Quantum phase transitions in a new exactly solvable quantum spin biaxial model with multiple spin interactions
aa r X i v : . [ c ond - m a t . s t r- e l ] J u l Quantum phase transitions in a new exactly solvable quantum spin biaxial model withmultiple spin interactions
A.A. Zvyagin
1, 2 Max-Planck-Institut f¨ur Physik komplexer Systeme,N¨othnitzer Str., 38, D-01187, Dresden, Germany B.I. Verkin Institute for Low Temperature Physics and Engineering of the NationalAcademy of Sciences of Ukraine, Lenin Ave., 47, Kharkov, 61103, Ukraine (Dated: November 5, 2018)The new integrable quantum spin model is proposed. The model has a biaxial magnetic anisotropyof alternating coupling between spins together with multiple spin interactions. Our model givesthe possibility to exactly find thermodynamic characteristics of the considered spin chain. Theground state of the model can reveal spontaneous values of the total magnetic and antiferromagneticmoments, caused by multiple spin couplings. Also, in the ground state, depending on the strengthof multiple spin couplings, our model manifests several quantum critical points, some of which aregoverned by the external magnetic field.
PACS numbers: 75.10.Pq,75.40.Cx
Integrable models of quantum physics of magnetismare, unfortunately, rare. Such models, however, are veryimportant for theorists, because they permit to comparethe results of standard for theoretical physics of real sys-tems perturbative approaches with exact ones. For ex-ample, the seminal Ising model served as a basis for manypowerful methods of modern physics, like the scaling,renormalization group, etc. Quantum integrable mod-els in one space dimension (1D) are developed relativelymore, comparing to the 2D or 3D ones (in fact there existonly few examples of quantum integrable models in 2Dand 3D), due to the relative simplicity of their study. Onthe other hand, according to the Mermin-Wagner theo-rem, any nonzero temperature in 1D (and 2D) destroyslong-range magnetic ordering. That theorem reveals, infact, the enhancement of quantum and thermal fluctua-tions in low-dimensional systems, due to peculiarities intheir densities of states.On the other hand, the interest in quantum spin sys-tems, where spin-spin interactions along one or two spacedirections are much stronger than couplings along otherdirections, has considerably grown during last decade.Such interest to low-dimensional quantum spin systemsis motivated, first of all, by the progress in the prepara-tion of real substances with well defined one-dimensionalsubsystems. On the other hand, modern technologiespermit to compose artificial one-dimensional quantumsystems, like quantum wires and rings, which propertiesare created to be similar to theoretically known mod-els. Devices, based on such especially prepared quantum1D spin systems are very promising in the modern nano-technology, in the development of spintronics, or even inthe quantum computation.1D quantum spin systems often manifest quantumphase transitions, i.e. those, which take place in theground state, and which are governed by other than thetemperature parameters, like an external magnetic field,external or internal (caused by chemical substitutions)pressure, etc. In the past most of exactly solvable quantum spin mod-els were related to the class of models with only pairspin-spin interactions between nearest-neighbor spins. Last years more attention of physicists was paid to the-oretical studies of quantum spin models with not onlynearest-neighbor spin-spin interactions, but also withnext-nearest neighbor ones, multiple spin exchange mod-els (e.g., a ring exchange), etc.
