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Journal of Physics C Solid State Physics

Conductivity of a one-dimensional system of interacting fermions in a random potential

W. Apel

Published Mar 30, 1982• 56 citations
Publisher

Abstract

Impurity scattering is introduced in the Tomonaga model of an interacting one-dimensional Fermi system. The dynamic (frequency-dependent) conductivity is averaged over the random potential resulting from the impurities. To study this function the author exploits connections to two different two-dimensional classical models for which a Kosterlitz-Thouless-like transition is obtained. Thus, the system's conductivity changes from the usual insulating form to the ideal conducting behaviour of free fermions. Whether the system insulates or conducts depends on the sign and the strength of the fermionic interaction in qualitative agreement with the behaviour of the density response function and the pairing susceptibility of the pure system.

DOI
10.1088/0022-3719/15/9/018

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Impurity scattering is introduced in the Tomonaga model of an interacting one-dimensional Fermi system. The dynamic (frequency-dependent) conductivity is averaged over the random potential resulting from the impurities. To study this function the author exploits connections to two different two-dimensional classical models for which a Kosterlitz-Thouless-like transition is obtained. Thus, the system's conductivity changes from the usual insulating form to the ideal conducting behaviour of free fermions. Whether the system insulates or conducts depends on the sign and the strength of the fermionic interaction in qualitative agreement with the behaviour of the density response function and the pairing susceptibility of the pure system.
We study the temperature-dependent conductivity $\ensuremath{\sigma}(T)$ and spin susceptibility $\ensuremath{\chi}(T)$ of the two-dimensional disordered Hubbard model. Calculations of the current-current correlation function using a quantum Monte Carlo method show that repulsion between electrons can significantly enhance the conductivity, and at low temperatures change the sign of $d\ensuremath{\sigma}/\mathrm{dT}$ from positive (insulating behavior) to negative (conducting behavior). This result suggests the possibility of a metallic phase, and consequently a metal-insulator transition, in a two-dimensional microscopic model containing both interactions and disorder. The metallic phase is a non-Fermi liquid with local moments as deduced from $\ensuremath{\chi}(T)$.
A brief introduction to the bosonization method for interacting one-dimensional fermion systems is given. Using these results, the long-distance decay of correlation functions in the one-dimensional Hubbard model is determined exactly for arbitrary bandfilling and correlation strength, using the exact solution of Lieb and Wu. For infinite U the results are generalized to the case of nonzero nearest-neighbour interaction. The behaviour of thermodynamic quantities, of the frequency-dependent conductivity, and of the thermopower is also discussed, in particular in the proximity of the metal-insulator transitions occurring for half- and quarter-filling. The one-dimensional Luttinger liquid is shown to be unstable in the presence of interchain hopping. The results for the metal-insulator transition are compared with other scenarios developed in higher dimensions.
A brief introduction to the bosonization method for interacting one-dimensional fermion systems is given. Using these results, the long-distance decay of correlation functions in the one-dimensional Hubbard model is determined exactly for arbitrary bandfilling and correlation strength, using the exact solution of Lieb and Wu. For infinite U the results are generalized to the case of nonzero nearest-neighbour interaction. The behaviour of thermodynamic quantities, of the frequency-dependent conductivity, and of the thermopower is also discussed, in particular in the proximity of the metal-insulator transitions occurring for half- and quarter-filling. The one-dimensional Luttinger liquid is shown to be unstable in the presence of interchain hopping. The results for the metal-insulator transition are compared with other scenarios developed in higher dimensions.
The conductivity for half-filled disordered Hubbard model in different interaction regimes is studied using a typical medium dynamical mean-field theory. The Kubo formula with the geometrically averaged Green function is used to calculate the optical and dc conductivity of the system. The dependence of the conductivities in the model on the random potential and temperature in the weak and intermediate interaction regimes is investigated numerically. It is shown that in the weak interaction regime at a fixed temperature the optical conductivity at low frequencies as well as the dc conductivity get decreased as disorder strength increases. Whereas in the intermediate interaction regime, the results show a reentrant effect of the insulating phases. Our dc conductivity of a non-interacting disordered system is consistent with those known in the literature and our other results are in agreement with the phase diagram obtained by using typical medium theory with different impurity solvers.
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.
We analyze the uniform conductivity of a one-dimensional degenerate fermion system placed in a random disorder potential so smooth that backward scattering can be neglected. We use the nonlinear Luttinger liquid model to consider effects of both interaction and the curvature of fermionic dispersion. The finite frequency conductivity, calculated in the lowest order of disorder potential, consists of two parts. First one is the elastic contribution, largely independent of temperature and interaction. Second one is the inelastic contribution, strongly dependent on temperature and frequency and appearing upon simultaneous presence of curvature, disorder and interaction. We argue that apart from such finite frequency conductivity, there should always remain the δδδ-function peak of conductivity at zero frequency, whose weight is weakly dependent on the disorder.