Research project is (co) funded by the Slovenian Research Agency.

**UL Member:** Faculty of Mathematics and Physics**Code:** J1-7279**Project:** Thermodynamics of dissipative nanosystems**Period:** 1.1.2016 - 31.12.2018**Range per year:** 0,72 FTE, category D**Head:** Žnidarič Marko**Research activity:** Natural sciences and mathematics**Research Organisations:** link on SICRIS**Researchers:** link on SICRIS**Citations for bibliographic records:** link on SICRIS

**Project description:**

Thermodynamics is the theory that describes behavior of macroscopic observables in systems that usually have many degrees of freedom. Quantum mechanics is on the other hand the theory of small systems with microscopic, in general non-commutative, observables, and tensor-product structure of the Hilbert space. The language and the scale at which they work best is seemingly different, although, in principle, one should be able to "derive" statistical laws that underlay thermodynamics purely from microscopic quantum mechanical laws.

Within classical systems there is a well established theory that tries to do that and is known as the theory of dynamical systems (or the ergodic or chaos theory). One of the important findings there is that already with a few-degree of freedom toy systems one can often capture generic chaotic behavior of a many-particle system. Because few-body systems are easier to analyze classical theory of dynamical systems is relatively mature and established field. However, microscopic dynamics is in principle quantum and in in quantum mechanics, due to a tensor-product structure, things are not that simple. Quantum systems can be qualitatively different than classical - it is quantum information theory that exploits that - and very importantly, complexity of a many-body quantum system grows exponentially with its size, rendering its analysis in general very difficult. As a consequence, quantum many-body physics is less explored, with many famous old problems outstanding (e.g., superconductivity, transport in the 1D Heisenberg model, etc.).

Several developments in recent years intensified our quest to understand many-body physics of quantum systems. On one hand there is experimental motivation: physics experiments are getting increasingly more advanced, being able to manipulate individual quantum systems. On the other hand there is also a technology-driven top-down impetus -for instance, with increasingly miniaturized electronic components one has to understand interaction of nanoscopic systems with its environment, and in addition one would like to design devices that would exploit all the riches of quantum world. Not least, there is also purely theoretical and old motivation, to simply understand how and which statistical laws are obeyed by many-body quantum systems. Because such systems are usually exposed to the environment, either unavoidably or intentionally, one in fact has to deal with dissipative systems.

The aim of the project is to study statistical properties of many-body dissipative systems. We would like to stress that most of the questions that we plan to study are rather unexplored for dissipative many-body systems. We deem several questions to be very important -- judged by their famous counterparts studied extensively in closed Hamiltonian systems. We are confident that such pioneering research is a guarantee for interesting new discoveries. In particular, we shall deal with spectral properties of dissipative propagators, like the famous spectral gap question. Being a non-hermitian linear operator this opens up a completely new field, as in closed systems one usually deals with hermitian operators. Another important topic that we plan to study is computational power with limited resources. An important example are limits on cooling techniques with given (realistic, e.g., local) couplings. We are also going to study emergent laws in many-body systems, like transport, rectification, or thermoelectric coefficients, as well as analyze fluctuations in simple quantum machines, with a particular stress on the validity or violations of thermodynamical laws. Finally, we are also going to search for new states and phases of matter as well as phenomena that can arise in many-body systems and out of equilibrium. One particularly appealing and hot topic is many-body localization.

**Work packages:**

There are three types of questions that shall be addressed.

1) *Spectral properties of dissipative systems*

Dissipative dynamics is in principle determined by dissipative Liouvillian generator whose properties are rather unexplored. We are going to study the following questions:

(a) scaling of relaxation time in systems without disorder (already realized).

(b) scaling of relaxation time in disordered systems and possible connection to transport.

(c) properties of eigenvectors, i.e., of relaxation modes.

2) *Quantum engineering*

While it is well known that dissipative dynamics is universal, much less in known about the power of limited resources. Given a set of resources, what is the optimal way to preform a given task? We are going to study the following questions:

(a) locality of interactions is frequently enforced. How can we prepare a given state using only local evolution? One instance of such a question has already been tackled and solved.

(b) analysis of small »quantum« machines that operate with only a few particles, and analysis of their efficiency.

3) *Emergence*

Manyparticle systems can display properties that are solely due to interactions and a large number of constituent parts. Interesting questions are the following.

(a) transport laws in clean systems, e.g., in the XXZ Heisenberg model.

(b) transport in systems with disorder, for instance, in so-called many-body localized systems. There are many open questions where dissipative formulation is likely to be very efficient. We already realized a preliminary study along those lines.

(c) the effects of coupling with the environment on interesting phases of matter. One example, namely that of a many-body localized systems has been already explored.