Such models some-times appear to be closer to the real situation in quasi-one-dimensional magnets, comparing to the ones withonly nearest-neighbor couplings between only pairs ofspins. For example, such additional interactions are oftenpresent in oxides of transition metals, where the directexchange between magnetic ions is complimented by thesuperexchange between magnetic ions via nonmagneticones. Terms, involving the product of three and fourspin operators or more, were only recently recognizedto be important for the theoretical description of manyphysical systems, despite the fact that multiple spin ex-change models were introduced by Thouless already in1965. For example, multiple spin exchanges were usedfor the description of the magnetic properties of solid He. Later similar models were used to study somecuprates and spin ladders. Often quantum spin mod-els with antiferromagnetic interactions and multiple spininteractions manifest the spin frustration, i.e. the low-est in energy state is highly degenerate. Last but notthe least, such models often reveal quantum phase tran-sitions. For instance, many transition metal compounds,like copper oxides, are believed to manifest features, char-acteristic for quantum phase transitions. However, it isknown that for many quantum spin models the standardquasi-classical theoretical description, based on the quan-tization of small deviations of classical vectors of magne-tization of magnetic sublattices, yield incorrect results,especially in the vicinity of quantum critical points. Thisis why, quantum exactly solvable spin models with multi-ple spin exchange interactions, even being rather formal,and, sometimes, non-realistic, are of great importance:They provide the possibility to check approximate theo-retical methods, used for the description of more realisticphysical models of quantum spin systems with spin frus-tration.In this paper we propose a new integrable model ofquantum spins with nearest-neighbor interactions andmultiple spin exchange. The aim of this work is tostudy a model, that, on the one hand, contains multiplespin interactions, which usually produce incommensuratemagnetic structures. Second, the proposed model con-sists alternating exchange interactions between nearestneighbors, which can be the reason for the spin gap forlow-lying excitations. Finally, the model has the biax-ial magnetic anisotropy, which is believed to be the keyproperty of transition metal compounds with strong spin-orbit coupling. The 2D counterpart of the model can berelated to the plaquette model of p + ip superconduct-ing arrays. On the other hand, the model is relativelysimple, because the Hamiltonian of the model can be ex-actly transformed to the one of the free fermion latticegas, and, hence, most of thermodynamic characteristicscan be calculated explicitly.The Hamiltonian of our exactly solvable quantum spinmodel with alternating nearest-neighbor couplings andthree-spin interactions, which permits exact solution, hasthe form: H = − H X n ( µ S zn, + µ S zn, ) − X n ( J x S xn, S xn, + J y S yn, S yn, ) − X n ( J x S xn, S xn +1 , + J y S yn, S yn +1 , ) − J X n ( S xn, S zn, S xn +1 , + S yn, S zn, S yn +1 , ) − J X n ( S xn, S zn +1 , S xn +1 , + S yn, S zn +1 , S yn +1 , ) , (1)where S x,y,zn, , are the operators of the spin- projectionsof the spin in the n -th cell, which belongs to the sub-lattice 1 or 2, µ , are effective magnetons of the sublat-tices, H is the external magnetic field, directed along z axis, J , ,x,y are the alternating exchange coupling con-stants between nearest neighbor spins in the cell and be-tween cells, and J , are alternating coupling constantsfor three-spin interactions. Notice that the model re-veals the biaxial magnetic anisotropy, i.e. the exchangeinteractions (in the spin subspace) along x , y , and z di-rections are different. In the case J , x = J , y , i.e. inthe case of only uniaxial magnetic anisotropy, the modelcontains, as a special case, the model, studied in Ref. 18.On the other hand, the special case J = J = 0 of themodel is known for many years. Finally, in the absenceof the magnetic field, H = 0, and three-spin couplings, J = J = 0 the model can be related to the so-called1D quantum compass model (in the special case J x = α , J y = 1 − α , J x = 0 J y = 1). Terms, in the Hamilto-nian, which describe three-spin couplings, obviously vi-olate time-reversal and parity symmetries of the systemseparately, but the combination of both symmetries is preserved.After the Jordan-Wigner transformation S zn, , = 12 σ n, , = 12 − a † n, , a n, , ,S + n, ≡ S xn, + iS yn, = Y m 12 ( µ J + µ J ) H cos k + 14 [ J J cos k − J x J y − J x J y − ( J x J x + J y J y ) cos k ] (cid:19) + 116 ( J x J x − J y J y ) sin k . (7)Using the standard particle-hole transformation one canget only positive eigenvalues of the Hamiltonian (6). Onecan check that in the case J , = 0 the spectrum coin-cides with the one of Ref. 19, while for J , x = J , y itreproduces the spectrum from Ref. 