The assumption that quantum systems relax to a stationary (time-independent) state in the long-time limit underpins statistical physics and much of our intuitive understanding of scientific phenomena. For isolated systems this follows from the eigenstate thermalization hypothesis. When an environment is present the expectation is that all of phase space is explored, eventually leading to stationarity. However, real-world phenomena, from life to weather patterns are persistently non-stationary. We will discuss simple algebraic conditions that lead to a quantum many-body system never reaching a stationary state, not even a non-equilibrium one. This unusual state of matter characterized by persistent oscillations has been recently called a time crystal. We show that it’s existence can be, counter-intuitively, induced through the dissipation itself. We further present necessary and sufficient conditions for the occurrence of persistent oscillations in an open quantum system. Finally, we also discuss how our framework allows for open quantum many-body system displaying complex dynamical behaviour, usually found in macroscopic classical systems, such as synchronization.

References:
B Buca, J Tindall, D Jaksch. Nat. Comms. 10 (1), 1730 (2019)
M Medenjak, B Buca, D Jaksch. arXiv:1905.08266 (2019)
B Buca, D Jaksch. Phys. Rev. Lett. 123, 260401 (2019)
J Tindall, B Buca, J R Coulthard, D Jaksch. Phys. Rev. Lett. 123, 030603 (2019)
J Tindall, C Sanchez Munoz, B Buca, D Jaksch. New J. Phys. 22 013026 (2020)
C Booker, B Buca, D Jaksch. arXiv:2005.05062 (2020)

* Berislav Buča, Department of Physics, University of Oxford

The seminar will be online via Zoom (ID: 930 5528 5938, Password: 373842)

https://fmf-uni-lj-si.zoom.us/j/93055285938?pwd=WHI5RW5aUlVRU3JYd2RLVHpFbjBhUT09

I will discuss how to use Bethe Ansatz techniques for studying the properties of certain systems experiencing loss. First of all, I will describe the general approach to obtain the Liouvillian spectrum of a wide range of experimentally relevant models. This includes any integrable model with particle number conservation experiencing the single particle bulk loss throughout the system. Following the general discussion, I will address different aspects of the XXZ spin chain driven at the single boundary. In particular, I will consider the scaling of Liouvillian gap, the dynamical dissipative phase transition, and the physics of the boundary bound states. The existence of infinitely many boundary bound states translates into the formation of a stable domain wall in the easy-axis regime despite the presence of loss.

[1] B. Buca, C. Booker, M. Medenjak, D. Jaksch, arXiv:2004.05955

* Marko Medenjak, Institut de Physique Theorique Philippe Meyer, Ecole Normale Superieure, Paris

The seminar will be online via Zoom (ID: 925 5819 1511, Password: 669133)

https://zoom.us/j/92558191511?pwd=Q3hyL1N6S21SSHRpS0lURHkzMGVTdz09

We examine novel links between the theory of braces and set theoretical solutions of the Yang-Baxter equation, and fundamental concepts from the theory of quantum integrable systems. More precisely, we make connections with Hecke algebras, we identify quantum groups associated to set-theoretic solutions coming from braces and we also derive new classes of symmetries for the corresponding periodic transfer matrices.

* Anastasia Doikou, Heriot-Watt University, Edinburgh

The seminar will be online via Zoom (ID: 917 7027 2241, Password: 820642).

We define and study an integrable G-invariant dynamics of a field subject to a nonlinear constraint on a 1+1 dimensional discrete space-time lattice. The model allows for efficient numerical simulations, which suggest superdiffusion and Kardar-Parisi-Zhang physics in the entire family of models (arXiv:2003.05957). Further, I will present some recent results on extending the model onto other symmetric spaces and more general symmetry groups. Lastly I will discuss a recent surprising observation of conic sections in the correlation tensor of (non)-integrable G-invariant models of Landau-Lifshitz type.

* Žiga Krajnik, FMF, Ljubljana

The seminar will be online via Zoom (ID: 969 2016 1126, Password: 432541).

Rule 54 reversible cellular automaton (RCA54) is a 1-dim lattice model of solitons that move with fixed velocities and undergo nontrivial scattering. In the past years, numerous exact results have been found, ranging from the exact non-equilibrium steady state and exact large deviation treatment of the boundary driven setup to the matrix product form of time evolution of local observables. In the talk I will discuss recent progress. In particular I will present the matrix product form of multi-time correlation functions (arXiv:1912.09742) and the formulation of space-like evolution of such time-states (arXiv:2004.01671).

* Katja Klobas, FMF, Ljubljana

The seminar will be online at https://zoom.us/j/483251248

Meeting ID: 483 251 248

Thursday 5.3.2020, 11:15h in Kuščerjev seminar, Jadranska 19.

* Sašo Grozdanov, FMF, Ljubljana

Tuesday 3.3.2020, 13:15h in Kuščerjev seminar, Jadranska 19.

The Luttinger model with finite-range interactions is an exactly solvable model in 1+1 dimensions somewhere between conformal and Bethe-ansatz integrable ones. I will show how exact analytical results can be computed for the time evolution of this non-local Luttinger model following an inhomogeneous quench from initial states defined by smooth temperature and chemical-potential profiles. These results demonstrate that the finite-range interactions give rise to dispersive effects that are not present in the conformal case of point-like interactions. Combing the same methods with the recent proposal of generalized hydrodynamics, one finds that this model allows for fully explicit yet non-trivial solutions of the resulting Euler-scale hydrodynamic equations. These results are shown to emerge from the exact analytical ones at the relevant time and length scales. As such, the non-local Luttinger model provides a simple tractable example to analytically study the emergence of hydrodynamics in a quantum many-body system.

