤_1^†â_𝐤_2^†â_𝐤_3â_𝐤_4×
       ×∑_Δe^-i(𝐤_2-𝐤_3-𝐤_4)·Δ
        =J/4N∑_𝐤_1,𝐤_2,𝐤_3,𝐤_4δ_𝐤_1+𝐤_2,𝐤_3+𝐤_4â_𝐤_1^†â_𝐤_2^†â_𝐤_3â_𝐤_4∑_Δe^i𝐤_1·Δ .

The last sum is simply a function of 𝐤_1, which can
be given explicitly once the lattice is known. For example for a cubic
lattice of lattice constant a 

    γ(𝐤_1)=∑_Δe^i𝐤_1·Δ    =e^ia𝐤_1x+e^-ia𝐤_1x+e^ia𝐤_1y+e^-ia𝐤_1y+e^ia𝐤_1z+e^-ia𝐤_1z
        =2[cos(a𝐤_1x)+cos(a𝐤_1y)+cos(a𝐤_1y)] .

The term â_𝐤_1^†â_𝐤_2^†â_𝐤_3â_𝐤_4
corresponds to two magnons with momentum 𝐤_3 and 𝐤_4
respectively being annihilated, and two magnons with momentum 𝐤_1
and 𝐤_2 being created in the interaction process. The
Kronecker delta δ_𝐤_1+𝐤_2,𝐤_3+𝐤_4
ensures conservation of momentum, 𝐤_1+𝐤_2=𝐤_3+𝐤_4.
The other 4-magnon interaction terms in <ref>
can be treated analogously. One can easily see that for long wavelength
magnons (i.e. k small), the scattering cross section of such processes
go as (ka)^4 and is therefore small. Further interaction
terms (6-magnon, etc) are suppressed by factors of increasing order
in 1/S <cit.>.

Including dipole-dipole interactions has two main effects: (i) it
modifies the dispersion relation ω(𝐤), which in that
case depends on the angle between the wavevector 𝐤 and
the equilibrium direction of the saturated spins, since the dipole-dipole
interaction is anisotropic. This gives rise to a spin-wave manifold.
(ii) New 3-magnon momentum-conserving interaction terms, e.g. â_𝐤_1^†â_𝐤_2^†â_𝐤_3
or â_𝐤_1â_𝐤_2â_𝐤_3^†
are allowed: one magnon can split into two, and vice-versa.



 §.§ Check points



  * Why do we get magnon-magnon interactions? 

  * What conservation rules do they fulfill and where do they come from? 




CHAPTER: MAGNETO-OPTICAL EFFECTS


In this chapter we will explore the interaction between light and
magnetism in magnetic insulators. The coupling mechanism is the Faraday
effect, in which the plane of polarization of the light rotates as
it goes through a magnetized material. In turn, the light exerts a
very tiny effective " magnetic field"
on the spins: this is called the inverse Faraday effect and
it is an example of backaction. In what follows we will go
back to the classical realm to obtain the coupling term. This will
allows us to, by proper quantization of the classical coupling energy
term, obtain a coupling Hamiltonian between magnons and the quanta
of light, photons.



§ ELECTROMAGNETIC ENERGY AND ZERO-LOSS CONDITION 


We go back now to the full Maxwell equations in matter, in contrast
to the magnetostatic approximation we used throughout Chapter <ref>

    ∇×𝐇    =∂𝐃/∂ t+𝐣_ F
    ∇×𝐄    =-∂𝐁/∂ t
    ∇·𝐃    =ρ
    ∇·𝐁    =0 .

These equations describe completely an electromagnetic system once
we give the constitutive equations 

    D_i    =ε_ijE_j
    
    B_i    =μ_ijH_j

where we used the Einstein convention of summation over repeated indices.
These constitutive equations assume an instantaneous response of the
system: the response does not depend on time and the system has no
memory. This is referred to as a dispersionless. For an isotropic
system, the permittivity and permeability tensors are diagonal and
proportional to the identity, ε_ij=ε_0ε_rδ_ij,
μ_ij=μ_0μ_rδ_ij and Eqs. <ref>
reduce to the scalar versions, 𝐃=ε_0ε_r𝐄
and 𝐁=μ_0μ_r𝐇.

We will argue that we can represent the coupling of light and magnetization
just by using the permittivity tensor. Our aim now is to obtain symmetry
conditions on the permittivity tensor ε_ij in the presence
of a static magnetization in the material where the light propagates.
For that we will use conservation of electromagnetic energy, given
by a continuity equation involving the energy flux density. The instantaneous
electromagnetic power per unit area is given by the Poynting vector

    𝐏=𝐄×𝐇 .

If we consider volume V bounded by a surface 𝒮, the
energy per unit time entering the volume is given by 

    -∮_𝒮𝐏· d𝐬

where d𝐬 is an area element with vector pointing
outwards. This power can be stored in the volume in the form of an
energy density W, or dissipated - we denominate the dissipated
power P_d. We can hence write 

    -∫_V∇·𝐏 d^3r=∫_V∂ W/∂ t d^3r+∫_VP_d d^3r

where on the LHS we have used the divergence theorem. Since the volume is
arbitrary, we have 

    ∇·𝐏+∂ W/∂ t+P_d=0

which has the form of a continuity equation. It remains to identify
the terms W and P_d in terms of the electromagnetic fields.
For that we look at the Maxwell equations and we see that by taking
the scalar product of <ref> with 𝐄, of <ref>
with 𝐇, and subtracting <ref> from <ref>,
we can obtain an equation for the Poynting vector 𝐏 by
using the vector identity 

    ∇·(𝐀×𝐂)=𝐂·(∇×𝐀)-𝐀·(∇×𝐂) .

Putting all together we obtain the Poynting theorem 

    ∇·(𝐄×𝐇)+𝐇·∂𝐁/∂ t+𝐄·∂𝐃/∂ t+𝐄·𝐣_ F    =0 
    ∇·(𝐄×𝐇)+H_i∂(μ_ijH_j)/∂ t+E_i∂(ε_ijD_j)/∂ t+𝐄·𝐣_ F    =0 ,

where in the last line we have used Eqs. <ref>.
For dispersionless, isotropic media, we obtain 

    ∇·(𝐄×𝐇)+μ_0μ_r/2∂ H^2/∂ t+ε_0ε_r/2∂ E^2/∂ t+𝐄·𝐣_ F=0 ,

from where we can identify 

    P_d    =𝐄·𝐣_ F
    
    W     =μ_0μ_r/2H^2+ε_0ε_r/2E^2

as the dissipated power density and instantaneous energy density stored
in the magnetic and electric fields, respectively.

