CERN-TH/97-72 Anisotropic Correlated Electron Model Associated With the Temperley-Lieb Algebra Angela Foerster (;y;a), Jon Links (2;b) and Itzhak Roditi (3;z;c) () Institut fur Theoretische Physik, Freie Universitat Berlin, WE2, Arnimallee 4, D 495 Berlin, Germany (2) Department of Mathematics, University of Queensland, Queensland, 4072, Australia (3) Theory Division, CERN, CH-2 Geneva 23, Switzerland Abstract We present an anisotropic correlated electron model on a periodic lattice, constructed from an R-matrix associated with the Temperley-Lieb algebra. By modication of the coupling of the rst and last sites we obtain a model with quantum algebra invariance. y Permanent address: Instituto de Fsica da UFRGS, Av. Bento Goncalves, 9500, Porto Alegre, 950-970, Brazil z Permanent address: Centro Brasileiro de Pesquisas Fsicas - CBPF, Rua Dr. Xavier Sigaud, 50, 22290-80 { Rio de Janeiro, RJ { Brazil e-mail address: (a) angela@if.ufrgs.br, (b) jrl@maths.uq.oz.au, (c) roditi@cat.cbpf.br. CERN-TH/97-72 April 997
Since the discovery of high T c superconductivity there has been a great interest in the area of integrable highly correlated electron systems. Notably for manyyears the Hubbard and the supersymmetric t J models, which are both exactly solvable in one dimension, stood as the prototypes for such types of models. Subsequently other correlated electron models have been formulated [{3]. In this letter we present a new q-deformed integrable electronic model following the approach of [4], which consists of using the Temperley-Lieb (TL) algebra to obtain solutions of the Yang-Baxter equation. In [3] this was achieved utilizing a 4-dimensional module of Lie superalgebra g`(2=), here we proceed along the same lines by investigating the corresponding module of U q (g`(2=)) in order to obtain a representation of the TL algebra. The procedure adopted in [3] involved a symmetry breaking transformation so that the resulting Hamiltonian, dened on a one-dimensional periodic lattice, was not invariant with respect to g`(2=) but rather its even subalgebra g`(2) u(). For the present situation, the usual impositon of periodic boundary conditions has the eect of also breaking the U q (g`(2)) u() symmetry due to the non-cocommutativity ofthe quantum algebra generators. However we will show, following the methodology of [5{8] that a quantum algebra invariant closed system can be dened by the introduction of an operator coupling the rst and last sites into the expression for the Hamiltonian. Let fjxig 4 x= be an orthonormal basis for a four-dimensional U q (g`(2=)) module V. The quantum superalgebra U q (g`(2=)) obtained by deforming g`(2=) has simple generators feig i 3 i= UfEi i+ ;Ei+ i g 2 i= which act on this module according to E i ijji = E 3 3jji = i j + 4 j jji ; i =;2; j=;2;3;4 2 +j 4 j jji; j=;2;3;4 E 2jji = j 3j2i ; j=;2;3;4 E 2 jji = j 2j3i ; j=;2;3;4 () E 2 3ji = E 2 3j3i = E 3 2j2i = E 3 2j4i =0 E 2 3j2i = E 3 2ji = 2 =2 q 2 =2 q ji ; j2i ; E 2 3j4i= E 3 2j3i= 2 =2 q j3i 2 =2 q j4i 2
where [x] q = qx q x q q ; x 2 C : The quantum superalgebra U q (g`(2=)) carries the following parity [Ej]=([i]+[j]) i (mod:2) ; (2) where [] = [2] = 0; [3] =, and by consistency the parity of the basis vectors is chosen to be [ji] =[j4i]=0 [j2i]=[j3i]= (3) Associated to U q (g`(2=)) there is also a co-product structure given by :U q (g`(2=))! U q (g`(2=)) U q (g`(2=)) (E i i) = I E i i + E i i I i =;2;3 (E i j) = E i j q (E E 2 2 )=2 + q (E E 2 2 )=2 E i j i; j =;2 (4) (E k` ) = E k` q (E2 2 +E3 3 )=2 + q (E2 2 +E3 3 )=2 E k` k; ` =2;3 Everywhere we shall use the graded-tensor product law, dened by. (a b)(c d) =( ) [b][c] (ac bd) Following a strategy analogous to the one employed in ref. [3] to contruct an hermitian Hamiltonian, we now consider the operator T = j ih j where j i in an unnormalized vector of V V dened by and j i = q =2 j4iji+q +=2 jij4i (5) + q =2 j3ij2i q =2 j2ij3i h j = q =2 h4jhj+q +=2 hjh4j (6) + q =2 h3jh2j+q +=2 h2jh3j 3
