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arXiv:cond-mat/0203303 · AIP conference proceedings

Transport of Strongly Correlated Electrons

Peter Prelovšek

Published Jan 1, 2002• 5 citations
arXivHTMLPDF

Abstract

Lectures deal with the theory of electronic transport, in particular with the electrical conductivity, in systems dominated by strong electron ‐ electron repulsion. The concept of charge stiffness is introduced to distinguish conductors and insulators at T = 0, but as well usual resistors, possible ideal conductors and ideal insulators at finite temperature. It is shown that the latter singular transport appears in many integrable systems of interacting fermions, the evidence coming from the relation with level dynamics, from the existence of conserved quantities as well as from numerical studies and exact results. Then, exact duagonalization approaches for the calculation of static and dynamical quantities in small correlated systems are described, with the emphasis on the finite‐temperature Lanczos method applicable to transport quantities. Finally, anomalous dynamical conductivity within the planar single‐band model is discussed in relation with experiments on cuprates.

DOI
10.1063/1.1509143

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Journal of Physics A Mathematical and Theoretical•Jun 9, 2014•1 citation
Lectures deal with the theory of electronic transport, in particular with the electrical conductivity, in systems dominated by strong electron ‐ electron repulsion. The concept of charge stiffness is introduced to distinguish conductors and insulators at T = 0, but as well usual resistors, possible ideal conductors and ideal insulators at finite temperature. It is shown that the latter singular transport appears in many integrable systems of interacting fermions, the evidence coming from the relation with level dynamics, from the existence of conserved quantities as well as from numerical studies and exact results. Then, exact duagonalization approaches for the calculation of static and dynamical quantities in small correlated systems are described, with the emphasis on the finite‐temperature Lanczos method applicable to transport quantities. Finally, anomalous dynamical conductivity within the planar single‐band model is discussed in relation with experiments on cuprates.
We study the effect of conservation laws on the finite-temperature transport properties in one-dimensional integrable quantum many-body systems. We show that the energy current is closely related to the first conservation law in these systems and therefore the thermal transport coefficients are anomalous. Using an inequality on the time decay of current correlations we show how the existence of conserved quantities implies a finite charge stiffness (weight of the zero-frequency component of the conductivity) and so ideal conductivity at finite temperatures.
Using numerical diagonalization techniques we analyze the finite temperature/frequency conductance of a one-dimensional model of interacting spinless fermions. Depending on the interaction, the observed finite temperature charge stiffness and low-frequency conductance indicate a fundamental difference between integrable and nonintegrable cases. The integrable systems behave as ideal conductors in the metallic regime and as ideal insulators in the insulating one. The nonintegrable systems are, as expected, generic conductors in the metallic regime and activated ones in the insulating regime. \textcopyright{} 1996 The American Physical Society.
One-dimensional conductors are a long-standing topic of research with direct applications to organic conductors and mesoscopic rings. The discovery of the ceramic high-temperature superconductors has revitalized the interest in low-dimensional charge and spin fluctuations of highly correlated electron systems. Several mechanisms proposed to explain the high-T c superconductors invoke properties of the two-dimensional Hubbard model, but probably also some one-dimensional aspects are relevant. Numerous one-dimensional models for correlated electrons have been studied with various approximate, asymptotically exact and exact methods. These results lead to the concept of Luttinger liquid for interacting electron gases without excitation gaps (metallic systems). Characteristic of Luttinger liquids are the charge and spin separation, marginal Fermi liquid properties, e.g. the absence of quasiparticles in the vicinity of the Fermi surface, nonuniversal power-law singularities in the one-particle spectral function and the related absence of a discontinuity in the momentum distribution at the Fermi level, the power-law decay of correlation functions for long times and large distances, persistent currents in finite rings, etc. Due to the peculiarities of the phase space in one dimension some of the models have sufficient conserved currents