Influence of XY anisotropy on a magnetoelectric effect in spin-1/2 XY chain in a transverse magnetic field
aa r X i v : . [ c ond - m a t . s t r- e l ] J u l Regulararticle In fl uence of X Y anisotropy on a magnetoelectriceffect in spin-1/2
X Y chain in a transverse magnetic fi eld Vadim Ohanyan Laboratory of Theoretical Physics, and Joint Laboratory of Theoretical Physics – ICTP A ffi liated Centre inArmenia, Yerevan State University, 1 Alex Manoogian Str., 0025 Yerevan, Armenia July 27, 2020
A magnetoelectric effect according to Katsura-Nagaosa-Balatsky mechanism in spin-1/2 XY chain in transversemagnetic fi eld is considered. A spatial orientation of the electric fi eld is chosen to provide an exact solutionof the model in terms of free spinless fermions. The simplest model of quantum spin chain demonstratingmagnetoelectric effect, a zero temperature case of the spin-1/2 X X chain in a transverse magnetic fi eld withKatsura-Nagaosa-Balatsky mechanism, is considered. The model has the simplest possible form of the magne-tization, polarization and susceptibility functions, depending on electric and magnetic fi elds in a most simpleform. For the case of arbitrary XY anisotropy a non-monotonic dependence of the magnetization on the XY anisotropy parameter is fi gured out. This non-uniform behaviour is governed by the critical point, which isconnected with possibility to drive the system gapless or gapped by the electric fi eld. Singularities of the mag-netoelectric susceptibility at the critical value of system parameters are shown. Key words:
KNBmechanism,Magnetoelectriceffect, XY chain,freespinlessfermions PACS:
1. Introduction
Magnetoelectrics are materials having both dielectric polarization and magnetization in a single phaseand exhibiting a magnetoelectric effect (MEE), a vast class of phenomena of intercoupling of magneti-zation and polarization in matter [1–4]. These materials are particularly important for their applicationin spintronic devices [5, 6]. The MEE is a class of phenomena in solids, which can be detected as mag-netic field dependance of dielectric polarization and electric field dependance of magnetization. In mostinteresting cases of non-trivial MEE the magnetization (dielectric polarization) can be induced by onlyapplying an electric (magnetic) field. Nowadays, several microscopic mechanisms of the MEE are known[1–4]. One of these mechanisms is based on so-called spin-current model or inverse Dzyaloshinskii-Moriya (DM) model and have been proposed in a seminal paper by Katsura, Nagaosa and Balatsky[7]. The Katsura-Nagaosa-Balatsky (KNB) mechanism establishes a connection between the dielectricpolarization of the crystal structure unit consisting of two magnetic ions chemically bonded to one ormore p -elements and the spin states of the ions [7, 8]. The dielectric polarization, induces into the bondbetween two spins in this model is given by the following expression: P ij = µ e ij × S i × S j , (1.1)here e ij is the unit vector pointing from site i to site j and µ is a microscopic constant characterizing thequantum chemical features of the bond between two metallic ions and p -element(s) [7, 8]. S i and S j arethe spin operators of the corresponding ion states. The simplest case of the KNB mechanism is the linear This work is licensed under a CreativeCommonsAttribution4.0InternationalLicense. Further distributionof this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. ?????-1 adim Ohanyan arrangement of magnetic ions (spins), the geometrically linear spin chain. If we suppose the chain to bedirected toward the x -axis then the local polarization according to equation (1.1) acquires the followingcomponents: P xj , j + = , (1.2) P y j , j + = µ (cid:16) S y j S xj + − S xj S y j + (cid:17) , P zj , j + = µ (cid:16) S zj S xj + − S xj S zj + (cid:17) . To a large family of magnetoelectric materials belong those, which feature a one-dimensional arrange-ment of exchange-interaction paths between Cu + ions, with ferromagnetic nearest-neighbour ( J < J >
