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Physical Review B

Hubbard model with infinite-range attractive interaction

Carlindo Vitoriano, K. Rocha, Mauricio D. Coutinho-Filho

Published Oct 19, 2005• 9 citations
Publisher

Abstract

The ground state and low-temperature properties of the Hubbard model with infinite-range attractive interaction in the presence of a magnetic field are considered. This model is exactly solvable in any dimension and is mapped onto an ideal gas of three species of exclusons. The system is found to behave as a noninteracting Fermi gas whenever the correlation function $C(\mathbf{k})=⟨{n}_{\mathbf{k}\ensuremath{\uparrow}}{n}_{\mathbf{k}\ensuremath{\downarrow}}⟩\ensuremath{-}⟨{n}_{\mathbf{k}\ensuremath{\uparrow}}⟩⟨{n}_{\mathbf{k}\ensuremath{\downarrow}}⟩$ is negligible, otherwise it displays nontrivial correlated behavior which manifests as fractional statistical effects. The corresponding effective Hamiltonians and free energies governing the distinct regimes are presented. In one dimension, our results are analyzed in light of those derived using the on-site attractive Hubbard chain.

DOI
10.1103/physrevb.72.165109

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The ground state and low-temperature properties of the Hubbard model with infinite-range attractive interaction in the presence of a magnetic field are considered. This model is exactly solvable in any dimension and is mapped onto an ideal gas of three species of exclusons. The system is found to behave as a noninteracting Fermi gas whenever the correlation function C(k)=⟨nk\ensuremath↑nk\ensuremath↓⟩\ensuremath−⟨nk\ensuremath↑⟩⟨nk\ensuremath↓⟩C(\mathbf{k})=⟨{n}_{\mathbf{k}\ensuremath{\uparrow}}{n}_{\mathbf{k}\ensuremath{\downarrow}}⟩\ensuremath{-}⟨{n}_{\mathbf{k}\ensuremath{\uparrow}}⟩⟨{n}_{\mathbf{k}\ensuremath{\downarrow}}⟩C(k)=⟨nk\ensuremath↑​nk\ensuremath↓​⟩\ensuremath−⟨nk\ensuremath↑​⟩⟨nk\ensuremath↓​⟩ is negligible, otherwise it displays nontrivial correlated behavior which manifests as fractional statistical effects. The corresponding effective Hamiltonians and free energies governing the distinct regimes are presented. In one dimension, our results are analyzed in light of those derived using the on-site attractive Hubbard chain.
We show that the Hubbard model with infinite-range Coulomb coupling is equivalent to an ideal gas of three species of particles obeying fractional exclusion statistics. A full appreciation of this mapping requires an extension of the pertinent formalism. This very simple, but rather peculiar model is exactly solvable in any dimension and exhibits a Mott metal-insulator transition, whose universality class is shown to be that of a free spinless Fermi gas. A modified version of the Luttinger theorem is shown to apply in any dimension. We also characterize the metallic and insulating phases by obtaining the electronic band structure as well as the interacting density of states. The fractional statistics manifests itself on the amplitudes of several thermodynamic quantities and, in particular, the Pauli spin susceptibility is subdominant in all metallic phases, and a Curie-type of response appears.
The ground-state properties of the Hubbard model with attractive local pairing interactions are investigated in the framework of lattice density-functional theory. A remarkable correlation is revealed between the interaction-energy functional $W[\mathbit{\ensuremath{\eta}}]$ corresponding to the Bloch-state occupation-number distribution ${\ensuremath{\eta}}_{\ensuremath{\kappa}\ensuremath{\sigma}}$ and the entropy $S[\mathbit{\ensuremath{\eta}}]$ of a system of noninteracting fermions having the same ${\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}$. The relation between $W[\mathbit{\ensuremath{\eta}}]$ and $S[\mathbit{\ensuremath{\eta}}]$ is shown to be approximately linear for a wide range of ground-state representable occupation-number distributions ${\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}$. Taking advantage of this statistical analogy, a simple explicit ansatz for $W[\mathbit{\ensuremath{\eta}}]$ of the attractive Hubbard model is proposed, which can be applied to arbitrary periodic systems. The accuracy of this approximation is demonstrated by calculating the main ground-state properties of the model on several 1D and 2D bipartite and nonbipartite lattices and by comparing the results with exact diagonalizations.
