In recent years dual-unitary circuits have emerged as minimal models for chaotic quantum many-body systems in which the dynamics of correlations and entanglement remains tractable. The building blocks of these circuits are gates that are unitary in both time and space. After an introduction to these circuits I will extend the notion of dual-unitarity to biunitarity, which allows for a richer variety of building blocks and circuit dynamics, as well as unifying different notions of ‘dual-unitarity’. The resulting interactions are governed by quantum combinatorial data, which I will precisely characterize using a graphical 2-categorical framework based on the ‘shaded calculus’. These generalized circuits remain exactly solvable and we show that they retain the attractive features of the original dual-unitary models, with exact results for correlations functions, maximal entanglement growth and exact thermalization.

# Viktor Eisler: Entanglement spreading across a defect

# Jaš Bensa: Phantom eigenvalues

In this talk, we investigate the behavior of purity and out-of-time-ordered correlations in random quantum circuits. We show that the time evolution of both quantities can be described by a Markov chain, and their relaxation towards their asymptotic values is not governed by the second largest eigenvalue of the transfer matrix, as one could expect. The exponential relaxation is instead given by an “eigenvalue”, which is not in the spectrum of the transfer matrix at all — a phantom eigenvalue. We shall explore this phenomenon and find that it is rooted in the non-Hermiticity of the transfer matrix and in the locality of the dynamics.

*Jaš Bensa: University of Ljubljana

# Mile Vrbica: Pole-skipping and hidden structure of perturbed four-dimensional black holes

Pole-skipping is a generic feature of black hole perturbation theory that amounts to the inability of imposing the ingoing boundary condition at the event horizon at certain points in the Fourier space. As a consequence, various quantities, such as time evolution Green’s functions, take the indeterminate value of “0/0”, which can elucidate some of their structure from horizon analysis alone. A complete classification of all such points in the four dimensional maximally symmetric case will be presented with the aid of the hidden structure of perturbation equations that takes the form of Darboux symmetries. This structure relates the solutions of two independent Schrodinger-type master equations of the perturbation theory. Implications for physical black holes will be discussed, as well as the relation to dual holographic QFTs.

* Mile Vrbica: University of Edinburgh – Scotland.

# Gergely Zarand: Matrix product state simulations for interacting systems with non-Abelian symmetries

We apply the non-Abelian time evolving block decimation (TEBD) approach to study out of equilibrium properties of interacting many-body systems. We first show how we can use this approach to capture dynamical composite particle formation in SU(3) Hubbard models, where a large class of initial states is shown to develop into a negative temperature gas of strongly interacting ‘hadrons’. Then we extend non-Abelian TEBD to open systems with Lindbladian time evolution. As an illustration, we study the one-dimensional SU(2) Hubbard model on a semi-infinite lattice with localized particle loss at one end. We observe a ballistic front propagation with strongly renormalized front velocity, and operator entanglement is found to propagate faster than the depletion profile, preceding the latter.

* Gergely Zarand: BUTE, Budapest

# Gunter Schuetz: Dynamical universality classes: Recent results and open questions

Universality asserts that, especially near phase transitions, the macroscopic properties of a physical system do not depend on its details such as the precise form of microscopic interactions. We show that the two best-known examples of dynamical universality classes, the diffusive and Kardar-Parisi-Zhang-classes, are only part of an infinite discrete family. The members of this family have dynamical exponents which surprisingly can be expressed by the Kepler ratio of consecutive Fibonacci numbers. This strongly indicates the existence of a simpler but still unknown underlying mechanism that determines the different classes.

* Gunter Schuetz: Institute of Complex Systems (ICS), Forschungszentrum Jülich · Germany

# Felix Fritzsch: Universal Spectral Correlations in Bipartite Chaotic Quantum Systems

The emergence of random matrix spectral correlations in interacting quantum systems is a defining feature of quantum chaos. We study such correlations in terms of the spectral form factor in coupled bipartite chaotic quantum systems and obtain all moments of the spectral form factor exactly in the semiclassical limit of large Hilbert space dimension. Extrapolating those results to finite Hilbert space dimension we find a universal dependence of the spectral form factor on a single scaling parameter for times larger than the subsystem’s Heisenberg time. We complement our analysis by a perturbative approach covering the small coupling regime. Our results are derived in a random matrix model adapted to the bipartite nature of our setting in which we find excellent agreement between analytical results and extensive numerical studies. Moreover, we demonstrate that our results apply equally well to actual bipartite chaotic quantum systems by accurately describing the spectral form factor and its universal dependence on the scaling parameter in quantized coupled kicked rotors. Ultimately, we discuss an extension of our results to the many-body setting.

* Felix Fritzsch, FMF, University of Ljubljana

# Bruno Bertini: Rényi Entropies and Charge Moments from Space-Time Duality

Rényi entropies are conceptually valuable and experimentally relevant generalisations of the celebrated von Neumann entanglement entropy. After a quantum quench in a clean quantum many-body system they generically display a universal linear growth in time followed by saturation. While a finite subsystem is essentially at local equilibrium when the entanglement saturates, it is genuinely out-of-equilibrium in the growth phase. In particular, the slope of the growth carries vital information on the nature of the system’s dynamics, and its characterisation is a key objective of current research. In the talk I will show that the slope of Rényi entropies can be determined by means of a spacetime duality transformation. I will argue that the slope coincides with the stationary density of entropy of the model obtained by exchanging the roles of space and time. Therefore, very surprisingly, the slope of the entanglement can be expressed as an equilibrium quantity. I will use this observation to find an explicit formula for the slope of Rényi entropies in all integrable models treatable by thermodynamic Bethe ansatz and evolving from integrable initial states. I will then show that the duality approach can be generalised to the computation of full-counting statistics and symmetry resolved entanglement.

* Bruno Bertini, University of Nottingham, United Kingdom

# Yusuf Kasim: Dual unitary circuits in random geometries

Recently introduced dual unitary brickwork circuits have been recognised as paradigmatic exactly solvable quantum chaotic many-body systems with tunable degree of ergodicity and mixing. In this talk we show that regularity of the circuit lattice is not crucial for exact solvability. We consider a circuit where random 2-qubit dual unitary gates sit at intersections of random arrangements of straight lines in two dimensions (mikado) and analytically compute the variance of the spatio-temporal correlation function of local operators. Note that the average correlator vanishes due to local Haar randomness of the gates. The result can be physically motivated for two random mikado settings. The first corresponds to the thermal state of free particles carrying internal qubit degrees of freedom which experience interaction at kinematic crossings, while the second represents rotationally symmetric (random euclidean) space-time.

arXiv:2206.09665

* Yusuf Kasim, Faculty for Mathematics and Physics, University of Ljubljana

# Timotej Lemut: Reconstruction of spectra and an algorithm based on the theorems of Darboux and Puiseux

Assuming only a known dispersion relation of a single mode in the spectrum of a two-point function in some quantum field theory, we investigate when and how the reconstruction of the complete spectrum of physical excitations is possible. In particular, we develop a constructive algorithm based on the theorems of Darboux and Puiseux that allows for such a reconstruction of all modes connected by level-crossings. For concreteness, we focus on theories in which the known mode is a gapless excitation described by the hydrodynamic gradient expansion, known at least to some (preferably high) order. We first apply the algorithm to a simple algebraic example and then to the transverse momentum excitations in the holographic theory that describes a stack of M2 branes and includes momentum diffusion as its gapless excitation.

* Timotej Lemut, Faculty for Mathematics and Physics, University of Ljubljana