Strong entanglement of spins inside a quantum domain wall
H. Y. Yuan Man-Hong Yung X. R. Wang
cond-mat.mes-hall quant-ph





Department of Physics, South University of Science and
Technology of China, Shenzhen 518055, Guangdong, China


[Electronic address: ]yung@sustc.edu.cn
Institute for Quantum Science and Engineering and Department
of Physics, South University of Science and Technology of China, Shenzhen 518055,
Guangdong, China


[Electronic address: ]phxwan@ust.hk
Department of Physics, The Hong Kong University of
Science and Technology, Clear Water Bay, Kowloon, Hong Kong
HKUST Shenzhen Research Institute, Shenzhen 518057, China



Magnetic domain walls (DWs) are widely regarded as classical objects
in physics community, even though the concepts of electron spins and
spin-spin exchange interaction are quantum mechanical in nature.
One intriguing question is whether DWs can survive at the quantum
level and acquire the quantum properties such as entanglement.
Here we show that spins within a DW are highly entangled in their
quantum description. The total magnetization of a magnetic DW is
nonzero, which is a manifestation of the global entanglement of the
collective spin state. These results significantly deepen our
understanding of magnetic DWs and enable the application of DWs in
quantum information science. The essential physics can be
generalized to skyrmions so that they can also play a role in
quantum information processing.


Strong entanglement of spins inside a quantum domain wall
    X. R. Wang
    December 30, 2023
=========================================================

Entanglement is a measure of quantum correlations between two or among
more than two quantum particles. It is a natural resource for quantum
computing and quantum information processing. Finding/generating,
extracting and utilizing this resourse is an important task in quantum
information science<cit.>. Entangling tens particles have been realized
experimentally <cit.>. Remarkably, in superconducting circuits and
trapped ions, it is now possible to control and to tune the coupling
strength between two spins from ferromagnetic to antiferromagnetic
interactions. These developments open intriguing possibilities for
studying the quantum properties of magnetic structures, such as DWs 
<cit.>, vortices <cit.> and even skyrmions 
<cit.>with a 
finite number of controllable spins.
To initiate this interdisciplinary field, it is essential to connect
magnetic structures with the measurable quantities of quantum information,
such as purity of spins and global entanglement of the system <cit.>.

Here we study a quantum magnetic wire with two magnetic domains and a DW
in between. We show that all spins in the DW are highly entangled.
The net magnetization of DWs does not depend on the nanowire length and
it can be well characterized by the global entanglement of the system.
In addition, we find that global entanglement of the system is a natural
indictor of the phase transition between quantum DWs and domains.






Results

Model of quantum domain walls. Let us consider the
transverse Ising model of N spin-1/2 particles on a one dimensional
(1D) lattice, subject to boundary fields h, the Hamiltonian reads

    ℋ=J∑_⟨ ij ⟩σ_i^z ·σ_j^z
    - g∑_i=1^N σ_i^x -h (σ_1^z + σ_N^z),

where σ_i^x, σ_i^z are the Pauli matrices on the i-th
site, J is the exchange coupling and g is the anisotropy energy.
Here ⟨ ij ⟩ denotes nearest neighbor spins.
The ground-state energy E_0 and the corresponding ground state
|0⟩ is calculated using the standard Lancoz algorithm
<cit.> (See Methods for details), which is double checked by the exact
diagonization of the Hamiltonian ℋ for N ≤ 8.

In the following, we shall focus on the antiferromagnetic (AFM)
coupling  (J>0) for an illustration and discussion of our results.
The conclusions drawn below can be generalized to the ferromagnetic
coupling as well. In an AFM domain, the quantum version of the
magnetization order m_i^z and Néel order n_i^z
<cit.> is
given by the following expectation values:

    m_i^z = 1/2⟨ 0| σ_i^z + σ_i+1^z |0⟩,
    n_i^z = 1/2⟨ 0| σ_i^z - σ_i+1^z |0⟩,

where i=1,3,5,...N-1. Without ambiguity, the number of spins is
assumed  even and J=1.0 if it is not stated otherwise.







Domain wall properties. Let us first look at
the numerical results, in the regime with weak anisotropy g ≪ J.
Figure <ref>a shows the spatial variation of m_i^z and n_i^z
for N=12 (triangles), 16 (circles), 20 (squares), respectively.
These profiles are similar to those of classical AFM DWs. (1)
The Néel order n_z has a typical classical DW profile whose
value varies from 1 on the left hand side (LHS) of the chain to -1
on the right hand side (RHS). (2) The magnetization order m_z is
zero in the two domains and reaches its maximum at the DW center.
As the system becomes larger, the magnetic moments near the center
become smaller, but, by summing all the magnetic moments in the DW,
the net magnetization is m_t^z = 1/2, independent of the system
size (N) as shown in the inset of Fig. <ref>a.