to be completely integrable. We review exact results derived within the framework of Bethe's ansatz for integrable one-dimensional models of correlated electrons. The Bethe-ansatz method is presented by explicitly showing the steps leading to the solution of the N-component electron gas interacting via a δ-function potential (repulsive and attractive interaction), which is probably the simplest model of correlated electrons. Emphasis is given to the procedure to extract the groundstate properties, the classification of states, the excitation spectrum, the thermodynamics and finite size effects, such as critical exponents of correlation functions and persistent currents. The method is then applied to numerous other models, e.g. (i) a two-band model involving attractive and repulsive potentials and crystalline fields splitting the bands, (ii) the traditional Hubbard chain with attractive and repulsive U, (iii) the degenerate Hubbard model with repulsive U, which displays a metal–insulator transition at a finite U, (iv) a two-band Hubbard model with repulsive U, (v) the traditional supersymmetric t–J model (vi) a two-band supersymmetric t–J model with band-splitting and (vii) the N-component supersymmetric t–J model. Finally, results for models with long-range interactions, in particular r-2 and sinh -2(r) potentials, are briefly reviewed.
We analyze the high-temperature conductivity in one-dimensional integrable models of interacting fermions: the $t\text{\ensuremath{-}}V$ model (anisotropic Heisenberg spin chain) and the Hubbard model, at half-filling in the regime corresponding to insulating ground state. A microcanonical Lanczos method study for finite size systems reveals anomalously large finite-size effects at low frequencies while a frequency-moment analysis indicates a finite dc conductivity. This phenomenon also appears in a prototype integrable quantum system of impenetrable particles, representing a strong-coupling limit of both models. In the thermodynamic limit, the two results could converge to a finite dc conductivity rather than an ideal conductor or insulator scenario.
Integrable and nonintegrable systems have very different transport properties. In this paper, we highlight these differences for specific one-dimensional models of interacting lattice fermions. The technique used is a finite temperature numerical calculation of the adiabatic stiffness (also called the Drude weight or charge stiffness) and isothermal stiffness (also called the ``Meissner'' stiffness) in electrical and energy transport, and the momentum dependent dynamical conductivities $\ensuremath{\sigma}(q,\ensuremath{\omega})$ and $\ensuremath{\kappa}(q,\ensuremath{\omega})$. We apply a flux twist to break the Kramers degeneracy; thus, allowing us to focus on the effect of the dynamical degeneracies in the integrable system. In this situation, we show that the isothermal stiffness goes to zero rapidly with the system size for both types of systems even at high temperatures; while the adiabatic stiffness appears to go to zero in the nonintegrable system and to a finite value in the integrable one. We analyze this difference in terms of the statistics of the current matrix elements and the degeneracies of the systems, and show that in the integrable system, despite the presence of degeneracies, the dominant contribution to the adiabatic stiffness comes from large-current-carrying nondegenerate states. We also show that energy transport at nonzero $\ensuremath{\omega}$ and qqq occurs within a banded continuum in the integrable system indicative of ballistic transport while the nonintegrable system shows diffusion but with the existence of overdamped excitations at large values of momentum.
Finite-temperature T > 0 transport properties of integrable and nonintegrable one-dimensional (1D) many-particle quantum systems are rather different, showing ballistic and diffusive behavior, respectively. The repulsive 1D Hubbard model is a prominent example of an integrable correlated system. For electronic densities n ≠ 1 (and spin densities m ≠ 0) it is an ideal charge (and spin) conductor, with ballistic charge (and spin) transport for T ⩾ 0. In spite of the fact that it is solvable by the Bethe ansatz, at n = 1 (and m = 0) its T > 0 charge (and spin) transport properties are an issue that remains poorly understood. Here we combine this solution with symmetry and the explicit calculation of current-operator matrix elements between energy eigenstates to show that for on-site repulsion U > 0 and at n = 1 the charge stiffness D η ( T ) vanishes for T > 0 in the thermodynamic limit. A similar behavior is found by such methods for the spin stiffness D s ( T ) for U > 0 and T > 0, which vanishes at m = 0. This absence of finite temperature n = 1 ballistic charge transport and m = 0 ballistic spin transport are exact results that clarify long-standing open problems.