Spectral characteristics of the antiferromagnetic spin-1/2 Heisenberg model on the square lattice in a magnetic field
P. S. Savchenkov A. F. Barabanov
cond-mat.str-el


Meissner Currents Induced by Topological Magnetic Textures in Hybrid Superconductor/Ferromagnet Structures
    Samme M. Dahir, Anatoly F. Volkov, and Ilya M. Eremin
    December 30, 2023
==========================================================================================================



	We predict that spin-waves in an ordered square quantum antiferromagnet in a transverse magnetic field (h) may demonstrate three modes of spin excitations. Starting from the self-consistent rotation-invariant Green’s function method, a new mean-field theory is constructed for h0. The method preserves the translational and the axial symmetries, and provides exact fulfillment of the single-site  constraint for each of the three modes. We examine the dynamical structure factors S^αα(𝐤,ω ),α= x, y, z.
	It is shown, that the introduction of h leads to the hybridization of two degenerate spin modes due to the appearance of a nondiagonal on α, β spin-spin Green's functions.  The comparison of the theory with  the exact diagonalization study and with results on inelastic neutron scattering experiments is discussed at T = 0. We discuss also the correspondence of the theory to the existing theories, which allow only two spin excitations modes for the total S(𝐤,ω ).
	
	
	





§ INTRODUCTION

The consideration of a two dimensional quantum antiferromagnet (2D-AFM) is usually based on various versions of the spin-1/2 Heisenberg square-lattice model.

This model continues to be studied <cit.>. It is used to describe superconducting cuprates and related compounds, including recently synthesized molecular 2D AFM<cit.>. 

The model demonstrates the strong influence of quantum fluctuations on the properties of spin excitations. The inclusion of the magnetic field h should enhance the role of fluctuations and rearrange the spin properties of the system.  This problem is often studied based on the simplest Hamiltonian (without introducing spin frustration and spin anisotropy).


    H=J+h^z=1/2J∑ _𝐧,𝐠
    𝐒_𝐧𝐒_𝐧+𝐠-h∑_
    	𝐧S_𝐧^z

where 𝐒_𝐧 - is the quantum spin-1/2 operator on the site n, g - the vectors of the nearest neighbours on a square lattice in the XY plane. J is the AFM exchange parameter. The magnetic field h is applied perpendicular to the XY-plane. 

Most analytical considerations are based on the introduction of Bose operators using the 1/S expansion in the Holstein – Primakov  method <cit.>, or using the Dyson – Maleev method <cit.>.  
Such a consideration leads to the existence of one branch of Bose excitations <cit.> in the case of low fields (h<0.7h_sat, h_sat=4J  is the saturation field,  at which spins become fully aligned).
The second mode of Bose excitations in the <cit.>  occurs only at h>0.7h_sat when the decay of spin waves becomes important.

However, in the exact diagonalization method (ED) <cit.>, the total dynamic structure factor S(k,ω) has a three-peak structure even in low fields. 
This kind of S(k,ω) cannot be described by the theory <cit.>.

 Let us mention the consideration of the model within the Tyablikov decoupling approximation. The approximation  leads to the presence of two modes of spin excitations <cit.>.
 
Recent inelastic neutron scattering experiments also demonstrate the presence of several peaks of spin excitations in low magnetic fields.
In particular,  spin excitations branches splitting was established for 2D AFM Ba_2MnGe_2O_7 <cit.> even by h=0.3h_sat.

Similar measurements were carried out for a new class of molecular 2D-AFM <cit.> and quasi two dimensional AFM <cit.>.
The presence of several branches of spin excitations is experimentally observed in low magnetic fields, as in <cit.>. Branch splitting at the point k=(π,π) increases linearly with h.


This work aims to establish that upon the introduction of h 0, three branches of spin excitations can be observed in the system. This physical picture qualitatively corresponds to the ED results <cit.>.

Our consideration is based on a spherically symmetric self-consistent approach (SSSA) <cit.>. Three degenerate branches of spin excitations are realized in the SSSA at h = 0. With the introduction of h0 (the transition from O (3) symmetry to O (2) symmetry), a self-consistent theory for retarded Green's functions (GF) is developed.
Translational symmetry is preserved. A distinctive feature of the approach is the fulfillment of the spin one site constraint condition for each of the three branches of spin excitations.