0) interactions. The corresponding model of one-dimensional spin-1/2 J − J spin chain is usually referred to as multiferroic spin chain. A list of magneticmaterials successfully describing by this model is quite broad: LiCuO [9–12], LiCuVO [13–15], CuCl [16], CuBr [17, 18], PbCu(SO )(OH) [19, 20], CuCrO [21], SrCuTe O [22] just to mention few ofthem.Last few years were marked by an interest toward exact and numerical investigation of one-dimensionalquantum spin models with KNB mechanism. Exact description (for some models supplemented withnumerics) for the MEE are available, so far, only for simplified models, such as strictly linear integrable X X Z chain [23], the same system but in both longitudinal and transverse fields [24], spin-1/2
XY Z chain[25], the spin-1/2 XY chain with three-spin interaction [26–28], generalized quantum compass modelwith magnetoelectric coupling [30], spin-1/2 Heisenberg-Ising ladder [31]. However, exact results arevery helpful for understanding general features of the phenomena. Moreover, some of them can serve asa mean-field approximation for the more realistic models. The latter case is typical for a class of exactlysolvable spin chain models, where spins interact to each other via two coplanar components (usuallytaken as S x and S y ).In the present paper we focused on the MEE in a XY chain, which was introduced in seminal paperof Lieb, Schulz and Mattis [32]. As KNB mechanism is essentially affected by the physical form of thelattice, the simplest case corresponds to the linear arrangement of spins, which features the polarizationgiven in equation (1.2). Then, seeking for exactly solvable case we have to chose an electric field to bepointed in y -direction. This leads to a model of XY chain with DM terms, when DM-vector is parallel to z -axis. This model is well known [33–50], however, for the last half-century quite restricted amount ofpapers have been devoted to it. The paper is organized as follows: in the second Section the formulationof the model and its exact solution in terms of Jordan-Wigner fermionization is given, in the next Sectionthe zero temperature MEE for the simplest model of MEE in quantum spin chains are described, thenthe finite temperature MEE and effects of the XY anisotropy γ are analyzed. The paper is ended with aConclusion.
2. The model and its exact solution
Let us consider spin-1/2 XY chain which has linear form and spin dependent polarization due to KNBmechanism. Supposing the chain to be collinear with the x -axis and the electric field to be pointed in y -direction according to equation (1.2) we arrive at the following Hamiltonian: H = J N Õ j = n ( + γ ) S xj S xj + + ( − γ ) S y j S y j + o + E N Õ j = (cid:16) S xj S y j + − S y j S xj + (cid:17) − B N Õ j = S zj , (2.1)where S α j are the spin-1/2 operators at lattice site j , E is the magnitude of the electric field written inproper units (with coefficient µ absorbed in it) and B is an external magnetic field pointing in z -direction.Various aspects of this model have been considered in a series of papers in last decades [33–50]. In thepresent paper we are interested in the MEE in this model, and particularly, in the effects of XY anisotropy γ . The model is exactly solvable within the Jordan-Wigner fermionization. To proceed we first should ?????