A straightforward calculation shows that T 2 = [2(q + q )]T (7) and, using the fact that j i spans a -dimensional submodule of U q (g`(2=)) (see [3, 4]) (T I)(I T )(T I) = T I (8) (I T )(T I)(I T ) = I T ; such that T provides a representation of the TL algebra. This can be used to obtain an R-matrix by the transformation [4] R(u) =PR(u)=I+ sinh(u) sinh( u) T ; (9) where cosh() = (q + q ) and P is the Z 2 -graded permutation operator dened by P (jxijyi)=( ) [jxi][jyi] jyijxi ; 8x ; y4. It is easy to check that it satises the Yang-Baxter equation (I R(u)( R(u + v) I)(I R(v)) = ( R(v) I)(I R(u + v))( R(u) I) (0) A local Hamiltonian can be dened by [9] H i;i+ = sinh() d du R(u) i;i+ u= = T i;i+ ; () where on the N-fold tensor product space we denoted R(u) i;i+ = I (i ) R(u) I (N i ) : Finally in view of the grading the basis vectors of the module V can be identied with the eletronic states as follows ji j+ i = c + + c+ j0i ; j2i j i = c + j0i ; j3i j+i=c + +j0i; j4ij0i allowing H i;i+ to be expressed in terms of the canonical fermion operators as H i;i+ = qn i;+ n i; ( n i+;+ )( n i+; )+q ( n i;+ )( n i; )n i+;+ n i+; + q n i;+ ( n i; )n i+; ( n i+;+ )+qn i; ( n i;+ )n i+;+ ( n i+; ) S + i S i+ S i S + i+ + c + i;+ c+ i; c i+; c i+;+ + c + i+;+ c+ i+; c i c i+ (2) + qc + i;+ c i+;+n i; ( n i+; )+h:c: c + i; c i+; n i;+ ( n i+;+ )+h:c: + c i; c + i+; n i+;+ ( n i;+ )+h:c: q c i;+ c + n i+;+ i+; ( n i; )+h:c: 4
where the c (+) i are spin up or down annihilation (creation) operators, the S 0 is spin matrices and the n 0 is occupation numbers of electrons at lattice site i. Thus we obtain a Hamiltonian describing electron pair hopping, correlated hopping and generalized spin interactions. isotropic Hamiltonian discussed in [3]. We notice that in the limit q! we recover the By means of the quantum inverse scattering method [9, 0] it is possible to show that the model is integrable. Basically, by this procedure, the Hamiltonian (2) is related to the transfer matrix of a graded vertex model [] (see also [2]), constructed from the spectral parameter dependent R-matrix. The associated Yang-Baxter algebra implies the commutativity of the transfer matrix for dierent spectral parameters, which reects the integrability of the model. The global Hamiltonian takes the form H = NX i= H i;i+ + H N : (3) It is not invariant with respect to U q (g`(2)) u() since H N 6= H N reecting the noncocommutativity of the co-product. However, by modifying the above Hamiltonian we can obtain a quantum algebra invariant model as follows [c.f. 5-8]. Let denote the braid generator dened by which satises the braid relations = lim u! R(u) = I e T (4) ( I)(I )( I) =(I)( I)(I ) : (5) Setting G = 2 23 N N, then it follows from the TL relations that GH i;i+ G = H i+;i+2 ; i =;N 2 We now dene H o = GH N ;N G (6) which can be shown to satisfy GH o G = H 2 5
and set our new Hamiltonian to be H = NX i= H i;i+ + H o (7) satisfying [H; G] = 0 and additionally invariance with respect to the quantum algebra U q (g`(2)) u(). The details about the Bethe ansatz solution of the present model will be developed elsewhere. ACKNOWLEDGEMENTS A.F. would like to thank the Institute fur Theoretische Physik { FUB for its kind hospitality, particularly M. Karowski. She also thanks SFB 288 { Dierentialgeometrie und Quantenphysik for nancial support. J.L. is supported by an Australian Research Council Postdoctoral Fellowship. I.R. would like to thank V. Rittenberg for interesting discussions. He also thanks CERN for its warm hospitality and the CNPq for nancial support. 6
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