In the discussion, the obtained spectra, as well as dynamical structure factors, are compared with the results on the ED <cit.> and <cit.> theory.
Note that the proposed theory allows one to consider the case of finite temperatures T0.

 



§ GREEN'S FUNCTIONS AT A FINITE MAGNETIC FIELD. 


Here, we discuss the retarded GF <cit.> G_𝐧𝐦^αα=⟨S_𝐧^α|S_𝐦^α⟩ _z=ω +iη, η→ +0  and their Fourier transform G_𝐤^αα(ω)=⟨S_𝐤^α|S_-𝐤^α⟩_z, α = x, y, z to calculate the dynamic properties and the spectrum of spin excitations for the system with the Hamiltonian (<ref>). 

At h=0 the system   with O(3)-symmetry Hamiltonian (<ref>) is in states with an average spin value at the site ⟨𝐒_𝐧^α⟩ =0 at an arbitrarily low temperature. Due to spherical symmetry G_𝐧𝐦^αα is independent of α. The spectrum of spin excitations is threefold degenerate <cit.>.

The presence of h in (<ref>) distinguishes the z direction. Therefore one should  consider G_𝐧𝐦^xx=G_𝐧𝐦^yy≠ G_𝐧𝐦^zz.

Let us introduce the relations between the dynamical structure factor S^αβ(𝐤,ω ), the static structure factor S^αβ(𝐤), and spin-spin correlation functions ⟨S_𝐧+𝐥^αS_𝐧^β⟩
=c_l^αβ:


    S^αβ(𝐤,ω )=-1/π(m(ω )+1) Im G^αβ_𝐤(ω );


    S^αβ(𝐤)=∫_-∞^∞dω S^αβ(𝐤,ω );


    c_l^αβ=1/N∑_𝐤e^i𝐤𝐥S^αβ(𝐤),

where  l=g,d,2g -  vectors to first, second and third nearest neighbours, m(ω ) is the Bose distribution function.

Further, for GF G_𝐧𝐦^αα we will require the conditions of constraint on each site, namely c_𝐥=0^αα =1/4. This is equivalent to the normalization conditions for static structure factors:


    N^-1∑_𝐤S^αα(𝐤)=1/4.



We introduce the chain of equations for GF's G_𝐧𝐦^zz,  G_𝐧𝐦^xx(yy) and  G_𝐧𝐦^yx, in accordance with <cit.>. Our approach implies the exact consideration of one- and two-site excitation operators  that arise when commutating S_𝐧^α with the Hamiltonian (<ref>). The approximation is required to close the chain of equations. In our case, it will correspond to decoupling only the three-site excitation operators S_𝐧^α related to different sites. The decoupling will preserve translational and O(2) symmetries. In particular, this means that c_𝐥≠ 0^xy=0.

The first equation for the G_𝐧𝐦^zz is:

    zG_𝐧𝐦^zz=⟨S_𝐧^z;Ĵ]|S_𝐦^z⟩_z

The approach needs to consider the second step - the differentiation of the right side of (<ref>).
Taking into account that the commutator [[S_𝐧^α;Ĵ];ĥ]=0, we have the following exact expression for G_𝐤^zz(ω):

    z^2G_𝐤^zz(ω)=-4J(1-γ _𝐤)(c_𝐠^xx+c_𝐠^yy)+2J^2(1-γ _𝐤)G_𝐤^zz(ω)+J^2⟨𝐃_𝐧^z|S_𝐦^z⟩ _𝐤,z
    𝐃_𝐧^α=∑_β{∑_𝐠;𝐚δ_𝐚;𝐠[ (S_𝐧+𝐠^βS_𝐧^βS_𝐧+𝐠+𝐚^α-S_𝐧+𝐠^αS_𝐧^βS_𝐧+𝐠+𝐚^β)] -
    -∑_𝐠;𝐚δ_𝐚;𝐠[ (S_𝐧^βS_𝐧+𝐠^βS_𝐧+𝐚^α-S_𝐧^αS_𝐧+𝐠^βS_𝐧+𝐚^β)] },

where γ_𝐤=1/2(cos k_x+cos k_y),δ _𝐥;𝐦 - is  the Kronecker symbol, δ_𝐥;𝐦=1-δ _𝐥;𝐦, 𝐚(𝐠) -  vectors of nearest neighbours, g=-g.

