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Physical review. B, Condensed matter

Nonperturbative results for few electrons in the two-dimensional Hubbard model

Michele Fabrizio, Alberto Parola, Erio Tosatti

Published Jul 15, 1991• 38 citations
Publisher

Abstract

Based on a generalization of the equation which solves exactly the two-electron problem, we have found a method which allows the nonperturbative, asymptotically exact solution for the ground state of any finite number n of electrons in the low-density limit (n/N\ensuremath{\rightarrow}0) of the N-site two-dimensional Hubbard model. The three-electron case is checked against the exact numerical diagonalization and the total energy is shown by finite-size scaling to agree well, to leading order of an expansion in powers of 1/lnN, with the asymptotic solution. We then indicate how to solve for an arbitrary electron number, including both closed and open shells. The case of four electrons is explicitly presented as an application. More generally, the closed-shell d=2 ground state is a nondegenerate singlet with zero total momentum. Similar to d=1, all observables differ from their U=0 value by a correction which is U independent. Unlike d=1 and similar to d=3, however, the ground state is a linear combination of the unperturbed U=0 Fermi sea with the set of two-particle excitations. Moreover, the momentum distribution n(k) can be evaluated explicitly, and shows a finite Fermi jump Z=1-4 ln2/${\mathrm{ln}}^{2}$(n/N). This result appears to argue in favor of a Fermi-liquid-like behavior for d=2 in the zero-density limit n/N\ensuremath{\rightarrow}0. The open-shell ground states display an interesting degeneracy between a nondegenerate singlet and high-spin states.