We are interested however in time-averaged quantities. To proceed
further we consider for simplicity monochromatic fields in complex
notation 

    𝐄(t)     = Re{𝐄(ω)e^-iω t}
    𝐇(t)     = Re{𝐇(ω)e^-iω t} .

It can be easily shown that the time average of the product of two
oscillating fields A(t)=A_0cos(ω t), B(t)=B_0cos(ω t+ϕ)
over one period T=2π/ω is given simply by 

    ⟨ A(t)B(t)⟩_T=1/2 Re{ÃB̃^*}

where 

    A(t)     = Re{ A_0e^-iω t} = Re{Ãe^-iω t}
    
    B(t)     = Re{ B_0e^-iϕe^-iω t} = Re{B̃e^-iω t} .

As a rule, one works with the complex fields and takes the real part
at the end of the calculation. In an abuse of notation, the tilde
notation is dropped. We will use Eq. <ref> to obtain
a time average of the Poynting theorem given in Eq. <ref>.
For that we write Maxwell equations in frequency space, in particular

    ∇×𝐇    =-iω𝐃+𝐣_ F
    ∇×𝐄    =iω𝐁 .

To obtain the Poynting theorem in complex form, we take the complex
conjugate of Eq. <ref> and perform the scalar product
with 𝐄, and take the scalar product of Eq. <ref>
with 𝐇^*. Subtracting the resulting equations and using
again the vector identity <ref> we obtain 

    ∇·(𝐄×𝐇^*)+iω(𝐄·𝐃^*-𝐇^*·𝐁)+𝐄·𝐣_ F^*=0 ,

form where the time average is easily obtained as 

    Re{∇·(𝐄×𝐇^*)+iω(𝐄·𝐃^*-𝐇^*·𝐁)+𝐄·𝐣_ F^*} =0 .

For lossless media, ⟨∇·(𝐄×𝐇^*)⟩_T
must vanish, since all power that enters a volume must leave within
one cycle. Moreover, if there are no free currents, the dissipated
power 𝐄·𝐣_ F^* is also zero. Therefore,
in lossless media 

    Re{ iω(𝐄·𝐃^*-𝐇^*·𝐁)} =0 .

Moreover, in the frequency domain dispersive effects are included
in a simple way by frequency-dependent permittivity and permeability
tensors 

    𝐃(ω)     =ε̅(ω)·𝐄(ω)
    𝐁(ω)     =μ̅(ω)·𝐇(ω)

where the bar indicates that ε̅(ω), μ̅(ω)
are matrices. Eq. <ref> then can be written
as <cit.> 

    iω/2[𝐄^*·(ε̅^†-ε̅)·𝐄+𝐇^*·(μ̅^†-μ̅)·𝐇]=0

where † indicates complex conjugate and transpose: (ε_ij)^†=ε_ji^*
(note that the same expression can be obtained directly from Eq. <ref>
by replacing the real fields using Re{ z} =(z+z^*)/2
and noting that ⟨ zz⟩_T=⟨ z^*z^*⟩_T=0).
We deduce therefore that for lossless media 

    ε̅^†    =ε̅
    μ̅^†    =μ̅ ̅,̅

that is, the permittivity and permeability must be Hermitian matrices.
Note that if the material is isotropic and ε̅(ω),
μ̅(ω) can be written as in principle complex scalars
ε(ω)=ε'(ω)+iε”(ω),
μ(ω)=μ'(ω)+iμ”(ω), the zero-loss condition
implies that the imaginary parts ε”(ω) and μ”(ω)
must vanish.

To define the average electromagnetic energy density in the presence
of dispersion is a little bit more subtle <cit.>.
We give here for completeness the corresponding expression without
proof 

    ⟨ W⟩_T=1/4[𝐄^*·∂(ωε̅)/∂ω·𝐄+𝐇^*·∂(ωμ̅)/∂ω·𝐇] .

If ε̅ and μ̅ are independent of frequency,
this expression reduces to 

    ⟨ W⟩_T=1/4[𝐄^*·ε̅·𝐄+𝐇^*·μ̅·𝐇]

as expected. 


  * Exercise: Prove Eq. <ref> starting
from <ref>. 




 §.§ Check points



  * Obtain Eq. <ref> from Maxwell equations 

  * What is the zero loss condition? 

  * What does it tell us about the symmetries of the permittivity tensor? 




§ PERMITTIVITY TENSOR AND MAGNETIZATION


In the following section we will use the permittivity tensor ε̅
to describe the Faraday effect in a magnetized sample. For that we
will use the symmetry properties of ε̅ in the presence
of a permanent magnetization, which breaks time reversal invariance.
If we write the permittivity tensor explicitly separating the real
and imaginary parts 

    ε_ij=ε'_ij+iε”_ij ,

the zero-loss condition <ref> tells us that real
and imaginary parts are respectively symmetric and antisymmetric matrices:

    ε'_ij(𝐌)     =ε'_ji(𝐌)
    ε”_ij(𝐌)     =-ε”_ji(𝐌) ,

where we have made explicit a possible dependence on the magnetization
𝐌. On the other hand, Onsager reciprocity relations for
response functions dictate how the permittivity transforms under time
reversal symmetry 

    ε'_ij(𝐌)     =ε'_ji(-𝐌)
    ε”_ij(𝐌)     =ε”_ji(-𝐌) ,

where the time reversed form of ε̅ consists in transposing
the matrix and at the same time inverting the magnetization vector.
We see therefore that the real and imaginary parts are also symmetric
and antisymmetric in the magnetization. Putting all together we obtain

    ε'_ij(𝐌)     =ε'_ji(𝐌)=ε'_ji(-𝐌)
    ε”_ij(𝐌)     =-ε”_ji(𝐌)=ε”_ji(-𝐌) .

In linear response, the permittivity depends linearly on the magnetization.
This is valid as long as the effect of the magnetization on the permittivity
is small. To fulfill conditions <ref> to first order
in the magnetization we write <cit.> 

    ε_ij(𝐌)=ε_0(ε_rδ_ij-ifϵ_ijkM_k)

where we have assumed the material is isotropic, and f is a small
material-dependent parameter related to the Faraday rotation as we
show below. A material for which the permittivity takes this form
is denominated gyrotropic. 


  * Exercise: Prove that Eq. <ref> fulfills <ref>.
 




 §.§ Check points



  * Explain how Eq. <ref> is obtained 




§ FARADAY EFFECT


For optical frequencies one can usually safely take the permeability
of a dielectric as the vacuum permeability μ_0, even for a
magnetic material. This amounts to neglecting the interaction of the
small oscillating magnetic field part of the optical electromagnetic
field with the material. In turn, the interaction between the electric
part of the optical field and the magnetization in the sample is modeled
by the permittivity given in Eq. <ref>. The magnetization
𝐌, even if it has a time dependence, it it much slower
than the optical fields and therefore it is well defined.

