PHYSICAL REVIEW B VOLUME 56, NUMBER 13 1 OCTOBER 1997-I Exact solution of the biquadratic spin-1 t-J model in one dimension F. C. Alcaraz and R. Z. Bariev* Departamento de Fi´sica, Universidade Federal de Sa o Carlos, 13565-905, Sa o Carlos, Sao Paulo, Brazil Received 28 May 1997 A generalization of the t-J model with a nearest-neighbor hopping is formulated and solved exactly by the Bethe ansatz method. The model describes the dynamics of spin-S fermions with isotropic or anisotropic interactions. In the case S 1 the magnetic interaction is biquadratic in the spin operators. In contrast to the SU(N) generalization of the t-J model, studied previously in the literature, the present model possesses beyond a massless excitation also a massive one. The physical properties indicate the existence of Cooper-type pairs with finite binding energy. S0163-1829 97 05538-0 The t-J model has emerged as a paradigm for studying where now S x ,Sy ,Sz) are spin-1 Pauli operators lo- the low-energy electronic properties of the copper-oxide- j (S j j j cated at site j. We show that this model is exactly integrable based high-temperature superconductors.1,2 Although high- at the special point t J. Actually the above Hamiltonian is Tc cuprates are at least two-dimensional systems the one- the isotropic version of a family of anisotropic models de- dimensional version of the model and its generalizations are scribing the dynamics of spin-S fermions with Hamiltonian also intensively studied since in this case exact results can be derived.3­12 The t-J model describes the dynamics of spin- 12 fermionic particles with Hamiltonian given by L S H P c j,sc j 1,s c j 1,sc j,s P j 1 s S H t P c j, c j 1, c j 1, c j, P j, L S u sutc j,sc j,tc j 1, sc j 1, t J j 1 s,t S S j*S j 1 njnj 1/4 , 1 j 1 1 cosh n jn j 1 , 3 where cj, is the standard fermion creation operator, S j 1 2 j is the particle-spin operator and n j is the particle- number operator at site j. The projection operator P ex- where L is the lattice size, , cludes the double occupation at each site. Unfortunately the 1 1 and the parameters us , which play the role of anisotropies should satisfy u exact integrability of Eq. 1 is obtained only at the super- s 1/u 2 symmetric point J 2t.3­6 At this point the model has no s (s S, S 1, . . . , S) and 2 cosh u S 2 2 gap and the critical exponents governing the long-distance u S 1 *** uS . The particular case S 12 and 1 1 behavior of correlation functions were calculated.7 These re- is the anisotropic version of the supersymmetric t-J model. sults show that for any density of holes the spin-spin corre- The biquadratic t-J model, at t J, given in Eq. 2 is ob- lation functions dominate the superconducting ones, and as a tained by choosing in Eq. 3 S 1, 1 1 and u 1 consequence the model has no superconducting properties. u0 u1 1. For general spin S the magnetic interactions These results were extended to the SU(N) generalization of can be written as a polynomial of degree 2S in the spin the t-J model of fermions of arbitrary spin S.8­10 The inte- operators. grability of an anisotropic generalization of the SU(N) su- The exact integrability of these models, from a math- persymmetric t-J model has been shown13,14 and the critical ematical point of view, comes from the fact that the Hamil- exponents of the correlation functions have been tonian density in Eq. 3 is related to the generators of Hecke calculated.15,16 algebras,17 with deformation parameter q given by the rela- In this paper we present a set of models of strong- tion q 1/q 2 cosh . correlated electrons which are exactly solvable. The first ex- The eigenstates and eigenvalues of Hamiltonian 3 can ample of these models is the spin-1 biquadratic t-J model be obtained exactly within the framework of the Bethe- with Hamiltonian given by ansatz method.18­21 The structure of the Bethe-ansatz equa- tions follows from the solution of the two-particle problem. The