Anomalous thermal conduction in one dimension: a quantum calculation

Abstract

In this paper, we study the thermal conductivity of an anharmonically coupled chain of atoms. Numerical studies using classical dynamics have shown that the conductivity of a chain with nearest neighbor couplings diverges with chain length L as L^α; earlier studies found α≈0.4 under a range of conditions, but a recent study on longer chains claims α=1/3. Analytically, this problem has been studied by calculating the relaxation rate Γ_q of the normal modes of vibration as a function of its wave vector q. Two theoretical studies of classical chains, one using the mode-coupling formulation and the other the Boltzmann equation method, led to Γ_q?q^5/3, which is consistent with α=0.4. Here we study the problem for a quantum anharmonic chain with quartic anisotropy. We develop a low-temperature expansion for Γq and find that, in the regime ω_q«k_BT, Γ_q∝q^5/3T², where ω_q is the frequency of the mode. In our analysis, the relaxation arises due to umklapp scattering processes. We further evaluate the thermal conductivity of the chain using the Kubo formula, which enables us to take into account the transport relaxation time through vertex corrections for the current-current correlator. This calculation also yields α=0.4.