# 25th Rencontres Itzykson - Many Body Chaos, Scrambling and Thermalization in Interacting Quantum Systems

Europe/Paris
Amphitheatre Claude Bloch (Institut de Physique Théorique)

### Amphitheatre Claude Bloch

#### Institut de Physique Théorique

Orme des Merisiers Batiment 774 F-91191 Gif-sur-Yvette France
Description

The Itzykson Meeting is held every year in Saclay to honour the memory of Claude Itzykson. The 25th edition of this meeting will take place on June 2-4 2021

Thermalization and out of equilibrium dynamics of many body strongly interacting quantum systems have been the subject of a recent burst of research activities. In particular the dynamics of operator and information spreading (scrambling) as well as its chaotic behavior have been recently investigated starting from models originated in the context of strongly correlated many body systems and have found application to black holes physics. This has been triggered by the appearance of a new class of exactly soluble large N quantum field theories, such as the celebrated Sachdev-Ye-Kitaev model. The aim of this workshop is to review the recent advances in the field and to stimulate further research in this direction which lies at the interface between statistical and condensed matter physics and high-energy/mathematical physics.

Due to the current situation, the conference will take place online.

The location is sent by email each day. If you registered and did not see any message, please check the spam folder.

Organizing commitee: Monica Guica, Marco Schirò, Pierfrancesco Urbani, Laure Sauboy (Secretary)

The workshops is supported by

Contact the organizers
• Wednesday, June 2
• 1
Opening
• 2
Quantum gravity meets statistical physics

Recent work on quantum gravity has revealed deep connections with subjects like quantum information, statistical physics and quantum chaos. In particular, low-energy effective field theories that include gravity turn out to have more access to high-energy degrees of freedom than their non-gravitational Wilsonian counterparts. While precise microscopic high-energy information is inaccessible, certain statistical high-energy information does manifest itself in an interesting way at low energies. I will describe some recent work trying to make this connection more precise, and explain how it connects to issues like wormholes, averaging over theories and the black hole information paradox.

Speaker: J. de Boer
• 3
Many-body delocalisation as symmetry breaking

I will give an overview of recent work on minimal models for quantum chaos and many-body localisation. The models are Floquet quantum circuits for lattice spin systems, in which time evolution is generated by unitary gates that couple neighbouring sites. In particular, I will discuss the circumstances in which a version of the so-called diagonal approximation (originally developed for the semiclassical limit in low-dimensional chaotic systems) can be applied to these systems. Within this framework I will show that the many-body delocalisation transition can be seen as a form of symmetry breaking transition, having many of the features generally associated with conventional phase transitions in classical statistical mechanical models. Joint work with Sam Garratt: arXiv:2008.01697 and arXiv:2012.11580

Speaker: J. Chalker
• 3:20 PM
Break
• 4
Exact results on dynamics of dual unitary circuits and their perturbations

I will review the recent results on the proof of random matrix spectral form factor and explicit computation of correlation functions of local observables in the so-called dual-unitary brickwork circuits (including integrable, non-ergodic, ergodic and chaotic cases). Further I will show how these results can be extended to another quantum-circuit platform, specifically to unitary interactions round-a-face (IRF). I will argue that correlation functions of these models are generally perturbatively stable with respect to breaking dual-unitarity, and describe a simple rigorous result within this framework.

Speaker: T. Prosen
• 5
Delocalization of chaotic eigenmodes

In this talk I will review different mathematical results on the
delocalization of eigenmodes of the Laplacian in closed billiards or compact Riemannian manifolds, assuming the geometry generates a chaotic ray (geodesic) dynamics.
The focus will be on the high frequency regime, where semiclassical / microlocal methods can be applied. In particular I will recall the Quantum Ergodicity Theorem, which concerns "almost all" the eigenmodes, and also give more recent "absolute" delocalization results, especially valid in the case of strongly chaotic (Anosov) flows.

Speaker: S. Nonnenmacher
• 5:20 PM
Break
• 6
When is the onset of quantum chaos?

In a chaotic quantum system matrix elements of local operators exhibit statistical properties captured by the Eigenstate Thermalization Hypothesis. It describes the equilibrium but not the approach to it. To describe equilibration dynamics correlations between matrix elements should be taken into account. At late times the correlations can be neglected: matrix elements with small energy differences behave as independent random variables, giving rise to Random Matrix Theory description. This marks the onset of quantum chaos. We show that corresponding timescale is parametrically longer than thermalization time, in a sharp distinction with the timescale marking the onset of RMT behavior of the energy spectrum.