To understand how the magnetization dependent permittivity in Eq.
<ref> encapsulates the Faraday effect we will first derive
the Fresnel equation for the optical field, starting from the Maxwell
equations <ref> and <ref> in the absence
of free currents, 𝐣_ F=0. We are interested in
light propagating through a material, we therefore write the electric
and magnetic fields as plane waves of frequency ω and wavevector
𝐤 

    𝐄(t,𝐫)     =𝐄e^-i(ω t-𝐤·𝐫)
    𝐇(t,𝐫)     =𝐇e^-i(ω t-𝐤·𝐫) .

In momentum and frequency representation, Eqs. <ref>
and <ref> read 

    𝐤×𝐄    =μ_0ω𝐇
    𝐤×𝐇    =-ω𝐃 .

Inserting Eq. <ref> into <ref> , using 𝐃=ε̅·𝐄
and the product rule 𝐚×(𝐛×𝐜)=𝐛(𝐚·𝐜)-𝐜(𝐚·𝐛)
one obtains 

    k^2/μ_0ω^2[𝐄-𝐤(𝐤·𝐄)/k^2]=ε̅·𝐄 .

This is a form of the Fresnel equation, and it determines the dispersion
relation of the electromagnetic wave (that is, ω(𝐤))
by imposing the determinant of its coefficients to be zero. In components

    k^2/μ_0ω^2[δ_ijE_j-k_ik_jE_j/k^2]=ε_ijE_j

and therefore 

    det{k^2/μ_0ω^2[δ_ij-k_ik_j/k^2]-ε_ij} =0 .


The term 𝐤(𝐤·𝐄)/k^2 gives us simply
the projection of the electric field along the propagation direction
𝐤/k . Whereas in vacuum the electric field 𝐄
is purely transverse, in a medium the purely transverse field is actually
𝐃, see Eq. <ref>. We will work now however
with the particular form of the permittivity given in Eq. <ref>,
and consider the direction of propagation 𝐤/k to coincide
with the magnetization axis, 𝐌=Mẑ, in which case
the 𝐄 field is also transverse as one can easily verify
by using the resulting form of the permittivity tensor 

    ε̅=ε_0([  ε_r -ifM    0;  ifM  ε_r    0;    0    0  ε_r ]) .

The Fresnel equation reduces therefore to 

    |([ k^2/μ_0ω^2-ε_0ε_r               ifM;              -ifM k^2/μ_0ω^2-ε_0ε_r ])|=0

with solutions 

    k_±^2=(ω/c)^2(ε_r∓ fM)

wehere we have used that c=1/√(ε_0μ_0). Inserting
these solutions back into Eq. <ref> with ε̅
given by Eq. <ref> (in this case only two dimensional,
since E_z=0) we obtain 

    E_x=∓ iE_y ,

with ∓ corresponding to k_±^2. We have therefore obtained
two solutions for the propagating wave along ẑ, with the
same amplitude and cicularly polarized in the xy plane, but with
opposite polarizations for k_+ and k_-: 

    𝐄_±(z,t)=E_0 Re{(𝐞̂_x± i𝐞̂_y)e^i(k_±z-ω t)} .


To derive the Faraday rotation we consider now an EM wave propagating
in the medium such that at z=0 it is linearly polarized along x̂
with amplitude E_0, 

    𝐄(z=0,t)=E_0𝐞̂_xe^-iω t .

The linear combination of Eqs. <ref> 

    𝐄(z,t)=1/2[𝐄_+(z,t)+𝐄_-(z,t)]

fulfills the condition <ref>. We therefore have as
solution for the propagating wave 𝐄(z,t)= Re{ E_x𝐞̂_x+E_y𝐞̂_y}

    E_x    =E_0/2(e^ik_+z+e^ik_-z)
    
    E_y    =iE_0/2(e^ik_+z-e^ik_-z) .

After the wave propagated a distance L through the material 

    E_y/E_x=tan[(k_--k_+/2)L] ,

which indicates that the plane of polarization of the light rotated
by an amount θ_ FL, where 

    θ_ F=k_--k_+/2

is the Faraday rotation per unit length. This is depicted schematically
in Fig. <ref>. 


Using Eq. <ref> and fM≪ε_r we obtain 

    θ_ F=ω/2c√(ϵ_r)fM .





 §.§ Check points



  * What is the Faraday effect? 




§ MAGNETO-OPTICAL ENERGY


In the previous sections we saw that the magnetization in a medium
modifies the permittivity tensor, which acquires an antisymmetric
imaginary term due to the breaking of time reversal symmetry. As light
goes through the material, it experiences this effective permittivity
which we saw leads to the Faraday effect. Light and magnetization
in the sample are therefore coupled. To obtain this coupling, we look
at the electromagnetic energy obtained from Eq. <ref>
by using the permittivity Eq. <ref>. We see that the magnetization-dependent
part of the permittivity introduces a correction to the usual electromagnetic
energy expression, given by 

    U_ MO=-i/4fε_0∫ d^3r𝐌(𝐫)·[𝐄^*(𝐫)×𝐄(𝐫)] .

One can easily prove that this term is real. In terms of the Faraday
rotation per unit length this can be rewritten as 

    U_ MO=θ_ F√(ε_r/ε_0)∫ d^3r𝐌(𝐫)/M·ε_0/2iω[𝐄^*(𝐫)×𝐄(𝐫)] .

The term 

    𝐒_ light(𝐫)=ε_0/2iω[𝐄^*(𝐫)×𝐄(𝐫)]

is called the optical spin density and it is related to the
helicity of light. For example, for circularly polarized light 𝐒_ light
points perpendicular to the plane of polarization with a direction
given by the right-hand-rule.

We know from the previous section that the magnetization causes the
plane of polarization of the light to rotate. From Eq. <ref>
we see that the light itself acts as an effective magnetic field on
the magnetization (compare with the usual expression for the Zeeman
energy). This gives rise to the inverse Faraday effect, which
takes into account the effect of the light on the magnetization dynamics
of the sample. This effect is usually small, one can show that the
effective light-induced magnetic field for YIG (Yttrium Iron Garnet,
a magnetic insulator, widely used in both technical applications and
current experiments) is of the order of 10^-11T per photon/μ^3.
For comparison, the earth's magnetic field is of the order of 10^-6T!
We will see however in the following chapter that this value can be
enhanced by using an optical cavity, effectively " trapping"
photons. 


  * Exercise: Derive Eq. <ref> starting from
<ref>. 




 §.§ Check points



  * Obtain the correction to the electromagnetic energy if a medium is
magnetized. 




