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Journal of the Physical Society of Japan

Thermodynamic Quantities of the One-Dimensional Hubbard Model at Finite Temperatures

Tatsuya Usuki, Norio Kawakami, Ayao Okiji

Published Apr 15, 1990• 62 citations
Publisher

Abstract

The thermodynamic quantities of the 1D Hubbard model are investigated exactly by means of the Bethe Ansatz solution at finite temperatures. For an arbitrary electron concentration, the magnetic susceptibility, the charge susceptibility and the specific heat are calculated numerically as a function of temperature. On the basis of the obtained results, the characteristic behavior, which appears in the charge excitation near half-filling, is discussed, and is understood to be closely related to the gap formation of the Mott-type insulator.

DOI
10.1143/jpsj.59.1357

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The thermodynamic quantities of the 1D Hubbard model are investigated exactly by means of the Bethe Ansatz solution at finite temperatures. For an arbitrary electron concentration, the magnetic susceptibility, the charge susceptibility and the specific heat are calculated numerically as a function of temperature. On the basis of the obtained results, the characteristic behavior, which appears in the charge excitation near half-filling, is discussed, and is understood to be closely related to the gap formation of the Mott-type insulator.
We analyse the thermodynamic Bethe Ansatz equations of the one-dimensional half-filled Hubbard model in the ‘spin-disordered regime’, which is characterized by the temperature being much larger than the magnetic energy scale, but small compared to the Mott–Hubbard gap. In this regime the thermodynamics of the Hubbard model can be thought of in terms of gapped charged excitations with an effective dispersion and spin degrees of freedom that only contribute entropically. In particular, the internal energy and the effective dispersion become essentially independent of temperature. An interpretation of this regime in terms of a putative interacting-electron system at zero temperature leads to a metal–insulator transition at a finite interaction strength above which the gap opens linearly. We relate these observations to studies of the Mott–Hubbard transition in the limit of infinite dimensions.
On the basis of the Bethe ansatz solution, the characteristic properties of charge excitations are investigated for the one-dimensional Hubbard model by inserting a charged hole into the Mott insulator. The density of states for the charged hole shows divergent behavior in the lower-energy region, whose connection is discussed analytically with the compressibility and magnitude of the Hubbard gap. Momentum-dependent susceptibility is also investigated within the framework of the quasiparticle formalism in the Bethe ansatz method. The logarithmic anomaly of the susceptibilty for the charged hole near 4kF{k}_{F}kF​ (kF{k}_{F}kF​ being the Fermi momentum in the noninteracting case) is considerably enhanced in the vicinity of the Mott insulating phase, while it is suppressed at the insulating phase due to the formation of the Hubbard gap.
We investigate the one-dimensional (1D) Hubbard model with an additional bond-charge interaction, recently considered in the description of compounds that exhibit strong 1D features above the temperature of ordered phases. The partition function of the model is exactly calculated for a value of the bond-charge coupling; the behavior of the specific heat and spin susceptibility as a function of temperature is derived at arbitrary filling, and particularly discussed across the occurring metal-insulator transition. The results show that the bond-charge terms weaken the spin excitations of the system.
The exact phase diagram and the ground-state properties of the one-dimensional Hubbard model with arbitrary on-site interaction of electrons are calculated over a wide range of magnetic field and electron concentrations by means of the Bethe-ansatz formalism. The ground-state properties, including the total energy, the average spin (magnetization) and spin (magnetic) susceptibility are investigated for both signs of the interaction strength U/t. The critical behavior near the onset of magnetization and magnetic saturation are also analyzed. At the onset of magnetization and near the magnetic saturation the spin susceptibility χ diverges at all U/t for half-filling case n=1, whereas for n≠1 it is always finite. The reverse susceptibility χ-1(U) exhibits anomalous hump, which increases with h or n, and shows discontinuity as U/t→±0 at infinitesimal h→0. The analytical results for the ground-state properties in strong and weak interaction limits are in full agreement with our numerical calculations.
We investigate the temperature region in which a Tomonaga-Luttinger liquid (TLL) description of the charge sector of the one-dimensional Hubbard model is valid. By using the thermodynamic Bethe ansatz method, electron number is calculated at finite temperatures and fixed chemical potential. We observe maximum electron number as a function of temperature close to the chemical potential of the upper critical value that corresponds to half filling. As the chemical potential approaches the upper critical value from below, the temperature (TM)(T_{\rm M})(TM​) at which the electron number shows its maximum asymptotically approaches a universal relation. We show that, below the energy corresponding to TMT_{\rm M}TM​, the charge excitation spectrum nearly obeys a linear dispersion relation. The results demonstrate that TMT_{\rm M}TM​ marks the important temperature below which TLL is realized.
The phase diagram and thermodynamic properties of the Hubbard model in one dimension are calculated numerically within the generalized self-consistent approach over a wide range of temperature T, magnetic field h, interaction strength U/t and electron concentration n. The temperature variation of the transverse magnetization, double occupied sites and chemical potential at various n, U/t and h is also analyzed. The wave vector q for the magnetic order parameter in the vicinity of the half-filling in the metallic phase shows a crossover from incommensurate magnetic phase (0<q<π) into the state with strong antiferromagnetic correlations (q=π) as temperature is increased. Overall our numerical results in the approximate theory at finite temperatures are in good quantitative and qualitative agreement with the Bethe-ansatz results.