Note that ∑_𝐧𝐃_𝐧^z=0. This means that in the limit when k→0 the exact equation (<ref>) is closed. 
The spin excitation mode ω_𝐤^z corresponding to the G_𝐤^zz is gapless at k=0. Such a mode must be present at any accurate approach. 

The operator 𝐃_𝐧^z contains three-site terms at different sites and, unlike the right side of (<ref>), allows one to decouple (<ref>) in the site representation.

Such a decoupling corresponds the approximation of the following form:

    δ_𝐚;𝐠(S_𝐧+𝐠^xS_𝐧^x)S_𝐧+𝐠+𝐚^z≈δ_𝐚;𝐠⟨S_𝐧+𝐠^xS_𝐧^x⟩S_𝐧+𝐠+𝐚^z

 Decoupling (<ref>) leads to the replacement in (<ref>) of the operator 𝐃_𝐧^z by 𝐃_n^z, which has the form:

    𝐃_𝐧^z=2α
    ^zz{4(c_2g^xx+2c_d^xx)S_𝐧^z-(c_2g^xx+2c_d^xx)Σ _𝐠S_𝐧+𝐠^z+
    +c_g^xx[(∑ _𝐠S_𝐧+2𝐠^z+2∑ _𝐝S_𝐧+𝐝^z)-3∑ _𝐠S_𝐧+𝐠^z]}.

It is taken into account that c_g(d,2g)^xx=c_g(d,2g)^yy due to axial symmetry.

 The simplest vertex correction α ^zz is introduced for decoupling (<ref>). The value of α ^zz is determined self-consistently from the requirement of the constraint c_0^zz=⟨S_𝐧^zS_𝐧^z⟩ =1/4.
 The decoupling (<ref>) preserves the exact condition ω _𝐤→ 0^z=0.

We emphasize that the theory developed here uses unambiguous decoupling in the site representation which is performed after the second stage of differentiation of GF. The approach differs significantly from the recently presented mean-field approach <cit.>. In  <cit.> the averages are taken in the momentum representation at the level of the Hamiltonian.

As a result,  one can derive the following expression for the G_𝐤^zz(ω) after decoupling:

    G_𝐤^zz(ω)=F_𝐤^zz/z^2-(ω _𝐤^z)^2;
    
    F_𝐤^zz=-4J(c_g^yy+c_g^xx)(1-γ _𝐤);
    
    (ω _𝐤^z)^2=-32J^2α ^zzc_g^xx(1-γ _𝐤)(1+γ _𝐤+ρ ^zz);
    ρ ^zz=1+4α^zz(c_2g^xx+2c_d^xx+12c_g^xx)/-16α^zzc_g^xx.

The value of ρ ^zz is directly connected with the gap value ω _𝐤=(π ,π )^z.

G_𝐤^zz does not explicitly depend on the magnetic field h. The field influence is manifested over the values of c_g,d,2g^xx  which are calculated self-consistently.


Let us now represent exact equations for the GF G_𝐤^xx(ω). The equations have the form:


    zG_𝐤^xx(ω)=⟨S_𝐧^x;Ĵ]|S_𝐦^x⟩ _z,𝐤+ihG_𝐤^yx(ω)


    z^2G_𝐤^xx(ω)=-4J∗ c_𝐠^zz+c_𝐠^yy](1-γ _𝐤)+2J^2(1-γ _𝐤)G_𝐤^xx(ω)+J^2⟨𝐃_𝐧^x|S_𝐦^x⟩ _z𝐤+ihzG_𝐤^yx(ω)

The form of Eqs.(<ref>, <ref>) are close to Eqs.(<ref>, <ref>). But here the term with h and nondiagonal in αβ indexes GF G_𝐤^yx(ω) arises for the first time. 