-2 n fl uence of XY anisotropy on a magnetoelectric effect in spin-1/2 XY chain in a transverse magnetic fi eld perform a Jordan-Wigner transformation from spin operators to the creation and annihilation operatorsof lattice spinless fermions: S − j = e i π Í j − l = c + l c l c j , S + j = ( S − j ) + , S zj = c + j c j − / , (2.2)where S ± j = S xj ± iS y j . In terms of Fermi operators the Hamiltonian reads: H = N Õ j = (cid:26) J + iE c + j c j + − J − iE c j c + j + J γ ( c + J c + j + − c j c j + ) − B ( c + j c j − / ) (cid:27) . (2.3)Here periodic or anti-periodic boundary conditions are assumed, depending on the number of spinlessfermions which is a conserved quantity. For even (odd) particle number the anti-periodic (periodic)boundary conditions for Fermi operators is imposed, c j + N = − c j ( c j + N = c j ). The further step towardthe diagonalization of the Hamiltonian is a Fourier transformation, c j = √ N Õ k e − ijk c k , c k = √ N Õ j e ijk c j , (2.4)here k takes N values in the first Brillouin zone, − π ≤ k < π , and is equal to π N n for periodic boundary conditions or π N ( n + / ) for the antiperiodic ones. Here n = − N / , − N / + , ... N / − N iseven and n = −( N − )/ , −( N − )/ + , ... ( N − )/ − N is odd. Then, the Hamiltonian takesthe appropriate matrix-form, which is straightforward for diagonalization: H = Õ − π ≤ k <π (cid:0) c + k , c − k (cid:1) (cid:18) ε ( k ) − iJ γ sin kiJ γ sin k − ε (− k ) (cid:19) (cid:18) c k c + − k (cid:19) , (2.5)where ε ( k ) = J cos k + E sin k − B . Let us perform Bogoliubov transformation to new Fermi creationand annihilation operators, (cid:18) c k c + − k (cid:19) = (cid:18) iu k v k − v k − iu k (cid:19) (cid:18) β k β + − k (cid:19) , (cid:18) β k β + − k (cid:19) = (cid:18) − iu k − v k v k iu k (cid:19) (cid:18) c k c + − k (cid:19) , (2.6)where u k + v k =
1. Then, putting u k = √ s + J cos k − B λ k , v k = sgn ( J γ sin k ) √ s − J cos k − B λ k ,λ k = q ( J cos k − B ) + J γ sin k , we finally obtain the diagonal form of the Hamiltonian expressed in terms of free spinless fermions: H = Õ − π ≤ k <π E γ ( k ) (cid:0) β + k β k − / (cid:1) , (2.7) E γ ( k ) = E sin k + sgn ( J cos k − B ) λ k . For the isotropic
X X -chain with DM-terms ( γ =
0) one can easily see that the Hamiltonian (2.5) isalready diagonal in c k operators: H XX = Õ − π ≤ k <π E ( k ) (cid:0) c + k c k − / (cid:1) , (2.8) E ( k ) = p J + E cos ( k − φ ) − B , φ = arcsin E √ J + E . ?????-3 adim Ohanyan
3. Zero-temperature properties and MEE
Let us first describe zero-temperature properties of the spin-1/2
X X chain in presence of the electricand magnetic fields. The simplest quantum chain model exhibiting MEE via KNB mechanism is thesystem described by the Hamiltonian (2.8). The free-fermion picture here is quite simple. The DM-termbreaks a time-reversal symmetry, E (− k ) , E ( k ) , and the two Fermi points are not symmetric withrespect to k =
0. They are given by k , = φ ∓ arccos BB c , B c = p J + E , (3.1)when − B c < B < B c . For B ≤ − B c and B ≥ B c all N free-fermion states in the system are occupied andempty, respectively. The ground state energy per one site, thus, is given by e = − B , B ≥ B c − B + π (cid:16) B arccos BB c − p B c − B (cid:17) , − B c ≤ B ≤ B cB , B ≤ − B c . (3.2)Using standard relations, m = − ∂ e ∂ B and p = − ∂ e ∂ E , one can find zero-temperature asymptotic values ofthe magnetization and polarization: m = , B ≥ B c − π arccos BB c , − B c ≤ B ≤ B c − , B ≤ − B c . , p = , B ≥ B cE √ J + E − B π ( J + E ) , − B c ≤ B ≤ B c , B ≤ − B c . (3.3)The common feature of the free-fermion models with KNB mechanism is that the dielectric polarizationbecomes zero for empty as well as full fermionic filling. Besides the magnetic and dielectric susceptibil-ities, χ = ∂ m ∂ B and χ P = ∂ p ∂ E , the magnetoelectric systems have one more important quantity to describethe response, a magnetoelectric or mixed susceptibility, which in general case is defined by the followingrelation: α ij = (cid:18) ∂ M i ∂ E j (cid:19) T , B = (cid:18) ∂ P j ∂ B i (cid:19) T , E , (3.4)where M i ( P j ) and B i ( E j ) are components of the magnetization (polarization) vector of the sample andexternal magnetic (electric) fields, respectively. For our case all susceptibilities are non-zero only within − B c ≤ B ≤ B c and are given by χ = π p B c − B , χ P = J ( B c − B ) + E B π B c p B c − B , α = − E B π B c p B c − B , (3.5)respectively. Important feature of the simplest spin-1/2 X X chain accounting for