18. The energies ofeigenstates of the first branch are non-negative for allparameters of the model. The energies of eigenstates,belonging to the second branch can be equal to zero, de-pending on the values of the parameters of the model.Figs. 1-2 represent the zero field dispersion relations forboth branches as functions of three-spin interactions forthe homogeneous three-spin couplings and for the alter-nating three-spin couplings, respectively, for the quan-tum compass model with α = 0 . F = − T X k X j =1 ln (cid:16) ε k,j T (cid:17) . (8) J k k,1,2 ε −3−2−10123−3 −2 −1 0 1 2 30.20.40.60.811.21.41.6 FIG. 2: The same as in Fig. 1, but for J = 0. Obviously, the z -projection of the average magnetizationof the system is M z = 12 X k X j =1 ∂ε k,j ∂H tanh (cid:16) ε k,j T (cid:17) . (9)From this formula it is easy to show that M z is zerofor H = 0 for any nonzero temperature, in accordancewith the Mermin-Wagner theorem. The low temperaturebehavior of the magnetic susceptibility, χ = 12 X k X j =1 (cid:20) ∂ ε k,j ∂H tanh (cid:16) ε k,j T (cid:17) + (cid:18) ∂ε k,j ∂H (cid:19) h T cosh (cid:16) ε k,j T (cid:17)i − (cid:21) − , (10)and the specific heat, c = X k X j =1 ε k,j T cosh ( ε k,j / T ) (11)depend on the values of coupling constants J x,y , , J , ,the effective magnetons, µ , , and the value of the ex-ternal magnetic field H , see below. One can check thatthere is no ordering and, therefore, none of thermody-namic characteristics of the considered system has pecu-liarities at any nonzero temperature. On the other hand,as it will be shown below, in the ground state sponta-neous magnetic ordering can take place. In the cases,where elementary excitations of the model are gapped,the low-temperature magnetic susceptibility and the spe-cific heat reveal exponential in T dependencies in theabsence of spontaneous magnetization. If the model re-veals the spontaneous magnetic moment, the magneticsusceptibility at low temperatures is divergent. On theother hand, for gapless situation of low-energy states ofthe model, the magnetic susceptibility is finite at lowtemperatures for the absence of spontaneous magneticordering at T = 0, while the specific heat is linear in T . At the critical lines of quantum phase transitions(see below) our model manifests either square root, orlogarithmic in T and magnetic field behaviors of the spe-cific heat and the magnetic susceptibility. In the case,where interaction constants are very different from eachother (or, to be more precise, when two branches of eigen-states are characterized by very different energy scales),the specific heat and the magnetic susceptibility can re-veal two-maxima temperature dependencies.The most important properties of the one-dimensionalspin system are manifested in the ground state. Theground state energy of our model can be written as E = − √ X k q F k + p G k . (12)Then the z -projections of each total spin moment of acell in the ground state can be written as: S z , ≡ ∂E ∂µ , H = 14 √ X k √ G k x , ,k + y , ,k √ G k p F k + √ G k , (13)where x , ,k = µ , H − J , cos k ,y , ,k = (cid:18) µ µ H − ( µ J + µ J ) H cos k + 12 [ J J cos k − J x J y − J x J y − ( J x J x + J y J y ) cos k ] (cid:19) × ( µ , H − J , cos k ) . (14)The sum of the z -projections of spin moments can be con-sidered as the ground state vector of magnetism, or mag-netization of the model, M z = µ S z + µ S z , while thedifference describes the vector of antiferromagnetism, L z = µ S z − µ S z , or the staggered magnetization of themodel. From these expressions one immediately sees thatin the ground state the model can have nonzero spon-taneous magnetic and antiferromagnetic moments (i.e.magnetic ordering for H = 0), caused by nonzero three-spin interactions. We would like to turn attention thatthe signs of J , do matter. Namely, depending on theirsigns, the spontaneous magnetization of the model in theground state can be positive or negative, with respect tothe direction of the magnetic field. It is different fromthe behaviors of other known exactly solvable spin chainmodels. The reason for the onset of the spontaneousmagnetic and antiferromagnetic moments in the groundstate of our model is related to the violation of the time-reversal symmetry by three-spin coupling terms. It is interesting to notice that the equality G k = 0means that ε k, = 0. As it is shown below, namely thecondition G k = 0 determines the features in the behaviorof all ground state characteristics of the spin chain. Onecan see, that G k = 0 either at sin k = 0 (i.e. for k = 0 , π ),or, for any k , if J x J x = J y J y (it turns out that thiscondition does not depend on the magnetic field and onthe values of three-spin couplings).Let us consider first the case with J x J x = J y J y .Notice that the limiting case of the quantum compassmodel belongs to the situation. The first branch of eigen-states is ever positive, but the second one can reach zeroonly for two values of the quasimomenta ( k = 0 , π ). Thenit is simple to show that