The exact equation for the G_𝐤^yx(ω) gives:

    z^2G_𝐤^yx(ω)=-izJ⟨S_𝐧^z⟩ +2J^2(1-γ _𝐤)G_𝐤^yx(ω)+J^2⟨𝐃_𝐧^y|S_𝐦^x⟩ _z𝐤-ihzG_𝐤^xx(ω),

where ⟨S_𝐧^z⟩ is the average value of the operator S^z_n on the site n.  ⟨S_𝐧^z⟩ does not depend on n. 

In the absence of a magnetic field h G_𝐤^xx(yy)(ω)=G_𝐤^zz(ω) and G_𝐤^yx(ω)=0. The spectrum of the spin excitations has a solution with three degenerate modes.

After (<ref>)-type decoupling  𝐃_𝐧^α→𝐃_𝐧^α in (<ref>, <ref>) the system of equations (<ref>-<ref>) is becomes closed for G_𝐤^xx(ω) and G_𝐤^yx(ω). Here and below we assume that α ^xx=α ^yy. 

The solution for G_𝐤^xx(ω) has the form:


    G_𝐤^xx(ω)=F_𝐤^xx(z^2-(ω _𝐤^x)^2)+z^2h⟨S_𝐧^z⟩/(z^2-(ω _𝐤^x)^2)^2-z^2h^2; 
    
    F_𝐤^xx=-4J( c_𝐠^zz+c_𝐠^yy)(1-γ _𝐤)
    
    (ω _𝐤^x)^2=-16J^2α
    ^xx(c_g^xx+c_g^zz)(1-γ _𝐤)(1+γ _𝐤+ρ ^xx);
    ρ ^xx=1+2α^xx(c_2g^xx+2c_d^xx+c_2g^zz+2c_d^zz+12c_g^xx+12c_g^zz)/-8α ^xx(c_g^xx+c_g^zz).

Significantly, that the denominator of G_𝐤^xx(ω) has the form of two degenerate spin excitations with a frequency ω _𝐤^x, which are hybridized due to the term  z^2h^2. As a result, the denominator of G_𝐤^xx(ω) (<ref>) describes two modes of excitations with:

    ω _𝐤^±=√(h^2/4+(ω _𝐤^x)^2)±h/2


The expression for the  G_𝐤^xx(ω) (<ref>) leads to the following form of the dynamical structure factor (<ref>):

    S^xx(𝐤,ω )=S^+(𝐤,ω )+S^-(𝐤,ω )
    
    S^+(𝐤,ω )=I^+ (𝐤)[(m(ω _𝐤^+)+1)δ (ω -ω _𝐤^+)+m(ω _𝐤^+))δ (ω +ω _𝐤^+)]
    
    S^-(𝐤,ω )=I^- (𝐤)[(m(ω _𝐤^-)+1)δ (ω -ω _𝐤^-)+m(ω _𝐤^-))δ (ω +ω _𝐤^-)]

here I^+(𝐤) and I^-(𝐤) are intensities of spin excitations  with ω _𝐤^+ and ω _𝐤^-:

    I^+(𝐤) =F_𝐤^xx/2√(h^2/4+(ω _𝐤^x)^2)+h⟨S_𝐧^z⟩/8√(h^2/4+(ω _𝐤^x)^2)+⟨S_𝐧^z⟩/4
    
    I^-(𝐤)=F_𝐤^xx/2√(h^2/4+(ω	_𝐤^x)^2)+h⟨S_𝐧^z⟩/8√(h^2/4+(ω _𝐤^x)^2)-⟨S_𝐧^z⟩/4

 It turns out that G_𝐤^yy(ω)=G_𝐤^xx(ω) and S^yy(𝐤,ω )=S^xx(𝐤,ω ).

One can show that the explicit equation of G_𝐤𝐤^xy(ω) (<ref>) satisfies the exact condition c_0^xy=⟨S_𝐧^yS_𝐧^x⟩ =1/N∑_𝐤S^yx(𝐤)=i/2⟨S_𝐧^z⟩ as well as c_𝐥≠ 0^xy=0. 

