Lower bounds for the conductivities of correlated quantum systems

We show how one can obtain a lower bound for the electrical, spin, or heat conductivity of correlated quantum systems described by Hamiltonians of the form $H=H_0+g{H}_1$. Here, $H_0$ is an interacting Hamiltonian characterized by conservation laws which lead to an infinite conductivity for $g=0$. The small perturbation $g{H}_1$, however, renders the conductivity finite at finite temperatures. For example, $H_0$ could be a continuum field theory, where momentum is conserved, or an integrable one-dimensional model, while $H_1$ might describe the effects of weak disorder. In the limit $g\to 0$, we derive lower bounds for the relevant conductivities and show how they can be improved systematically using the memory matrix formalism. Furthermore, we discuss various applications and investigate under what conditions our lower bound may become exact.

Figure 1

For $g=0$, a Drude peak shows up in the conductivity, resulting from exact conservation laws. For $g\ne 0$, the Drude peak broadens and the dc conductivity becomes finite.

Figure 2

Comparison of the result of a single mode memory matrix calculation (solid line) [Eq. (C3)] to the full numerical solution of the Boltzmann equation (dotted line) for $T=0.05$ and $N=500$. The memory matrix is always a lower bound to the Boltzmann result and converges toward it as the disorder strength $g$ is reduced, as shown in the inset (ratio of the single mode approximation to the Boltzmann result).