To further prove the result of m_t^z = 1/2, we look at the structure
of a quantum DWs of N spins. The boundary spins are aligned along
the +z direction by the external fields. Given g ≪ J, the AFM
exchange interaction dominates the Hamiltonian (<ref>).
Thus, the lowest energy states (i.e. ground-state) are those states
with only one pair of neighboring spins left alone (sketched as a
pair of parallel spins in Fig. <ref>b) while rest nearest
neighboring spins are anti-parallel with each other forming a zero spin
singlet state. For N=6, there are five such configurations of same
energy as sketched in Fig. <ref>b. In general, there are N-1
such states of energy -(N-3)J. These states can be further classified
into |φ^S=1_i ⟩ for S=1 and |φ^S=0_i ⟩
for S=0, N/2 for S=1 and the remaining for S=0, where S is the
net magnetization of the system.
The true ground state under the boundary conditions is the linear
combination of this states. The corresponding eigenstate of the
Hamiltonian (<ref>) is

    |0_th⟩ = ∑_i=1^N/2 a_i|φ^S=1_i ⟩
    +∑_i=1^N/2-1 b_i|φ^S=0_i⟩

where a_i,b_i are the superposition coefficients.
By rearranging the basis states such that S=1 and S=0 states
are ordered alternatively, the Hamiltonian (<ref>) can be recasted
as a tridiagonal-Toeplitz matrix, where the ground state energy is found
to be E_0 = -(N-3)J - 2g cos ( π / N ) with the wave
function

    |0_th⟩ = √(2/N) ( sinπ/N,
    sin2π/N,..., sin(N-1)π/N ),

Note that the anisotropy term g only gives a first-order correction
of the ground state energy. As a result, the magnetic moments distribution
is given by m_i^z=⟨ 0_th| σ_i^z
+ σ_i+1^z |0_th⟩/2 = u_i,
where u_i = 2/Nsin^2 iπ/N. The net magnetization can be calculated as

    m^z_t= ∑_odd  i^N-1 u_i=1/2,

which does not depend on N, in agreement with numerical results.
The essential physics is that the ground state is a superposition of
total spin S=1 and S=0 configurations with equal contributions
(∑ u_2i = ∑ u_2i+1=1/2) so that the average spin number is always 1/2.


In fact, the above result is also true when the Dzyaloshinsiki-Moriya interaction
(DMI) <cit.> H_DM=-𝐃·∑_⟨ ij⟩σ_i ×σ_j is added to Hamiltonian
(<ref>), where the ground state of the revised Hamiltonian is still
solvable analytically. Specifically, for 𝐃=Dŷ (D>0),
the ground state is given by,

    |0_th⟩ = √(2/N) ( τ_1 sinπ/N,
    τ_2 sin2π/N,...,
    τ_N-1sin(N-1)π/N ),

where the prefactor τ_i =1-2 mod ( ⌈ i/2 ⌉,2),
⌈ x ⌉ is the ceiling function of x.
Note that τ_i^2=1, the magnetization distribution
m_i^z = 2/N sin^2 (iπ/N), is not changed by the DMI.
However, the chirality of DW does depend on the sign of D.
For D>0, the system always prefers a clockwise Néel wall.
However, if D reverses the sign, the ground state becomes
counter-clockwise. This is consistent with classical case <cit.>.





Entanglement. In quantum information science, the
purity of the i-th spin is defined as P_i =tr(ρ_i^2)
where ρ_i is the density matrix of the i-th spin obtained by
tracing all the other spins in the full density matrix |0⟩⟨0|.
It quantifies the distance of a state relative to a pure state; for example,
it takes the value 1 for a separable pure state, but 1/2 for a maximally
entangled states such as Bell states and Greenberg-Horne-Zelinger state
(GHZ) states <cit.>.

Figure <ref>a shows the space distribution of purity as colored
rectangles for N=12 (top inset), 16 (middle inset), and 20 (bottom inset),
respectively. Regardless of the system size, the purity takes 1 near
the boundary of spin chain and then decreases to form a symmetric dip
around the chain center at i/N=1/2. The magnetization distribution
of DWs takes on a peak centered at the dip position and its space
dependence is strongly correlated with the purity. Physically, this
correlation originates from the antiferromagnetic nature of nearest
neighbour interaction. For the spins near the boundary, their directions
are strongly bounded by the fixed orientations of the boundary spins
and have a larger probability to be in Néel-state-like configurations.
Consequently, the magnetization is close to zero and these spins are not
entangled with the others (i.e., close to a pure state).
The spins near the center are influenced lightly by the boundary spins
and their directions become much more uncertain, hence the purity of
these spins become smaller. The purity of spins close 1/2 at the wire center
 suggests that the spins inside the domain walls are highly entangled.



