Integrable anisotropic spin-ladder model

Transcription

1 PHYSICA REVIEW B, VOUME 64, Interable anisotropic spin-ladder model Arlei Prestes Tonel,, * Anela Foerster,, Jon inks,,2, and Andre uiz Malvezzi 3, Instituto de Física da UFRGS, Avenida Bento Gonçalves 9500, Porto Alere, RS-Brazil 2 Department of Mathematics, The University of Queensland, Queensland, 4072, Australia 3 Departamento de Física da UNESP, Avenida En. uiz Edmundo Carrijo Coube, s/n, Bauru, SP-Brazil Received 3 December 2000; published 6 July 200 We present an interable spin-ladder model, which possesses a free parameter besides the run couplin J. Wan s system based on the SU(4) symmetry can be obtained as a special case. The model is exactly solvable by means of the Bethe ansatz method. We determine the dependence on the anisotropy parameter of the phase transition between apped and apless spin excitations and present the phase diaram. Finally, we show that the model is a special case of a more eneral Hamiltonian with three free parameters. DOI: 0.03/PhysRevB With the discovery of hih-temperature superconductivity in doped copper oxide or cuprate materials, a tremendous effort has been made to understand the physics underlyin this phenomenon. In the absence of dopin these compounds are reasonably approximated by the two-dimensional Heisenber model 2,3 or some suitable eneralization describin spin-exchane-type interaction. It is well known that in one dimension the Heisenber model is exactly solvable via Bethe ansatz methods and from this solution the spectrum of elementary spin excitation is apless. On the other hand, the existence of the spin ap is critical for the observed phenomenon of superconductivity to occur under dopin. To maximize the interaction between theory and experiment, much work is now focused on quasi-one-dimensional models known as ladders. The introduction of these ladder systems has brouht about a sinificantly increased understandin of the physics of the cuprate compounds. By introducin the concept of the ladder model the apparent contradiction in the excitation spectrum is resolved, since the ladder allows for the formation of sinlet states alon the runs which are responsible for the formation of the spin ap. However, the usual Heisenber ladder model cannot be solved. In order to ain some results in the theory of spin ladder systems, many authors have considered eneralized models, which incorporate additional interaction terms that uarantee exact solvability. Remarkably, such eneralized models still exhibit realistic physical properties such as the existence of a spin ap 4 and the manetization plateaus at fractional values of the total manetization. 5 This approach has been used to derive quasi-onedimensional systems usin the well-established theories from the one-dimensional case. 4 5 In all cases cited above, no free parameters are present, other than the run interaction couplin and applied manetic field, due to the strict conditions of interability. With the presence of free parameters it is reasonable to expect that the solution may provide better test models for describin the various behaviors associated with ladder systems. The purpose of this paper is to present an interable eneralized spin ladder with one extra parameter, characterizin anisotropy, without violatin interability. This model is exactly solvable by the Bethe ansatz and it reduces to the model introduced by Wan 4 for a special limit of this extra parameter. The situation here is akin to the eneralization of PACS numbers: 75.0.Jm, 7.0.Fd, Fd the XXX chain to the anisotropic XXZ version. The introduction of the additional free parameter in the present case allows for an example of a model with a critical line varyin continuously with the anisotropy. More specifically, the size of the ap in the massive reion depends explicitly on the anisotropy parameter, which in turn shows dependence of the anisotropy parameter for the points at which the ap closes that define the phase transition. et us bein by introducin the eneralized spin-ladder model, whose Hamiltonian reads h j, 2 J j j, H () h j, 4 z z j j t j z z j j j j z z j. 2 t j 2 z z j t j t j Above j and j are Pauli matrices actin on site j of the upper and lower les, respectively, J is the strenth of the run couplin we will consider only the case J0 in the subsequent analysis correspondin to antiferromanetic couplin, and t is a free parameter representin an anisotropy in the les and interchain interaction. Throuhout, is the number of runs equivalently, the lenth of the ladder and periodic boundary conditions are imposed. By settin t in Eq., Wan s model based on the SU(4) symmetry 4 can be recovered. Strictly speakin, it is SU(4) invariant in the absence of the run interactions. The Hamiltonian is invariant under interchane of the les; i.e., j j. Moreover, under spin inversion for both le spaces the Hamiltonian is invariant with the interchane t t. For this reason we see that the parameter t plays the role of spin anisotropy. The enery eienvalues of the Hamiltonian are iven by E j 2 /4 2J 2J, /200/645/ /$ The American Physical Society