Six spin correlation functions c_g,d,2g^zz,c_g,d,2g^xx are calculated self-consistently using the equations for G_𝐤^zz(ω)  (<ref>) and G_𝐤^xx(ω)  (<ref>). Vertex corrections α ^zz and α ^xx  are determined self-consistently from the constraint condition (<ref>). 


The present approach allows us to consider both finite temperatures T > 0 and the case T=0, which will be considered below. At T=0 one should introduce Bose condensation part <cit.> of the spin correlation functions (<ref>).

The average   ⟨S_𝐧^z⟩ is taken as  ⟨S_𝐧^z⟩ =χ^zz(h)h, where the magnetic susceptibility χ^zz(h) coincides with numerical calculations presented in <cit.>. This will allow us to compare the results of the presented approach with the ED-results <cit.>, which uses the same form of χ^zz(h) <cit.>.

We restrict our attention to the low magnetic field case (h<2J), when the spin excitations instability is insignificant <cit.>.




§ RESULTS AND DISCUSSIONS

Let us discuss the form of the total dynamical structure factor S(𝐤,ω ) at T=0:

    S(𝐤,ω )=S^zz(𝐤,ω )+S^xx(𝐤,ω)+S^yy(𝐤,ω )=S^zz(𝐤,ω )+2(S^+(𝐤,ω )+S^-(𝐤,ω ))

It has the form:

    S(𝐤,ω )=I^z(𝐤)δ (ω -ω _𝐤^z)+2I^+(𝐤)δ (ω -ω _𝐤^+)+2I^-(𝐤)δ (ω -ω _𝐤^-)

here intensities I^+(𝐤) and I^-(𝐤) are defined by the expression (<ref>), I^z(𝐤)=F_𝐤^zz/ω _𝐤^z.

Equation (<ref>) gives that S(𝐤,ω ) have a three-fold structure at h0. Intensities of the peaks  I^+(𝐤) and I^-(𝐤) explicitly depend on h.




Fig. 1 represents S(𝐤,ω ) for three points of the first Brillouin zone: 𝐤_0=(0;0), 𝐤_0=(π/2;π/2), 𝐤_0=(3π/4;3π/4). An artifical broadening with Lorentzian type (δ=0.1J) is introduced into the delta functions in (<ref>).

Firstly, let us discuss S(𝐤_0=0,ω ), see Fig.1a.  The peaks related to ω _𝐤_0=0^z and ω _𝐤_0=0^- are absent (I^-(𝐤_0=0)=0 and I^z(𝐤_0=0)=0), ω _𝐤_0=0^-=ω _𝐤_0=0^z=0. Accordingly to (<ref>) ω _𝐤_0=0^+=h. 
 
 S(𝐤_0=0,ω ) was investigated by ED methods in <cit.>. At Fig. 2 we compare the h dependence of ω _𝐤_0=0^+ (<ref>) and I^+(k=0)(<ref>) with the ED results (see Fig.13 in <cit.>). It may be seen that the presented approach reproduces the ED results: linear increase of  ω _𝐤_0=0^+ with h and the increase of intensity.
 
Figs. 1b and 1c represent S(𝐤_0,ω ) for typical points of Brillouin zone  corresponding to the essential  contribution to the constraint condition (<ref>). The splitting of the spin excitations bands into three peaks with frequencies ω _𝐤_0^z,ω _𝐤_0^+,ω _𝐤_0^- is observed. Such a splitting is similar to the ED results. For example, one can distinguish these modes at Fig.13(a,b) in <cit.> (𝐤_0=(π/2;π/2)). According to (<ref>) splitting between modes ω _𝐤^+ and ω _𝐤^- increases with h:  ω _𝐤^+ -ω _𝐤^-=h. It turns that ω _𝐤^z satisfies the condition ω _𝐤^-<ω _𝐤^z<ω _𝐤^+  for h≠ 0, 𝐤≠ 0 . 