The one-dimensional (1D) Hubbard model, describing electrons on a lattice with an on-site repulsive interaction, provides a paradigm for the physics of quantum many-body phenomena. Here, by solving the thermodynamic Bethe ansatz equations, we study the universal thermodynamics, quantum criticality, and magnetism of the 1D attractive Hubbard model. We show that the compressibility and the susceptibility of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-like state obey simple additivity rules at low temperatures, indicating an existence of two free quantum fluids. The magnetic properties, such as magnetization and susceptibility, reveal three physical regions: quantum fluids at low temperatures, a non-Fermi liquid at high temperatures, and the quantum fluid to non-Fermi liquid crossover in between. The lattice interaction is seen to significantly influence the nature of the FFLO-like state in 1D. Furthermore, we show that the dimensionless Wilson ratio provides an ideal parameter to map out the various phase boundaries and to characterize the two free fluids of the FLLO-like state. The quantum scaling functions for the thermal and magnetic properties yield the same dynamic critical exponent z=2z=2z=2 and correlation critical exponent $\ensuremath{\nu}=1/2$ in the quantum critical region whenever a phase transition occurs. Our results provide a rigorous understanding of quantum criticality and free fluids of many-body systems on a 1D lattice.
The exact phase diagrams of the one-dimensional Hubbard model, both attractive and repulsive, in the presence of an arbitrary magnetic field h for various electron concentrations n and on-site interaction strengths U <0 or U >0 are calculated and investigated. The exact ground-state properties, namely, the ground-state energy, the average spin (magnetization), the concentration of the doubly occupied sites, the kinetic energy, the chemical potential, the spin (magnetic) susceptibility and the charge compressibility, are calculated and examined over a wide range of interaction strengths U for various h and n . It is found that the spin susceptibility at half-filling is non-analytic and changes discontinuously as U →0. The exact theory shows the absence of a charge energy gap in the U ⩽0 region for all n and provides, for the chemical potential, the rigorous upper and lower bounds for half-filled and empty bands respectively. The analytical results derived for the weak-coupling limit and asymptotic expansions for strong coupling are in full agreement with the numerical calculations.
Abstract. We study the influence of on-site disorder on the magnetic properties of the ground state of the infinite-U one-dimensional (1D) Hubbard model. We find that the ground state is not ferromagnetic. This is analysed in terms of the algebraic structure of the spin dependence of the Hamiltonian. A simple explanation is derived for the 1=N periodicity in the persistent current for this model. The Hubbard model is the simplest model one can study to examine the effects of correlations between electrons in narrow energy bands. The Hamiltonian consists of a nearest neighbour hopping term and an electron–electron repulsion, U, which acts only when two electrons are on the same site. The Hubbard model is also the canonical model for the study of itinerant ferromagnetism. The strong coupling regime is of special importance for the study of ferromagnetism, since a theorem by Nagaoka [1] states that in theU D 1 limit, the ground state (GS) is ferromagnetic given some connectivity property of the lattice (which holds in most cases for d> 1). The model is solvable in one dimension and it was shown [2] that for open boundary conditions (BCs) the GS for finiteU is a singlet, i.e. there can be no ferromagnetism unless one postulates explicitly spin- or velocity-dependent forces. For infinite U, the GS of
The Hubbard model is a "highly oversimplified model" for electrons in a solid which interact with each other through extremely short ranged repulsive (Coulomb) interaction. The Hamiltonian of the Hubbard model consists of two pieces; H_hop which describes quantum mechanical hopping of electrons, and Hint which describes nonlinear repulsive interaction. Either H_hop or H_int alone is easy to analyze, and does not favor any specific order. But their sum H=H_hop+H_int is believed to exhibit various nontrivial phenomena including metal-insulator transition, antiferromagnetism, ferrimagnetism, ferromagnetism, Tomonaga-Luttinger liquid, and superconductivity. It is believed that we can find various interesting "universality classes" of strongly interacting electron systems by studying the idealized Hubbard model. In the present article we review some mathematically rigorous results on the Hubbard model which shed light on "physics" of this fascinating model. We mainly concentrate on magnetic properties of the model at its ground states. We discuss Lieb-Mattis theorem on the absence of ferromagnetism in one dimension, Koma-Tasaki bounds on decay of correlations at finite temperatures in two-dimensions, Yamanaka-Oshikawa-Affleck theorem on low-lying excitations in one-dimension, Lieb's important theorem for half-filled model on a bipartite lattice, Kubo-Kishi bounds on the charge and superconducting susceptibilities of half-filled models at finite temperatures, and three rigorous examples of saturated ferromagnetism due to Nagaoka, Mielke, and Tasaki. We have tried to make the article accessible to nonexperts by describing basic definitions and elementary materials in detail.