Large deviations, beyond standard results – LABS

Large deviation functions (LDFs) have emerged as a central tool to understand models of statistical physics. They constitute a natural framework not only to formulate equilibrium thermodynamics, but also to study the distributions of dynamical observables. Physical examples range from current fluctuations far from equilibrium to dynamical heterogeneities in glasses. Theoretical developments have identified singularities in LDFs, corresponding to dynamical phase transition and dynamical coexistence.

We aim at going beyond standard results pertaining to the symmetries and singularities of LDF, by using the peculiar mathematical structure of specific large-deviation problems which allows one to map these to another physical problem, where the large deviations of the original system correspond to typical configuration in the new system. The correspondence is based on a mathematical remark: after a Legendre transform, the parameters which condition a stochastic problem to take an atypical value of an observable (for instance the current) becomes a physical field in a quantum problem. Such correspondence will provide us with new physical informations on phenomena occurring in the target systems of this mapping. In our proposal, we focus on two problems belonging to quantum-matter physics: spin chains and quantum localization; in a first task, from exclusion processes to XXZ quantum spin chains and in a second task, from directed polymers in random media to quantum localization problems. Our aim is to import and export techniques between the classical and the quantum problems in order to gain novel knowledge on the phenomena underlying these systems.

The physical gain of using such mapping will be twofold: (i) an understanding of the scaling behavior of transitions in these systems in regimes which are not attainable by the usual tools of condensed matter, but which can be understood by the tools of the corresponding LDF problem in statistical mechanics via a mapping; (ii) the determination of finite-size and finite-time or -temperature scaling laws that are crucial when considering possible experimental realizations of the systems at hand. Of particular interest also is the existence of dynamical phase transitions which are present in the LDFs of the stochastic problem that become quantum phase transitions which in a scaling regime that has not been understood yet using tools of condensed matter. For the second task, new physical behavior are directly expected for the quantum localization transition as a manifestation of the third-order phase transition present in the large-deviation of the Kardar-Parisi-Zhang field probability distribution. We will consider possible experimental manifestations of the results we have obtained. Condensed matter offers realizations of such spin chains in electronic transport on crystal surfaces, where the chain is not isolated (the total magnetization is not conserved). We will use instead another realization of such spin chains that is available in cold atom systems.

Besides, we will focus on a transverse third task, more methodological, aimed at building consistent field theories for the systems considered in the first two tasks. The gain will not only be a better understanding of this problem for the specific systems that we consider, but also a novel construction of the path-integral representation of processes (effectively) described by Langevin equations. Generically, such constructions indeed turn out to present unsuspected issues (for instance, when performing changes of variables) as recently shown, and that we intend to elucidate using an original point of view coming from signal analysis and non-Gaussian noise theory. Understanding these issues will help us to elucidate crucial field-theoretical aspects of the two first tasks. The goal of this transverse task is to identify new physical behavior induced by the non-standard terms appearing in the action when following such a procedure.