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Lower Bounds for the Ground-State Energies of the 2D Hubbard and t-J Models

Joachim Stolze, Roser Valentí, P. J. Hirschfeld

Published Jan 1, 1991• 5 citations
Publisher

Abstract

We present simple lower bounds on the ground-state energy of the two-dimensional (2D) Hubbard and t-J models for arbitrary values of band filling and coupling constant. For the Hubbard model we derive two types of bounds, both based on decomposing the model Hamiltonian into a sum of sub-Hamiltonians. For a decomposition into local cluster sub-Hamiltonians, we perform a generalized Legendre transform on previously derived bounds for the grand-canonical potential. For a decomposition into spin-up and spin-down parts, previous results on the spinless Falicov-Kimball model may be used to obtain bounds for the Hubbard model, generalizing a result of Langer and Mattis to arbitrary filling. For the 2D t-J model we have only considered the decomposition into clusters. The 1D Hubbard model is used as a test case. The bounds may be improved by diagonalizing the Hamiltonian for larger clusters.

DOI
10.17877/de290r-3489

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We present simple lower bounds on the ground-state energy of the two-dimensional (2D) Hubbard and t-J models for arbitrary values of band filling and coupling constant. For the Hubbard model we derive two types of bounds, both based on decomposing the model Hamiltonian into a sum of sub-Hamiltonians. For a decomposition into local cluster sub-Hamiltonians, we perform a generalized Legendre transform on previously derived bounds for the grand-canonical potential. For a decomposition into spin-up and spin-down parts, previous results on the spinless Falicov-Kimball model may be used to obtain bounds for the Hubbard model, generalizing a result of Langer and Mattis to arbitrary filling. For the 2D t-J model we have only considered the decomposition into clusters. The 1D Hubbard model is used as a test case. The bounds may be improved by diagonalizing the Hamiltonian for larger clusters.
We present simple lower bounds on the ground-state energy of the two-dimensional (2D) Hubbard and t-J models for arbitrary values of band filling and coupling constant. For the Hubbard model we derive two types of bounds, both based on decomposing the model Hamiltonian into a sum of sub-Hamiltonians. For a decomposition into local cluster sub-Hamiltonians, we perform a generalized Legendre transform on previously derived bounds for the grand-canonical potential. For a decomposition into spin-up and spin-down parts, previous results on the spinless Falicov-Kimball model may be used to obtain bounds for the Hubbard model, generalizing a result of Langer and Mattis to arbitrary filling. For the 2D t-J model we have only considered the decomposition into clusters. The 1D Hubbard model is used as a test case. The bounds may be improved by diagonalizing the Hamiltonian for larger clusters.
We show that the thermodynamic potential E-\ensuremath{\mu}N of an itinerant interacting Fermi system is bounded below by the sum of the thermodynamic potentials of small subsystems into which the infinite system is partitioned. Exact lower bounds for the energy in the thermodynamic limit are therefore easily obtainable by diagonalization of small clusters. In one dimension, these bounds are remarkably close to Bethe-ansatz results for the Hubbard model. We present lower bounds for the two-dimensional Hubbard and t-J Hamiltonians. Such bounds may serve as rigorous tests for approximate treatments of this class of problems.
Ground-state properties of the asymmetric Hubbard model are studied on one-dimensional clusters (rings) with N=6, 8, and 10 sites. The ground-state energy, correlation functions, and phase diagrams are determined for various sets of model parameters. Computations are performed both by exact diagonalization (for N=6) and by an approximate method by which correlation effects can be examined on larger clusters than exact diagonalization allows. In the limiting cases (one limit corresponds to the spinless Falicov-Kimball model and the other to the Hubbard model) our results agree quite well with those obtained analytically for infinite systems.
We report on a numerical study of the two-dimensional Hubbard model and describe two new algorithms for the simulation of many-electron systems. These algorithms allow one to carry out simulations within the grand canonical ensemble at significantly lower temperatures than had previously been obtained and to calculate ground-state properties with fixed numbers of electrons. We present results for the two-dimensional Hubbard model with half- and quarter-filled bands. Our results support the existence of long-range antiferromagnetic order in the ground state at half-filling and its absence at quarter-filling. Results for the magnetic susceptibility and the momentum occupation along with an upper bound to the spin-wave spectrum are given. We find evidence for an attractive effective d-wave pairing interaction near half-filling but have not found evidence for a phase transition to a superconducting state.