2 TONE, FOERSTER, INKS, AND MAVEZZI PHYSICA REVIEW B FIG.. Eneries E i (i0,...,8)versus run couplin J for different values of the anisotropy t. Notice that there is mainly competition between E 0 and E to be the lowest-enery level of the model. In addition, the critical value of J above which E 0 is the round-state enery varies with t reachin its minimum value when t. j are solutions to the Bethe ansatz Eqs. 3 below. The Bethe ansatz equations arise from the exact solution of the model throuh the nested alebraic Bethe ansatz method and read M ji/2 j l i 2 t (2 ) j i/2 l j j l i j i/2 j i/2, t (2 ) i i j i/2 j i/2 i/2 i/2, t (2 2 ) i i i/2 i/2. We remark that althouh the anisotropy parameter t does not appear explicitly in the enery expression 2, the solutions j for the Bethe ansatz equation do depend on t as will be illustrated later. The exact diaonalization of the two-site Hamiltonian shows that for J 2 (t/t) the unique round state 3 assumes the form of the product of the sinlets with enery E 0 24J and the eneries of the excitations are iven by E 2Jt/t, E 3 t/t, E 2 2Jt/t, E 4 t/t, E 5 2J2, E 6 2J2, E 7 2, E 8 2. A sample of these numerical results are presented in Fi. above. For sites it follows that the round state is still iven by a product of run sinlets when J 2 (t/t) and the enery is (2J). This is in fact the reference state used in the Bethe ansatz calculation and corresponds to the case 0 for the Bethe ansatz Eq. 3. To describe an elementary spin- excitation, we take and 0 in Eq. 3, which leads to the imainary solution for the variable strictly, the lattice lenth is assumed to be even, i 2 t t,

3 INTEGRABE ANISOTROPIC SPIN-ADDER MODE PHYSICA REVIEW B By solvin 0 for J we find the critical value J c 2 (t/t), indicatin the critical line at which the quantum phase transition from the dimerized phase to the apless phase occurs. The phase diaram in Fi. 2a assumes a simpler form after a suitable reparametrization. We introduce a new parameter K iven by K(t/t)/2. In Fi. 2b the phase diaram is represented in terms of K and J. The phase boundary is now a straiht line iven by JK. As a preliminary attempt to characterize the apless phase we have studied by numerical diaonalization the enery spectra of Eq. on ladders of sizes up to eiht runs and several values of J and K. As mentioned above, if JK the spectra is apped and the round state a product of sinlets on each run. As we cross the phase transition line a state with finite manetization becomes the round state. By further decreasin J, with K fixed, the round-state manetization initially increases and then, if J is made small enouh, drops to zero. This behavior resembles that of the one-dimensional anisotropic Heisenber model, also called the XXZ chain, in the presence of a manetic field. 6 In the XXZ chain with no manetic field applied, the anisotropy can be tuned to brin the system into a massive antiferromanetic AF phase, the round state is a Néel state. By tunin the manetic field inside this AF phase we observe a behavior similiar to the one found here in terms of the parameter J. In particular, the Pokrovsky-Talapov 6,7 phase transition appearin in the phase diaram of the XXZ chain has many features in common with the phase transition found here, with J and K playin the roles of the manetic field and the anisotropy, respectively. Based on this analoy, we conjecture that the round state with null manetization found in our model for small values of J suests the presence of an AF apped phase Néel phase. Therefore, another phasetransition line is expected to exist below the one presented here. The interability of this model can be shown by the fact that it can be mapped see Eq. 7 below to the followin Hamiltonian, which can be derived from an R matrix obeyin the Yan-Baxter alebra for J0, while for J0 the run interactions take the form of a chemical-potential term. FIG. 2. a Run couplin J versus anisotropy t. This raphic represents the phase diaram and the dotted line shows Wan s point. The curve J(t/t)/2 divides the apped and apless phases. b Run couplin J versus reparametrization parameter K. This raphic shows a reparametrization of the curve J(t /t)/2 in terms of K(t/t)/2. In this parameterization, the phase boundary is a straiht line. ivin the minimal excited-state enery. The enery ap can easily be calculated usin the exact Bethe ansatz solution and has the form 3 ĥ j, 0 Ĥ () ĥ j, 2JX j 00, X j X X 20 j X 02 X 02 j X 20 X 3 3 j X X 3 j X 3 tx 0 j X 0 X 2 j X 2 X j X X 23 j X 32 t X 0 j X 0 X 2 j X 2 X j X 6 2 J 2 t t. 5 X 32 j X 23. Above X j j j are the Hubbard operators with j the orthoonalized eienstates of the local operator j j, as in Wan s case. 4 The local Hamiltonians and 6 are related throuh the followin basis transformation:, /2,, ),, /2,, ), The followin R matrix,,,,,,

