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 Taking into account the experience acquired so far, more general models are discussed and unpublished results presented, with exact predictions in completely generic (non integrable) one dimensional lattice systems.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).

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*

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*

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 Department, IST, Universidade de Lisboa.*

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

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*

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*

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*

Out of Time Ordered Correlators (OTOCs) have been suggested as a probe of scrambling (generically referred as the delocalization of quantum information) and as a measure of chaos in quantum many-body systems. We explore scrambling in connection to entanglement dynamics in generic long-range systems, and in particular in the infinite-range Ising model. We study both bipartite and multipartite entanglement dynamics and we compare the results with the OTOCs of collective spin operators.

We argue that scrambling and entanglement growth are two distinct phenomena, characterized by two different time scales. While entanglements saturate at a time $t_{Ehr}\sim \sqrt N$ at which the semi-classical approximation breaks, the OTOCs keep growing in time up to $N$. Furthermore, by expanding in spin waves on top of the classical solution, we are able to device an approximated semi-analytic method that predicts the behavior of the OTOC up to $t_{Ehr}$. This method seems to be generic for long-range interacting hamiltonians and can be extended adding interaction to the hamiltonian.

*** ***Silvia Pappalardi, SISSA, Trieste*

The ability to prepare a physical system in a desired quantum state is central to many areas of physics such as nuclear magnetic resonance, cold atoms, and quantum computing. Preparing states quickly and with high fidelity remains a formidable challenge. In this work we implement cutting-edge Reinforcement Learning (RL) techniques to find short, high-fidelity driving protocols from an initial to a target state in non-integrable single-particle and many-body quantum spin systems. The quantum state preparation problem, viewed as an optimization problem, is shown to exhibit examples of prototypical equilibrium phase transitions in classical macroscopic systems: both first and second order phase transitions, a glass phase, and symmetry breaking, as a function of the protocol duration. These control phase transitions, present even in low-dimensional clean quantum systems, are classical yet of non-equilibrium nature, and carry far-reaching consequences for manipulating quantum states.

*** ***Marin Bukov,*** ***Department of Physics, Boston University*