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.

In typical Hamiltonian systems regions of regular and chaotic motion coexist within a mixed phase space. While they are strictly separated in classical mechanics, dynamical tunneling allows quantum states to penetetrate classical inaccessible regions in phase space. We present a semiclassical picture for this regular-to-chaotic tunneling process, which also captures the resonance-assisted enhancement due to a nonlinear resonance chain in the classical phase space. Within an integrable approximation with one nonlinear resonance chain we identify complex paths for direct and resonance-assisted tunneling. Using WKB techniques we obtain tunneling rates from this paths. In particular the resonance-assisted contribution can be evaluated analytically and leads to a prediction based on just a few properties of the classical phase space. We illustrate our approach with the paradigmatic model of the standard map where excellent agreement with numerically determined tunneling rates is observed.

* Felix Fritzsch, TU Dresden

Márton Mestyán: Molecular dynamics simulation of entanglement spreading in generalized hydrodynamics (arXiv:1905.03206)

The so-called flea gas is an elementary yet very powerful method that allows the simulation of the out-of-equilibrium dynamics after quantum quenches in integrable systems. We show that, after supplementing it with minimal information about the initial state correlations, the flea gas provides a versatile tool to simulate the dynamics of entanglement-related quantities. The method can be applied to any quantum integrable system and to a large class of initial states. Moreover, the efficiency of the method does not depend on the choice of the subsystem configuration. We implement the flea gas dynamics for the gapped anisotropic Heisenberg XXZ chain, considering quenches from globally homogeneous and piecewise homogeneous initial states. We compute the time evolution of the entanglement entropy and the mutual information in these quenches, providing strong confirmation of recent analytical results obtained using the Generalized Hydrodynamics approach. The method also allows us to obtain the full-time dynamics of the mutual information after quenches from inhomogeneous settings, for which no analytical results are available.

Lev Vidmar: Quantum chaos challenges many-body localization (arXiv:1905.06345)

Characterizing states of matter through the lens of their ergodic properties is a fascinating new direction of research. In the quantum realm, the many-body localization (MBL) was proposed to be the paradigmatic nonergodic phenomenon, which extends the concept of Anderson localization to interacting systems. At the same time, random matrix theory has established a powerful framework for characterizing the onset of quantum chaos and ergodicity (or the absence thereof) in quantum many-body systems. Here we study a paradigmatic class of models that are expected to exhibit MBL, i.e., disordered spin chains with Heisenberg-like interactions. Surprisingly, we observe that exact calculations show no evidence of approaching MBL while increasing disordered strength in the ergodic regime. Moreover, a scaling analysis suggests that quantum chaotic properties survive for any disorder strength in the thermodynamic limit. Our results are based on calculations of the spectral form factor, which provides a powerful measure for the emergence of many-body quantum chaos.

Tiago Mendes-Santos: Entanglement guided search for parent Hamiltonians

We introduce a method for the search of parent Hamiltonians of input wave-functions based on the structure of their reduced density matrix. The two key elements of our recipe are an ansatz on the relation between reduced density matrix and parent Hamiltonian that is exact at the field theory level, and a minimization procedure on the space of relative entropies, which is particularly convenient due to its convexity. As examples, we show how our method correctly reconstructs the parent Hamiltonian correspondent to several non-trivial ground state wave functions, including conformal and symmetry-protected-topological phases, and quantum critical points of two-dimensional antiferromagnets described by strongly coupled field theories. Our results show the entanglement structure of ground state wave-functions considerably simplifies the search for parent Hamiltonians.

Spyros Sotiriadis: Quantum dynamics in the sine-Gordon model

The study of dynamics in quantum many-body systems is one of the main challenges of modern theoretical physics. While a large amount of intuition has been drawn from weakly interacting systems, linear response theory or semiclassical approximations, the dynamics of strongly interacting systems far from equilibrium remains largely unexplored. We study the dynamics of an interacting Quantum Field Theory, the sine-Gordon model, after an abrupt change of the parameters, a protocol known as “Quantum Quench”. Our focus is on the time evolution of correlation functions, which we study using a combination of numerical and analytical techniques.

Jacopo Sisti: Entanglement entropy in higher dimensional CFTs and holography

Entanglement entropy is a quantity of great interest in diverse fields of physics like condensed matter, statistical and high energy physics. In this talk, I will review some aspects of entanglement entropy in higher dimensional quantum field theories mainly focusing on CFTs that admit a holographic dual. In such theories, the Ryu–Takayanagi formula identifies the entanglement entropy of a spatial region with the area of the minimal hypersurface anchored to the entangling surface and that extends along the holographic direction of the space-time. The holographic duality is a useful tool also to study CFTs with boundary. In particular, I will show some analytical and numerical results on minimal surfaces in space-times dual to (2+1)-dimensional BCFTs.