MEE is vanishing α whenany of two fields, electric or magnetic, becomes zero. This is an example of trivial MEE, when magnetic(electric) field affects polarization (magnetization) but can not induce it unless the other field is non zero.It can be easily seen that at critical magnetic field, B = ± B c , all susceptibilities have inverse square-rootsingularities, which is the universal properties of XY -type chains. The zero temperature magnetizationcurve around critical field, B = ± B c has square-root behavior. Although, finite-temperature MEE in X X chain was briefly described in Ref. [23], as a limiting case of
X X Z chain, the zero-temperatureMEE is also worth studying, as this is the simplest example of the MEE, described by simple analyticexpressions. In figure 1 zero-temperature polarization and magnetization of the spin-1/2
X X chain withKNB mechanism are presented as functions of electric field. The polarization curves, p ( E ) , demonstratethree different regime of polarization processes close to E =
0: linear, square-root and plateau with furtherquadratic behavior. The regime of polarization curve depends on the value of the magnetic field. It isvery simple to see from the equation (3.3) that polarization curve initially has linear behavior for B < J ,which becomes quadratic at B = J and then changed to plateau with square-root for B > J . Interestingly, ?????-4 n fl uence of XY anisotropy on a magnetoelectric effect in spin-1/2 XY chain in a transverse magnetic fi eld E p E m Figure 1.
Zero-temperature polarization (left panel) and magnetization (right panel) dependance onelectric field for the spin-1/2
X X chain with KNB mechanism. Three regimes of initial polarization p ( E ) is presented: linear ( B < J ), quadratic ( B = J ) and plateau with further square root ( B > J ) atsmall E . The same picture can be seen in the magnetization dependence on the electric field. For bothpanels J =
1, red solid line corresponds to B = .
5, blue dashed line to B = B = . the simplest model of KNB magnetoelectric to great extent reproduces three of four qualitative shapes of polarization curves for more complicated spin-1/2 X X Z chain with KNB mechanism [23]. The rightpanel of the figure 1 demonstrates magnetization dependence on the electric field for the same threevalues of magnetic field as in the left panel, B = . B = B = . M = / B > J (black dot-dashed line). When B is decreasingthe length of the plateau becomes smaller and riches zero at critical value B = J (blue dashed line). Thered solid line demonstrates monotonous decrease of magnetization with increasing the electric field for B < J . The plots of zero-temperature magnetization and polarization dependence on a magnetic field arepresented in figure 2. Here the magnetization curves, m ( B ) , for different values of the electric field havethe same standard form. Also the polarization dependence on magnetic field is uniform with plateau at p =
0, which corresponds to the fully polarized spin state realized at strong enough magnetic fields. B p B m Figure 2.
Zero-temperature polarization (left panel) and magnetization (right panel) dependance onmagnetic field for the spin-1/2
X X chain with KNB mechanism at J =
1. For both panels red solid linecorresponds to E = .
5, blue dashed line to E = E = . Zero-temperature description of a more general spin-1/2 XY chain with γ , XY chain with DM terms, given byequation (2.7) the one-particle excitations are gapless when E ≥ J γ and B ≤ E + J ( − γ ) , orin case of E < J γ for B = J . In the latter case all fermionic one-particle states are still empty, butthe spectrum touches zero at a single point. The general property of the MEE in free-fermion models ?????-5 adim Ohanyan is a vanishing polarization in both cases of fully filled or empty system. Thus, in case of non-zero γ the polarization is non-zero only in the region of ( E , B ) -plane, given by the conditions, E ≥ J γ and B ≤ E + J ( − γ ) .
