This review deals with the nonequilibrium Green’s function (NEGF) method applied to the problems of energy transport due to atomic vibrations (phonons), primarily for small junction systems. We present a pedagogical introduction to the subject, deriving some of the well-known results such as the Laudauer-like formula for heat current in ballistic systems. The main aim of the review is to build the machinery of the method so that it can be applied to other situations, which are not directly treated here. In addition to the above, we consider a number of applications of NEGF, not in routine model system calculations, but in a few new aspects showing the power and usefulness of the formalism. In particular, we discuss the problems of multiple leads, coupled left-right-lead system, and system without a center. We also apply the method to the problem of full counting statistics. In the case of nonlinear systems, we make general comments on the thermal expansion effect, phonon relaxation time, and a certain class of mean-field approximations. Lastly, we examine the relationship between NEGF, reduced density matrix, and master equation approaches to thermal transport.

We propose a thermodynamically consistent minimal model to study synchronization which is made of driven and interacting three-state units. This system exhibits at the mean-field level two bifurcations separating three dynamical phases: a single stable fixed point, a stable limit cycle indicative of synchronization, and multiple stable fixed points. These complex emergent dynamical behaviors are understood at the level of the underlying linear Markovian dynamics in terms of metastability, i.e., the appearance of gaps in the upper real part of the spectrum of the Markov generator. Stochastic thermodynamics is used to study the dissipated work across dynamical phases as well as across scales. This dissipated work is found to be reduced by the attractive interactions between the units and to nontrivially depend on the system size. When operating as a work-to-work converter, we find that the maximum power output is achieved far from equilibrium in the synchronization regime and that the efficiency at maximum power is surprisingly close to the linear regime prediction. Our work shows the way towards building a thermodynamics of nonequilibrium phase transitions in conjunction with the bifurcation theory.

A novel scheme for the steady state solution of the standard Redfield quantum master equation is developed which yields agreement with the exact result for the corresponding reduced density matrix up to second order in the system-bath coupling strength. We achieve this objective by use of an analytic continuation of the off-diagonal matrix elements of the Redfield solution towards its diagonal limit. Notably, our scheme does not require the provision of yet higher order relaxation tensors. Testing this modified method for a heat bath consisting of a collection of harmonic oscillators we assess that the system relaxes towards its correct coupling-dependent, generalized quantum Gibbs state in second order. We numerically compare our formulation for a damped quantum harmonic system with the nonequilibrium Green's function formalism: we find good agreement at low temperatures for coupling strengths that are even larger than expected from the very regime of validity of the second-order Redfield quantum master equation. Yet another advantage of our method is that it markedly reduces the numerical complexity of the problem; thus, allowing to study efficiently large-sized system Hilbert spaces.

Symmetries play a crucial role in ubiquitous systems found in Nature. In this work, we propose an elegant approach to detect symmetries by measuring quantum currents. Our detection scheme relies on initiating the system in an anti-symmetric initial condition, with respect to the symmetric sites and using a probe that acts like a local noise. Depending on the position of the probe the currents exhibit unique signatures such as a quasi-stationary plateau indicating the presence of metastability and multi-exponential decays in case of multiple symmetries. The signatures are sensitive to the characteristics of the probe and vanish completely when the timescale of the coherent system dynamics is much longer than the timescale of the probe. These results are demonstrated using a 4-site model and an archetypal example of the para-benzene ring and are shown to be robust under a weak disorder.

We describe a method to obtain the reduced density matrix (RDM) correct up to second order in system-bath coupling in nonequilibrium steady-state situations. The RDM is obtained via a scheme based on analytic continuation, using the time-local Redfield-like quantum master equation, which was earlier used by the same authors [J. Chem. Phys. 136, 194110 (2012)] to obtain the correct thermal equilibrium description. This nonequilibrium modified Redfield solution is then corroborated with the exact RDM obtained via the nonequilibrium Green's function technique for the quantum harmonic oscillator. Lastly, the scheme is compared to different quantum master equations (QMEs), namely the time-local Redfield-like and the Lindblad-like QMEs, in order to illustrate the differences between each of these approaches.