CHAPTER: MODERN TOPICS: CAVITY OPTOMAGNONICS


In this last chapter we will put together all concepts we learned
so far to derive the basics of a topic of current research: Cavity
Optomagnonics. In these systems, a magnetic insulator material forms
a cavity for the light, which is used to enhance the magnon-photon
coupling. We call this the optomagnonic coupling. As we pointed
out in the previous chapter, the Faraday effect is usually a small
effect, which depends on the Faraday rotation constant of the material
and also on the path's length of the light inside of the material.
Since in a cavity the light is " trapped" ,
this effectively enhances the path's length and therefore the coupling.
This is an intuitive way of seeing the enhancement, we will see this
more formally in the next sections. For that we need first to learn
about optical cavities and the quantization of the electromagnetic
field.



§ QUANTIZATION OF THE ELECTROMAGNETIC FIELD


We start by quantizing a single-mode field in a cavity formed by perfectly
conducting walls at z=0 and z=L. We summarize here the most
important concepts, a more thorough discussion can be found for example
in Refs. <cit.>.
We assume further the electric field to be polarized along x, 𝐄=E_x(z,t)𝐞̂_x.
The boundary condition therefore implies 

    E_x(z=0,t)=E_x(z=L,t)=0 .

From Maxwell equations in vacuum and no sources, 

    ∇×𝐄    =-∂𝐁/∂ t   ∇·𝐁    =0
    ∇×𝐁    =μ_0ε_0∂𝐄/∂ t   ∇·𝐄    =0

we obtain a wave equation for the electric field 

    ∇^2𝐄=μ_0ε_0∂^2𝐄/∂ t^2

which in terms of 𝐄=E_x(z,t)𝐞̂_x simplifies
to 

    ∂^2E_x(z,t)/∂ z^2-1/c^2∂^2E_x(z,t)/∂ t^2=0 .

The solution of Eq. <ref> satisfying the boundary
conditions in Eq. <ref> is simply 

    E_x(z,t)=√((2ω_n^2/Vε_0))q(t)sin(k_nz)

with 

    ω_n/c=k_n=π n/L .

In Eq. <ref>, q(t) has units of length and V=LS
is the volume of the cavity, where S is its cross-section. From
the left Eq. in <ref> one obtains 𝐁=B_y(z,t)𝐞̂_y
with 

    B_y(z,t)=μ_0ε_0/k_n√((2ω_n^2/Vε_0))q̇(t)cos(k_nz) .

Inserting Eqs. <ref> and <ref> into the electromagnetic
energy 

    E_ EM=1/2∫ dV(ε_0𝐄^2+1/μ_0𝐁^2)

one obtains 

    E_ EM=1/2(ω_n^2q^2+p^2)

where we have defined p=q̇ the canonical momentum of a "
particle" of unit mass. Eq. <ref> is the
energy of a harmonic oscillator of unit mass. We can now proceed to
quantize the theory by 

    q    →q̂
    
    p    →p̂

and imposing the commutator [q̂,p̂]=iħ.
It is convenient to introduce the bosonic creation and annihilation
operators 

    â    =1/√(2ħω_n)(ω_nq̂+ip̂)
    â^†    =1/√(2ħω_n)(ω_nq̂-ip̂)

which satisfy [â,â^†] in terms of
which the electric and magnetic field can be expressed as 

    Ê_x(z,t)     =√((ħω_n/Vε_0))(â+â^†)sin(k_nz)
    B̂_y(z,t)     =-i√((ħω_n/Vε_0))(â-â^†)cos(k_nz) .

The energy Eq. <ref> gives rise to the usual harmonic-oscillator
Hamiltonian 

    Ĥ_ EM=ħω_n(â^†â+1/2) .

The time dependence of the ladder operators can be obtained from the
Heisenberg equation of motion and is given by 

    â(t)     =â(0)e^-iω t
    â^†(t)     =â^†(0)e^iω t .

Note that an eigenstate of the number operator n̂=â^†â,
n̂|n⟩=n|n⟩, is an energy eigenstate but the electric
field operator's expectation value vanishes 

    ⟨ n|Ê_x|n⟩=0

and therefore it is not well defined. The expectation value squared
field ⟨ n|Ê_x^2|n⟩ is however finite, as
one can easily prove. This reflects the uncertainty in the phase of
the electric field, which is conjugate to the number operator. We
call the excitation with energy ħω_n a photon.

We now proceed to the quantization of multimode fields in a 3D cavity.
For that, it is convenient to use the Coulomb gauge 

    ∇·𝐀(𝐫,t)=0 ,

in which both electric and magnetic field can be expressed in terms
of the vector potential 𝐀(𝐫,t) 

    𝐄(r,t)     =-∂𝐀(𝐫,t)/∂ t
    𝐁(r,t)     =∇×𝐀(𝐫,t) .

From Maxwell's equations, one obtains a wave equation for the vector
potential 

    ∇^2𝐀-1/c^2∂^2𝐀/∂ t=0

which can be solved by separating time end space variables. It is
customary to split the time dependence 

    𝐀(𝐫,t)=𝐀^+(𝐫,t)+𝐀^-(𝐫,t)

such that 

    𝐀^+(𝐫,t)     =∑_kc_k𝐮_k(𝐫)e^-iω_kt
    𝐀^-(𝐫,t)     =∑_kc_k^*𝐮_k^*(𝐫)e^iω_kt

where ω_k≥0. The mode-functions 𝐮_k are
solutions of 

    (∇^2+ω_k^2/c^2)𝐮_k(𝐫)=0

satisfying orthonormality 

    ∫ dV𝐮_k^*(𝐫)𝐮_k'(𝐫)=δ_k,k'

and the appropriate boundary conditions. The index k indicates
both the mode and the polarization vector. Moreover, due to the Coulomb
gauge 

    ∇·𝐮_k(𝐫)=0 .

Guided by our one-mode example, we quantize replacing the amplitudes
in the sum over modes by creation and annihilation operators 

    𝐀̂(𝐫,t)=∑_k√(ħ/2ω_kε_0)[â_k𝐮_k(𝐫)e^-iω_kt+â_k^†𝐮_k*(𝐫)e^iω_kt]

from which, using Eq. <ref>, we obtain the electric field
operator 

    𝐄̂(𝐫,t)=𝐄̂^+(𝐫,t)+𝐄̂^-(𝐫,t)=i∑_k√(ħω_k/2ε_0)[â_k𝐮_k(𝐫)e^-iω_kt-â_k^†𝐮_k*(𝐫)e^iω_kt]

with the same convention for 𝐄̂^±(𝐫,t)
as for 𝐀̂^±(𝐫,t). Using Eq. <ref>
we can obtain the corresponding expression for the magnetic field.
The bosonic ladder operators â_k, â_k^†
as defined are dimensionless and satisfy the usual commutation relations

    [â_k,â_k']     =[â_k^†,â_k'^†]=0
    [â_k,â_k'^†]     =δ_k,k'

and the corresponding Hamiltonian is that of a collection of non-interacting
harmonic oscillators 

    H_ EM=∑_kħω_k(â_k^†â_k+1/2) .