4. Thermodynamics and MEE
In the more general case of spin-1/2 XY chain with non-zero γ it is much more simpler to deal withthermodynamics of the model than with zero-temperature expressions, which are quite cumbersome andcomplicated even for E =
0, when the system is always gapped or gapless with zero occupation [51]. Inorder to investigate the finite-temperature features of the MEE in the model we need to start from freeenergy (per one spin), which is given by the following integral over the first Brillouin zone: f = − T π π ∫ − π log (cid:18) (cid:18) E γ ( k ) T (cid:19) (cid:19) dk , (4.1)here T is the temperature and the Boltzmann constant was set to unity ( k B =
1) for the sake of simplicity.Using standard relations one can easily obtain expressions for magnetization and polarization of thesystem: m = π π ∫ − π tanh (cid:18) E γ ( k ) T (cid:19) B − J cos k λ k dk , p = π π ∫ − π tanh (cid:18) E γ ( k ) T (cid:19) sin k dk . (4.2)Also, the mixed magnetoelectric susceptibility is useful for figuring out important properties of the MEE: α = π T π ∫ − π ( B − J cos k ) sin k λ k cosh (cid:16) E γ ( k ) T (cid:17) dk . (4.3)Particularly, we are going to figure out an effect of XY anisotropy parameter γ on the MEE. In case ofvanishing electric field, the spectrum of the model is always non-negative, thus, the system is alwaysempty (in terms of Bogoliubov quasi-particles), and increasing XY anisotropy always decreases themagnetization. Polarization in this case is zero. In virtue of DM terms and electric field the spin-1/2 XY chain with KNB mechanism features non-monotonic behavior of magnetization as a function of γ . In thefigure 3 the polarization and magnetization dependence on XY anisotropy γ are exhibited. As in the caseof finite γ the system can have gapless spectrum as well as gapped one depending on the mutual relationbetween electric field, magnetic field and XY anisotropy. Therefore, the behavior of local observables isalso non-monotonic. In contrast to the E = γ (figure 3(right panel)) within the gapless phase. Once the value of XY anisotropy crosses the critical value, | γ c | = | J | p J + E − B , (4.4)a gap opens and magnetization starts to decrease (figure 3, right panel, blue dashed and black dot-dashedlines). If the value of magnetic field is greater than √ E + J (figure 3, right panel, magenta dotted line)there is no gapless phase and magnetization exhibits monotonous decrease with increasing γ . Behaviorof the polarization as a function of γ (figure 3, left panel) shares much in common with the effect ofmagnetic field, the monotonous decrease in gapless phase with plateau at zero for gapped phase. Redsolid and blue dashed lines go to zero at the same value of γ , as for B ≤ J the value of γ at whichthe free-fermion states start to fill up is the same, J γ = E . In figure 4 polarization (left panel) andmagnetization (right panel) dependence on the magnetic field are illustrated. Though, the behaviour ofmagnetization of the XY chain is well known and understood, here an additional feature can be pointedout. For E = γ besides the absence of the saturation field there is only one phasewithout any features on the magnetization curve. In case of finite E it is possible to have both smooth ?????-6 n fl uence of XY anisotropy on a magnetoelectric effect in spin-1/2 XY chain in a transverse magnetic fi eld γ p γ m Figure 3.
Effect of XY anisotropy parameter γ on polarization (left panel) and magnetization (right panel)in appropriate units ( J =
1) and T = . γ is clearly seen in the right panel. For both panels E = . B = .
05 (red solid line), B = B = . B = B (cid:0) B (cid:1) Figure 4.
Low-temperature polarization (left panel) and magnetization (right panel) dependance onmagnetic field for the spin-1/2 XY chain with KNB mechanism at J = γ = . E = E = E = . E = magnetization curve for E < J γ (figure 4 right panel, red solid line) as well as curve with a cuspcorresponding to transition from gapless regime to gapped one (figure 4 right panel, blue dashed, blackdot-dashed and magenta dotted lines). Interestingly, the magnetization curves are exactly the same for allvalues of E < J γ , as in all these cases the spectrum touches zero at one single point. For the valuesof electric field E ≥ J γ the magnetization curves have a cusp at B c = p E + J ( − γ ) separatinggapless regime from gapped one (figure 4 right panel, blue dashed, black dot-dashed and magenta dottedlines). The left panel of the figure 4 demonstrates magnetic-field effect on the polarization. Four curvesare presented for four different constant values of the electric field. All three curves share the samequalitative trend, monotonous decrease from maximal values at B = B c = p E + J ( − γ ) .Electric-field dependence of polarization and magnetization is presented in the figure 5. Here againone can distinguish two parts of the curve, corresponding to gapless and gapped regimes, respectively.The transition takes place at E c = p B − J ( − γ ) . Appearance of the critical point brings to thethermal singularity in the behavior of susceptibilities. Considering magnetoelectric susceptibility givenby equations (3.4) and (4.3) one can see well pronounced peaks at the corresponding values of γ givenby equation (4.4) (See figure 6). Left panel shows the γ dependence of the magnetoelectric susceptibilityfor E = B = . T = .
