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Physical review. B, Condensed matter

Renormalized pseudoparticle description of the one-dimensional Hubbard model thermodynamics

J. M. P. Carmelo, Peter Horsch, P.-A. Bares, Alexander A. Ovchinnikov

Published Nov 1, 1991• 58 citations
Publisher

Abstract

It is shown that the Hubbard chain in a magnetic field of arbitrary strength H can be treated as a liquid of interacting pseudoparticles which at T=0 has only forward scattering. The excitation spectra can be understood in terms of particle-hole processes in the pseudoparticle renormalized bands. Within this framework, we are able to study in detail the low-temperature thermodynamics of the Hubbard model. Explicit expressions for the specific heat ${\mathit{c}}_{\mathit{v}}$ for a nearly half-filled band and for values of the magnetic field close to the ferromagnetic saturated state are derived. The renormalized charge and spin masses which regulate the crossover regime to the exponential behavior of several physical quantities close to the metal-insulator and ferromagnetic transitions, respectively, are calculated.

DOI
10.1103/physrevb.44.9967

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It is shown that the Hubbard chain in a magnetic field of arbitrary strength H can be treated as a liquid of interacting pseudoparticles which at T=0 has only forward scattering. The excitation spectra can be understood in terms of particle-hole processes in the pseudoparticle renormalized bands. Within this framework, we are able to study in detail the low-temperature thermodynamics of the Hubbard model. Explicit expressions for the specific heat cv{\mathit{c}}_{\mathit{v}}cv​ for a nearly half-filled band and for values of the magnetic field close to the ferromagnetic saturated state are derived. The renormalized charge and spin masses which regulate the crossover regime to the exponential behavior of several physical quantities close to the metal-insulator and ferromagnetic transitions, respectively, are calculated.
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 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.
Even though the Hubbard model is one of the most fundamental models of highly correlated electrons, analytical and numerical data describing its thermodynamics at nonzero magnetization are relatively scarce. We present a detailed investigation of the thermodynamic properties for the one-dimensional repulsive Hubbard model in the presence of an arbitrary magnetic field for all values of the filling fraction and temperatures as low as $T\ensuremath{\sim}0.005t.$ Our analysis is based on the system of integral equations derived in the quantum transfer matrix framework. We determine the critical exponents of the quantum phase transitions and also provide analytical derivations for some of the universal functions characterizing the thermodynamics in the vicinities of the quantum critical points. Extensive numerical data for the specific heat, susceptibility, compressibility, and entropy are reported. The experimentally relevant double occupancy presents an interesting doubly nonmonotonic temperature dependence at intermediate values of the interaction strength and also at large repulsion and magnetic fields close to the critical value. The susceptibility in zero magnetic field has a logarithmic singularity at low temperatures for all filling factors similar to the behavior of the same quantity in the spin-1/2 isotropic Heisenberg model. We determine the density profiles for a harmonically trapped system and show that while the total density profile seems to depend mainly on the value of chemical potential at the center of the trap the distribution of phases in the inhomogeneous system changes dramatically as we increase the magnetic field.
We use the Landau-liquid character of the low-energy eigenstate spectrum of the Hubbard chain in a magnetic field to investigate its two-particle spectral properties. The picture that emerges is that of a gas of two coupled fluids of pseudoparticles. The kinetic equations that govern the flow of pseudoparticles are derived. The macroscopic currents are expressed in terms of the elementary pseudoparticle currents. As a central result we evaluate in the limit of zero momentum the wave function for the electronic excitations obtained by applying the up-spin and down-spin density-fluctuation operators on the ground state. This wave function is calculated on the basis of the pseudoparticle single-pair eigenstates. It is defined by the microscopic two-particle matrix elements of the corresponding fluctuations taken between the ground state and the elementary excitations. The on-site interaction U, electronic density n, and magnetic field H dependence of these matrix elements, which play a major role in the two-particle spectral functions, provide relevant information on the microscopic foundation of the one-dimensional Landau-liquid theory. This leads to the introduction of an adiabatic continuity principle, which is valid in the Hilbert subspace of the pseudoparticle single-pair eigenstates of lowest energy and momentum. The non-Landau-liquid character of the zero-magnetic-field L\"uttinger phase, where such an adiabatic continuity principle is not valid, is explained.Exact expressions for the two-pseudoparticle forward-scattering amplitudes and for the charge-charge and spin-spin correlation functions at small momentum and low frequency are obtained. The dependence of the charge and spin stiffnesses of the conductivity spectra on U, n, and H, which is determined by the pseudoparticle transport masses, is investigated in detail. The study of these masses and of the effective charge and spin projection of the pseudoparticles gives information on their structure in the electronic representation. It is shown that the decoupling that characterizes the low-energy excitation spectrum does not correspond in the case of finite magnetic fields to a real charge-spin separation. Our results provide a method for the study of the low-frequency dynamics of interacting systems solvable with the Bethe ansatz, and we identify the spectral parameters that control the nonclassical critical exponents with microscopic two-electron matrix elements.
We study the dynamical response of the half-filled one-dimensional Hubbard model for a range of interaction strengths UUU and temperatures TTT by a combination of numerical and analytical techniques. Using time-dependent density matrix renormalization group computations we find that the single-particle spectral function undergoes a crossover to a spin-incoherent Luttinger liquid regime at temperatures $T\ensuremath{\sim}J=4{t}^{2}/U$ for sufficiently large U>4tU>4tU>4t. At smaller values of UUU and elevated temperatures the spectral function is found to exhibit two thermally broadened bands of excitations, reminiscent of what is found in the Hubbard-I approximation. The dynamical density-density response function is shown to exhibit a finite-temperature resonance at low frequencies inside the Mott gap, with a physical origin similar to the Villain mode in gapped quantum spin chains. We complement our numerical computations by developing an analytic strong-coupling approach to the low-temperature dynamics in the spin-incoherent regime.