the ground state magnetisationis a continuous function of the external magnetic field, ex-cept of at H = 0 for µ J = µ J and µ J = µ J ,see Eqs. (13-14). For the latter the spontaneous magneti-zation appears, and, therefore, the ground state magneticsusceptibility is divergent there at H = 0. The magneticsusceptibility for nonzero values of H can have peculiar-ities, proportional to ln | H − H c,i | ( i = 1 , , , H c, , = 14 µ µ (cid:18) ( µ J + µ J ) ± (cid:20) ( µ J − µ J ) + µ µ ( J x + J y )( J y + J x ) (cid:21) / ,H c, , = 14 µ µ (cid:18) − ( µ J + µ J ) ± (cid:20) ( µ J − µ J ) + µ µ ( J x − J y )( J y − J x ) (cid:21) / (cid:19) . (15)at which second order quantum phase transitions cantake place, see Figs. 3-4.Such quantum critical points exist, naturally, only ifthose critical values of the field are real and non-negative.They are real if | µ µ ( J x ± J y )( J y ± J x ) | ≤ ( µ J − µ J ) , (16)and the first two critical values are non-negative for µ µ > 0, if( µ J + µ J ) ≥ (cid:20) ( µ J − µ J ) + µ µ ( J x + J y )( J y + J x ) (cid:21) / > , (17)or the second two critical values are non-negative, if − ( µ J + µ J ) ≥ (cid:20) ( µ J − µ J ) + µ µ ( J x − J y )( J y − J x ) (cid:21) / > . (18)For µ µ < J J 13 23c 1,2 H −3−2−10123 −202−2−1012 FIG. 3: Critical values of the magnetic fields H c , for the ex-actly solvable spin model as functions of parameters of three-spin interactions J and J . We used µ = 1 . µ = 0 . J x = 1, J y = 1 . J x = 2, and J y = 0 . c 3,4 H J J −3 −2 −1 0 1 2 3 0 2−1.5−1−0.500.511.5 FIG. 4: Critical values of the magnetic fields H c , for the ex-actly solvable spin model as functions of parameters of three-spin interactions J and J . The set of parameters is thesame as in Fig. 3. phase transitions, governed by the external magneticfield, take place in the system. If one of them is sat-isfied, and the other isn’t, then only up to two quantumphase transitions can happen. If the conditions Eqs. (17),or (18), are not satisfied, then only one or two quantumphase transitions, governed by the field, take place. Ifone of the effective magnetons is zero (i.e. one of theions, which form elementary cell, is non-magnetic), butthree-spin interaction constants are not, then the quan- tum phase transitions take place at the values of the mag-netic field H c , = ± J J + ( J x ± J y )( J y ± J x )2 µ , J , , (19)at which the magnetic susceptibility has logarithmic sin-gularities. Naturally, only positive values of H c , matter.Finally, if both of effective magnetons are zero, then, ob-viously, there are no quantum phase transitions, governedby the magnetic field. It turns out that at nonzero tem-peratures thermodynamic characteristics of the model re-veal logarithmic in T features at the critical values of themagnetic field.Consider now the situation, in which J x J x = J y J y .The energies of the eigenstates in this case can be writtenas ε k, , = (cid:20)(cid:0) (cid:20) ( µ + µ ) H − 12 ( J + J ) cos k (cid:21) + | B k | (cid:1) / ± (cid:0) (cid:20) ( µ − µ ) H − 12 ( J − J ) cos k (cid:21) + | A k | ] (cid:1) / (cid:21) / , (20)where A k = 12 [ J +1 + J +2 exp( − ik )] ,B k = 12 [ J − − J − exp( − ik )] . (21)It is obvious that ε k, is positive for any parameters ofthe model. On the other hand, for the lower branch forsome ranges of the quasimomentum k and external mag-netic field H , the first term under the square root sign inEq. (20) can be smaller than the second one. It impliesthat eigenstates for lower branch can exist only for someranges of k , depending on the value of the external field H . The analysis of this situation is similar to the above(except the fact that one has to take into account nonzeroFermi seas, i.e. totally filled states with negative energiesfor some ranges of k depending on the value of the ex-ternal field; the critical value of k is determined from thecondition ε ,k c = 0). One can see that there exist fourcritical values of the magnetic field, at which quantumphase transitions can take place, see Eqs.(15)-(19). Thedifference, comparing to the case with J x J x = J y J y ,is in the more strong features of the magnetic suscep-tibility at critical fields ∼ / p | H − H c,i | in the groundstate, and, therefore, in square root peculiarities in T of thermodynamic characteristics of the model, like themagnetic susceptibility and specific heat, at critical val-ues of the magnetic field.Let us consider the homogeneous limiting case of ourmodel J x = J x = Jx , J y = J y = J y , J = J = J , µ = µ = µ . In this case the Hamiltonian can be writtenas H = X k ε k (cid:18) b † k b k − (cid:19) , (22) H M z FIG. 5: The ground state dependencies of the magnetizationas a function of the magnetic field for the homogeneous limitof the exactly solvable model for J x = 1, J y = 0 . J = − where ε k = (cid:20) µH − 12 ( J cos(2 k ) + ( J x + J y ) cos( k ) (cid:21) + 14 ( J x − J y ) sin ( k ) . (23)One can see that the energy (23) is non-negative. It canbe equal to zero only for J x = J y , or, if J x = J