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Physical review. B, Condensed matter

Rigorous lower bounds on the ground-state energy of correlated Fermi systems

Roser Valent�, P. J. Hirschfeld, J C Anglès d'Auriac

Published Aug 15, 1991• 9 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.44.3995

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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.
We show that upper and lower bounds on the ground-state energy of models describing correlated Fermi systems may be combined to produce bounds on the ground-state magnetization and chemical potential. Such bounds are obtainable through standard variational techniques and through recently developed methods involving exact diagonalization of finite-size clusters. For the Hubbard model on the square lattice, we give rigorous bounds for the magnetization at nonzero magnetic field B and for the chemical potential at nonzero hole density 1-n. The quality of these bounds degrades as B\ensuremath{\rightarrow}0 and n\ensuremath{\rightarrow}1, precluding rigorous statements about the stability of the ferromagnetic state or the existence of a Mott-Hubbard gap. Nevertheless, the tendency towards large-U ferromagnetism and localization is evident. We discuss ways of improving these bounds, including the use of kinetic frustration, nonuniform clusters, and averaging over boundary conditions.
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.
A simple, physically motivated, scaling hypothesis, which becomes exact in important limits, yields estimates for the ground-state energy of large, composed, systems in terms of the ground-state energy of its building blocks. The concept is illustrated for the electron liquid, and the Hubbard model. By means of this scaling argument the energy of the one-dimensional half-filled Hubbard model is estimated from that of a two-site Hubbard dimer, obtaining quantitative agreement with the exact one-dimensional Bethe-Ansatz solution, and the energies of the two- and three-dimensional half-filled Hubbard models are estimated from the one-dimensional energy, recovering exact results for $U\ensuremath{\rightarrow}0$ and $U\ensuremath{\rightarrow}\ensuremath{\infty}$ and coming close to quantum Monte Carlo data for intermediate UUU.
In this work, we study two models of strongly correlated fermions: Hubbard and FalicovKimball. We present a proof of the existence of real and ordered solutions to the nested Bethe Ansatz equations for the one dimensional Hubbard model on a finite lattice, with periodic boundary conditions. The existence of a continuous set of solutions extending from any U > 0 to U = ∞ is also shown, where U is the strength of the interaction between fermions at the same site. We use this continuity property, combined with the proof that the norm of the wavefunction obtained with the generalized Bethe Ansatz is not zero, to prove that the solution gives us the ground state of the finite system, as assumed by Lieb and Wu. For the absolute ground state at half-filling, we show that the solution converges to a distribution in the thermodynamic limit. This limit distribution satisfies the integral equations that led to the Lieb-Wu solution of the 1D Hubbard model. We also obtain a lower bound for the ground state energy of the Falicov-Kimball model. This bound is given by a bulk term, plus a term proportional to the boundary. A numerical value for the coefficient of the boundary term is determined. The explicit derivation is important in the proof of the conjecture of segregation of the two kinds of fermions in the Falicov-Kimball model, for sufficiently large interactions.