Quantum Chaos and Holography

In the last few years, operator complexity has emerged as a powerful probe for chaos in quantum and semiclassical systems. There are many useful measures for quantifying operator growth. One such measure is Krylov complexity. According to the "Universal Operator Growth hypothesis", the behaviour of K-complexity with time is conjectured to be able to distinguish between quantum integrable and chaotic systems. There are some subtle differences between scrambling and chaos, which we have explored in light of the operator growth hypothesis. Here we will discuss some fundamental notions regarding quantum and classical chaos, as well as the notion of operator complexity. We will then go over the concept and machinery associated with Krylov complexity and the universal operator growth hypothesis. In semi-classical systems, the exponential growth of the out-of-timeorder correlator (OTOC) is believed to be the hallmark of quantum chaos. However,on several occasions, it has been argued that, even in integrable systems, OTOC can grow exponentially due to the presence of unstable saddle points in the phase space. We probe such an integrable system exhibiting saddle dominated scrambling through Krylov complexity and the associated Lanczos coefficients. In the realm of the universal operator growth hypothesis, we demonstrate that the Lanczos coefficients follow the linear growth, which ensures the exponential behavior of Krylov complexity at early times. The linear growth arises entirely due to the saddle, which dominates other phase-space points even away from itself. Our results reveal that the exponential growth of Krylov complexity can be observed in integrable systems with saddle-dominated scrambling and thus need not be associated with the presence of chaos.

The dynamics of a quantum kicked rotor(QKR), unlike that of its classical counterpart, is known to be non-ergodic. This is due to the emergence of a dynamically localized wave-function in the angular momentum space, courtesy of interference affects. In this work, we analyze the dynamics of a quantum rotor kicked with a binary Fibonacci sequence of two distinct drive amplitudes. While the dynamics at low drive frequencies is found to be diffusive, a long-lived pre-ergodic regime emerges in the other limit. The dynamics in this pre-ergodic regime mimics that of a regular QKR and can be associated with the onset of a dynamical quasi-localization. We establish that this peculiar behavior arises due to the presence of localized eigenstates of an approximately conserved effective Hamiltonian, which drives the evolution at Fibonacci instants. However, the effective Hamiltonian picture does not persist indefinitely and the dynamics eventually becomes ergodic after asymptotically long times.

Co-author/co-presenter: Stylianos Apollonas Matsoukas-Roubeas In an isolated system, the time evolution of a given observable in the Heisenberg picture can be efficiently represented in Krylov space. In this representation, an initial operator becomes increasingly complex as time goes by, a feature that can be quantified by the Krylov complexity. We introduce a fundamental and universal limit to the growth of the Krylov complexity by formulating a Robertson uncertainty relation, involving the Krylov complexity operator and the Liouvillian, as generator of time evolution. We further show the conditions for this bound to be saturated and illustrate its validity in paradigmatic models of quantum chaos.

The scaling of the Thouless time with system size is of fundamental importance to characterize dynamical properties in quantum systems. In this work, we study the scaling of the Thouless time in the Anderson model on random regular graphs with on-site disorder. We determine the Thouless time from two main quantities: the spectral form factor and the power spectrum. Both quantities probe the long-range spectral correlations in the system and allow us to determine the Thouless time as the time scale after which the system is well described by random matrix theory. We find that the scaling of the Thouless time is consistent with the existence of a sub-diffusive regime anticipating the localized phase. Furthermore, to reduce finite-size effects, we break energy conservation by introducing a Floquet version of the model and show that it hosts a similar sub-diffusive regime.

An intriguing regime of universal charge transport at high entropy density has been proposed for periodically driven interacting one-dimensional systems with Bloch bands separated by a large single-particle band gap. For weak interactions, a simple picture based on well-defined Floquet quasiparticles suggests that the system should host a quasisteady state current that depends only on the populations of the system's Floquet-Bloch bands and their associated quasienergy winding numbers. Here we show that such topological transport persists into the strongly interacting regime where the single-particle lifetime becomes shorter than the drive period. Analytically, we show that the value of the current is insensitive to interaction-induced band renormalizations and lifetime broadening when certain conditions are met by the system's non-equilibrium distribution function. We show that these conditions correspond to a quasisteady state. We support these predictions through numerical simulation of a system of strongly interacting fermions in a periodically-modulated chain of Sachdev-Ye-Kitaev dots. Our work establishes universal transport at high entropy density as a robust far from equilibrium topological phenomenon, which can be readily realized with cold atoms in optical lattices.

ETH gives successful predictions for the behavior of operator expectation values and their variances. However, due to the general formulation and its ignorance of system specific properties, it fails to give an accurate description for higher order quantities as for example OTOCs or the operator entanglement entropy. We tackle this question by analyzing the implications of locality on correlations between different eigenstates. We extend the picture of random eigenvectors to recover these nontrivial correlations. This allows us to provide a joint probability distributions of eigenvectors which recovers the correct behavior of operator entanglement entropy and other fourth order quantities.