Giuliano Giudici: Measuring von Neumann entanglement entropies without wave functions

It is nowadays a well known fact that the von Neumann entropy (VNE) of the ground state is a powerful tool to characterize many-body quantum systems, since it provides distinctive information such as length of correlations and universal data of critical systems. Despite its central role as a diagnostic tool for low-energy properties of many-body Hamiltonians, its measurement has so far been elusive both from an experimental and — beyond one dimension — numerical point of view. Here we propose a method to compute the ground state VNE without accessing the many-body wave function. The method is based on the knowledge, from quantum field theory, of the entanglement Hamiltonian of the ground state. We benchmark our technique on critical quantum spin chains, and apply it to several two-dimensional quantum magnets, where we are able to unambiguously determine the onset of area law, together with logarithmic corrections independent of the geometry of the bipartition. We finally focus on one-dimensional critical systems whose large distance behaviour is conformally invariant. We investigate to which extent it is possible to extract the central charge by computing the entanglement capacity, which is simply related to the expectation value of the energy density and thus easily accessible in experiments.

Isolated systems consisting of many interacting particles are generally assumed to relax to a stationary equilibrium state whose macroscopic properties are described by the laws of thermodynamics and statistical physics. Time crystals, as first proposed by Wilczek, could defy some of these fundamental laws and for instance display persistent non-decaying oscillations. They can be engineered by external driving or contact with an environment, but are believed to be impossible to realize in isolated many-body systems. Here, we will demonstrate analytically and numerically that the paradigmatic model of quantum magnetism, the Heisenberg XXZ spin chain, does not relax to stationarity and hence constitutes a genuine time crystal that does not rely on external driving or coupling to an environment. We will trace this phenomenon to the existence of periodic extensive quantities and find their frequency to be a no-where continuous (fractal) function of the anisotropy parameter of the chain.

* Marko Medenjak, ENS Paris

Using an open-quantum system description, we revisit the Josephson effect in hybrid junctions made of the topological superconductor (TS) and normal metal (N) wires. We consider an X-Y-Z configuration for the junctions where X, Y, Z = TS, N. We assume the wires X and Z being semi-infinite and in thermal equilibrium. We connect the wires X and Z through the short Y wire at some time, and numerically study time-evolution of the full device. For TS-N-TS device, we find a persistent, oscillating electrical current at both junctions even when there is no phase or thermal or voltage bias. The amplitude and period of the oscillating current depend on the initial conditions of the middle N wire indicating the absence of thermalization. This zero-bias current vanishes at a long time for any of X and Z being an N wire or a TS wire near a topological phase transition. Employing properties of different bound states within the superconducting gap, we develop a clear understanding of the oscillating currents.

* Dibyendu Roy, Raman Research Institute, India

In my talk, I will discuss the recently-discovered phenomenon of pole-skipping in thermal correlators of energy and momentum operators. In the presence of low-energy hydrodynamic modes, pole-skipping provides a precise relation between hydrodynamics and the underlying microscopic quantum chaos, as diagnosed by an out-of-time-ordered correlator (OTOC). In the absence of hydrodynamics, as for example in a 1+1 dimensional quantum critical theory (a conformal field theory), pole-skipping can still be used to compute the rate of the exponential growth of the OTOC.

* Sašo Grozdanov, MIT

The temporal evolution of certain aggregates of particles — of neutrons in a multiplying medium, of electron-photon cascades in cosmic rays, say — can be successfully modeled via so-called branching processes. A brief overview of the most popular classes of branching processes is given. In particular the continuous-time Bienaymé-Galton-Watson processes and continuous-state branching processes are described as time-changed Lévy processes that are “skip-free downwards”.

We study the transport properties of the Yang–Gaudin model – a one-dimensional, integrable, spinful Fermi gas – after a junction of two semi-infinite subsystems held at different temperatures. The ensuing dynamics is studied by analyzing the space-time profiles of local observables emerging at large distances x and times t, as a function of ζ=x/t. At equilibrium, the system displays two distinct species of quasiparticles, naturally associated with different physical degrees of freedom. By employing the generalized hydrodynamic approach, we show that when the temperatures are finite no notion of separation can be attributed to the quasiparticles. In this case, the profiles can not be qualitatively distinguished by those associated to quasiparticles of a single species that can form bound states. On the contrary, signatures of separation emerge in the low-temperature regime, where two distinct characteristic velocities appear. In this regime, we analytically show that the profiles display a piecewise constant form and can be understood in terms of two decoupled Luttinger liquids.

* Márton Mestyán, SISSA, Trieste

[1] M. M., B. Bertini, L. Piroli, P. Calabrese, Phys. Rev. B 99, 014305 (2019)