Limited validity of the Kubo formula for thermal conduction in modular quantum systems

Abstract

The Kubo formula describes a current as a response to an external field. In the case of heat conduction there is no such external field. We analyze why and to what extend it is nevertheless justified to describe heat conduction in modular quantum systems by the Kubo formula. "Modular" we call systems that may be described as consisting of weakly coupled identical subsystems. We explain in what sense this description applies to a large class of systems. Furthermore, we numerically evaluate the Kubo formula for some finite modular systems. We compare the results with data obtained from the direct numerical solution of the corresponding time-dependent Schrödinger equation.

Energy flux operator, current conservation and the formal Fourier's law

“…While previous works have typically relied on specific quantum states, calculating only expectation values, see for example [34,35,36,37,38], the results presented here essentially depend only on operator calculations. Possible extensions include generalization of the heat current definition to time dependent situations, and exploration of the necessary condition for the applicability of the Fourier's law of heat conduction in 1D chains [30].…”

“…We follow here a symmetry-limited definition of the heat flux operator often adopted in the literature [29,36,37]. The generic 1D Hamiltonian includes local potentials and inter-site interactions…”

“…[29,36,37,38], In order to increase generality, Ref. [36] further suggests a 'symmetrized local flux' that has the same form as j (2) s→s+1 . The heat flux operator was also extensively examined in 1D chains in the absence of an on-site energy term (h 0 s = 0), e.g.…”

“…We next assume that a continuity equation for h 0 s holds, based on the approximation that the local energy is conserved [36] …”

Theory of quantum energy transfer in spin chains: Superexchange and ballistic motion

“…This was done in the context of vibrational energy transfer [16] and thermal transport in spin chain [17][18][19], for both classical and quantum systems, using, e.g., molecular dynamics simulations [16], the quantum master equation method [19][20][21], the Green-Kubo formula [22,23], and the density matrix renormalization group method [24]. In the off-resonant regime simula-tions on purely harmonic systems indicated on a tunneling dynamics of heat transfer [10].…”