The sum over modes has no cutoff and therefore the factor 1/2 leads
to a divergence. In these notes we will not worry about that, since
we will always work with differences of energies, where this factor
cancels out. We will therefore simply omit this term in the following.
The energy eigenstates are also eigenstates of the number operator
n̂_k=â_k^†â_k 

    n̂_k|n_k⟩    =n_k|n_k⟩
    
    E_n_k    =ħω_k(n_k+1/2) .

The states |n_k⟩ form an orthonormal basis of the Hilbert
space and are called number or Fock states, where n_k
gives the number of photons in state k. A general multi-mode state
can be written as 

    |ψ⟩=∑_n_1,n_2,...c_n_1,n_2,...|n_1,n_2,...⟩ .


In the simplest example, one uses periodic boundary conditions in
a cubic box of side length L. In this case, 

    𝐮_k(𝐫)=1/L^3/2𝐞̂^λe^i𝐤·𝐫

with 𝐤=2π/L(n_x,n_y,n_z), n_i=0,±1,±2,...
and λ=1,2 indicating the polarization, which from Eq. <ref>
must fulfill 

    𝐞̂^λ·𝐤=0 .

One can analogously use reflecting boundary conditions, where the
solutions are standing waves as in the single mode we studied above.
Note that in this case the normalization factor of 𝐄̂(𝐫,t)
and the quantization of 𝐤 will be different. 


  * Exercise: Derive Eq. <ref> 

  * Exercise: Derive Eq. <ref> 




 §.§ Check points



  * Write a general expression for the quantized electric field 




§ OPTICAL CAVITIES AS OPEN QUANTUM SYSTEMS




Cavities are usually open systems, in contact with an environment
both because, for example, the mirrors are not perfect, allowing contact
to a bath of photons or/and phonons, and because we want to have access
to the cavity by means of an external probe. The environment (also
called bath, or reservoir) is assumed to be very large and in thermal
equilibrium, and it is modeled as a collection of harmonic oscillators.
The simplest Hamiltonian of the cavity plus bath is written as 

    Ĥ=Ĥ_S+Ĥ_R+Ĥ_I

where Ĥ_S , Ĥ_R, and Ĥ_I are the
cavity, reservoir, and interaction Hamiltonians respectively 

    Ĥ_S    =ħΩâ^†â
    Ĥ_R    =∑_kħω_kb̂_k^†b̂_k
    Ĥ_I    =ħ∑_k(g_kâ^†b̂_k+g_k^*b̂_k^†â) ,

where we have taken for simplicity only one mode for the cavity â
with frequency Ω. The interaction Hamiltonian represents an
excitation in the cavity being converted into one in the reservoir
and vice-versa, with coupling constant g_k.

Our aim is to obtain an effective equation of motion for the cavity
mode â which encapsulates the effect of the bath <cit.>.
This procedure is denominated to integrate out the bath, and
it means that, since we are not interested in the dynamics of the
bath per se, we want to eliminate these degrees of freedom and just
retain the ones we are interested in, in this case, the single cavity
mode. We begin by writing the Heisenberg equations of motion for both
the cavity mode and the reservoir modes 

    â̇(t)     =-iΩâ(t)-i∑_kg_kb̂_k(t)
    ḃ̂̇_k(t)     =-iω_kb̂(t)-ig_k^*â(t) .

We can integrate formally Eq. <ref> to obtain 

    b̂_k(t)=b̂_k(0)e^-iω_kt-ig_k^*∫_0^t dt'â(t')e^-iω_k(t-t') ,

where the first term corresponds to the free evolution of b̂_k
and the second one is due to the interaction with the cavity. Substituting
Eq. <ref> into <ref> we obtain 

    â̇(t)=-iΩâ(t)-i∑_kg_kb̂_k(0)e^-iω_kt-∑_k|g_k|^2∫_0^t dt'â(t')e^-iω_k(t-t') .

We now transform the cavity operators to a rotating frame with frequency
Ω 

    Â(t)=â(t)e^iΩ t .

We see that this transformation preserves the bosonic commutation
relations 

    [Â(t),Â(t)]     =[Â^†(t),Â^†(t)]=0
    [Â(t),Â^†(t)]     =1

and removes the free, fast rotating term from the equation of motion:

    Â̇(t)=-i∑_kg_kb̂_k(0)e^i(Ω-ω_k)t-∑_k|g_k|^2∫_0^t dt'Â(t')e^i(Ω-ω_k)(t-t') .


The first term in Eq. <ref> is denominated the noise
operator  

    F̂(t)=-i∑_kg_kb̂_k(0)e^i(Ω-ω_k)t .

We see that this operator is composed of many different frequencies
and therefore oscillates rapidly in time. Its effect on the cavity
mode is that of exerting random " quantum kicks"
. Its expectation value for a reservoir in thermal equilibrium is
easily shown to be zero 

    ⟨F̂(t)⟩_R=0

and therefore this operator is the quantum analog to the noise due
to the environment responsible for the Brownian motion of a classical
particle. The second term in Eq. <ref> 

    B̂_ba=-∑_k|g_k|^2∫_0^t dt'Â(t')e^i(Ω-ω_k)(t-t')

is due to backaction: changes in the cavity mode affect slightly
the bath, which in turn acts back onto the cavity. We will see in
the following that this term leads to decay of the cavity mode, which
corresponds to dissipation of energy from the cavity into the environment.

To analyze the second term in Eq. <ref> we use that
the environment volume is large, which allows us to take the continuum
limit for the bath modes. We write the sum over modes directly in
terms of a density of states (DOS) which we do not specify,
this will depend on the details of the bath. The DOS 𝒟(ω_k)
gives the number of modes with frequency between ω_k and
ω_k+ dω_k. In terms of the DOS, the second term
in Eq. <ref> is 

    B̂_ba=-∫_0^∞ dω_k𝒟(ω_k)|g(ω_k)|^2∫_0^t dt'Â(t')e^i(Ω-ω_k)(t-t') .

