We investigate universal signatures of quantum chaos in the presence of time reversal symmetry (TRS) in generic many body quantum chaotic systems (gMBQCs). We study three classes of minimal models of gMBQCs with TRS, realized through random quantum circuits with (i) local TRS, (ii) global TRS, and (iii) TRS combined with discrete time-translation symmetry. In large local Hilbert space dimension $q$, we derive the emergence of random matrix theory (RMT) universality in the spectral form factor (SFF) at times larger than the Thouless time $t_{\mathrm{Th}}$, which diverges with system sizes in gMBQCs. At times before $t_{\mathrm{Th}}$, we identify universal behaviour beyond RMT by deriving explicit scaling functions of SFF in the thermodynamic limit. In particular, in the simplest non-trivial setting - preserving global TRS while breaking time translation symmetry - we show that the SFF is mapped to the partition function of an emergent classical ferromagnetic Ising model, where the Ising spins correspond to the time-parallel and time-reversed pairings of Feynman paths, and external magnetic fields are induced by TRS-breaking mechanisms. Without relying on the large-$q$ limit, we develop a second independent derivation of the Ising scaling behaviour of SFF using space-time duality and parity symmetric non-Hermitian Ginibre ensembles. Moreover, we show that many body effects originating from time-reversed pairings of Feynman paths manifest in the two-point autocorrelation function (2PAF), the out-of-time-ordered correlator (OTOC), and the partial spectral form factor - quantities sensitive to both eigenvalue and eigenstate correlations. We establish that the fluctuations of 2PAF are governed by an emergent three-state Potts model, leading to an exponential scaling with the operator support size, at a rate set by the three-state Potts model. [See full abstract in the paper]
We investigate universal signatures of quantum chaos in the presence of time reversal symmetry (TRS) in generic many body quantum chaotic systems (gMBQCs). We study three classes of minimal models of gMBQCs with TRS, realized through random quantum circuits with (i) local TRS, (ii) global TRS, and (iii) TRS combined with discrete time-translation symmetry. In large local Hilbert space dimension q, we derive the emergence of random matrix theory (RMT) universality in the spectral form factor (SFF) at times larger than the Thouless time tTh, which diverges with system sizes in gMBQCs. At times before tTh, we identify universal behaviour beyond RMT by deriving explicit scaling functions of SFF in the thermodynamic limit. In particular, in the simplest non-trivial setting - preserving global TRS while breaking time translation symmetry - we show that the SFF is mapped to the partition function of an emergent classical ferromagnetic Ising model, where the Ising spins correspond to the time-parallel and time-reversed pairings of Feynman paths, and external magnetic fields are induced by TRS-breaking mechanisms. Without relying on the large-q limit, we develop a second independent derivation of the Ising scaling behaviour of SFF using space-time duality and parity symmetric non-Hermitian Ginibre ensembles. Moreover, we show that many body effects originating from time-reversed pairings of Feynman paths manifest in the two-point autocorrelation function (2PAF), the out-of-time-ordered correlator (OTOC), and the partial spectral form factor - quantities sensitive to both eigenvalue and eigenstate correlations. We establish that the fluctuations of 2PAF are governed by an emergent three-state Potts model, leading to an exponential scaling with the operator support size, at a rate set by the three-state Potts model. [See full abstract in the paper]
Abstract We study the consequences of having translational invariance in space and time in many-body quantum chaotic systems. We consider ensembles of random quantum circuits as minimal models of translational invariant many-body quantum chaotic systems. We evaluate the spectral form factor as a sum over many-body Feynman diagrams in the limit of large local Hilbert space dimension q. At sufficiently large t, diagrams corresponding to rigid translations dominate, reproducing the random matrix theory (RMT) behaviour. At finite t, we show that translational invariance introduces additional mechanisms via two novel Feynman diagrams which delay the emergence of RMT. Our analytics suggests the existence of exact scaling forms which describe the approach to RMT behavior in the scaling limit where both t and L are large while the ratio between L and LTh(t), the many-body Thouless length, is fixed. We numerically demonstrate, with simulations of two distinct circuit models, that the resulting scaling functions are universal in the scaling limit.
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.
