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Physical Review Letters

Conductivity of One-Dimensional Interacting Fermions

Daniel Charles Mattis

Published Apr 1, 1974• 53 citations
Publisher

Abstract

Using an exactly soluble model, the decay rate of a current-carrying state of one-dimensional fermions is calculated in the presence of random scatterers at finite temperature and the dc conductivity thereby inferred. For interacting fermions it is modified by a factor ${(\frac{T}{T}+\ensuremath{\theta})}^{g}$, where $g$ is a positive (negative) coupling constant for repulsive (attractive) two-body forces. While the conductivity could be greatly enhanced for $g<0$ and $T\ensuremath{\ll}\ensuremath{\theta}$, one-dimensional superconductivity appears ruled out at any finite temperature.

DOI
10.1103/physrevlett.32.714

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Using an exactly soluble model, the decay rate of a current-carrying state of one-dimensional fermions is calculated in the presence of random scatterers at finite temperature and the dc conductivity thereby inferred. For interacting fermions it is modified by a factor (TT+\ensuremathθ)g{(\frac{T}{T}+\ensuremath{\theta})}^{g}(TT​+\ensuremathθ)g, where ggg is a positive (negative) coupling constant for repulsive (attractive) two-body forces. While the conductivity could be greatly enhanced for g<0g<0g<0 and T\ensuremath≪\ensuremathθT\ensuremath{\ll}\ensuremath{\theta}T\ensuremath≪\ensuremathθ, one-dimensional superconductivity appears ruled out at any finite temperature.
The exact energies and wavefunctions for a system of (N − 1) one-dimensional fermions all of the same spin and one fermion of the opposite spin are calculated in the large volume, finite density limit, when the particles interact via an attractive delta function potential. It is found that the attractive potential gives rise to a bound state, but, in spite of the presence of this bound state, all of the physical properties which are calculated (ground-state energy, effective mass of a certain class of excitations, etc.) are analytic continuations in the coupling constant of the corresponding results in the repulsive case. In addition, it is possible to have eigenstates which do not have the bound state present. These excited states are also discussed and are found to exhibit a negative effective mass and to modify the particle density at very large distances from the different particle.
We study the transport properties of one-dimensional (1D) interacting fermions through a barrier by numerically calculating the Kohn charge stiffness constant and the relative Drude weight. We find that the transport properties of the 1D Hubbard model are quite different from those of the 1D spinless fermion model. For example, the presence of the attractive interaction between electrons in the 1D Hubbard model actually suppresses the dc conductance, while a small repulsive interaction enhances the dc conductance. These results show that the spin degree of freedom plays an important role in the transport properties of the 1D interacting fermion systems.
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.
The frequency and temperature dependence of the conductivity of a one-dimensional fermion system with attractive interactions is studied by using a renormalization-group technique. At half filling the real part of the conductivity has both a \ensuremath{\delta}(\ensuremath{\omega}) part and a divergent frequency behavior ${\mathrm{\ensuremath{\omega}}}^{\mathrm{\ensuremath{-}}\ensuremath{\nu}}$ at finite frequencies, where \ensuremath{\nu} is a nonuniversal exponent depending on the interactoins. For the particular case of the attractive Hubbard model, logarithmic corrections appear and the conductivity behaves as 1/[\ensuremath{\omega} log102{\mathrm{log}}_{10}^{2}log102​(\ensuremath{\omega})], plus a \ensuremath{\delta}(\ensuremath{\omega}) part. Away from half filling the conductivity has a \ensuremath{\delta}(\ensuremath{\omega}) part and a gap up to a critical frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$, where ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$ is proportional to the doping with a prefactor depending on the interactions. The results obtained for the fermion model can be straightforwardly extended to the conductivity of an interacting one-dimensional boson model.
Dynamical conductivity in a disordered one-dimensional model of interacting fermions is studied numerically at high temperatures and in the weak-interaction regime in order to find a signature of many-body localization and vanishing dc transport coefficients. On the contrary, we find in the regime of moderately strong local disorder that the dc conductivity ${\ensuremath{\sigma}}_{0}$ scales linearly with the interaction strength while being exponentially dependent on the disorder. According to the behavior of the charge stiffness evaluated at the fixed number of particles, the absence of the many-body localization seems related to an increase in the effective localization length with the interaction.
The dc conductivity for the t-J model is exactly obtained at t=J and for the Hubbard model in one dimension. Our calculation is carried out for arbitrary band filling by use of the finite-size-scaling method based on the Bethe-ansatz solution under twisted boundary conditions. The effective transport mass defined as the inverse of the conductivity serves as a sensitive probe of the metal-insulator transition. The effective mass is extremely enhanced near half-filling both for the t-J model at t=J and the repulsive Hubbard model, which is a direct evidence of the metal-insulator transition due to the electron correlation. We observe that the large antiferromagnetic coupling J in the t-J model increases the effective mass. On the other hand, the conductivity in the attractive Hubbard model exhibits quite different properties from the repulsive models.