To perform this integral we have to resort to approximations that
rely on the physics of the system. We will work with what is known
as the Weisskopf-Wigner approximation, which is at its core a Markovian
approximation: the evolution of the system of interest is local in
time. This implies a separation of time scales between the bath, which
we assume to be the fast, and the system (in our case, the cavity
mode) which is slow. The information from the system that goes into
the reservoir is lost, since the bath fluctuates very rapidly. The
system therefore is said to have no memory. The timescale of
the bath is defined by the inverse bandwidth 1/W. If the rate of
variation Â(t) is slow compared to this timescale, we can
replace Â(t')→Â(t) in Eq. <ref>,
and extend the integral from t to ∞ 

    B̂_ba≈-∫_0^∞ dω_k𝒟(ω_k)|g(ω_k)|^2Â(t)∫_0^∞ dτ e^i(Ω-ω_k)τ

where we defined τ=t-t'. We use now that 

    ∫_0^∞ dτ e^i(Ω-ω_k)τ=πδ(ω_k-Ω)-i𝒫(1/ω_k-Ω) ,

where the last term indicates the principal part. We neglect this
term for the moment, since it leads to a frequency shift (note that
its contribution is proportional to the cavity operator, and has an
i in front). Evaluating the Delta function we obtain 

    B̂_ba≈-Â(t)π𝒟(Ω)|g(Ω)|^2

and therefore the effective equation of motion for the cavity mode
in the rotating frame is 

    Â̇(t)=-γ/2Â(t)+F̂(t) ,

with 

    γ=π𝒟(Ω)|g(Ω)|^2

the cavity decay rate. Eq. <ref> is a quantum
Langevin equation and the decay rate γ and the noise operator
F̂(t) can be shown to fulfill the fluctuation-dissipation
theorem  

    γ=1/n̅∫_-∞^∞ dτ⟨F̂^†(τ)F̂(0)⟩_R

in equilibrium, with 

    n̅=⟨b̂^†(Ω)b̂(Ω)⟩_R=1/e^ħβΩ-1

the thermal occupation of the bath at the cavity frequency. In the
Markov approximation the noise correlators fulfill 

    ⟨F̂^†(t')F̂(t”)⟩_R    =γn̅δ(t'-t”)
    ⟨F̂(t')F̂^†(t”)⟩_R    =γ(n̅+1)δ(t'-t”) .

In particular for the vacuum one obtains 

    ⟨0|F̂(t')F̂^†(t”)|0⟩_R=γδ(t'-t”) .

This delta-correlated noise shows clearly that the dynamics of the
bath is fast compared to that of the system of interest, and that
it has no memory since every " quantum kick"
is uncorrelated with the previous one. Usually the noise operator
is normalized to an operator Â_ in such that 

    ⟨0|Â_ in(t')Â_ in^†(t”)_|0⟩_R=δ(t'-t”)

and the equation of motion is written as 

    Â̇(t)=-γ/2Â(t)+√(γ)Â_ in(t)

or, in the original frame, 

    â̇(t)=-iΩâ(t)-γ/2â(t)+√(γ)â_ in(t) .

For the decay rate sometimes κ=γ/2 is used.



 §.§ Check points



  * Write the total Hamiltonian of a cavity coupled to an environment,
and explain each term 

  * How does one obtain an effective equation of motion for the cavity
mode? What approximations are involved? 

  * Write the effective equation of motion for the cavity mode 

  * What is the meaning of κ (or γ)? 




§ THE OPTOMAGNONIC HAMILTONIAN


We will now put together all the elements from the previous sections
to derive the optomagnonic Hamiltonian, that is, the Hamiltonian for
a system in which optical photons couple to magnons. For that, we
will quantize the interaction term given by the Faraday effect, Eq.
<ref>. This section and the next follow the recent work
in Ref. <cit.>.

We can quantize the electric field following Sec. <ref>,
Ê^+(𝐫,t)=∑_β𝐄_β(𝐫)â_β(t)
and correspondingly Ê^-(𝐫,t)=∑_β𝐄_β^*(𝐫)â_β^†(t),
where 𝐄_β(𝐫) indicates the β^ th
eigenmode of the electric field (eigenmodes are indicated with greek
letters in what follows). The magnetization requires more careful
consideration, since 𝐌(𝐫) depends on the local
spin operator which, in general, cannot be written as a linear combination
of bosonic modes. There are however two simple cases: (i) the spin-wave
approximation, which is valid for small deviations of the spins from
equilibrium and, as we saw in Sec. <ref>,
the Holstein-Primakoff representation can be truncated to linear order
in the bosonic magnon operators, and (ii) considering the homogeneous
Kittel mode [The Kittel mode is a spin wave with 𝐤=0, so that all spins
precess in phase and can be replaced by a precessing macrospin.] 𝐌(𝐫)=𝐌, for which we can work simply
with the resulting macrospin 𝐒. In the following we treat
this second case. Although it is valid only for the homogeneous magnon
mode, it allows us to capture the nonlinear dynamics of the spin.

From Eq. <ref> we obtain the coupling Hamiltonian 

    Ĥ_MO=ħ∑_jβγŜ_jG_βγ^jâ_β^†â_γ

with coupling constants 

    G_βγ^j=-iε_0f M_s/4ħ Sϵ_jmn∫ d𝐫E_β m^*(𝐫)E_γ n(𝐫) ,

where we replaced M_j/M_s=Ŝ_j/S, with S the extensive
total spin (scaling like the magnetic mode volume). G^j are hermitian
matrices which in general cannot be simultaneously diagonalized. For
simplicity, in the following we treat the case of a strictly diagonal
coupling to some optical eigenmodes (G_ββ^j≠0 but
G_αβ^j=0).

As an example, we consider circular polarization (R/L) in the y-z-plane.
In this case, the optical spin density is perpendicular to this plane,
and therefore G^x is diagonal while G^y=G^z=0. The Hamiltonian
H_MO is then diagonal in the the basis of circularly polarized
waves, 𝐞_R/L=1/√(2)(𝐞_y∓ i𝐞_z).
We choose moreover the magnetization axis along the ẑ
axis. This setup is shown schematically in Fig. <ref>.


The rationale behind choosing the coupling direction perpendicular
to the magnetization axis, is to maximize the coupling to the magnon
mode, that is to the deviations of the magnetization with respect
to the magnetization axis. The light field hence only couples to the
x component of the spin operator, Ŝ_x.

Again for simplicity, we consider the case of plane waves for quantizing
the electric field. Therefore 

    Ê^+(-)(𝐫,t)=+(-)i∑__j𝐞_j√(ħω_j/2ε_0ε V)â_j^(†)(t)e^+(-)i𝐤_𝐣· 𝐫 ,

where V is the volume of the cavity, 𝐤_j the wave
vector of mode j and we have identified the positive and negative
frequency components of the field as 𝐄→𝐄̂^+,
𝐄^*→𝐄̂^-. In the normalization
of the fields we have used the relative permittivity ε
of the magnetic insulator, since the electric fields considered live
in the cavity formed by the material. The factor of ε_0ε
in the denominator ensures the normalization ħω_j=ε_0ε⟨ j|∫ d^3𝐫|𝐄(𝐫)|^2|j⟩-ε_0ε⟨0|∫ d^3𝐫|𝐄(𝐫)|^2|0⟩,
which corresponds to the energy of a photon in state |j⟩
above the vacuum |0⟩. For two degenerate (R/L) modes at frequency
ω, using Eq. <ref> we see that the frequency
dependence cancels out and we obtain the simple form for the optomagnonic
Hamiltonian 

    H_MO=ħ GŜ_x(â_L^†â_L-â_R^†â_R)

with 

    G=1/Sc θ_F/4√(ε).

