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Supersymmetric approach to the infinite U Hubbard Model

Piers Coleman, C. Pépin

Published Jan 1, 2001• 4 citations
OpenAlex

Abstract

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

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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
We construct a new U(1) slave-spin representation for the single-band Hubbard model in the large-U limit. The mean-field theory in this representation is more amenable to describe both the spin-charge-separation physics of the Mott insulator at half-filling and the strange metal behavior at finite doping. By employing a dynamical Green’s function theory for slave spins, we calculate the single-particle spectral function of electrons. The result is comparable to that in dynamical mean field theories. We then formulate a dynamical t/U expansion for the doped Hubbard model that reproduces the mean-field results at the lowest order of expansion. To the next order of expansion, it naturally yields an effective low-energy theory of a t–J model for spinons self-consistently coupled to an XXZ model for the slave spins. We show that the superexchange J is renormalized by doping, in agreement with the Gutzwiller approximation. Surprisingly, we find a new ferromagnetic channel of exchange interactions which survives in the infinite U limit, as a manifestation of the Nagaoka ferromagnetism.
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.
The authors determine the ground-state phase diagram of the Hubbard model on the square lattice allowing for homogeneous spiral, antiferromagnetic, ferromagnetic and paramagnetic phases in the overall parameter range. This is obtained from a saddle-point approximation of a spin-rotation-invariant form of the slave-boson representation introduced by Kotliar and Ruckenstein (1986). In addition they determine analytically the transition point for infinite repulsive interaction strength between the Nagaoka state and the paramagnetic state on any bipartite lattice.
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.
We present the simplest model to which one can apply the supersymmetric Hubbard operators recently introduced. For the atomic model, H=−EdX00H=-E_d X_{00}H=−Ed​X00​, where X00=∣0><0∣X_{00} = |0><0|X00​=∣0><0∣ is a Hubbard operator and EdE_dEd​ is the energy of the localized spin level, we show how one can develop exact solutions for the entropy and heat capacity as a function of temperature. With this gold standard we are able to develop a controlled approximation scheme to field theoretically treat the susy approximation at the level of mean field + gaussian corrections and test its accuracy against the widely used slave boson and slave fermion approximations. We find that in addition to slave boson and slave fermion solutions, a new class of solutions exists in the physical case Q = 1, N = 2 which can be properly treated by neither previously existing approach. The phase diagram generated by the mean field saddle-point bears a superficial resemblance to the V-shaped phase diagram common to systems close to a quantum critical point and may provide a natural starting point for investigations of strongly correlated models capturing this physics.
A new representation for electrons is introduced, in which the electron operators are written in terms of a spinless fermion and the Pauli operators. This representation is canonical, invertible, and constraint-free. Importantly, it simplifies the Hubbard interaction. On a bipartite lattice, the Hubbard model is reduced to a form in which the exchange interaction simply emerges by decoupling the Pauli subsystem from the spinless fermion bath. This exchange correctly reproduces the large UUU superexchange. Also derived, for $U=\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$, is the Hamiltonian to study Nagaoka ferromagnetism. In this representation, the infinite-UUU Hubbard problem becomes elegant and easier to handle. Interestingly, the ferromagnetism in the Hubbard model is found to be related to the gauge invariance of the spinless fermions. A generalization of this representation for the multicomponent fermions, a new representation for bosons, the notion of a ``soft-core'' fermion, and some interesting unitary transformations are introduced and discussed in the appendixes.