Statistical theory of transport by strongly interacting lattice fermions

We present a study of electric transport at high temperature in a model of strongly interacting spinless fermions without disorder. We use exact diagonalization to study the statistics of the energy eigenvalues, eigenstates, and the matrix elements of the current. These suggest that our nonrandom Hamiltonian behaves like a member of a certain ensemble of Gaussian random matrices. We calculate the conductivity $\sigma \left(\omega \right)$ and examine its behavior, both in finite-size samples and as extrapolated to the thermodynamic limit. We find that $\sigma \left(\omega \right)$ has a prominent nondivergent singularity at $\omega =0$ reflecting a power-law long-time tail in the current autocorrelation function that arises from nonlinear couplings between the long-wavelength diffusive modes of the energy and particle number.

Figure 1

(Top) The average density of states in each symmetry sector. (Bottom) The probability distribution of the scaled level spacing for the $L=18$ system (data points) and the GOE form for this distribution (line), showing the excellent agreement.

Figure 2

(Top) Drude weight as a function of $L$. The Drude weight is expected to vanish exponentially with $L$. (Bottom) Conductivity as a function of frequency for $L=18$, $L=16$, and $L=14$. The agreement among different system sizes is good at high frequencies but significant finite-size effects can be seen at low frequency. (Inset) Closer view of the low frequency regime. The dip in the conductivity around $\omega =0$ is due to level repulsion. This dip rapidly narrows with increasing system size. The bins used here have a width equal to the mean level spacing at the maximum of the density of states for that $L$, except in the case of the main panel for $L=14$, where the bins widths are instead chosen proportional to $\omega$ in order to reduce the scatter.

Figure 3

(Top) Conductivity plotted vs $\sqrt{\omega }$ for $L=16$ and $L=18$. The straight line is a linear fit on this plot. The conductivity appears linear in $\sqrt{\omega }$ at low frequencies before it rounds off at very low frequency due to finite-size effects. The finite-size effects are, as expected, more pronounced for $L=16$ than $L=18$. (Bottom) Autocorrelation function of the total current vs time on a log-log plot. There is a long-time tail with $⟨J\left(0\right)J\left(t\right)⟩\sim t^{-3∕2}$.