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arXiv:1407.2050 · Condensed Matter Physics

On the particle-hole symmetry of the fermionic spinless Hubbard model in D=1

M. T. Thomaz, Corrêa Silva, Onofre Rojas

Published Jan 1, 2014• 3 citations
arXivHTMLPDF

Abstract

We revisit the particle-hole symmetry of the one-dimensional (D = 1) fermionic spinless Hubbard model, associating that symmetry to the invariance of the Helmholtz free energy of the one-dimensional spin-1/2 X X Z Heisenberg model, under reversal of the longitudinal magnetic field and at any finite temperature.Upon comparing two regimes of that chain model so that the number of particles in one regime equals the number of holes in the other, one finds that, in general, their thermodynamics is similar, but not identical: both models share the specific heat and entropy functions, but not the internal energy per site, the first-neighbor correlation functions, and the number of particles per site.Due to that symmetry, the difference between the first-neighbor correlation functions is proportional to the z-component of magnetization of the X X Z Heisenberg model.The results presented in this paper are valid for any value of the interaction strength parameter V , which describes the attractive/null/repulsive interaction of neighboring fermions.

DOI
10.5488/cmp.17.23002

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We revisit the particle-hole symmetry of the one-dimensional (D = 1) fermionic spinless Hubbard model, associating that symmetry to the invariance of the Helmholtz free energy of the one-dimensional spin-1/2 X X Z Heisenberg model, under reversal of the longitudinal magnetic field and at any finite temperature.Upon comparing two regimes of that chain model so that the number of particles in one regime equals the number of holes in the other, one finds that, in general, their thermodynamics is similar, but not identical: both models share the specific heat and entropy functions, but not the internal energy per site, the first-neighbor correlation functions, and the number of particles per site.Due to that symmetry, the difference between the first-neighbor correlation functions is proportional to the z-component of magnetization of the X X Z Heisenberg model.The results presented in this paper are valid for any value of the interaction strength parameter V , which describes the attractive/null/repulsive interaction of neighboring fermions.
We found that when the spinless model is off the half-filling regime (µ V ), the Helmholtz free energy (HFE) can be written as two β-expansions: one expansion comes from the half-filling configuration and another one that depends on the parameter x = µ -V .We show numerically that the chemical potential as a function of temperature satisfies a relation similar to the one derived from the particle-hole symmetry of the fermionic spinless model.We extend the β-expansion of the HFE of the one-dimensional fermionic spinless Hubbard model up to order β 8 .
We make use of a recently developed diagrammatic theory to calculate correlation functions for interacting fermions of Hubbard-type models in terms of the Gutzwiller wave function. Of the eleven nontrivial correlation functions involving the spin, density, empty and doubly occupied sites, and local Cooper pairs, four are shown to be independent. They are expressed as a power series in a suitably chosen correlation parameter, whose terms are represented diagrammatically. In one dimension these terms may be evaluated to arbitrary order by employing symmetry relations. This allows for an analytic, approximation-free calculation of the correlation functions for arbitrary momentum, particle density, and interaction strength. In the atomic limit the momentum-dependent spin-correlation function shows an antiferromagnetic divergence at half filling in all dimensions. In one dimension the behavior is in very good agreement with all exact analytic and numerical results for the antiferromagnetic Heisenberg chain. Hole-hole correlations also compare very well with exact results. However, correlations between holes and doubly occupied sites appear insufficient. Superconducting correlations involving on-site, singlet Cooper pairs are suppressed. The results allow for an analytic evaluation of the ground-state energy of a large class of extended Hubbard models in terms of the Gutzwiller wave function. Thus they provide exact upper bounds for their ground-state energies.
Correlation effects in the one-dimensional Hubbard model are studied by application of the L-resolvent method. The calculations are well beyond one-particle mean field approximation. All correlations involving two neighboring sites are included exactly and higher order correlations are approximated. The effects of varying temperature, band filling, Coulomb repulsion and external magnetic field are investigated. Two-site correlations for non-half-filled band fillings are calculated for the first time. Some exact relations between correlation functions are derived by electron-hole symmetry. Results indicate that short-range spin correlations always favor antiferromagnetic order. Large Coulomb repulsion induces a gap in the excitation spectrum and a metal-insulator transition is predicted as the Coulomb repulsion parameter is changed.
The authors show with the aid of a canonical transformation that separates charge and spin degrees of freedom that the strongly correlated one-dimensional Hubbard model with infinite (U= infinity ) on-site repulsion is related to a corresponding spinless free-fermion lattice model. Thermodynamic properties of the two models are shown to be closely related whereas the single-particle two-site function for the U= infinity model is shown to be related to a modified many site function for the spinless model which involves many-point correlations. The latter is shown to be expressible in terms of the inverse of a certain matrix which should be amenable to numerical and analytical analysis.
By making use of some symmetry properties of the relevant Hamiltonian, two fundamental relations between the ferromagnetic magnetization and a spin correlation function are derived for the d(=1,2,3)d(=1,2,3)d(=1,2,3)-dimensional Hubbard model at finite temperatures. These can be viewed as a kind of Ward-Takahashi identity. The properties of the magnetization as a function of the applied field are discussed. The results thus obtained hold true for both repulsive and attractive on-site Coulomb interactions, and for arbitrary electron fillings.
In this Rapid Communication, we show that low-energy macroscopic properties of the one-dimensional (1D) attractive Hubbard model exhibit two fluids of bound pairs and of unpaired fermions. Using the thermodynamic Bethe ansatz equations of the model, we first determine the low-temperature phase diagram and analytically calculate the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing correlation function for the partially polarized phase. We then show that for such an FFLO-like state in the low-density regime the effective chemical potentials of bound pairs and unpaired fermions behave like two free fluids. Consequently, the susceptibility, compressibility, and specific heat obey simple additivity rules, indicating the ``free'' particle nature of interacting fermions on a 1D lattice. In contrast to the continuum Fermi gases, the correlation critical exponents and thermodynamics of the attractive Hubbard model essentially depend on two lattice interacting parameters. Finally, we study scaling functions, the Wilson ratio and susceptibility, which provide universal macroscopic properties and dimensionless constants of interacting fermions at low energy.