Quantum collisional thermostats

Collisional reservoirs are becoming a major tool for modelling open quantum systems. In their simplest implementation, an external agent switches on, for a given time, the interaction between the system and a specimen from the reservoir. Generically, in this operation the external agent performs work onto the system, preventing thermalization when the reservoir is at equilibrium. One can recover thermalization by considering an autonomous global setup where the reservoir particles colliding with the system possess a kinetic degree of freedom. The drawback is that the corresponding scattering problem is rather involved. Here, we present a formal solution of the problem in one dimension and for flat interaction potentials. The solution is based on the transfer matrix formalism and allows one to explore the symmetries of the resulting scattering map. One of these symmetries is micro-reversibility, which is a condition for thermalization. We then introduce two approximations of the scattering map that preserve these symmetries and, consequently, thermalize the system. These relatively simple approximate solutions constitute models of quantum thermostats and are useful tools to study quantum systems in contact with thermal baths. We illustrate their accuracy in a specific example, showing that both are good approximations of the exact scattering problem even in situations far from equilibrium. Moreover, one of the models consists of the removal of certain coherences plus a very specific randomization of the interaction time. These two features allow one to identify as heat the energy transfer due to switching on and off the interaction. Our results prompt the fundamental question of how to distinguish between heat and work from the statistical properties of the exchange of energy between a system and its surroundings.

Trace spectrum of 1D Transfer Matrices for wave propagation in layered media

The Transfer Matrix formalism is ubiquitous when considering wave propagation in various stratified media, applications ranging from Seismology to Quantum Mechanics. The relation between variables at two points in the laminate can be established via a matrix, termed (global) transfer matrix (product of "atomic'' single-layer matrices). As a matter of convenience, we focus on 1D phononic structures, but our derivation can be extended to other fields where the formalism applies. We present exact expressions for entries of the global propagator for N layers. When the layering corresponds to a representative repeated cell in an otherwise infinitely periodic medium, the trace of the cumulative multi-layer matrix is known to control the dispersion relation. We show how this trace has a discrete spectrum made up of distinct 2**(N-1) harmonics (not necessarily orthogonal to each other in any sense) which we characterize exactly both in terms of the periods that they contain and their amplitudes; we also show that the phase shift among harmonics is either zero or pi. This definite appraisal of the spectrum of the trace opens the path for rational design of band gaps, going beyond parametric or sensitivity studies.

Transfer Matrix of the Two-dimensional Ising Model

This chapter deals with the exact solution of the two-dimensional Ising model as it is achieved through the transfer matrix formalism. It discusses the crucial role played by the commutative properties of the transfer matrices, which lead to a functional equation for their eigenvalues. The exact free energy of the Ising model and its critical point can be identified by means of the lowest eigenvalue. The chapter covers Baxter's approach, the Yang–Baxter equation and its relation to the Boltzmann weights, the R-matrix, and discusses activity away from the critical point, the six-vertex model, as well as functional equations and symmetries.

Quantum Field Theory

Chapter 7 covers the main reasons for adopting the methods of quantum field theory (QFT) to study the critical phenomena. It presents both the canonical quantization and the path integral formulation of the field theories as well as the analysis of the perturbation theory. The chapter also covers transfer matrix formalism and the Euclidean aspects of QFT, the field theory of the Ising model, Feynman diagrams, correlation functions in coordinate space, the Minkowski space and the Legendre transformation and vertex functions. Everything in this chapter will be needed sooner or later, since it highlights most of the relevant aspects of quantum field theory.

Interfacial engineering and optical coupling for multicolored semitransparent inverted organic photovoltaics with a record efficiency of over 12%

Guided by finite-difference time-domain (FDTD) and optical transfer matrix formalism (TMF) simulation, the contradiction between PCE and AVT was solved, and multicolored ST-OSCs with record high efficiency were achieved.

Greybody Factors for Schwarzschild Black Holes: Path-Ordered Exponentials and Product Integrals

In earlier work concerning the sparsity of the Hawking flux, we found it necessary to re-examine what is known regarding the greybody factors of black holes, with a view to extending and expanding on some old results from the 1970s. Focusing specifically on Schwarzschild black holes, we have re-calculated and re-assessed the greybody factors using a path-ordered-exponential approach, a technique which has the virtue of providing a pedagogically useful semi-explicit formula for the relevant Bogoliubov coefficients. These path-ordered-exponentials, being based on a variant of the “transfer matrix” formalism, are closely related to so-called “product integrals”, leading to quite straightforward and direct numerical evaluation, while side-stepping any need for numerically solving the relevant ordinary differential equations. Furthermore, while considerable analytic information is already available regarding both the high-frequency and low-frequency asymptotics of these greybody factors, numerical approaches seem better adapted to finding suitable “global models” for these greybody factors in the intermediate frequency regime, where most of the Hawking flux is actually concentrated. Working in a more general context, these path-ordered-exponential techniques are also likely to be of interest for generic barrier-penetration problems.