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
Phenomena in quantum chaotic systems such as spectral fluctuations are known to be described remarkably well by the random matrix theory (RMT) in small energy intervals and over large time scales. However, in exchange for analytical tractability, RMT has been constructed to leave out an important element of realistic physical systems { the notion of locality. Since quantum information propagates at a finite speed in a spatially-extended quantum chaotic system, there exists time scales before which RMT is effective. How do we describe the non-equilibrium quantum dynamics at intermediate timescales which are much longer than the local relaxation time but too short for RMT to apply? What are the signatures of many-body quantum chaotic systems beyond the description of RMT? In this thesis, we address these questions by analysing minimal models that incorporate a notion of locality and at the same time retain some analytical tractability of RMT. More specifically, we study Floquet quantum circuits that are random in space but periodic in time. These circuits consist of local and random quantum gates that act on a q-state `spin' at each site. After a brief review of classical chaos, quantum chaos, and quantum circuits in Chapter 1, we present three main results. In Chapter 2, we formulate a diagrammatical approach for computing ensemble averages of observables over random unitary circuits, and derive exact results on spectral statistics and dynamical observables for a Floquet circuit in the large-q limit. In Chapter 3, we further investigate the spectral statistics in spatiallyextended many-body quantum chaotic systems by computing the spectral form factor K(t) analytically and numerically. We show that K(t) follows the RMT description at times longer than the Thouless time tTh and identify the scaling of tTh at the large-q limit of a Floquet circuit. In Chapter 4, we discuss eigenstate correlations for spatially-extended many-body quantum chaotic systems, in terms of the statistical properties of matrix elements of local observables. While the eigenstate thermalization hypothesis (ETH) is known to give an excellent description of these quantities, the phenomenon of scrambling and the butterfly effect imply structure beyond ETH. We determine analytically and verify numerically the universal form of this structure at long distances and small eigenvalue separations for Floquet systems.
We numerically study the spectral statistics of open quantum many-body systems (OQMBS) as signatures of quantum chaos (or the lack thereof), using the dissipative spectral form factor (DSFF), a generalization of the spectral form factor to complex spectra. We show that the DSFF of chaotic OQMBS generically displays the quadratic ramp-plateau behaviour of the Ginibre ensemble from random matrix theory, in contrast to the linear ramp-plateau behaviour of the Gaussian ensemble in closed quantum systems. Furthermore, in the presence of many-body interactions, such RMT behaviour emerges only after a time scale τdev, which generally increases with system size for sufficiently large system size, and can be identified as the non-Hermitian analogue of the many-body Thouless time. The universality of the random matrix theory behavior is demonstrated by surveying twelve models of OQMBS, including random Kraus circuits (quantum channels) and random Lindbladians (Liouvillians) in several symmetry classes, as well as Lindbladians of paradigmatic models such as the Sachdev-Ye-Kitaev (SYK), XXZ, and the transverse field Ising models. We devise an unfolding and filtering procedure to remove variations of the averaged density of states which would otherwise hide the universal RMT-like signatures in the DSFF for chaotic OQMBS. Beyond chaotic OQMBS, we study the spectral statistics of non-chaotic OQMBS, specifically the integrable XX model and a system in the many-body localized (MBL) regime in the presence of dissipation, which exhibit DSFF behaviours distinct from the ramp-plateau behaviour of random matrix theory. Lastly, we study the DSFF of Lindbladians with the Hamiltonian term set to zero, i.e. only the jump operators are present, and demonstrate that the results of RMT universality and scaling of many-body Thouless time survive even without coherent evolution.
We show the emergence of random matrix theory (RMT) spectral correlations in the chaotic phase of generic, periodically kicked, interacting, quantum many-body systems by analytically calculating the spectral form factor (SFF), K ( t ) , up to two leading orders in time, t . We explicitly consider the presence or absence of time-reversal ( T ) symmetry to investigate all three Dyson symmetry classes, namely, the circular orthogonal ensemble (COE) for T -invariant systems with T 2 = 1 , the circular unitary ensemble (CUE) in the absence of T symmetry, and the circular symplectic ensemble (CSE) for T -invariant systems with T 2 = − 1 . Our derivation applies the random phase approximation to perform ensemble averages, and we give its justification for generic many-body interactions. We express the SFF as a sum of contributions of certain diagrams. While the number of diagrams that contribute to the SFF in the first and second orders in t for the COE and CUE classes is finite, the number of such diagrams is exponentially large in t for the CSE class. Thus, we have developed a new diagrammatic technique that uses reduced diagrams to incorporate contributions from many diagrams. For the COE class, we show that beyond Thouless time t * , the SFF takes the form K ( t ) ≃ 2 t − 2 t 2 / N up to second order in time, where N is the Hilbert space dimension. For the CUE class, we show that K ( t ) ≃ t beyond t * , and there is no universal term in the second order, unlike the COE case. For the CSE class, we show that K ( t ) ≃ 2 t + 2 t 2 / N up to two orders in time beyond t * . The above findings of the SFF up to second order in time are in agreement with the RMT predictions for these classes. In all three cases, the scaling of t * with system size L is determined by the eigenvalues of a doubly stochastic matrix M . For strongly interacting fermionic chains, M is S U ( 2 ) invariant in all three cases, leading to t * ∝ L 2 in the presence of U ( 1 ) symmetry. In the absence of U ( 1 ) symmetry, we find t *