y for k = 0 , π . In the later case there are two critical values ofthe magnetic field, at which quantum phase transitionscan take place H hc, , = (2 µ ) − [ J ± ( J x + J y )] . (24)Obviously, quantum phase transitions take place if valuesof the critical field are non-negative, i.e. they take placefor µ > 0, if J ± ( J x + J y ) ≥ J = ± ( J x + J y ) are the conditions of the quan-tum phase transition, governed by the three-spin cou-pling. The ground state magnetic susceptibility has log-arithmic features ∼ ln | h − H hc , | at critical values of thefield.Figs. 5 and 6 show the ground state behavior of themagnetization of our model for the homogeneous case asa function of the magnetic field. Fig. 5 presents the be-havior for J < − ( J x + J y ), and Fig. 6 demonstrates themagnetic field behavior for the region J > ( J x + J y ).One can see that for both regions there is a spontaneousmagnetization, but its sign (with respect to the directionof the field) depends on the sign of three-spin interac-tions. Also, there are two quantum phase transitions for H M FIG. 6: The ground state dependencies of the magnetizationas a function of the magnetic field for the homogeneous limitof the exactly solvable model for J x = 1, J y = 0 . J = 2. H M z FIG. 7: The ground state dependencies of the magnetizationas a function of the magnetic field for the homogeneous limitof the exactly solvable model for J x = 1, J y = 0 . J = 0 . J > ( J x + J y ), while for J < − ( J x + J y ) the groundstate magnetization is a smooth function of H .Figs. 7 and 8 present the magnetic field behavior of themagnetization for − ( J x + J y ) < J < ( J x + J y ) for positiveand negative values of J , respectively. One can see thatin this region there is no spontaneous magnetization, andonly one second order quantum phase transition takesplace.On the other hand, if J x = J y , the eigenstates of theHamiltonian can be negative for some ranges of k depend-ing on the value of the magnetic field. Negative energies HM z FIG. 8: The ground state dependencies of the magnetizationas a function of the magnetic field for the homogeneous limitof the exactly solvable model for J x = 1, J y = 0 . J = − . imply the nonzero Fermi sea, where eigenstates with neg-ative energies are totally filled, and the ones with positiveenergies are empty. In that case quantum phase transi-tions yield square root singularities ∼ / q | H − H hc, , | of the magnetic susceptibility, cf. Refs. 8,9.It is important to point out that the quasiclassical de-scription of our model (when one replaces spin operatorsby classical vectors, and quantizing small deviations fromthe classical minimal energy state) does not reproduce ex- act results for the inhomogeneous (dimerized) situation.Namely, in the classical description of the model withoutbiaxial anisotropy one of the branches of eigenstates isobviously gapless, unlike the exact result.In conclusion, motivated by recent experiments onquasi-1D quantum spin systems and recent theories ofquantum compass model, we proposed the integrablemodel, in which exchange interactions between neighbor-ing spins is accompanied by the multiple spin exchangewith the biaxial magnetic anisotropy. The model is sim-ple (due to the exact mapping to the problem of the lat-tice free fermion gas), and, therefore, permits to obtainexactly thermodynamic characteristics of the consideredquantum spin chain. The most important behavior ofthe model is in the ground state. Our model manifestsa ferrimagnetic-like ordering in the ground state. De-pending on the signs of the parameters of three-spin cou-plings, the spontaneous magnetic moment of the systemin the ground state can be positive or negative (with re-spect to the direction of the magnetic field). The systemcan undergo several second order quantum phase tran-sitions, governed by the external magnetic field and thethree-spin couplings strengths (the later can be causedby an external or internal pressure). Despite some artifi-cial structure of our model, we expect that more realisticquantum biaxial spin systems with multiple exchange in-teractions and the alternation of the exchange betweennearest neighbor spins, will show similar to our simplemodel behavior, i.e. our exact solution has generic fea-tures for this class of quantum systems.Partial support from the Institute of Chemistry ofV.N. Karasin Kharkov National University is acknowl-edged. See, e.g., A. A. Zvyagin Finite Size Effects in CorrelatedElectron Models: Exact Results , Imperial College Press,London, 2005. V. M. Kontorovich and V. M. Tsukernik, Zh. Eksp. Teor.Fiz. , 1167 (1967) [Sov. Phys. JETP , 687 (1968)]. A. M. Tsvelik, Phys. Rev. B , 779 (1990); H. Frahm, J.Phys. A , 1417 (1992). A. E. Borovik, A.A. Zvyagin, V. Yu. Popkov, andYu. M. Strzhemechny, Pis’ma Zh. Eksp. Teor. Fiz. , 292(1992) [JETP Lett. , 292 (1992)]; A. A. Zvyagin Fiz.Nizk. Temp. , 1029 (1992) [Sov. J. Low Temp. Phys. , 723 (1992)]. V. Yu. Popkov and A. A. 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