Speaker: A. Dymarsky
• 7
Non-unitary dynamics via spacetime duality

The addition of non-unitary ingredients to many-body quantum dynamics has led to a series of exciting developments in recent years, including new out-of-equilibrium entanglement phases and phase transitions enabled by quantum measurements. I will present recent work [1] in which we show that a duality transformation between space and time on one hand, and unitarity and non-unitarity on the other, can be used to realize non-unitary evolutions whose steady states exhibit a rich variety of behavior in the scaling of their entanglement with subsystem size — from logarithmic to extensive to fractal. These fractally entangled states add a qualitatively new entry to the families of many-body quantum states that have been studied as energy eigenstates or dynamical steady states, whose entropy almost always displays either area-law, volume-law or logarithmic scaling. The range of steady-state entanglement scalings for the non-unitary evolution are closely related to the question of entanglement growth in time under different kinds of unitary dynamics, from localized to chaotic. This connection is sharpened by an exact mapping to unitary evolution with edge decoherence, in which information is irreversibly “radiated away” from one edge of the system. Finally, I will discuss how these ideas could be experimentally realized with present-day or near-term quantum technologies, and how spacetime duality allows us to mitigate (or eliminate altogether) the overhead from "postselection" of random measurement outcomes [2].

[1] Matteo Ippoliti, Tibor Rakovszky, Vedika Khemani, arxiv:2103.06873
[2] Matteo Ippoliti, Vedika Khemani, PRL 126, 060501 (2021)

Speaker: V. Khemani
• Thursday, June 3
• 8
Ergodic phase in many body quantum chaos

Chaotic many body quantum systems can be in a phase of (many body) localized or extended eigenstates. In recent years, the seemingly less enigmatic delocalized phase has become a subject of controversy. It has been suggested that quantum many body eigenstates may be Non Ergodic yet Extended (NEE) in Hilbert space, with exotic multifractal distributions. In this talk I discuss how a blend of concepts developed in different fields --- including matrix integral techniques pioneered by the French school of field theory, lessons learned from the SYK model, and concepts of quantum information applied to random many body states --- may shed light on the situation.
Our bottom line will be that (i) the delocalized quantum states of chaotic systems (subject to long range interactions) are ergodically distributed over (ii) an energy shell nontrivially interlaced into Hilbert space. We do not see room for the emergence of an NEE phase. At the same time (iii), the ergodic states of many body systems show entanglement exceeding that of thermal states, and of the (Page) entanglement of pure random states. We will argue that these entanglement signatures are sensitive probes into the many body physics of chaotic chaos.

Speaker: A. Altland
• 9
Influence matrix approach to ergodic and non-ergodic quantum dynamics

Dynamical properties of a many-body system are determined by its properties as a quantum bath: the systems that thermalize act as an efficient bath, while integrable and many-body localized (MBL) systems fail to do so. I will describe a new approach to quantum many-body dynamics, inspired by the notion of the Feynman-Vernon influence functional (IF). I will consider interacting spin systems, and formulate an equation satisfied by their influence functionals. While difficult in general, this equation can be solved exactly for a class of many-body systems – perfect dephasers – which act as Markovian baths on their subsystems. More generally, I will show that, viewed as a fictitious wave function in the temporal domain, influence functional can be described by tensor-network methods. The efficiency of this approach is based on the behavior of temporal entanglement of the IF, which surprisingly remains relatively low in very different physical regimes, including fast thermalization, integrability, and many-body localization. IF approach offers a new lens on many-body non-equilibrium phenomena, both in ergodic and non-ergodic regimes, connecting the theory of open quantum systems to quantum statistical physics.

Based on: [1] Lerose, Sonner, Abanin, Phys. Rev. X 11, 021040 (2021); arXiv:2012.00777; arXiv:2104.07607

Speaker: D. Abanin
• 3:20 PM
Break
• 10
Strange metals and the time reparameterization soft mode

A strange metal has a resistivity which is smaller than h/e^2, with a linear dependence on temperature at low temperatures. I will describe recent progress in realizing such metals in theoretical models. Numerical studies of a random t-J model show strange metal behavior - I will argue that this is connected to a time reparameterization Schwarzian mode, similar to that found in the SYK model.This same mode has been connected in earlier work by others to maximal chaos.

Speaker: S. Sachdev
• 11
Planckian metal and SYK physics at a quantum critical metal with spin 1/2 fermions.

I will present our recent results on a model of itinerant SU(2) electrons with random exchange. This model hosts a quantum critical point separating two distinct metallic phases as a function of doping: a Fermi liquid with a large Fermi surface volume and a low-doping phase with local moments ordering into a spin-glass. This quantum critical point has non-Fermi liquid properties characterized by T-linear Planckian behavior, ω/T scaling and slow spin dynamics of the Sachdev-Ye-Kitaev (SYK) type.