* Per Moosavi, ETH, Zurich

Thursday 6.2.2020, 14:15h in Kuščerjev seminar, Jadranska 19.

This talk covers the mathematical foundation of the Mittag-Leffler functions, fractional integrals and derivatives, and many recent novel definitions of generalized operators which appear to have many applications nowadays. We pay special attention to the analysis of the complete monotonicity of the Mittag-Leffler function which is a very important prerequisite for its application in modeling different anomalous dynamics processes. We will give a number of definitions and useful properties of different fractional operators, starting with those named as Riemann-Liouville fractional derivative and integral, Caputo fractional derivative, composite (or so-called Hilfer) derivative, and generalized integral operators which contain generalized Mittag-Leffler functions in the kernel. We present a generalization of Hilfer derivatives in which Riemann–Liouville integrals are replaced by more general Prabhakar integrals, called Hilfer-Prabhakar derivatives. Many useful properties and relations in fractional calculus that are used in modeling anomalous diffusion and non-exponential relaxation will be presented. The Cauchy-type problems of fractional differential equations and their solutions, existence and uniqueness theorems, and different methods for solving fractional differential equations (integral transform method, operational method), will be considered in this talk. Fractional PDEs are a useful tool for the modeling of many anomalous phenomena in nature. We’ll consider fractional diffusion equations and their connections with the continuous time random walk (ctrw) theory. Furthermore, we show some applications of Hilfer–Prabhakar derivatives in classical equations of mathematical physics such as the heat and the difference–differential equations governing the dynamics of generalized renewal stochastic processes.

* Zivorad Tomovski, Ss. Cyril and Methodius University, Skopje

Friday 11.10.2019, 12:15h in Kuščerjev seminar, Jadranska 19.

We will review the advances and challenges in the field of quantum combinatorial optimization and closely related problem of low-energy eigenstates and coherent dynamics in transverse field quantum spin glass models. We will discuss the role of collective spin tunneling that gives rise to bands of delocalized non-ergodic quantum states providing the coherent pathway for the population transfer (PT) algorithm: the quantum evolution under a constant transverse field that starts at a low-energy spin configuration and ends up in a superposition of spin configurations inside a narrow energy window. We study the transverse field induced quantum dynamics of the following spin model: zero energy of all spin configurations except for a small fraction of spin configuration that form a narrow band at large negative energy. We use the cavity method for heavy-tailed random matrices to obtain the statistical properties of the low-energy eigenstates in an explicit analytical form. In a broad interval of transverse fields, they are non-ergodic, albeit extended giving rise to a qualitatively new type of quantum dynamics. We argue that our approach can be applied to study PT protocol in other optimization problems with the potential quantum advantage over classical algorithms.

* Vadim N. Smelyanskiy, Google Quantum AI

A series of 10 lectures by Prof. Dmitry Shepelsky of the B. I. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, Mathematical division

Schedule (with slides):

Wednesday, 9.10.2019 (slides): 15:15-17:00
Thursday, 10.10.2019 (slides): 14:15-16:00
Friday, 11.10.2019 (slides): 14:15-16:00

Tuesday, 15.10.2019 (slides): 14:15-16:00
Wednesday, 16.10.2019 (slides): 15:15-17:00

Abstract:

Riemann-Hilbert (RH) problems are boundary-value problems for sectionally analytic functions in the complex plane. It is a remarkable fact that a vast array of problems in mathematics, mathematical physics, and applied mathematics can be posed as Riemann-Hilbert problems. These include radiation, elasticity, hydrodynamic, diffraction problems, orthogonal polynomials and random matrix theory, nonlinear ordinary and partial differential equations. In applications, the data for a RH problem depend on external parameters, which are physical variables (space, time, matrix size, etc), and, in turn, the solution depends on these parameters as well. It is this dependence that we are interested in, when speaking about the RH problem as a method for studying problems from one or another domain.

The representation of a solution to a nonlinear partial differential equation (PDE) in terms of a solution of the associated RH problem can be viewed as a nonlinear analogue of the contour integral representation for a linear PDE. It provides means to efficiently study not only the existence and uniqueness problems for a class of nonlinear PDE, but (i) to derive detailed asymptotics of solutions of initial value problems and initial boundary value problems for such equations and (ii) to accurately evaluate the solutions inside as well as outside asymptotic regimes. Recent literature on applications of the RH problem includes the monographs [1-3].

The main aim of the proposed course is to introduce the Inverse Scattering Transform method, in the form of the RH problem, for studying integrable nonlinear differential equations and to illustrate the fruitfulness of the method by studying the long-time asymptotics of solutions of such equations. As a prototype model, we will use the nonlinear Schrödinger equation, which is a basic models of nonlinear wave propagation (for instance, in the fiber optics).

References:

[1] P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, AMS, Providence, Rhode Island, 2000.
[2] A. S. Fokas, A. R. Its, A. A. Kapaev and V. Yu. Novokshenov, Painleve Transcendens: The Riemann-Hilbert Approach, Mathematical surveys and monographs 128, AMS, 2006.
[3] T. Trogdon and S. Olver, Riemann–Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special  Functions, SIAM, Philadelphia, 2016.