4 TONE, FOERSTER, INKS, AND MAVEZZI PHYSICA REVIEW B a t b 0 0 c b c tb c c 0 0 tb a tb 0 0 c b c c b c 0 0 t R b a tb 0 0 c c t b c b c 0 0 t b a, 8 with ax, bx, and c obeys the Yan-Baxter alebra P is the permutation operator. This model studied above represents one particular case of a more eneral Hamiltonian that has three free parameters and reads R 2 xyr 3 xr 23 yr 23 yr 3 xr 2 xy 9 and oriinates the Hamiltonian 6 for J0 by the standard procedure 8 H h j, 2 J j j, 0 ĥ j, P d dx Rx x0, h j, j t 4 z z j t 2 4 z z j t 3 j j j t 4 z z j t 2 4 z z j j t 3 j 4 z z j 2 z z j t j t j 4 z z j 2 z z j t 2 j t 2 j

5 INTEGRABE ANISOTROPIC SPIN-ADDER MODE PHYSICA REVIEW B This Hamiltonian can be mapped to see Eq. 7 Ĥ ĥ j, 2JX 00 j, 3 ĥ j, 0 X j X X 20 j X 02 X 02 j X 20 t X 0 0 j X X 2 j X 2 t 2 X 30 j X 03 X 32 j X 23 t 3 X 3 3 j X t X 0 j X 0 X 2 j X 2 t 2 X j X X 23 j X 32 t 3 X 3 j X 3. Usin the alebraic nested Bethe ansatz method, this model can be exactly solved. The enery eienvalues of the Hamiltonian 0 are also iven by Eq. 2 while the Bethe ansatz equations reads (M t 3 ) M t 3 M 2 t 3 3 ji/2 j i/2 l j j l i j l i (M t 3 ) M t 3 M 2 t 3 3 i i j i/2 j i/2 j i/2 j i/2, i/2 i/2, t ( ) t 2 ( ) t 3 ( ) i/2 i/2. i i The physics of the interable model presented here is expected to be much richer, since the presence of these extra parameters will certainly influence the phase diaram of the model. More details of this model are bein studied and the results will be shown in a future work. To summarize, we have introduced a eneralization of Wan s spin-ladder model based on the SU(4) symmetry. This was achieved by introducin one extra parameter into the system without violatin interability. The Bethe ansatz equations as well as the enery expression of the model are presented. We show also that the model has a ap that depends on the free parameter and the critical point, and the phase diaram was obtained. We note that our model with one free parameter is a special case of a more eneral interable Hamiltonian that has three free parameters. A comprehensive analysis of the eneral model will be undertaken in future work. J.. thanks the Fundação de Amparo a Pesquisa do Estado do Rio Grande do Sul and the Australian Research Council for financial support. He also thanks the Instituto de Física da UFRGS for their kind hospitality. A.P.T., A.F. and A..M. thank CNPq-Conselho Nacional de Desenvolvimento Científico e Tecnolóico for financial support. A..M. acknowledes the financial support from Fundação deam- paro à Pesquisa do Estado de São Paulo FAPESP-Brazil. A.F. wishes to acknowlede the kind hospitality of the Center for Mathematical Physics at the University of Queensland. * address: prestes@if.ufrs.br address: anela@if.ufrs.br address: jrl@maths.uq.edu.au address: malvezzi@fc.unesp.br J. B. Bednorz and K. A. Müller, Z. Phys. B: Condens. Matter 64, E. Daotto and T. M. Rice, Science 27, E. Daotto, Rep. Pro. Phys. 62, Y. Wan, Phys. Rev. B 60, J. de Gier and M. T. Batchelor, Phys. Rev. B 62, R S. Albeverio, S.-M. Fei, and Y. Wan, Europhys. ett. 47, M. T. Batchelor and M. Maslen, J. Phys. A 32, H. Frahm and A. Kundu, J. Phys.: Condens. Matter, M. T. Batchelor and M. Maslen, J. Phys. A 33, M. T. Batchelor, J. de Gier, J. inks, and M. Maslen, J. Phys. A 33, J. de Gier, M. T. Batchelor, and M. Maslen, Phys. Rev. B 6, J. inks and A. Foerster, Phys. Rev. B 62, A. Kundu, J. Math. Phys. 4, Y. Wan and P. Schlottmann, Phys. Rev. B 62, A. Foerster, K. E. Hibberd, J. R. inks, and I. Roditi, J. Phys. A 34, N. M. Booliubov, A. G. Izerin, and V. E. Korepin, Nucl. Phys. B 275, ; F. Woynarovich, H-P. Eckle, and T. T. Truon, J. Phys. A 22, V.. Pokrovsky and A.. Talapov, Zh. Éksp. Teor. Fiz. 78, Sov. Phys. JETP 5, V. E. Korepin, N. M. Booliubov, and A. G. Izerin, Quantum Inverse Scatterin Method, Correlation Functions and Alebraic Bethe Ansatz Cambride University Press, Cambride, Enland,