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Supersymmetric formulation of t-J model

Chi-Ho Cheng

Published Jan 1, 2000• 1 citation
Publisher

Abstract

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.

DOI
10.14711/thesis-b672006

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The supersymmetric formulation of the t - J model is studied in this paper at the mean-field level where the δ- T phase diagram is determined. We found that the slave-fermion–like spiral phase is stable at low doping concentration, and the slave-boson–like d -wave fermionic spin pairing state becomes energetically favourable when δ ⩾ 0.23. An improvement in free energy using Gutzwiller's method lowers the transition doping concentration to δ ≃ 0.06. We also point out the existence of new branches of excitations in the supersymmetric theory.

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Supersymmetric t–J model with 1/r2 interaction

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Effect of Exclusion of Double Occupancies in t-J Model: Extension of Gutzwiller Approximation

Masao Ogata, Akihiro Himeda

A new type of analytic estimation of the effect of strong correlation is developed for the two-dimensional t-J model. It is based on the Gutzwiller approximation which gives the renormalization of parameters, t and J, due to the Gutzwiller's projection operator excluding the double occupancies. The finite-range correlations are taken into account compared with the conventional Gutzwiller approximation where only the on-site expectation values are considered. It is shown that the essential point of the renormalization is its nonlinear dependence on the expectation values of Cooper pairs and antiferromagnetic moment. In particular the renormalization factor of J becomes anisotropic in the presence of antiferromagnetic moment, i.e., that for the z-component is enhanced compared with that for the xy-component. The physical origin of this enhancement is identified as the surrounding antiferromagnetic correlations of a bond. The self-consistency equations for the uniform variational wave functions are derived and solved numerically. Our result gives a reasonable estimate of antiferromagnetic order parameters near half filling, in contrast to the conventional slave-boson mean-field theory and the original Gutzwiller approximation. It is also found that, at half filling, the renormalization of J represents some of the quantum fluctuations of the Heisenberg spin system in a different manner from the spin-wave theory. For finite doping, our results have some similarity to SO(5) symmetric theory. Applications to the inhomogeneous systems such as the vortex state, around nonmagnetic impurities, and stripe state are discussed.

arXiv:cond-mat/0003465•Mar 29, 2000•0 citations
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.
The supersymmetric formulation of the t - J model is studied in this paper at the mean-field level where the δ- T phase diagram is determined. We found that the slave-fermion–like spiral phase is stable at low doping concentration, and the slave-boson–like d -wave fermionic spin pairing state becomes energetically favourable when δ ⩾ 0.23. An improvement in free energy using Gutzwiller's method lowers the transition doping concentration to δ ≃ 0.06. We also point out the existence of new branches of excitations in the supersymmetric theory.
The ground state of two-dimensional t – J model including superconductivity and antiferromagnetism is studied using a new type of analytic estimation of the effect of strong correlation. The method is an extension of the Gutzwiller approximation which gives renormalization of parameters, t and J , due to the Gutzwiller projection operator excluding double occupancies. It is shown that the essential point of the renormalization is its dependence on the expectation values of superconductivity and antiferromagnetic moment. In special the renormalization factor is anisotropic in spin space, which leads to a reliable estimate of antiferromagnetic order parameters in the phase diagram, in contrast to the conventional slave-boson mean-field theory. The obtained phase diagram shows the coexistence of antiferromagnetism and d x 2 - y 2 -wave superconductivity for the doping rate, δ, less than 10%. At half-filling, the renormalization of the J -term represents some aspects of quantum spin nature. For the finite doping case, there are two kinds of superconducting order parameters. The variational order parameter of the d-wave superconductivity increases as the doping rate decreases, which will correspond to the experimentally observed superconducting gap. In contrast, the actual expectation value of superconducting order parameter is proportional to δ in underdoped region and has a maximum near δ=0.11, which probably determines T c . We also speculate that the ground-state antiferromagnetic order, being destroyed at finite temperatures, gives the energy and temperature scale of the pseudogap. Applications to the inhomogeneous systems such as the vortex state and stripe state are discussed.
We develop a slave-boson theory for the $t\ensuremath{-}J$ model at finite doping that respects an SU(2) symmetry: a symmetry previously known to be important at half filling. The mean-field phase diagram is found to be consistent with the phases observed in the cuprate superconductors, which contain ddd-wave superconductor, spin-gap, strange metal, and Fermi-liquid phases. The spin-gap phase is best understood as the staggered flux phase, which is nevertheless translationally invariant for physical quantities. The physical electron spectral function shows small Fermi segments at low doping that continuously evolve into the large Fermi surface at high-doping concentrations. The close relation between the SU(2) and the U(1) slave-boson theory is discussed. The low-energy effective theory for the low-lying fluctuations is derived and additional lying modes [which were overlooked in the U(1) theory] are identified.
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 present a preliminary discussion on the use of supersymmetric representation of the Hubbard operator which unifies the slave boson and slave fermion representations into a single U(1) × SU(1|1) gauge theory to treat the physics of the infinite U Hubbard model. By looking for solutions to the Hamiltonian in which the spins can exist as both condensed ordered moments and as mobile charged carriers, we examine the possibility of a Nagaoka ferromagnetic phase at finite doping with a quantum critical point into the paramagnetic phase. Key words: Hubbard model, Supersymmetry, Ferromagnetism The infinite U Hubbard model[1] is a prototype model for strongly correlated electron systems. Interest in this venerable model has grown in recent times because of its link with one of the basic models of high temperature superconductivity, the t − J model.[2] The infinite U Hubbard model corresponds to the t − J model without antiferromagnetic interactions (J → 0). Despite the absence of antiferromagnetic interactions, evidence from finite temperature Lanzcos studies[3] suggests that many of high temperature properties associated with the cuprate metals are already present in the infinite U Hubbard model. The U = ∞ Hubbard hamiltonian is written HK = − t
A new type of analytic estimation of the effect of strong correlation is developed for the two-dimensional t-J model. It is based on the Gutzwiller approximation which gives the renormalization of parameters, t and J, due to the Gutzwiller's projection operator excluding the double occupancies. The finite-range correlations are taken into account compared with the conventional Gutzwiller approximation where only the on-site expectation values are considered. It is shown that the essential point of the renormalization is its nonlinear dependence on the expectation values of Cooper pairs and antiferromagnetic moment. In particular the renormalization factor of J becomes anisotropic in the presence of antiferromagnetic moment, i.e., that for the z-component is enhanced compared with that for the xy-component. The physical origin of this enhancement is identified as the surrounding antiferromagnetic correlations of a bond. The self-consistency equations for the uniform variational wave functions are derived and solved numerically. Our result gives a reasonable estimate of antiferromagnetic order parameters near half filling, in contrast to the conventional slave-boson mean-field theory and the original Gutzwiller approximation. It is also found that, at half filling, the renormalization of J represents some of the quantum fluctuations of the Heisenberg spin system in a different manner from the spin-wave theory. For finite doping, our results have some similarity to SO(5) symmetric theory. Applications to the inhomogeneous systems such as the vortex state, around nonmagnetic impurities, and stripe state are discussed.