In recent years, the field of theoretical physics has experienced a revival of interest in the exploration of integrable and quantum chaotic properties of physical systems. This workshop aims to bring together researchers from condensed matter, mathematical and high-energy physics communities to facilitate a cross-disciplinary discussion on these rapidly evolving developments.

The workshop will be held between 3 – 9 June, 2018, in Bled, Slovenia.

Maram Akila: Getting a Foot into Many-Body Quantum Chaos

In the limit of small ħ, e.g. short wavelengths, semi-classical theory provides a connection between the spectrum of a quantum mechanical system and the periodic orbits of its classical counter part. While this is well established for systems with few degrees of freedom the extension to many-body systems is challenging as the system dimension turns into an additional large parameter. For a long chain of interacting kicked tops short time spectral information may be obtained using a duality relation between the temporal and spatial evolution. This allows a direct comparison to the classical periodic orbits of the system. It reveals that this class of system possesses non-isolated orbit manifolds whose contributions to the trace formula are strongly dominant. For large N their presence leads to remarkably strong spectral fluctuations, which are not recovered by conventional semi-classical theory.

Maram Akila, Universität Duisburg-Essen

Alvise Bastianello: Moving Impurities & Quantum Sonic Bangs

It is a well known fact that the spreading of information in lattice quantum systems is not instantaneous, but it rather exists a maximum velocity dictated by the Lieb-Robinson bound.
The existence of such a lightcone deeply affects equilibrium properties as well as out-of-equilibrium ones, which have been a subject of outstanding interest in the recent years.
In this talk, a novel out-of-equilibrium protocol critically affected by the presence of a maximum velocity is proposed and discussed. Specifically, a one dimensional lattice model is considered, where a localized impurity is suddenly created and then dragged at a constant velocity.
Focussing on a simple, but far from trivial, free model the response of the system at late times is analyzed, with emphasis on its transport properties. The finite maximum velocity is responsible for a rich phenomenology, for which exact results are provided. Taking into account the experience acquired so far, more general models are discussed and unpublished results presented, with exact predictions in completely generic (non integrable) one dimensional lattice systems.

Alvise Bastianello, SISSA Trieste

[1] A. Bastianello, A. De Luca, Phys. Rev. Lett. 120, 060602 (2018).

Enej Ilievski: Thermodynamics and hydrodynamics in classical soliton theories

In this talk, we shall consider the classical field theory of the Heisenberg ferromagnet, which is the semi-classical limit of the Heisenberg spin-1/2 chain. Our primary motivation to study classical solvable dynamical system is to understand the similarities and differences between integrable quantum systems and their classical counterparts. Unfortunately, there seem to exist no efficient computational framework to achieve this goal. To this end, by starting from first principles and employing the finite-gap algebro-geometric integration technique, we identify a certain scaling procedure of the corresponding degenerate large-genus Riemann surfaces which provides a statistical description for a thermodynamic gas of magnetic solitons. Using simple arguments of semi-classical quantization, we use the classical S-matrix to derive a universal integral dressing equation for the spectral distribution function of soliton excitations which accounts for interactions with a non-trivial many-body vacuum. By lifting the conventional theory of Whitham modulations equations to the thermodynamic setting, we obtain a simple formula for the effective propagation velocity which is, remarkably, in formal agreement with that proposed recently for quantum integrable models.

Enej Ilievski, Universiteit van Amsterdam

Jacopo De Nardis: Generalized Gibbs Ensembles and Generalized Hydrodynamics in quantum integrable systems

I’ll give a short review of the recent theoretical progress to explicitly construct non-thermal steady states in quantum systems such as interacting bosons and spin chains. Moreover, I’ll present the recently introduced hydrodynamic description of such non-thermal steady states that allows to study (ballistic) transport properties of many-body systems and to construct non-equilibrium steady states with persistent energy or spin currents and stronger quantum correlations.

Jacopo De Nardis, École normale supérieure de Paris

Pedro Ribeiro: Thermodynamically unbalanced steady-states – A route to novel ordered phases

In this talk I will argue that novel ordered phases can be expected for interacting quantum systems away from thermal equilibrium. I will start by reporting a set of mean-field results concerning the effects of large bias voltages applied across an half-filled Hubbard chain. As a function of the applied voltage and temperature a rich set of phases can be found that is induced by the interplay between electron-electron interactions and non-equilibrium conditions. Taking a step back, I try to explain why such phases are possible (at least at the mean field level). This will motivate the characterization of the current-carrying steady-state that arises in the middle of a non-interacting metallic wire connected to macroscopic leads. Finally, I will comment on some ongoing work regarding the fate of the Peierls transition in a similar non-equilibrium setup.

* Pedro Ribeiro, CeFEMA & Physics DepartmentISTUniversidade de Lisboa.



