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*

A transformation that accounts for the universality found in Ref. [1] concerning the finite energy behavior of dynamical correlation functions of both integrable and non-integrable 1D correlated systems is used to generate from the pseudo-fermion dynamical theory of the integrable 1D Hubbard model [2] a corresponding renormalized theory with additional electron finite-range interactions [3,4]. The obtained renormalized theory is used to describe the experimental spectral lines in the angle resolved photoemission spectroscopy of the 1D quantum line defects in the 2D van der Waals layered semiconductor MoSe2 [3] and of the quasi-1D compound TTF-TCNQ [4]. The theoretical predictions refer to finite-energy ω windows in the vicinity of the cusps of the observed spectral lines. The dispersions and (k, ω)-plane weight distributions of such two systems are found to exactly follow those predicted by the non-integrable finite-range renormalized model with the exponent α for the density of states suppression, |ω| α, being given by α ≈ 0.73 − 0.78 for the MoSe2 line defects [3] and α ≈ 0.53 for TTF-TCNQ [4]. The latter value is thirteen times larger than that predicted by the simple 1D Hubbard model, α ≈ 0.04, for which α ∈ [0, 1/8] [5].

[1] A. Imambekov and L. I. Glazman Science 323 228 (2009).

[2] J. M. P. Carmelo and T. Cadez Nuclear Physics B 914, 461 (2017).

[3] Y. Ma, H. C. Diaz, J. Avila, C. Chen, V. Kalappattil, R. Das, M.-H. Phan, T. Cadez, J. M. P. Carmelo, M. C. Asensio, and M. Batzill Nature Communications 8, 14231 (2017).

[4] J. M. P. Carmelo, T. Cadez, M. Sing, and R. Claessen (Work in progress).

[5] M. Sing, U. Schwingenschlogl, R. Claessen, P. Blaha, J. M. P. Carmelo, L. M. Martelo, P. D. Sacramento, M. Dressel, and C. S. Jacobsen, Physical Review B 68 125111 (2003) .

***** *J. M. P. Carmelo, Universidade do Minho, Portugal*

I will first introduce formation probability as a quantity which can determine the universality class of a quantum critical system. In other words, by calculating this quantity one can find the central charge and critical exponents of a quantum system and determine the universality class uniquely. I will show that calculating this quantity boils down to finding Casimir energy of two needles. Then using boundary conformal field theory (BCFT) techniques we find exact results for the formation probabilities. Numerical results for transverse field Ising model will be presented to support the analytical results. Then we will briefly talk about Shannon mutual information as another quantity which can play similar role. We will present a conjecture which connects Shannon mutual information to the central charge of the underlying conformal field theory. We will support the conjecture with many numerical calculations. Finally, we will introduce post-measurement entanglement entropy as a tripartite measure of entanglement. We will show that this quantity is related to the Casimir energy of needles on Riemann surfaces and can be calculated exactly for conformal field theories. To do that we use a slightly different method than twist operator technique. Many analytical results, such as, Renyi entropy, entanglement Hamiltonian, distribution of the eigenvalues of entanglement Hamiltonian, the effect of the boundary and Affleck-Ludwig boundary entropy can be discussed naturally in our framework. Few numerical results regarding free bosons and transverse field Ising chain will be presented as support for analytical results.

***** *Mohammad Ali Rajabpour, Instituto de Fisica Universidade Federal Fluminense Niteroi, Rio de Janeiro*

We developed a novel perturbative expansion for the effective Hamiltonian governing the dynamics of periodically kicked systems in the small parameter of the kick strength. The expansion is based on the replica trick, and is formally equivalent to the infinite resummation of the Baker-Campbell-Hausdorff series in the non-kicked Hamiltonian, while taking finite order terms in the kicks. As an application of the replica expansion, we study the heating properties of a periodically kicked spin chain. We demonstrate that even away from the high frequency driving limit, the heating rate is at least stretched exponentially suppressed in the kick strength. This leads to a long pre-thermal regime, where the dynamics is governed by the effective Hamiltonian obtained from the expansion.

***** *Szabolcs Vajna, Faculty of Mathematics and Physics, University of Ljubljana*

*
* *
*