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Physical review. B, Condensed matter

Thermodynamics of a one-dimensional lattice system with long-range interelectron repulsion

Victor V. Slavin, A. A. Slutskin

Published Sep 15, 1996• 18 citations
Publisher

Abstract

The thermodynamic characteristics of one-dimensional lattice electron systems with long-range interelectron interactions are explicated and studied at arbitrary temperature T and pressure P. Only interactions between neighboring particles are taken into account. Electron-electron correlations are found to be controlled only by the ratio P/T, increases in which produce a continuous transition from a weakly nonideal gas to an electron ``crystal'' with filling factor 1/q,q=1,2, . . .. This is a remnant of the devil's staircase which obtains at T=O. It is shown that with a decrease in P the crystal periods, q, increase, the transitions occurring via narrow regions of a liquidlike state. Charge carriers in the ``crystals'' are discrete solitons with a fractional charge ${\mathit{e}}^{\mathrm{*}}$=\ifmmode\pm\else\textpm\fi{}e/q. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.8095

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Physical review. B, Condensed matter•May 15, 1996•6 citations
The thermodynamic characteristics of one-dimensional lattice electron systems with long-range interelectron interactions are explicated and studied at arbitrary temperature T and pressure P. Only interactions between neighboring particles are taken into account. Electron-electron correlations are found to be controlled only by the ratio P/T, increases in which produce a continuous transition from a weakly nonideal gas to an electron crystal'' with filling factor 1/q,q=1,2, . . .. This is a remnant of the devil's staircase which obtains at T=O. It is shown that with a decrease in P the crystal periods, q, increase, the transitions occurring via narrow regions of a liquidlike state. Charge carriers in the crystals'' are discrete solitons with a fractional charge e∗{\mathit{e}}^{\mathrm{*}}e∗=\ifmmode\pm\else\textpm\fi{}e/q. \textcopyright{} 1996 The American Physical Society.
We present an exactly solvable Hamiltonian which consists of nearest neighbor hopping and long-range interaction. The ground state and the thermodynamic quantities are analytically obtained in any dimensions. In a repulsive case, the Luttinger's theorem breaks down and the hypothesis of the adiabatic continuation no longer holds. At half filling, we show the existence of a metal-insulator transition as a function of the strength of the interaction. The zero energy excitation disappears at the critical value of the interaction when we increase the interaction. In this case, the system is an insulator. The charge compressibility behaves as κ≈|1-ρ| 1-2/ d near half filling in d dimensions where ρ is the density of electrons. At other fillings, there is always a zero energy excitation and the system is metallic. When the interaction is attractive, the ground state is composed of pairs of electrons with opposite spins. The low energy excitations, however, is not a particle-hole type of a non-interacting case but a pairwise excitation across the Fermi surface.
A one-dimensional gas of electrons interacting with long-range Coulomb forces [V(r)\ensuremath{\approxeq}1/r] is investigated. The excitation spectrum consists of separate collective charge and spin modes. For arbitrarily weak Coulomb repulsion density correlations at wave vector 4kF{\mathit{k}}_{\mathit{F}}kF​ decay extremely slowly and are best described as those of a one-dimensional Wigner crystal. Pinning of the Wigner crystal then leads to the nonlinear transport properties characteristic of charge density waves. The results allow a consistent interpretation of the plasmon and spin excitations observed in one-dimensional semiconductor structures, and suggest an interpretation of some of the observed features in terms of ``spinons.''
The Hubbard-Holstein model describes fermions on a discrete lattice, with on-site repulsion between fermions and a coupling to phonons that are localized on sites. Generally, at half-filling, increasing the coupling ggg to the phonons drives the system towards a Peierls charge density wave state, whereas increasing the electron-electron interaction UUU drives the fermions into a Mott antiferromagnet. At low ggg and UUU, or when doped, the system is metallic. In one dimension, using quantum Monte Carlo simulations, we study the case where fermions have a long-range coupling to phonons, with characteristic range $\ensuremath{\xi}$, interpolating between the Holstein and Fr\"ohlich limits. Without electron-electron interaction, the fermions adopt a Peierls state when the coupling to the phonons is strong enough. This state is destabilized by a small coupling range $\ensuremath{\xi}$ and leads to a collapse of the fermions, and, consequently, phase separation. Increasing interaction UUU will drive any of these three phases (metallic, Peierls, phase separation) into a Mott insulator phase. The phase separation region is once again present in the $U\ensuremath{\ne}0$ case, even for small values of the coupling range.
We study the influence of long-range Coulomb interactions on the properties of one-dimensional (1D) strongly correlated electron systems in the vicinity of the metal-insulator Mott-Hubbard phase transition. It is shown that, in the metallic phase, the standard square-root singularity of the compressibility at the transition point changes to a logarithmic one, due to the formation of a 1D Wigner crystal of solitons (holons). On increasing the soliton density in a finite-size chain, the behavior of the compressibilty reflects a sequence of crossovers between classical, low-density regimes of perfectly or nearly ordered Wigner-crystal states, and quantum regimes of a nearly free Fermi gas of solitons, followed (in the high-density limit) by a liquid phase of strongly correlated solitons. In a mesoscopic situation, where the screening length in a 1D chain is controlled by a massive electrode (gate) placed near the chain, there is a narrow region near the transition point where quantum fluctuations melt the Wigner crystal and recover the universal square-root singularity of the compressibility. Strong Coulomb interaction affects the formation of the charge excitations in the insulating phase, transforming the sine-Gordon solitons into quasiclassical Coulomb solitons. Multiplicative logarithmic renormalizations of the characteristic soliton size and rest energy are found.
The relationship among the Wigner crystal, charge ordering, and the Mott insulator is studied by the path-integral renormalization group method in two-dimensional systems with long-range Coulomb interaction. In contrast to the insensitivity of the Hartree-Fock results, the stability of the solid drastically decreases with the decrease in the lattice commensurability. The transition to liquid occurs at the electron gas parameter r(s) approximately 2 for the filling n=1/2, showing a large reduction from r(s) approximately 35 in the continuum limit. A correct account of quantum fluctuations is crucial to understanding the charge-order stability generally observed only at simple fractional fillings and the nature of quantum liquids away from them.
Whether Peierls distortion occurs for a system of one-dimensional spinless fermions is determined by the exponent g that characterizes the decay of correlations among the electrons in the undistorted state. Noninteracting electrons (g=1) is the marginal case, for which the Peierls effect just barely occurs. For repulsive interactions between the electrons (g1) the instability is enhanced and the leading dependence of the ground-state energy E on the lattice distortion \ensuremath{\delta} is E\ensuremath{\propto}-\ensuremath{\delta}${\mathrm{}}^{2/(2\mathrm{\ensuremath{-}}\mathit{g})}$. For attractive interactions (g\ensuremath{\gtrsim}1) a continuous phase transition between a normal metal and a Peierls insulator takes place for sufficiently strong electron-ion interaction. \textcopyright{} 1996 The American Physical Society.