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

**UL Member:** Faculty of Mathematics and Physics**Code:** N1-0025**Project:** Open Many-body Non-Equilibrium Systems**Period:** 1.10.2014 - 30.9.2017 **Range per year:** 2,89 FTE, category D**Head:** Tomaž Prosen **Research activity:** Natural sciences and mathematics**Research Organisations:** link on SICRIS**Researchers:** link on SICRIS**Citations for bibliographic records:** link on SICRIS

**Project description:**

We shall study non-equilibrium many-body quantum systems, considering local interactions in one or two spatial dimensions in situations where the generator of time evolution in the bulk of the system is unitary whereas the incoherent processes are limited to the system's boundaries. We plan to develop a mathematical theory of non-equilibrium quantum phase transitions and non-equilibrium quantum phases of matter with applications in the theory of quantum transport, thermoelectricity and nanoscale devices that manipulate heat, information, charge and magnetization.

We argue that our steady-state setup represents a fundamental paradigm of mathematical statistical physics which has been pioneered by the PI, who has - since 2008 - proposed several original and influential approaches. In [New J. Phys. 10, 043026 (2008)] he developed a formalism for canonical quantization of open many-body systems and in [Phys. Rev. Lett. 106, 217206 (2011); ibid.107, 137201 (2011)] he gave the first explicit solution for an open (boundary-driven) strongly interacting many-body problem (XXZ spin 1/2 chain) which answered a long debated question on strict positivity of the spin Drude weight at high temperature.

The main focus of the research project will be centered on exploring the following three pathways: Firstly and most importantly, we will develop a general frame for exact solutions of non-equilibrium quantum many-body models, in particular the so-called non-equilibrium steady states, and develop systematic extensions of the algebraic Bethe ansatz to non-equilibrium many-body density operator. We envision fundamentally new concepts, such as {\em infinitely dimensional Yang-Baxter equations}, and exact many-body states of tensor-product form with infinite bond dimensions. Secondly, we shall investigate the effect of exact solutions on the physics of generic models which are small perturbations of integrable models or, in the language of mathematical physics, we plan to systematically explore the problem of stability of local and quasi-local conserved quantities of extended integrable models under generic integrability-breaking perturbations. Thirdly, we shall formulate and study the problem of quantum chaos in clean lattice systems, in particular to establish a link between random matrix theory of level statistics and kinematic and dynamical features of lattice models with sufficiently strong integrability breaking.

Analytical investigations shall be corroborated with state of the art numerical simulations of the paradigmatic models. The second and third pathway will lead the ideas to applications that are near to experiments with spin chains, cold atoms and other fields reaching potentially as far as heat engineering.