Based on the recent discovery of the duality between JT quantum gravity and a double-scaled ensemble of random matrices, we show how in the universal RMT limit, consistence between the two sides of the duality imposes a set of constraints on the volumes of moduli spaces of Riemannian manifolds. These volumes are given in terms of polynomial functions, the Weil-Petersson volumes, solving a celebrated nonlinear recursion formula that are notoriously difficult to analyze. Since our results take the form of linear relations between the coefficients of the Weil-Petersson volumes, they therefore provide both a stringent test for their symbolic calculation and a possible way of simplifying their construction.

Quantum chaos cannot develop faster than $\lambda \leq 2 \pi/(\beta \hbar)$ for systems in thermal equilibrium [Maldacena et. al. JHEP (2016)]. This `MSS-bound' on the Lyapunov exponent is set by the width of the strip on which the regularized out-of-time-order-correlator is analytic. We show that similar analyticity constraints also bound the evolution of other dynamical quantities. We first find a family of functions that admit a universal bound inspired by the MSS bound, and then detail the case of the spectral form factor, which is the Fourier transform of the two-level correlation function and can be understood as the survival probability of the coherent Gibbs state. Specifically, the inflection exponent η that we introduce here is bounded as $\eta \leq \pi/(2 \beta \hbar)$. Importantly, the bound that we derive is universal and exists outside of the chaotic regime. We illustrate the results in systems with regular, chaotic, and tunable dynamics, namely the harmonic oscillator, a random matrix ensemble, and the quantum kicked top, and discuss the relation with known quantum speed limits.

We analyse near-extremal black brane configurations in asymptotically AdS_{4} 4spacetime with the temperature T, chemical potential μ, and three-velocity u^{ν}, varying slowly. We consider a low-temperature limit where the rate of variation is much slower than μ, but much bigger than T. This limit is different from the one considered for conventional fluid-mechanics in which the rate of variation is much smaller than both T, μ. We find that in our limit, as well, the Einstein-Maxwell equations can be solved in a systematic perturbative expansion. At first order, in the rate of variation, the resulting constitutive relations for the stress tensor and charge current are local in the boundary theory and can be easily calculated. At higher orders, we show that these relations become non-local in time but the perturbative expansion is still valid. We find that there are four linearised modes in this limit; these are similar to the hydrodynamic modes found in conventional fluid mechanics with the same dispersion relations. We also study some linearised time independent perturbations exhibiting attractor behaviour at the horizon — these arise in the presence of external driving forces in the boundary theory.

We consider a quantum many-body system - the Bose-Hubbard system on three sites - which has a classical limit, and which is neither strongly chaotic nor integrable but rather shows a mixture of the two types of behavior. We compare quantum measures of chaos (eigenvalue statistics and eigenvector structure) in the quantum system, with classical measures of chaos (Lyapunov exponents) in the corresponding classical system. As a function of energy and interaction strength, we demonstrate a strong overall correspondence between the two cases. In contrast to both strongly chaotic and integrable systems, the largest Lyapunov exponent is shown to be a multi-valued function of energy.

A quench experiment on a Rydberg-blockaded atom chain revealed a new (and subtle) form of ergodicity-breaking. Quantum scars in non-integrable models violate the (strong-)eigenstate thermalization hypothesis (ETH) and thus do not equilibrate. Despite extensive theoretical and numerical works on many models hosting scar states, the effective model of the experiment, the PXP model, has eluded a thorough understanding of the origin of the scar states. We unveil a hidden structure of the PXP model and show that the quasi-periodic motion ensuing from certain initial states is simply the projection onto the Rydberg-constraint subspace of the precession of a large pseudospin. The scar states are shown to arise from the system's close proximity to a Hamiltonian with a structure common to all known models hosting quantum scars.

The fundamental question of how information spreads in closed quantum many-body systems is often addressed through the lens of the bipartite entanglement entropy, a quantity that describes correlations in a comprehensive (nonlocal) way. Among the most striking features of the entanglement entropy are its unbounded linear growth in the thermodynamic limit, its asymptotic extensivity in finite-size systems, and the possibility of measurement-induced phase transitions, all of which have no obvious classical counterpart. Here, we show how these key qualitative features emerge naturally also in classical information spreading, as long as one treats the classical many-body problem on par with the quantum one, that is, by explicitly accounting for the exponentially large classical probability distribution. Our analysis is supported by extensive numerics on prototypical cellular automata and Hamiltonian systems, for which we focus on the classical mutual information and also introduce a `classical entanglement entropy'. Our study sheds light on the nature of information spreading in classical and quantum systems, and opens new avenues for quantum-inspired classical approaches across physics, information theory, and statistics.