5, 0 . . α is always negative (for positive fields). The development of peaks(negative) corresponding to the critical value of γ is well pronounced here. The peaks are graduallysmearing out with increasing temperature. Thus, one can see, that within the gapless phase the absolutevalue of the magnetoelectric susceptibility is growing with increasing γ reaching a peak at the transition ?????-7 adim Ohanyan from gapless regime to gapped one. The peak shows a tendency to achieve diverging singularity at T → XY anisotropy, corresponding to so-called quantumIsing chain [52] does not have any specific feature in the sense of MEE. All results concerning MEEare qualitatively similar to the ones presented for the spin-1/2 XY chain with γ ,
0. The quantitativedifference consists in the form of parameters region which corresponds to gapless regime (or non-zeropolarization region). In the case of γ = E ≥ J and B ≤ E . E (cid:2) E (cid:3) Figure 5.
Low-temperature polarization (left panel) and magnetization (right panel) dependance onelectric field for the spin-1/2 XY chain with KNB mechanism for J = 1, T=0.0001 and γ = .
5. For bothpanels, B = . B = B = - - - - - (cid:6)(cid:7)(cid:8) - (cid:9)(cid:10)(cid:11) - (cid:12)(cid:13)(cid:14) - - γ α - - - B α Figure 6.
Magnetoelectric susceptibility dependance on γ (left panel) and magnetic field (right panel)for J = E = B = . T = . T = . T = .
015 (black dot-dashed line). For the right panel γ = . T = . T = . T = .
015 (black dot-dashed line).
5. Conclusion
In the present paper we considered MEE in the exactly solvable spin-1/2 XY chain with KNBmechanism. Our main goal was to figure out the interplay between XY anisotropy γ and MEE. Itturned out that the main difference from the properties of underlying XY chain stems out from thefact that appearance of effective DM term in the Hamiltonian makes possible gapless structure of thespectrum in some region of parameter space, which is determined by electric field, magnetic field and XY anisotropy. Thus, even for the ordinary magnetization curve interplay between XY anisotropy andelectric field (effective DM term) brings essential modifications. It is well known that in case of E = XY ?????-8 n fl uence of XY anisotropy on a magnetoelectric effect in spin-1/2 XY chain in a transverse magnetic fi eld chain with KNB mechanism this is still the case for weak electric fields, but the situation changesdrastically for E > J γ when a transition point (cusp) appears in the magnetization curve. Thiscritical point corresponds to the transition from a gapless to gapped spectrum and emerges under theconditions B = E + J ( − γ ) and E > J γ . Furthermore, influence of the XY -anisotropy on thebehaviour of the magnetization curve is essentially different for gapless and gapped phases. As far as thespectrum is gapless magnetization is growing with increasing γ . For the gapped phase XY anisotropymakes opposite contribution to magnetization. As polarization is always zero for gapped situation in ourmodel, γ can affect polarization only within the gapless phase, where it is decreasing with increasing γ .Magnetoelectric susceptibility is shown to have a characteristic peak at a critical point closely associatedwith opening of a spin gap in the excitation spectrum. We also presented a zero-temperature descriptionof the MEE for the spin-1/2 X X chain being the particular case without XY anisotropy ( γ = Acknowledgements
The author expressed his deep gratitude to Taras Verkholyak and Artem Badasyan for helpful discus-sions.
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