In general however an overlap factor ξ≤1 appears, which takes
into account that there is a mismatch between the optical and magnonic
mode volumes. For example, current experiments couple optical whispering
gallery modes (WGM) in a YIG sphere to the magnetic Kittel mode <cit.>.
The Kittel mode is a bulk mode, and lives on the whole sphere, that
is, the magnetic mode volume equals the volume of the sphere. The
WGMs live however very close to the surface, and therefore its volume
is smaller than the magnetic one, leading to and overlap factor ξ<1,


    G_LL^x=-G_RR^x=G=1/Sc θ_F/4√(ε)ξ .


We now consider an incoming laser, which drives only one of the two
circular polarizations in the cavity. The total Hamiltonian of the
cavity optomagnonic system is therefore given simply by 

    H=-ħΔâ^†â-ħΩŜ_z+ħ GŜ_xâ^†â ,

where â^† (â) is the creation (annihilation)
operator for the cavity mode photon which is being driven. We work
in a frame rotating at the laser frequency ω_las, and Δ=ω_las-ω_cav
is the detuning vs. the optical cavity frequency ω_cav.
In Eq. <ref> we included that the dimensionless
macrospin S=(S_x,S_y,S_z) has a magnetization axis
along ẑ, and a Larmor precession frequency Ω
which can be controlled by an external magnetic field. 


  * Exercise: show that in the rotating frame the free Hamiltonian
for a cavity driven mode is given by ħΔâ^†â
. 




 §.§ Check points



  * Derive the optomagnonic coupling Hamiltonian 




§ COUPLED EQUATIONS OF MOTION AND
FAST CAVITY LIMIT


The coupled Heisenberg equations of motion are obtained by using [â,â^†]=1,
[Ŝ_i,Ŝ_j]=iϵ_ijkŜ_k.
We next focus on the classical limit, where we replace the operators
by their expectation values: 

    ȧ    =     -i(GS_x-Δ)a-κ/2(a-α_ max)
    𝐒̇    =    (Ga^*a 𝐞_x-Ω 𝐞_z)×𝐒+η_ G/S(Ṡ×𝐒) .

From Sec. <ref>, we know our optical cavity is an open
system and the optical fields are subject to a decay rate. Here we
introduced the cavity decay rate κ phenomenologically, its
value is in general determined by the scpecific experimental setup.
We also included the driving laser amplitude α_ max
for the optical mode. This gives the steady state amplitude of the
light field when it is not coupled to the magnetics and for zero detuning
of the driving laser. We also added an intrinsic damping for the spin
η_ G, which can be due to phonons and defects and it
is material dependent. This coefficient is denominated Gilbert
damping and, whereas it does not change the magnitude of the spin
vector, it causes a decay of the Larmor precession to the stable equilibrium
of the spin. The equation of motion for the spin without coupling
to the light reduces to 

    𝐒̇=-Ω 𝐞_z×𝐒+η_ G/S(Ṡ×𝐒) ,

which is known as the Landau-Lifschitz-Gilbert equation. We
have encountered this equation before, abeit without the damping term.

We see hence that the light acts as a kind of effective magnetic field
on the spin. Actually, since the field a depends on time, and the
spin-light dynamics is coupled, retardation effects cause also dissipation
for the spin, in a similar way in which an environment causes the
dissipation term κ for the optical field. In the following
we will obtain the effective equation of motion for the spin induced
by the light, " integrating out" the light
field. For that we have to resort to an approximation, which is denominated
the fast cavity limit, where the dynamics of the light is much
faster than that of the spin (sometimes it is also called the bad
cavity limit, since it implies that κ is large). That means
that the photons spend in average a very short time in the cavity,
during which they " see" the spin almost
as static.

The condition for the fast cavity limit to be valid is GṠ_x≪κ^2.
In that case we can expand the field a(t) in powers of Ṡ_x.
We write a(t)=a_0(t)+a_1(t)+…, where the subscript indicates
the order in Ṡ_x. From the equation for a(t), we find
that a_0 fulfills the instantaneous equilibrium condition 

    a_0(t)=κ/2α_ max1/κ/2-i(Δ-GS_x(t)) ,

from which we obtain the correction a_1: 

    a_1(t)=-1/κ/2-i(Δ-GS_x)∂ a_0/∂ S_xṠ_x .

To derive the effective equation of motion for the spin, we replace
|a|^2≈|a_0|^2+a_1^*a_0+a_0^*a_1 in Eq. <ref>
which leads to 

    𝐒̇=𝐁_ eff×𝐒+η_ opt/S(Ṡ_x 𝐞_x×𝐒)+η_ G/S(Ṡ×𝐒) .

Here 𝐁_ eff=-Ω𝐞_z+𝐁_ opt,
where 𝐁_ opt(S_x)=G|a_0|^2 𝐞_x
is the purely static contribution and acts as an optically induced
magnetic field. The second term is due to retardation effects, and
it reminiscent of Gilbert damping, albeit with spin-velocity component
only along 𝐱̂ due to the chosen geometry. These are
depicted in Fig. <ref>. 


Both the induced field 𝐁_ opt and dissipation coefficient
η_ opt depend explicitly on the instantaneous value of
S_x(t): 

    𝐁_ opt    =    G/[(κ/2)^2+(Δ-GS_x)^2](κ/2α_ max)^2𝐞_x
    η_ opt    =     -2Gκ S |𝐁_ opt| (Δ-GS_x)/[(κ/2)^2+(Δ-GS_x)^2]^2 .

These fields are highly non-linear functions of the spin. Note that
the optically induced dissipation can change sign! This leads to very
interesting dynamics. Two distinct solutions can be found: generation
of new stable fixed points (switching) and optomagnonic limit cycles
<cit.>. .


  * Exercise: fill in the steps of the derivation above. 




 §.§ Check points



  * What is the fast cavity limit? 

  * How do you obtain an effective equation of motion for the spin, and
what is the meaning of each term? 




§ LINEARIZED OPTOMAGNONIC HAMILTONIAN


As we mentioned in Sec. <ref>, one
can also treat the optomagnonic Hamiltonian in the limit of small
oscillations of the spins, by using a truncated Holstein Primakoff
expansion so that the spin ladder operators are replaced by linear
bosonic operators. This allows to study the behavior of the system
beyond the homogeneous magnetic Kittel mode studied in the previous
two sections, but restricts the analysis to small displacements of
the spins with respect of their equilibrium positions.

In order to proceed with the linearization, we consider spin wave
excitations on top of a possibly nonuniform static ground state 𝐦_0(𝐫),

    δ𝐦(𝐫,t)=𝐦(𝐫,t)-𝐦_0(𝐫) .

For small deviations |δ𝐦|≪1 we can express these
in terms of harmonic oscillators, which correspond to the magnon modes.
This is equivalent to a local Holstein Primakoff approximation. We
can quantize the spin wave as 

    δ𝐦(𝐫,t)→1/2∑_γ[δ𝐦_γ(𝐫)b̂_γe^-iω_γt+δ𝐦_γ^*(𝐫)b̂_γ^†e^iω_γt] .

In turn, the quantization of the optical fields can be written as

    𝐄(𝐫,t)→   ∑_β𝐄_β(𝐫)â_βe^-iω_βt
    𝐄^*(𝐫,t)→   ∑_β𝐄_β^*(𝐫)â_β^†e^iω_βt

From Eq. <ref> we obtain the coupling Hamiltonian linearized
in the spin fluctuations <cit.>

    Ĥ_MO=∑_αβγG_αβγ^â_α^†â_βb̂_γ+ h.c.

where 

    G_αβγ^=-iθ_ Fλ_n/4πε_0ε/2∫ d𝐫 δ𝐦_γ(𝐫)·[𝐄_α^*(𝐫)×𝐄_β(𝐫)]

is the optomagnonic coupling. Note that â correspond to photon
operators, while b̂ correspond to magnonic ones. The Greek
subindices indicate the respective magnon and photon modes which are
coupled. The information on the specific shape and normalization of
the magnon and optical modes is encoded in the respective mode functions
δ𝐦_γ(𝐫) and 𝐄_α(𝐫).
Eqs. <ref> and <ref> allow to treat arbitrary
geometries for the optical cavity, arbitrary magnetic ground states,
and arbitrary spin wave modes.

The Hamiltonian in Eq. <ref> is still nonlinear, since
it involves products of three bosonic operators. This Hamiltonian
therefore still contains " interacting"
three-particle terms. In particular, it describes scattering processes
in which a photon in mode β and a magnon in mode γ
are annihilated creating a photon in the mode α, and the complementary
process in which a photon α is annihilated creating a photon
β and a magnon γ. To bring this Hamiltonian into a
solvable, quadratic form, we linearize now the Hamiltonian Eq. <ref>
in the optical fields. For that we consider fluctuations of the optical
fields around a steady state solution ⟨â_α⟩,
⟨â_β⟩ 

    â_α=    ⟨â_α⟩+δâ_α
    â_β=    ⟨â_β⟩+δâ_β .

The steady state solutions ⟨â_α⟩, ⟨â_β⟩
satisfy ⟨â̇_α⟩=0, ⟨â̇_β⟩=0,
where the time evolution is given by the coupled equations of motion
dictated by the interaction Hamiltonian Eq. <ref> plus
driving and free terms, obtained by generalizing Eq. <ref>
to multiple modes. The average number of photons circulating in cavity
mode α in steady state is simply given by n_α=|⟨â_α⟩|^2
and it is related to the input laser power. To linear order in the
fluctuations defined by Eqs. <ref>, the Hamiltonian Eq.
<ref> reduces to 

    Ĥ_lin=∑_αβγG_αβγ^(√(n_α)δâ_βb̂_γ+√(n_β)δâ_α^†b̂_γ)+ h.c. ,

which is a Hamiltonian linear both in magnon and photon operators.
It contains two types of terms, denominated parametric amplifier,
corresponding to those terms which simultaneously create or annihilate
a photon and a magnon (δâ_βb̂_γ and
b̂_γ^†δâ_β^†), and
beam splitter, which converts a photon into a magnon (b̂_γ^†δâ_α),
and vice-versa (δâ_α^†b̂_γ).
Which of the two types of processes dominates depends on which of
them is in resonance, and can be tuned by the external laser driving.
We note that the optomagnonic coupling constant G_αβγ
is enhanced by the square root of the number of photons circulating
in the corresponding cavity mode. This is similar to optomechanics,
where light in an optical cavity couple to phonons <cit.>.



§ PROSPECTS IN CAVITY OPTOMAGNONICS


Cavity optomagnonic systems are the newest addition to a collection
of platforms being studied nowadays with the aim of manipulating,
processing, and storing quantum information. These systems, usually
of nano and micro scale dimensions, are called hybrid quantum
systems <cit.>, since they combine different degrees
of freedom (such as electronic, mechanical, photonic, or magnetic)
to enhance functionality. For example, whereas optical photons are
good carriers of information, they are not so good for information
processing. Further examples of hybrid systems are nanoelectromechanical
or optomechanical systems. A general underlying property of these
is that they use collective excitations (such as phonons or magnons)
whose properties can be engineered by proper design at the nanoscale.

A challenge in many of these platforms is that the coupling between
the different degrees of freedom is weak, even taking into account
the enhancement obtained by the use of a cavity. Strong coupling,
together with low losses, are required for quantum information applications.
This is because one should be able to process and transfer information
before it is lost to the environment. In particular for magnonic systems,
it has been shown that strong coupling to microwave photons is possible,
by using a microwave cavity <cit.>.
Note that in this case the coupling is resonant, meaning that the
frequencies from both microwave and magnonic excitations can be matched,
being both in the GHz range. The magnons in this case couple directly
to the slow, oscillating magnetic field, as in ferromagnetic resonance
experiments. We have not discussed this coupling in these notes, but
it can be shown it has the form 

    Ŝ^+â^†+Ŝ^-â

in terms of the spin ladder operators Ŝ^± and the microwave
photons â. This interaction converts a magnon into a photon
and vice-versa. The optomagnonic coupling instead is parametric in
the photon fields (coupling instead to terms of the form â^†â).
This is in general the case for non-resonant interactions, where the
frequency mismatch has to be accounted for (note that optical photons
have frequencies of hundreds of THz), and usually results in small
intrinsic coupling values.

The fact that magnons can couple coherently both to microwaves and
to light is however a big incentive to pursue the strong coupling
regime also in the optical domain. That would allow coherent transfer
of information from the microwave regime, where the information is
usually processed (e.g. with superconducting qubits <cit.>),
to the telecom regime, where information can be communicated through
long distances and at room temperature with the help of optical fibers.
We can expect that the next few years will bring many exciting advances
in this field.

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