Lenart Zadnik: Integrable Trotterization

I will discuss a general procedure to construct an integrable real–time trotterization of interacting lattice models. As an illustrative example we will consider a spin-$1/2$ chain, with continuous time dynamics described by the isotropic ($XXX$) Heisenberg Hamiltonian. I will derive local conservation laws from an inhomogeneous transfer matrix and construct a boost operator. In the continuous time limit these local charges reduce to the known integrals of motion of the Heisenberg chain.

In a simple Kraus representation I will examine the nonequilibrium setting, where our integrable cellular automaton is driven by stochastic processes at the boundaries.
We will see, how an exact nonequilibrium steady state density matrix can be written in terms of a staggered matrix product ansatz.

This simple trotterization scheme, in particular in the open system framework, could prove to be a useful tool for experimental simulations of the lattice models in terms of trapped ion and atom optics setups.

Lenart Zadnik, Faculty of Mathematics and Physics, University of Ljubljana

Matjaž Payrits: Embeddings of Hilbert spaces in superspaces induced by smooth overcomplete bases and their applications to diagonalizing few-mode many-body Hamiltonians

The properties of a Hilbert space may sometimes be usefully illuminated by expressing its states with respect to an overcomplete basis parameterized by the points of a smooth manifold. A prime example of the technique is the Segal-Bargmann representation wherein states of a single-mode bosonic Fock space are expanded in terms of the overcomplete basis of coherent states. The Fock space is then found to be isomorphic to the space of holomorphic functions of a certain finite norm. Furthermore, the creation and annihilation operators, and any function of them, can be expressed as functions of the complex position and complex derivative operators.

In this talk I will present the general theory of embedding a Hilbert space in a suitable space of functions over a smooth manifold that parameterizes an overcomplete basis in the original space.  In many cases, operators of theoretical interest in the original space may be mapped onto differential operators on the smooth manifold such that the spectrum of the mapped operator contains the spectrum of the original one. This, in particular, allows the use of calculus and geometrical reasoning when diagonalizing Hamiltonians in the original space.

I will demonstrate the technique on a general d-mode many-body system that may be mapped onto a single-particle problem on the (d-1)-sphere. I will finally review some low-d applications of the formalism, which have found utility in the context of few-site tight-binding Hamiltonians and Bose-Einstein condensates of spinful atoms.

Matjaž Payrits, Imperial College London

Marko Robnik: Application of the WKB method in 1D linear and nonlinear time-dependent oscillators

The WKB method is an important analytic tool for solving numerous problems in
mathematical physics of 1D systems, for example the stationary (time-independent)
Schrödinger equation in one dimension, or the classical dynamics of one-dimensional
time-dependent (nonautonomous) Hamilton oscillators. I shall review the standard
WKB method including the exact explicit solutions to all orders, published by Rob-
nik and Romanovski (2000), and applied in a series of papers. Among other results
we have shown that the application of the method in cases of the Schrödinger equa-
tion with exactly solvable potentials leads to an infinite series to all orders, that the
series converges and the sum reproduces the known exact eigenenergies. We shall
look in particular at the case of the time-dependent one-dimensional linear Hamilto-
nian oscillator, and then I shall present the approach towards generalizing the WKB
method for the case of one-dimensional time-dependent nonlinear Hamiltonian oscil-
lators having quadratic kinetic energy and homogeneous power law potential, which
includes e.g. the quartic oscillator, and of course also the linear oscillator. I will
show that the nonlinear method, although only in the leading approximation, is very
useful and accurate. We also shall touch upon possible generalizations.

Marko Robnik, CAMTP, Univerza v Mariboru

Carlos Pineda: Generic features of temporal evolution in hierarchical systems

Many complex systems are hierarchical in nature; social groups, economic and
biological ecosystems, transportation infrastructures, languages, they all
develop hierarchies of their constitutive elements that emerge from networked
interactions within the system and with the external world. These hierarchies
change in time according to system-dependent mechanisms of interaction, such as
selection in evolutionary biology, or rules of performance in human sports, and
reflect the relevance or ability of the element in performing a function in the
system. However, it is still unclear whether the temporal evolution of
hierarchies solely depends on the driving forces and characteristics of each
system, or if there are generic features of hierarchy stability that allow us
to model and predict patterns of hierarchical behaviour without considering the
particularities of the system. We explore this question by analysing over 30
datasets of social, nature, economic, infrastructure, and sports systems in a
wide range of sizes (10^2 − 10^5) and time scales (from days to centuries) and
find that, despite their various origins, the elements in these systems show
remarkably similar stability depending on their position in the hierarchy. By
classifying systems from closed to increasingly open, we manage to reproduce
their hierarchy evolution in a minimal model with no system-dependent
mechanisms of interaction. This allows us to make predictions on unobserved
data, such as the likelihood of an unknown element climbing high in the
hierarchy, or the time scale over which an element can maintain its relevance
in the system. Our results may be crucial in further understanding why
hierarchies evolve similarly in seemingly unrelated areas, and give clues on
how to promote stability in the complex socio-technical systems of our day.

Carlos Pineda, Instituto de Fisica, Universidad Nacional Autonoma de Mexico