Theoretically, the purity of the i-th spin is defined by the
reduced density operator of the  i-th spin obtained by tracing
all the spins except the i-th spin in density operator
|0_th⟩⟨ 0_th | as

    P_i^th = (∑_k=1^i-1 u_k  )^2
    + (∑_k=i^N-1 u_k )^2 + 2u_i u_i-1,

where i=1,2,...,N. The third term u_i u_i-1 is of the order
O(1/N^2), which is much smaller than the first two terms.
Considering ∑ u_i=1, P_i reaches its minimum value
(1/2) at i = N/2, i.e. the chain center, where
m_i^z = u_i ∝sin^2 (iπ/N) reaches its maximum value.

The correlations of the magnetization distribution and purity distribution
can be extended to two dimensions (2D) where rich magnetic structures
such as vortices, skyrmions can exist since the physics remain the same.
Nevertheless, it is a great challenge to find the ground states in 2D
both analytically and numerically. 




Practically, the correlations also allow us to measure a quantum DW, vortices
and even skyrmions by measuring the purity of each spin in a spin chain. 
The later can be realized by performing state tomography on each system 
qubit or by a more efficient technique that uses bitwise interactions 
between the system and identically prepared registers <cit.>.
A typical tomography of the density of states for a clockwise/counter-clockwise
DW is shown in Fig. <ref>b and <ref>c. The distinguishable
distributions of the density matrix allows us to classify clockwise and
counter-clockwise DWs.

Furthermore, the direct sum of magnetization recovers the net magnetization
of the spin chain while the proper average of the local purity can
give the global entanglement (E_g) of the spins. The original definition
of E_g is given by Meyer and Wallach <cit.> and then
reformulated in terms of the sum of local purity <cit.>, i.e.


    E_g ≡2/N∑_i=1^N (1-P_i).

Substituting Eq. (<ref>) into this definition, the leading order
of global entanglement is reduced to

    E_g = 2 - 2/N∑_k=1^N  [ (∑_i=1^k-1 u_i )^2
    + (∑_i=k^N-1 u_i  )^2  ].

The sum increases as N increases and saturates for N>20. The limiting
value is E_g = 2/3-5/2π^2 ≈ 0.41. This indicates that the
spins in a DW is still entangled in the macroscopic scale
(N→∞). This is quite different from a classic DW
that has zero entanglement since all its composite spins have definite
orientations.

As a comparison, for a quantum domain state without DWs such as the
superposition of two degenerated Néel states,

    |0_th⟩ = 1/√(2) (|↑↓↑↓...↑↓⟩
    +  |↓↑↓↑ ...↓↑⟩)

The global entanglement is E_g = 1 while its net magnetization is zero.
Based on this comparison, it becomes possible to distinguish domain and
DW states and the corresponding net magnetization by measuring the
global entanglement of the spin system.



















Phase transition. To know the size of the phase space
where the quantum DWs are stable, it is interesting to plot the phase
diagram of the spin model. In our numerical simulation, we fix the
exchange coupling and adjust the anisotropy g and boundary fields to
obtain various ground states in h-g plane as shown in Fig. <ref>.

Guided by the red line in Fig. <ref>a, the ground state changes
from a single domain state to a quantum DW at the critical field of
h_c = J for g=0. The critical field has a non-trivial
dependence on g and h. For simplicity, we focus on the regime
g ≪ 1, where the ground state is mainly determined by the
competition of exchange interaction and Zeeman interaction.
In this regime, the global entanglement (E_g) shows an abrupt
jump from 1.0 to a smaller value as shown in <ref>b and
Fig. <ref>c with g=0.2. This abrupt change is due to the
distinguishable quantum properties of the domains and DWs.
When h <h_c, the exchange interaction dominates the Hamiltonian
and the ground state is a superposition of two degenerated Néel
states Eq. (<ref>) with E_g=1. The numerical value of E_g
is a little smaller than the theoretical value 1, as shown in
Fig. <ref>c, due to the influence of the small anisotropy g
term, which tendency is to align all the spins to x direction and
reduce the entanglement. As the field increases above h_c,
the ground state becomes a DW state with finite entanglement 0.41
as discussed previously.