two-electron wave function can be written as a product H t P c of two factors: a coordinate wave function referring to the j, c j 1, c j 1, c j, P j, positions and momenta of the particles and a spin part, the global wave function being antisymmetric under the ex- J S change of two particles. The scattering matrix can be written j*S j 1 2 n jn j 1 , 2 j in the following form: 0163-1829/97/56 13 /7796 4 /$10.00 56 7796 © 1997 The American Physical Society 56 BRIEF REPORTS 7797 S m , 1 2 1 1 1 cosh 1 2 sin j i /2 L sin 1 m 1 j i /2 , sin j i /2 1 sin j i /2 , , 1u u 1 2 10 n m , , , 4 sin j i /2 sin i . where j 1 sin j i /2 1 sin i sin In the case 1 1 the first set of equations in Eq. 10 should be replaced by sin 5 i n and sin j ( j 1,2, . . . ,n) are suitable particle rapidities related sin j i /2 L j l i to the momenta k 1 m 1 j of the electrons by sin j i /2 l 1 sin j l i m sin j i /2 k . j j ; 12 , 1 1, 6 1 sin j i /2 j ; 12 , 1 1, with the function defined by The total energy and momentum of the model are given in terms of the particle rapidities j in the following form: ; 2 arctan cot tan ; , . n n 7 sinh2 E 2 cosk j 2 1 cosh , A necessary and sufficient condition for the applicability of j 1 j 1 cosh cos2 j 11 the Bethe ansatz method is the Yang-Baxter equation.18,21 In n our case the S-matrix satisfies these equations in the nonde- P k formed and q-deformed cases.17 The isotropic case corre- j . j 1 sponds for S 12 to the q-deformed case where us 1(s S, . . . ,S) and q 1/q 2S 1. The underlying Hecke al- Equations 10 and 11 have the same structure as those gebra of the model implies that differently from the super- appearing in the anisotropic t-J model15,16 provided a suit- symmetric t-J model we should have gapped spin excita- able definition of the parameter is given. It means that in tions for S 1. This model is an example of an integrable spite of the physical processes in the models with S 12 and model with the S-matrix of the form 4 which is connected S 12 being quite different there is a ``weak equivalence'' in with the Hecke algebra. The Hamiltonian 3 is diagonalized Baxter's sense25 between models with different values of by standard procedure by imposing periodic boundary con- spin S in the sector where m n/2. Of course in the general ditions on the Bethe function. These boundary conditions can case this equivalence does not exist. be expressed in terms of the transfer matrix of the nonuni- Although the models are exactly integrable for both signs form model which can be constructed on the basis of the of and 1 in Eq. 3 let us now restrict to the more physi- S-matrix 4 by using the quantum method of the inverse cally interesting case 1 and 1 1, where we have at- problem.22,23 The rapidities traction among pairs. In this case the ground state contains j that define a n-particle wave function are obtained by solving the equations m n/2 bound pairs characterized by a pair of complex elec- tron rapidities sinh j i /2 L 1 sinh 1 n 1 j , 8 j i /2 2 v i , v 2 . 12 where ( ) is the eigenvalue of the transfer matrix The second set of equations in Eq. 10 is fullfilled within n exponential accuracy whereas the first set can be treated in T l S l l 1 the similar way as in Refs. 15, 16. Inserting Eq. 12 in the l , n 1 1 . 9 l l l 1 l l first set of equations in Eq. 10 and introducing the density It is simple to verify that besides the number of particles n, function (v) for the distribution of v in the thermody- the magnetization z namic limit, we obtain the linear integral equation jS j and the number of paired electrons m are conserved quantities in the Hamiltonian 3 . Two elec- trons are paired if they are consecutive electrons with oppo- 2 v v; v v ; v dv , 13 site spins and have no unpaired electron between them. The I complete diagonalization of the transfer matrix 9 is not a where simple problem even in the simplest case S 1,n L see, for example, Ref. 24 . It is not difficult to convince ourselves sinh2 that in the interesting physical situation where we have low v; cosh2 cosh . 