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Journal of Zhejiang Normal University

The study on the phase transition of the one-dimensional Fermi-Hubbard model

Gao Xianlong

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

The phase diagram of the one-dimensional attractive Fermi-Hubbard model trapped in a harmonic potential was studied within the Tomas-Fermi approximation,which was based on the exactly solved Bethe-ansatz coupled equations.The compressibility,double occupancy,and its derivatives were calculated to measure the phase transition and the external potential induced by the system parameters.

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The phase diagram of the one-dimensional attractive Fermi-Hubbard model trapped in a harmonic potential was studied within the Tomas-Fermi approximation,which was based on the exactly solved Bethe-ansatz coupled equations.The compressibility,double occupancy,and its derivatives were calculated to measure the phase transition and the external potential induced by the system parameters.
We perform a numerical study of a one-dimensional Fermi-Hubbard model in harmonic traps within the Thomas-Fermi approximation based on the exact Bethe-ansatz solution. The $\ensuremath{\rho}\ensuremath{-}U/t$ phase diagram is shown for the systems of attractive interactions ($\ensuremath{\rho}$ is the characteristic density and U/tU/tU/t the interaction strength scaled in units of the hopping parameter). We study the double occupancy, the local central density, and their derivatives. Their roles in detecting the composite phases induced by the trapping potential are discussed in detail.
Ground-state properties of the attractive Hubbard model in one dimension are studied by means of both the exact Bethe-ansatz formalism and the self-consistent field (SCF) approach with renormalized chemical potential for general band fillings n and a wide range of coupling strength U/t.U/t.U/t. The energy, the concentration of double occupied sites, the kinetic energy and the chemical potential of the ground state are in a good numerical agreement with the exact results over a wide range of parameters U/tU/tU/t and n. The concentration of local pairs or double occupied sites in the Bethe-ansatz solution serves as a suitable parameter measuring the electron pairing correlations. The SCF theory provides a simple analytical relationship between the concentration of double occupied sites, the band filling and the BCS order parameter, valid for arbitrary U/tU/tU/t and n. The calculated energy gap, the BCS order parameter, the phase diagram and the compressibility are also discussed. The SCF theory in one dimension distinguishes the order parameter from the excitation gap and suggests a smooth crossover away from half-filling (at $n\ensuremath{\ne}1)$ from the BCS pairing to the Bose condensation regime under the variation of U/tU/tU/t and n.
The one-dimensional interacting Bose-Fermi mixtures, exhibiting quantum phase transitions at zero temperature, are particularly valuable for the study of quantum critical phenomena. In the present paper, we analytically study the quantum phase diagram, equation of state, and quantum criticality of the Bose-Fermi mixture using the thermodynamic Bethe ansatz equations. We show that thermodynamical properties display universal scaling behavior at quantum criticality. Furthermore, quantum criticality of the Bose-Fermi mixture in a harmonic trap is also studied within the local-density approximation. We thus demonstrate that the phase diagram and critical properties of the bulk system provide insight into understanding the universal features of many-body critical phenomena.
By investigating the compressibility of one-dimensional lattice fermions at various filling factors, we study the phase separation and re-entrant transition within the framework of the Bethe ansatz method. We model the system using the repulsive Hubbard model and calculate compressibility as a function of polarization for arbitrary values of chemical potential, temperature, and interaction strength. For filling factors 0<n<10<n<10<n<1, compressibility is a nonmonotonic function of polarization at all thermodynamic parameters. The compressibility reveals a phase transition into a phase-separated state for both low and intermediate temperatures at intermediate interactions as one increases the polarization. For certain filling factors, we find the re-entrant transition into the mixed phase at a higher polarization.
According to the exact analytical Bethe-ansatz solution of the one-dimensional spin-1/2 XXZ model, we perform a numerical study of one-dimensional spinless fermions in an optical lattice in the presence of harmonic potential by using a local density approximation. We study the density profile, and obtain the -u phase diagram (here =NfV2/t is the characteristic dimensionless density and u=V/t is the interaction strength scaled in units of the hopping parameter t). With the increases of particle density and nearest-neighbour interaction, the system undergoes five different quantum phases. Through calculating the thermodynamic stiffness S, we find that it can be used as an order parameter. Its singular points can measure the quantum phase transitions in such a confined system.
We revisit the one-dimensional attractive Hubbard model by using the Bethe-ansatz-based density-functional theory and density-matrix renormalization method. The ground-state properties of this model are discussed in details for different fillings and different confining conditions in weak-to-intermediate coupling regime. We investigate the ground-state energy, energy gap, and pair-binding energy and compare them with those calculated from the canonical Bardeen-Cooper-Schrieffer approximation. We find that the Bethe-ansatz-based density-functional theory is computationally easy and yields an accurate description of the ground-state properties for weak-to-intermediate interaction strength, different fillings, and confinements. In order to characterize the quantum phase transition in the presence of a harmonic confinement, we calculate the thermodynamic stiffness, the density-functional fidelity, and fidelity susceptibility, respectively. It is shown that with the increase in the number of particles or attractive interaction strength, the system can be driven from the Luther-Emery-type phase to the composite phase of Luther-Emery-type in the wings and insulatinglike in the center.