DOI
10.1103/physrevb.44.1033

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•Jan 1, 2008•0 citations
Based on a generalization of the equation which solves exactly the two-electron problem, we have found a method which allows the nonperturbative, asymptotically exact solution for the ground state of any finite number n of electrons in the low-density limit (n/N\ensuremath{\rightarrow}0) of the N-site two-dimensional Hubbard model. The three-electron case is checked against the exact numerical diagonalization and the total energy is shown by finite-size scaling to agree well, to leading order of an expansion in powers of 1/lnN, with the asymptotic solution. We then indicate how to solve for an arbitrary electron number, including both closed and open shells. The case of four electrons is explicitly presented as an application. More generally, the closed-shell d=2 ground state is a nondegenerate singlet with zero total momentum. Similar to d=1, all observables differ from their U=0 value by a correction which is U independent. Unlike d=1 and similar to d=3, however, the ground state is a linear combination of the unperturbed U=0 Fermi sea with the set of two-particle excitations. Moreover, the momentum distribution n(k) can be evaluated explicitly, and shows a finite Fermi jump Z=1-4 ln2/ln2{\mathrm{ln}}^{2}ln2(n/N). This result appears to argue in favor of a Fermi-liquid-like behavior for d=2 in the zero-density limit n/N\ensuremath{\rightarrow}0. The open-shell ground states display an interesting degeneracy between a nondegenerate singlet and high-spin states.
The one-dimensional Hubbard model is studied for general electron densities and large U. On the basis of the coupled integral equations of Lieb and Wu exact asymptotic expansions are derived to fifth order in ${U}^{\mathrm{\ensuremath{-}}1}$ for the ground-state energy, the kinetic energy, and the number of doubly occupied sites. A closed-form expression for the distribution functions of charge and spin degrees of freedom is proposed which yields the same large-U expansion as that derived directly from the Lieb-Wu equations. Our analytical results are in excellent agreement with previous numerical studies.
We present exact explicit analytical results describing the exact ground state of four electrons in a two-dimensional square Hubbard cluster containing 16 sites taken with periodic boundary conditions. The presented procedure, which works for arbitrary even particle number and lattice sites, is based on explicitly given symmetry adapted base vectors constructed in r space. The Hamiltonian acting on these states generates a closed system of 85 linear equations, providing by its minimum eigenvalue, the exact ground state of the system. The presented results, described with the aim to generate further creative developments, not only show how the ground state can be exactly obtained and what kind of contributions enter in its construction, but also emphasize further characteristics of the spectrum. On this line (i) possible explications are found regarding why weak coupling expansions often provide a good approximation for the Hubbard model at intermediate couplings, or (ii) explicitly given low-lying energy states of the kinetic energy, avoiding double occupancy, suggest new roots for pairing mechanism attracting decrease in the kinetic energy, as emphasized by kinetic energy driven superconductivity theories.
Working in a subspace with dimensionality much smaller than the dimension of the full Hilbert space, we deduce exact four-particle ground states in 2D samples containing hexagonal repeat units and described by Hubbard type of models. The procedure identifies first a small subspace in which the ground state is placed, than deduces by exact diagonalization in . The small subspace is obtained by the repeated application of the Hamiltonian on a carefully chosen starting wave vector describing the most interacting particle configuration, and the wave vectors resulting from the application of , till the obtained system of equations closes in itself. The procedure which can be applied in principle at fixed but arbitrary system size and number of particles is interesting on its own since it provides exact information for the numerical approximation techniques which use a similar strategy, but apply non-complete basis for . The diagonalization inside provides an incomplete image of the low lying part of the excitation spectrum, but provides the exact . Once the exact ground state is obtained, its properties can be easily analysed. The is found always as a singlet state whose energy, interestingly, saturates in the limit. The unapproximated results show that the emergence probabilities of different particle configurations in the ground state presents ‘Zittern’ (trembling) characteristics which are absent in 2D square Hubbard systems. Consequently, the manifestation of the local Coulomb repulsion in 2D square and honeycomb types of systems presents differences, which can be a real source in the differences in the many-body behaviour.
The problem of the ground state of the electronic system in the Hubbard model for U=∞ is discussed. The author investigates the normal (singlet or nonmagnetic) N state of the electronic system over the entire range of electron densities n⩽1. It is shown that the energy of the N state ɛ 0 (1) (n) in a one-particle approximation, such as (e.g.) the extended Hartree-Fock approximation, is lower than the energy of the saturated ferromagnetic FM state ɛ FM(n) for all n. The dynamic magnetic susceptibility is calculated in the random phase approximation, and it is shown that the N state is stable over the entire range of electron densities: The static susceptibility (ω=0) does not have a band singularity in the zero-wave vector limit q→0. A formally exact representation is obtained for the mass operator of the one-particle Green’s function, and an approximation of this operator is proposed: M k(E)⋍λF(E), where λ=n(1−n)/(1−n/2)z is the kinematic interaction parameter, z is the number of nearest neighbors, and F(E) is the total single-site Green’s function. For an elliptical density of states the integral equation for F(E) is solved exactly, ad it is shown that the spectral intensity rigorously satisfies the sum rule. The calculated energy of the strongly correlated N state ɛ 0(n)<ɛ FM(n) for all n, and in light of this relationship the author discusses the hypothesis that the ground state of the system is the normal (singlet) state in the thermodynamic limit. The electron distribution function at T=0 differs significantly from the Fermi step; it is “smeared” along the entire energy spectrum, and discontinuities do not occur in the region of the chemical potential m.
A Hubbard-like model with SU(4) symmetry for electrons with twofold orbital degeneracy is studied extensively. Exact solution in one dimension is derived by means of Bethe ansatz, where the sites are supposed to be occupied by at most two electrons. The features of ground state and excited states for repulsive coupling are shown. For finite NNN number of electrons, the configurations of quantum numbers are given explicitly and the spectra of excitations are obtained by solving Bethe-ansatz equation numerically. For infinite N,N,N, the ground state and various kinds of low-lying excitations are obtained on the basis of thermodynamics limit.
We study the one-dimensional Hubbard model for interacting electrons in a crystal lattice, using two methods: solution of the Lieb-Wu equations in the atomic limit of no intersite hopping, and variational techniques to obtain a lower bound for the ground state energy. From the Lieb-Wu equations we derive an exact expression for the ground state energy in the atomic limit. We obtain our result by exploiting the spin symmetry of the Hubbard Hamiltonian. Our energy is lower than that obtained by previous published results, and we are able to explain why our results differ. Our method also gives information about the magnetic ordering of the ground state as a function of the number of electrons. Our results are valid for arbitrary electron density values. The variational techniques we use are applied to the two-particle reduced-density matrix (2-RDM), because this quantity suffices to obtain expectation values of all one- and two-particle operators, and because it is less computationally intensive than obtaining the complete N -particle wave function. Because the 2-RDM must be positive semidefinite, we use an efficient programming algorithm known as the Semidefinite Programming Algorithm. A naive search through the space of two-particle matrices encounters a difficult problem, known as the " N -representability problem", which is to assure that the trial density matrices actually come from a wave function that has the required exchange symmetry. This problem is ameliorated by imposing constraints on the variational search that eliminate trial matrices with the incorrect exchange symmetry. These constraints express known physical properties of the ground state (for example its total spin and its translational symmetry) and the so-called D, Q , and G two-positivity conditions that express the required exchange symmetry. To calculate matrix elements, we use the two-particle eigenstates of the reduced Hubbard Hamiltonian; this choice of basis significantly decreases the number of matrix elements needed for energy calculations. After an optimal two-particle density matrix has been obtained, we calculate additional properties of the ground state, including spin correlation functions, chemical potentials, and the momentum distribution. We compare all of these properties to values obtained by exact numerical solution of the Schrödinger equation.