Quantum chaos is commonly associated with the emergence of random-matrix statistics in the spectra of quantum systems. A useful diagnostic is provided by the spectral form factor (SFF), which for random matrix ensembles displays a universal linear-growth regime (`ramp'). In the last decade, a landmark result by Bertini, Kos and Prosen (BKP) identified for the first time a class of geometrically local quantum dynamics of finite-dimensional particles where the SFF provably exhibits a random-matrix ramp: periodically driven (Floquet) qudit chains whose evolution is described by `dual-unitary' circuits. Here, building on the BKP result and on a recently proposed variant of the Feynman-Kitaev clock construction, we obtain a spectral ramp in a class of static, geometrically local Hamiltonians. