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arXiv:cond-mat/9802080 · Physical review. B, Condensed matter

Supersymmetry in models with strong on-site Coulomb repulsion: Application to the Heisenberg model

Tai Kai Ng, Chi-Ho Cheng

Published Mar 1, 1999• 8 citations
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Abstract

A supersymmetric way of imposing the constraint of no double occupancy in models with strong on-site Coulomb repulsion is presented in this paper. In this formulation the physical operators in the constrainted Hilbert space are invariant under local unitary transformations mixing boson and fermion representations. To illustrate, the formulation is applied to the Heisenberg, $t\ensuremath{-}J,$ and infinite-$U$ Anderson models. In particular, we study the Heisenberg model carefully in two dimensions where results on spin excitation spectrum are presented.

DOI
10.1103/physrevb.59.r6616

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Physical review. B, Condensed matter•Jul 1, 1993•9 citations
A supersymmetric way of imposing the constraint of no double occupancy in models with strong on-site Coulomb repulsion is presented in this paper. In this formulation the physical operators in the constrainted Hilbert space are invariant under local unitary transformations mixing boson and fermion representations. To illustrate, the formulation is applied to the Heisenberg, t\ensuremath−J,t\ensuremath{-}J,t\ensuremath−J, and infinite-UUU Anderson models. In particular, we study the Heisenberg model carefully in two dimensions where results on spin excitation spectrum are presented.
The exact and analytical low energy spectrum of the ferromagnetic Kondo lattice model on a complete graph extended with on-site repulsion UUU is obtained in terms of a dynamical spl(2,1) supersymmetry in the limit of infinitely strong Hund's coupling (double exchange). Furthermore, we show that for the particular value of U=JH/2U=J_H/2U=JH​/2 the supersymmetry is not constrained to infinite JH/tJ_H/tJH​/t only and we calculate the energy including the t2/JHt^2/J_Ht2/JH​ corrections analytically and we give numerical evidence which suggest that the Kondo Hamiltonian is itself supersymmetric for any JH/tJ_H/tJH​/t. On a NNN site graph, the ferromagnetic ground state is realized for 1 and NNN+1 electrons only. In the leading order in the value of the core spin, the quantum and semiclassical spectra are identical.
A new kind of symmetry, called Supersymmetry, is introduced in t-J model and its corresponding Supersymmetric Formulation is constructed. The new formulation unifies the so-called slave-fermion and slave-boson formulations to the model in a single framework and is able to incorporate the advantage of the two approaches. The formulation is analysed at mean-field level where we found that antiferromagnetism described by slave-fermion-like state is stable at low doping regime, and the slave-boson-like d-wave RVB superconducting state becomes favorable at optimal doping. The global phase diagram is determined. To remedy the relaxation of non-double-occupancy constraint in a simple mean-field theory, the Gutzwiller's approximation is applied. Surprisingly the resulting phase diagram is in rather close agreement with experiment. Several physical quantities such as spin-spin correlation lengths, electron spectral functions, magnetic dynamical structure factors and spin susceptibilities are also calculated at the mean-field level where the new effect coming from finite spin and charge excitations of opposite statistics allowed by supersymmetry is pointed out. Finally, future research direction of the formulation is discussed.
A one-dimensional electron has both spin and charge degrees of freedom. In the limit of strong on-site repulsion, double occupation of a site can be neglected since it has a large energy cost. If the number N e of electrons is less than the number N of lattice sites, vacant sites appear and the system acquires both spin and charge degrees of freedom. The simplest model to incorporate these degrees of freedom is called the t – J model, which is discussed in this chapter. We begin by reviewing the SU(2,1) supersymmetry (SUSY) as an extension of the SU(2) symmetry in the spin chain. With the supersymmetry, the hopping and exchange in the t – J model can be treated in a unified way as the graded permutation. Then the ground state of the SUSY t – J model is derived in Section 6.2 by the coordinate representation of wave functions by using the fully polarized state as the reference. The static structure factor at zero temperature is completely determined by the wave function at the ground state. The spin and charge components are derived using the determinant and its generalizations in Section 6.3. In Section 6.4, we proceed to derive the spectrum of the SUSY t – J model. The elementary excitations consist of spinons, holons, and their antiparticles. It is proved in Section 6.5 that these excitations span the complete set. The argument relies on a generalization of Young diagrams, which are called ribbon diagrams. At the same time, the degeneracy of energy levels beyond the global SU(2) symmetry is ascribed to the presence of a Yangian supersymmetry Y ( sl 2|1 ).
We develop a supersymmetric representation of the Hubbard operator which unifies the slave boson and slave fermion representations into a single $U(1)\ifmmode\times\else\texttimes\fi{}SU(1|1)$ gauge theory, a group with larger symmetry than the product of two U(1)U(1)U(1) gauge groups. These representations of the Hubbard operator can be used to incorporate strong Hund's interactions in multielectron atoms as a constraint. We show how this method can be combined with the SP(N)\mathrm{SP}(N)SP(N) group to yield a locally supersymmetric large-NNN formulation of the $t\ensuremath{-}J$ model.
In strongly correlated electron systems the constraint which prohibits the double electron occupation at local sites can be realized by either the infinite Coulomb interaction or the correlated hopping interaction described by the Hubbard operators, but they both render the conventional field theory inapplicable. Relaxing such the constraint leads to a class of correlated hopping models based on the deformed Hubbard operators which smoothly interpolate the locally free and strong coupling limits by a tunable interaction parameter 0≤λ≤10\leq λ\leq 10≤λ≤1. Here we propose a coherent state path integral approach appropriate to the deformed Hubbard operators for {\it arbitrary} λλλ. It is shown that this model system exhibits the correlated Fermi liquid behavior characterized by the enhanced Wilson ratio for all λλλ. It is further found that in the presence of on-site Coulomb interaction a finite Mott gap appears between the upper and lower Hubbard bands, with the upper band spectral weight being heavily reduced by λλλ. Our approach stands in general spatial dimensions and reveals an unexpected interplay between the correlated hopping and the Coulomb repulsion.