14 v density of holes the ground state will belong to the sector where we have zero magnetization and only pairs of elec- In order to minimize the ground-state energy trons. In this sector m n/2 and the diagonalization of the transfer matrix of the inhomogeneous model 9 gives for E0 2 cosh sinh v; v dv, 15 L 2 1 1, the following equations: I 7798 BRIEF REPORTS 56 the integration interval I in Eqs. 13 and 15 has to be lytically we find 2 for ( 0) and 12 for ( max chosen symmetrically around (I v0 ,2 v0 ). The pa- 1). This implies that for all nonzero values of the param- rameter v0 is determined by the subsidiary condition for the eters there is a density regime 0, total density 2m/L of electrons c where the system has dominating superconducting correlations. An analogous be- havior of correlation functions can also be observed in the 1 SU(N) generalization of the anisotropic t-J model where v dv superconducting properties are caused by the introduction of I 2 . 16 anisotropy in the interactions. However unlike these models To study the superconducting properties of the model un- the superconducting properties in the Hamiltonians 3 are der consideration we calculate the long-distance behavior of caused by both effects, the anisotropy and the value of the the correlation functions by finite-size studies and applica- spin S see definition of the parameter Eq. 3 . Moreover tion of conformal field theory see Refs. 26­28, and refer- in the present model for any value of N (N 2S 1) we ences therein . The results of this calculation are the follow- have bound pairs but not complexes of N bound particles as ing. The long-distance behavior of the density-density and in Ref. 16. the superconducting correlation functions are given by We conclude this paper with some remarks about the lat- tice vertex model counterpart of the quantum chain consid- r 0 2 A ered here. The quantum R matrix has 1 3N 2N2 nonzero 1r2 A2r cos 2kFr ; 2kF ; 17 Boltzmann weights, which are given by R00 0 0 00 1, R0 R 0 sinh /sinh 1 , r c r cr , 18 R0 0 0 R0 sinh /sinh 1 , 21 G r cr cr 1, c0, c1, Br . R The exponents and describing the algebraic decay are , , i u u ,N 1 ,N 1 calculated from the dressed charge function (v) which is sinh /sinh 1 , given by the solution of the integral equation where , 1,2, . . . ,N. The associated spin Hamiltonian can be found by taking the logarithmic derivative of the row- 1 v 1 to-row transfer matrix at 0. It gives the Hamiltonian 3 2 v v ; v dv , 19 I after a Jordan-Wigner transformation. Since we verified that Eqs. 21 satisfy the Yang-Baxter equations, the exact inte- and is given by grability of Eq. 3 is an immediate consequence. The above vertex model can be treated by the diagonal-to-diagonal 1 2 v0 2. 20 Bethe ansatz method,29,30 But this is not the aim of this work. In our one-dimensional system we have no superconduc- This work was supported in part by Conselho Nacional de tivity in the literal sense, since the model does not have finite Desenvolvimento Cienti´fico, CNPq, Brazil, by Fundac¸a o de off-diagonal long-range order. But we may say that in our Amparo a Pesquisa do Estado de Sa o Paulo, FAPESP, Bra- model there is a tendency to superconductivity since the su- zil, and by the Russian Foundation of Fundamental Investi- perconducting correlations have a longer range than the gations under Grant No. RFFI 97-02-16146. We would like density-density correlations. This happens when . Ana- to thank Dr. H. Babujian for discussions. *Permanent address: The Kazan Physico-Technical Institute of the Strongly Correlated Electrons World Scientific, Singapore, Russian Academy of Sciences, Kazan 420029, Russia. 1994 . 1 F. C. Zhang and T. M. Rice, Phys. Rev. B 37, 3759 1988 . 13 R. Z. Bariev, J. Phys. A 27, 3381 1994 . 2 P. W. Anderson, Phys. Rev. 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