We identify the chaotic phase of the Bose-Hubbard Hamiltonian by the energy-resolved correlation between spectral features and structural changes of the associated eigenstates as exposed by their generalized fractal dimensions. The eigenvectors are shown to become ergodic in the thermodynamic limit, in the configuration space Fock basis, and we demonstrate that the fluctuation of the generalized fractal dimensions among near-in-energy eigenstates is a rather sensitive probe of quantum chaos that exhibits a qualitative basis-independent behaviour [1]. We scrutinize the chaotic phase of the model in relation to the bosonic embedded random matrix ensemble, which mirrors the dominant few-body nature of many-particle interactions, and hence the Fock space sparsity of quantum many-body systems. The energy dependence of the chaotic regime is well described by the bosonic embedded ensemble, which also reproduces the Bose-Hubbard chaotic eigenvector features [2]. Despite this agreement, in terms of the fractal dimension distribution, these two models depart from each other and from the Gaussian orthogonal ensemble as Hilbert space grows [1,2]. These results provide further evidence of a way to discriminate among different many-body Hamiltonians in the chaotic regime. [1] L. Pausch. E. G. Carnio. A. Rodríguez, A. Buchleitner, Phys. Rev. Lett. 126, 150601 (2021). [2] L. Pausch, E. G. Carnio, A. Buchleitner, A. Rodríguez, New J. Phys. 23, 123036 (2021).

Spectral correlations are a powerful tool to study the dynamics of quantum many-body systems. For Hermitian Hamiltonians, quantum chaotic motion is related to random matrix theory spectral correlations. Based on recent progress in the application of spectral analysis to non-Hermitian quantum systems, we show that local level statistics, which probes the dynamics around the Heisenberg time, of a non-Hermitian $q$-body Sachdev-Ye-Kitev (nHSYK) model with $N$ Majorana fermions, and its chiral and complex-fermion extensions, are also well described by random matrix theory for $q>2$, while for $q=2$ it is given by the equivalent of Poisson statistics. For that comparison, we combine exact diagonalization numerical techniques with analytical results obtained for some of the random matrix spectral observables. Moreover, depending on $q$ and $N$, we identify $19$ out of the $38$ non-Hermitian universality classes in the nHSYK model, including those corresponding to the tenfold way. In particular, we realize explicitly $14$ out of the $15$ universality classes corresponding to non-pseudo-Hermitian Hamiltonians that involve universal bulk correlations of classes AI$^\dagger$ and AII$^\dagger$, beyond the Ginibre ensembles. These results provide strong evidence of striking universal features in non-unitary many-body quantum chaos, which in all cases can be captured by nHSYK models with $q>2$. Based on A. M. García-García, L. Sá, and J. J. M. Verbaarschot, arXiv:2110.03444 (to appear in Phys. Rev. X)

We establish a connection between many-body quantum chaotic (MBQC) systems and the Ginibre random matrix ensemble (GinUE) — namely that GinUE behaviors emerge in observables like the spectral form factor (SFF) in MBQC systems in suitable thermodynamic and scaling limits, just as the Gaussian unitary ensemble behaviours emerge in MBQC systems in late time. A many-body quantum system with translation invariance can be associated, via space-time duality, with a dual Floquet operator which is generically non-Hermitian with complex eigenvalues, the dual spectrum. We argue and demonstrate that the dual spectra of MBQC systems necessarily have non-trivial and universal correlation due to the existence of a linear ramp in the SFF. We show that, firstly, the spectral correlation of the dual spectra, probed by the dissipative spectral form factor, falls under the universality class of the GinUE for sufficiently large (complex) time, and in particular displays level repulsion at the scale of the mean level spacing in the complex plane. Secondly, that a scaling form of SFF of MBQC systems can be universally described by an analogous scaling form of GinUE. Lastly, we remark that the many-body nature in MBQC systems is necessary for the emergence of the GinUE in the thermodynamic limit. As a side result, we obtain an exact analytical expression of SFF (and its scaling form) for the GinUE

Based on the recent discovery of the duality between JT quantum gravity and a double-scaled ensemble of random matrices, we show how in the universal RMT limit, consistence between the two sides of the duality imposes a set of constraints on the volumes of moduli spaces of Riemannian manifolds. These volumes are given in terms of polynomial functions, the Weil-Petersson volumes, solving a celebrated nonlinear recursion formula that are notoriously difficult to analyze. Since our results take the form of linear relations between the coefficients of the Weil-Petersson volumes, they therefore provide both a stringent test for their symbolic calculation and a possible way of simplifying their construction.

Driven quantum systems may realize novel phenomena absent in static systems, but driving-induced heating can limit the time-scale on which these persist. We study heating in interacting quantum many-body systems driven by random sequences with n−multipolar correlations, corresponding to a polynomially suppressed low frequency spectrum. For n≥1, we find a prethermal regime, the lifetime of which grows algebraically with the driving rate, with exponent 2n+1. A simple theory based on Fermi's golden rule accounts for this behaviour. The quasiperiodic Thue-Morse sequence corresponds to the n→∞ limit, and accordingly exhibits an exponentially long-lived prethermal regime. Despite the absence of periodicity in the drive, and in spite of its eventual heat death, the prethermal regime can host versatile non-equilibrium phases, which we illustrate with a random multipolar discrete time crystal and anomalous random multipolar driven insulators. We also discuss how to suppress driving induced heating to higher bands.