Speaker: O. Parcollet
• 5:20 PM
Break
• 12
Entanglement island, miracle operators and the firewall

In this work, we obtain some general results on information retrieval from the black hole interior, based on the recent progress on quantum extremal surface formula and entanglement island. We study an AdS black hole coupled to a bath with generic dynamics, and ask whether it is possible to retrieve information about a small perturbation in the interior from the bath system. We derive a state reconstruction formula based on one norm. However, we show that a contradiction arises if we apply this result to a special situation when the bath dynamics includes a unitary operation that carries a particular measurement to a region A and send the result to another region W.Physically, the contradiction arises between transferability of classical information during the measurement, and non-transferability of quantum information which determines the entanglement island. We propose that the resolution of the contradiction is to realize that the state reconstruction formula does not apply to the special situation involving interior-information-retrieving measurements. This implies that the assumption of smooth replica AdS geometry with boundary condition set by the flat space bath has to break down when the particular measurement operator is applied to the bath. Using replica trick, we introduce an explicitly construction of such operator, which we name as miracle operators''. From this construction we see that the smooth replica geometry assumption breaks down because we have to introduce extra replica wormholes connecting with thesimulated blackholes'' introduced by the miracle operator. We study the implication of miracle operators in understanding the firewall paradox.

Speaker: X. Qi
• 13
Symmetry enriched phases of quantum circuits

Quantum circuits consisting of random unitary gates and subject to local measurements have been shown to undergo a phase transition, tuned by the rate of measurement, from a state with volume-law entanglement to an area-law state. I will argue that a much richer phase structure emerges if symmetries are imposed on the circuit. The classification of phases is governed in this case by an enlarged effective symmetry, which combines the physical circuit symmetry with dynamical symmetries associated with the ensemble of quantum trajectories. I'll give concrete examples for the establishment of steady states, which would not have been possible in thermal equilibrium in the presence of the circuit symmetry alone: (i) Topological states and measurement protected order in a 1+1 dimensional circuit with Z2 symmetry; (ii) A critical phase and measurement induced Kosterlitz-Thouless transition in a Gaussian Majorana circuit.

Speaker: E. Altman
• Friday, June 4
• 14
Eigenstate thermalization hypothesis: correlations and distributions of matrix elements

The Eigenstate Thermalization Hypothesis (ETH) represents a cornerstone in our understanding of the thermalization mechanism in quantum many body systems.
It deals with the matrix elements of physical observables in the energy eigenbasis and relies on ideas borrowed from quantum chaos and random matrix theory.
Inspired by the recent developments in the characterization of chaotic behavior in terms of out of time order correlations we argue that in order to have non trivial multi-point correlation functions one has to consider correlations between matrix elements previously neglected within ETH.
Moreover we show that generic rotationally invariant random matrix models satisfy a simple relation: the probability distribution of off-diagonal elements and the one of half the difference between any two diagonal elements coincide.
In the spirit of ETH we test in different models the hypothesis that the same relation holds in quantum systems that are non-localized, when one considers small energy differences. The relation provides a stringent test of ETH beyond the Gaussian ensemble.

Speaker: L. Foini
• 15
Simple models and low-temperature quantum chaos

(work with S Pappalardi)
In the past few years there has been considerable activity around a set of quantum bounds on transport coefficients (viscosity, conductivity) and on chaos (Lyapunov exponents), relevant at low temperatures. The interest comes from the fact that AdS/CFT Black-Hole models seem to saturate all of them. I shall discuss the simple case of bosonic systems whose lowest energy is a degenerate manifold, and in particular free motion on a curved space, the Hamiltonian being just the Laplace-Beltrami operator. Examples are quantum hard-sphere liquids and quantum spin liquids. In this context the bounds are approached and are consequences of the uncertainty principle, and one understands the mechanisms whereby quantum mechanics enforces them. For a system to saturate the bound, it appears as a necessary condition that at each temperature there are some degrees of freedom that are still classical, and some are on the verge of being affected by quantum effects.

Speaker: J. Kurchan
• 3:20 PM
Break
• 16
Spontaneous Symmetry Breaking in Coupled SYK or Tensor Models
Speaker: I. Klebanov
• 17
Quantum Extremal Surfaces in Isolated Black Holes

TBA

Speaker: N. Engelhardt
• 5:20 PM
Break
• 18
Wormholes without averaging

After averaging over fermion couplings, SYK has a collective field description that sometimes has “wormhole” solutions. We study the fate of these wormholes when the couplings are fixed. Working mainly in a simple model, we find that the wormhole saddles persist, but that new saddles also appear elsewhere in the integration space – “half-wormholes.” The wormhole contributions depend only weakly on the specific choice of couplings, while the half-wormhole contributions are strongly sensitive. The half-wormholes are crucial for factorization of decoupled systems with fixed couplings, but they vanish after averaging, leaving the non-factorizing wormhole behind. (Joint work with Phil Saad, Douglas Stanford and Shunyu Yao.)

Speaker: S. Shenker
• 19
Entanglement and Renyi entropies in chaotic systems: subleading corrections

The density matrix of a subsystem of a chaotic many-body system in an energy eigenstate can be modeled by a hermitian matrix that has been randomly selected from a suitable fixed-trace distribution; at leading order in system volume, the result is a thermal density matrix. I will discuss subleading corrections that arise at finite temperature, but which are usually absent at infinite temperature.

Speaker: M. Srednicki