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arXiv:cond-mat/9609107 · Physical review. B, Condensed matter

Some exact results for the multicomponent t−J model

Antimo Angelucci, Sandro Sorella

Published Nov 1, 1996• 5 citations
arXivHTMLPDF

Abstract

We present a generalization of the Sutherland multicomponent model. Our extension includes both the ferromagnetic and the antiferromagnetic $t\ensuremath{-}J$ model for any value of the exchange coupling $J$ and the hopping parameter $t$. We prove rigorously that for one-dimensional chains the ground state of the generalized model is nondegenerate. As a consequence, the ordering of energy levels of the antiferromagnetic $t\ensuremath{-}J$ model is determined. Our result rigorously proves and extends the analysis carried out by Sutherland in establishing the phase diagram of the model as a function of the number of components.

DOI
10.1103/physrevb.54.r12657

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We present a generalization of the Sutherland multicomponent model. Our extension includes both the ferromagnetic and the antiferromagnetic t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model for any value of the exchange coupling JJJ and the hopping parameter ttt. We prove rigorously that for one-dimensional chains the ground state of the generalized model is nondegenerate. As a consequence, the ordering of energy levels of the antiferromagnetic t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model is determined. Our result rigorously proves and extends the analysis carried out by Sutherland in establishing the phase diagram of the model as a function of the number of components.
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Two quantum spin models with bilinear-biquadratic exchange interactions are constructed on the checkerboard lattice. It is proved that, under certain sufficient conditions on the exchange parameters, their ground states consist of two degenerate Shastry-Sutherland singlet configurations. The constructions are studied for arbitrary spin SSS. The sufficient conditions for the existence of ferromagnetic ground state are also found exactly. The approximate quantum phase diagrams are presented using the exact results, together with a variational estimate for the N\'eel antiferromagnetic phase. A two-leg spin-1∕21∕21∕2 ladder model, based on one of the above constructions, is considered which admits exact solution for a large number of eigenstates. The ladder model is shown to have exact level crossing between the rung-singlet state and the AKLT state in the singlet ground state. Also introduced is the notion of perpendicularity for quantum spin vectors, which appears in the discussion on one of the two checkerboard models, and is discussed in the Appendix.
The Sutherland model has a number of variants. One of them is the U( K ) Sutherland model [71, 85, 86, 132]. This model describes N particles moving along a circle of perimeter L , and each particle possesses an internal degree of freedom with K possible values. This corresponds to spin with K = 2, and more generally a color. In the U( K ) Sutherland model, all particles obey common statistics: bosonic or fermionic. We can generalize the model further. The U( K B , K F ) Sutherland model [177] consists of bosons having K B possible colors and fermions having K F colors. The multi-component Sutherland model has a degeneracy in energy levels which is described in terms of a Yangian. The Yangian is an algebra related to quantum groups [43, 44]. The Yangian is nicely realized by variants of Jack polynomials which are modified so as to conform to the internal symmetry. Elementary excitations in the multi-component Sutherland model are described in a few alternative ways: interacting bosonic or fermionic particles, or non-interacting particles obeying generalized exclusion statistics. Furthermore, the lattice models such as the Haldane–Shastry models [77, 161] and 1/ r 2 supersymmetric t – J model [119] are obtained in the strong coupling limit of U(2) and U(2, 1) Sutherland models, respectively. The Sutherland models in the continuum space are much more tractable mathematically than the corresponding lattice models. Hence, the mapping to lattice models turns out to be useful to derive the explicit results on thermodynamics and dynamics in lattice models. In the present chapter, we extend our treatment for the single-component Sutherland model in order to include the internal degrees of freedom. We shall discuss the energy spectrum, thermodynamics, and dynamical correlation functions.
A multicomponent anisotropic $t\ensuremath{-}J$ model with a hard-core potential is formulated and solved by the Bethe ansatz method in one dimension for an arbitrary core radius $(\ensuremath{\Delta}+1)/2.$ The ground-state Bethe equations are analyzed and solved numerically for an arbitrary band filling, several values of color components, and the coupling constants. The ground-state energy, the Fermi velocity, and the critical exponents of the correlation functions have been calculated numerically for an arbitrary density of electrons. We discuss the effect of the hard-core potential. The Fermi velocity has a maximum as a function of band filling, vanishing for the empty and full bands. In comparison to the traditional anisotropic $t\ensuremath{-}J$ model in which $\ensuremath{\Delta}=0,$ the Fermi velocity is strongly enhanced in general. In the high-density limit, there is no dynamics of charges, the model exhibits a metal-insulator phase transition, and reduces to an antiferromagnetic spin chain with a new spacing parameter $\ensuremath{\Delta}+1.$