Dynamics of a many-body quantum system of strongly interacting constituents is traditionally considered in a hierarchy of approximations: mean field—quasiparticles—collective excitations—large amplitude motion. Incoherent collision-like processes can be included on the level of kinetics as the mechanism for thermalization. On the other hand, starting from some excitation energy, local spectral properties can always be practically described by random matrix theory. Deep interrelation between two seemingly opposite pictures is not fully understood. This is discussed below, including unresolved problems and possible ways to better theory. Along the road I show how the ideas of quantum chaos can serve as a powerful conceptual, theoretical, experimental and computational tool.
Dynamics of a many-body quantum system of strongly interacting constituents is traditionally considered in a hierarchy of approximations: mean field—quasiparticles—collective excitations—large amplitude motion. Incoherent collision-like processes can be included on the level of kinetics as the mechanism for thermalization. On the other hand, starting from some excitation energy, local spectral properties can always be practically described by random matrix theory. Deep interrelation between two seemingly opposite pictures is not fully understood. This is discussed below, including unresolved problems and possible ways to better theory. Along the road I show how the ideas of quantum chaos can serve as a powerful conceptual, theoretical, experimental and computational tool.
Many-body quantum chaos results from interparticle interactions in mesoscopic systems at sufficiently high level density. A short review shows that, going beyond generic properties of local spectral statistics, the ideas of many-body quantum chaos serve as a rich source of new theoretical approaches, computational and experimental tools. Chosen examples include the enhancement of weak perturbations, thermalization in finite systems, exponential convergence of large Hamiltonian matrices and correlations between the states of different symmetry.
Many-body quantum chaos at ℏ∼1: Analytic connection to random matrix theory
A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT). For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry (1985) within the so-called diagonal approximation of semiclassical periodic-orbit sums. Derivation of the full RMT spectral form factor K(t) from semiclassics has been completed only much later in a tour de force by Mueller et al (2004). In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming at the forefront even for simple many-body quantum systems, such as locally interacting spin chains. Such systems display two universal types of behaviour which are termed as `many-body localized phase' and `ergodic phase'. In the ergodic phase, the spectral fluctuations are excellently described by RMT, even for very simple interactions and in the absence of any external source of disorder. Here we provide the first theoretical explanation for these observations. We compute K(t) explicitly in the leading two orders in t and show its agreement with RMT for non-integrable, time-reversal invariant many-body systems without classical counterparts, a generic example of which are Ising spin 1/2 models in a periodically kicking transverse field.
A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT).Most prominent features of such RMT behavior with respect to a random spectrum, both encompassed in spectral pair correlation function, are statistical suppression of small level spacings (correlation hole) and enhanced stiffness of the spectrum at large spectral ranges.For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry [Proc.R. Soc.London, A400, 229 (1985)] within the so-called diagonal approximation of semiclassical periodic-orbit sums, while the derivation of the full RMT spectral form factor K(t) (Fourier transform of spectral pair correlation function), from semiclassics has been completed by Müller et al. [Phys.Rev. Lett.93, 014103 (2004)].In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming at the forefront even for simple many-body quantum systems, such as locally interacting spin chains.Such systems display two universal types of behaviour which are termed as the 'many-body localized phase' and 'ergodic phase'.In the ergodic phase, the spectral fluctuations are excellently described by RMT, even for very simple interactions and in the absence of any external source of disorder.Here we provide a clear theoretical explanation for these observations.We compute K(t) in the leading two orders in t and show its agreement with RMT for non-integrable, time-reversal invariant many-body systems without classical counterparts, a generic example of which are Ising spin 1/2 models in a periodically kicking transverse field.In particular, we relate K(t) to partition functions of a class of twisted classical Ising models on a ring of size t, hence the leading order RMT behavior K(t)2t is a consequence of translation and reflection symmetry of the Ising partition function.
We provide a pedagogical introduction to eigenstate thermalization. This phenomenon, which occurs in generic quantum many-body systems, allows one to understand why thermalization takes place in isolated systems under unitary dynamics. We motivate eigenstate thermalization using random matrix theory and discuss recent complementary results for the volume-law entanglement entropy of Haar-random states. We discuss numerical results that highlight the corresponding behaviors in quantum many-body systems.
A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT). Most prominent features of such RMT behavior with respect to a random spectrum, both encompassed in the spectral pair correlation function, are statistical suppression of small level spacings (correlation hole) and enhanced stiffness of the spectrum at large spectral ranges. For single-particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry [Proc. R. Soc. A 400, 229 (1985)] within the so-called diagonal approximation of semiclassical periodic-orbit sums, while the derivation of the full RMT spectral form factor K(t) (Fourier transform of the spectral pair correlation function) from semiclassics has been completed by Müller et al. [Phys. Rev. Lett. 93, 014103 (2004)]. In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming to the forefront even for simple many-body quantum systems, such as locally interacting spin chains. Such systems display two universal types of behaviour which are termed the "many-body localized phase" and "ergodic phase." In the ergodic phase, the spectral fluctuations are excellently described by RMT, even for very simple interactions and in the absence of any external source of disorder. Here we provide a clear theoretical explanation for these observations. We compute K(t) in the leading two orders in t and show its agreement with RMT for nonintegrable, time-reversal invariant many-body systems without classical counterparts, a generic example of which are Ising spin-1/2 models in a periodically kicking transverse field. In particular, we relate K(t) to partition functions of a class of twisted classical Ising models on a ring of size t; hence, the leading-order RMT behavior K(t)≃2t is a consequence of translation and reflection symmetry of the Ising partition function.Received 5 February 2018Revised 12 April 2018DOI:https://doi.org/10.1103/PhysRevX.8.021062Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum chaosQuantum statistical mechanicsStatistical PhysicsNonlinear DynamicsCondensed Matter, Materials & Applied Physics
Quantum chaos entails an entropic and computational obstruction to describing a system and thus is intrinsically difficult to characterize. An understanding of quantum chaos is fundamentally related to the mechanism of thermalization in many-body systems and the quantum nature of black holes. In this thesis we adopt the view that quantum information theory provides a powerful framework in which to elucidate chaos in strongly-interacting quantum systems. We first push towards a more precise understanding of chaotic dynamics by relating different diagnostics of chaos, studying the time-evolution of random matrix Hamiltonians, and quantifying random matrix behavior in physical systems. We derive relations between out-of-time ordered correlation functions, spectral quantities, and frame potentials to relate the scrambling of quantum information, decay of correlators, and Haar-randomness. We give analytic expressions for these quantities in random matrix theory to explore universal aspects of late-time dynamics. Motivated by our random matrix results, we define k-invariance in order to capture the onset of random matrix behavior in physical systems. We then refine our diagnostics in order to study chaotic systems with symmetry by considering Haar-randomness with respect to quotients of the unitary group, and in doing so we generalize our quantum information machinery. We further consider extended random matrix ensembles in the context of strongly-interacting quantum systems dual to black holes. Lastly, we study operator growth in classes of random quantum circuits.