Universal Thermodynamics of the One-Dimensional Attractive Hubbard Model

Abstract

The one-dimensional (1D) Fermi-Hubbard model describing interacting fermions on a lattice provides a paradigm of many-body physics, including spin-charge separation, fractional excitations, Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing, Mott insulating phase and phase transitions. Very recently, ultracold atoms trapped in optical lattices offer promising opportunities to test such traditional concepts. However, most of these studies were particularly restricted to repulsive interaction. The 1D attractive Hubbard model is a notoriously difficult problem and rarely studied in the literature due to the complicated bound states of multi-particles.

In this thesis, using the thermodynamic Bethe ansatz equations, we establish the equations of state in the strong coupling regime, by virtue of which the thermodynamic properties are analytically and numerically studied from the Luttinger liquid phase to the quantum critical region. Our analytical results show good agreement with the numerics, and could be used in fitting experimental data. We further derive the the uniform scaling relations at quantum criticality, and read off the critical exponents. In the partially polarized phase IV, we introduce two effective chemical potentials for the two Luttinger liquids, which help us to construct the additivity rules for thermodynamic properties, susceptibility and compressibility. The simple additivity rules demonstrate the macroscopic behavior of this phase, reminiscent of two ‘non-interacting Fermi liquids’, and thereby reveal a free Fermi liquid nature of the 1D attractive Hubbard model. Moreover, in this phase IV, we study the long-distance asymptotics of various correlation functions at zero temperature. The spatial oscillating behavior of pair correlation function and its singular peaks in momentum space theoretically confirm an analog of FFLO state in the 1D attractive Hubbard. Lastly we observe that in contrast to the continuum Fermi gases, the correlation critical exponents, thermodynamics, Luttinger parameter, and Wilson ratio of the attractive Hubbard model essentially depend on lattice interacting effect, all of which under the lattice-gas map can be reduced to the case of Fermi gas.

This thesis provides a precise understanding of the universal low energy physics of attractive fermions on a lattice, and benchmark physics for ultracold atom experiments.