In a neighborhood of 𝐤_0=(π,π ) three-peak structure may not be observed experimentally due to the proximity of frequencies ω _𝐤^z  and ω _𝐤^-. In this case, the peak structure corresponds to the results of neutron experiments for compounds: Cu(C_4H_4N_2)_2(ClO_4)_2 <cit.>, C_9H_18N_2CuBr_4 <cit.>.

Two modes of spin excitations are experimentally observed for Ba_2MnGe_2O_7 <cit.> at 𝐤=(π,π ). The splitting between these modes is linear with h. The splitting is observed even in low fields coincidentally to S(k,ω) (<ref>) with the condition of indistinguishability between ω _𝐤^z and ω _𝐤^-.


Let us discuss the difference between our theory and approaches based on the introduction of Bose operators <cit.> in the framework of <cit.>.

 

The expression for the dynamical structure factor of spin excitation S^SWT(k,ω) in <cit.> allows for the existence of two spin modes ω_1,k^SWT and ω_2,k^SWT =ω_1,k-Q^SWT at low magnetic fields.
 These modes coincide along the magnetic Brillouin zone boundary. As a result, the structure of S^SWT(k,ω) is a single-peak along this boundary. In contrast, our theory (as well in <cit.>) demonstrates a three-peak S(k,ω) structure (see Fig.1b). 
 
  It seems that the appearance of additional mode is possible in the approach [21]. In S^SWT(k,ω) cross-terms Green's functions were omitted as small numerically [16]. These terms can lead to hybridization splitting (15).
  
  Let us note, as mentioned in [16]  for the S=1/2 the quasiparticle weight redistribution is important. This redistribution is closely related to constraint condition (<ref>). The constraint condition is exactly fulfilled for S^xx(yy,zz)(𝐤,ω ). As to S^SWT(k,ω) this question remains open. 
  
To identify correspondence between theories  we represent the Fig. 3, where  ω _𝐤^z,ω _𝐤^+,ω _𝐤^-   are demonstrated, as well as ω_1,k^SWT. The modes are given along the symmetric direction of the Brillouin zone Γ((𝐤=(0 ,0 )))-S(𝐤=(π/2;π/2))-X(𝐤=(π ,π )) for h=0.4J. The intensities of modes (<ref>) are shown at inset.




The mode ω_1,k^SWT coincides  with the ω _𝐤^+ in a small neighborhood of Γ.
 At Γ only I^+(𝐤=0) is finite. In this sense, ω _𝐤^+  is the "leading"  mode. Both theories predict a linear dependence of these modes values on h.
 
 Values of ω _𝐤^z,ω _𝐤^+,ω _𝐤^-  are well separated and  ω_1,k^SWT  coincides to ω _𝐤^- in a wide region of the phase space (neighborhood of the S-point). But the intensities I^-(𝐤), I^z(𝐤), I^+(𝐤) are close to each other in this region.  

In a neighborhood of X ω_1,k^SWT reproduces the "leading" ω _𝐤^z mode. Both of them are linearly dependent on momentum.

One can conclude that ω_1,k^SWT coincides with the "leading" mode (if the "leading" mode can be distinguished).

Finally, we emphasize the main result of the present work. At h0 a three-peak structure of the total dynamical structure factor S(k,ω) should be observed over a wide part of the Brillouin zone. This can be most clearly observed by inelastic neutron scattering at the boundary of the magnetic Brillouin zone. 

Let's pay attention to the rare earth compound YbB_12. The results of an experiment on inelastic neutron scattering<cit.> can be interpreted as the presence of two-dimensional AFM spin correlations in the compound. The peak corresponding to these correlations is significantly broadened with h at 𝐤=(π ,π ). Perhaps this is evidence of a peak splitting, which is indistinguishable under the conditions of the experiment(h=10T, resolution equal to 0.5 meV). It seems important to study the structure of this peak at higher magnetic fields and with better resolution.

This work was supported by the Russian Foundation for Basic Research (project No.
19-02-00509).
This  work  was also supported by  the  Russian  Science  Foundation  (project  no.  18-12-00133).

The authors are grateful to P.A. Alekseev and  M.E. Zhitomirsky for useful discussions.

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