The global entanglement and its first derivative is continuous with h
near the critical field while its second order derivative is discontinuous,
as shown in the left inset of Fig. <ref>c. To verify whether this
discontinuity can be a good indictor of the phase transition, we first
plot the first derivative of E_g versus h in the left inset of Fig.
<ref>c for system size ranging from N=8, 10, 12, 14,16.
As N increases, dE_g/dh shows a clear divergence tendency at the critical field.
To eliminate the finite size effect, we do scaling analysis for finite systems
using the scaling ansatz <cit.>,

    dE_g/dh=ln N^ν |h-h_c|,

where ν is the critical exponent. As shown in the right inset of
Fig. <ref>c, ν =0.54 gives perfect scaling results for the
finite systems. This consistency shows that global entanglement is a
good measure of quantum DW/domain phase transition in this system.





Discussion

We have shown that an AFM DW in a 1D lattice has an intrinsic
magnetization of 1/2 independent of lattice length. The reason is that
the ground state is a superposition of S=1 states and S=0 states
of equal amplitude. The global entanglement of such a DW is non-zero
and even exists at a macroscopic scale. Moreover, the magnetization
profile is closely related to the local purity of spins such that the
DW width can be extracted from the purity profile that can be measured
through state tomography in quantum information science.
The typical energy gap of the ground state and the first excited state
in our model is 43  μeV (∼ 500 mK) for N=20.
Then the mixture of excited states and ground state can be neglected
at a temperature below 500 mK, which leaves a sufficient room to
experimentally verify our theoretical prediction.

Since our results are applicable to chiral DWs due to DMI, the main
conclusions can be generalized to two dimensional case for magnetic
skyrmions. This type of skyrmions has non-zero entanglement that is
very different from the classical skyrmions discussed in literature.
Further study of the quantum skyrmions may lead to
quantum skyrmion spintronics.




Methods

Lanczos algorithm.
Our numerical calculation of the ground states of the spin model are
based on the Lanczos method. First, a random initial state
| ψ_0 ⟩ is chosen in the 2^N dimensional Hilbert space.
Then we define a new state | ψ_1 ⟩ that is orthogonal to
| ψ_0 ⟩, realized by subtracting
ℋ | ψ_0 ⟩ over | ψ_0 ⟩ i.e.


    | ψ_1 ⟩ = ℋ | ψ_0 ⟩ -  a_0| ψ_0 ⟩

where a_0 = ⟨ψ_0 | ℋ | ψ_0 ⟩ / ⟨ψ_0|ψ_0 ⟩.
Next, we define the state |ψ_2⟩ that is orthogonal to both |ψ_0 ⟩ and
|ψ_1 ⟩,

    | ψ_2 ⟩ = ℋ | ψ_1 ⟩ - a_1 | ψ_1 ⟩ - b_0 | ψ_0 ⟩

where a_1 = ⟨ψ_1 | ℋ | ψ_1 ⟩ / ⟨ψ_1|ψ_1 ⟩,
b_0 = ⟨ψ_1|ψ_1 ⟩ /⟨ψ_0|ψ_0 ⟩.
Through iterations, the state |ψ_n+1⟩ is derived as

    | ψ_n+1⟩ = ℋ | ψ_n ⟩ - a_n | ψ_n ⟩ - b_n-1 | ψ_n-1⟩,

where the coefficients a_n = ⟨ψ_n | ℋ | ψ_n ⟩ / ⟨ψ_n|ψ_n ⟩,
b_n-1 = ⟨ψ_n|ψ_n ⟩ /⟨ψ_n-1|ψ_n-1⟩.

In terms of the mutual orthogonal basis spanned by {|ψ_0⟩, |ψ_1⟩,...|ψ_n⟩},
the Hamiltonian can be written in the form of a tridiagonal matrix
and then diagonalized through the standard subroutine.
The ground state energy E_n is obtained as the smallest eigenvalues
of the Hamiltonian. As n increases, the E_n will converge to the
real ground state energy of the system while the corresponding eigen-vector
is the wave function of ground states. The convergence criteria used in the
simulations is |E_n - E_n-1| ≤ 10^-14.














































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Acknowledgements

H.Y.Y. acknowledges the financial support from National Natural Science
Foundation of China (NSFC) Grant (No. 61704071).
MHY acknowledges support by NSFC Grant (No. 11405093), Guangdong
Innovative and Entrepreneurial Research Team Program (2016ZT06D348),
and Science, Technology and Innovation Commission of Shenzhen
Municipality (ZDSYS20170303165926217 and JCYJ20170412152620376).
XRW was supported by the NSFC Grant (No. 11774296) as well
as Hong Kong RGC Grants (Nos. 16300117 and 16301816).



Author contributions

H.Y.Y., M.H.Y. and X.R.W conceived the project. H.Y.Y. performed
the numerical and theoretical calculations.  All authors
analyzed and discussed the results and contributed to the writing of the
manuscript.



Additional information

The authors declare no competing financial interests.



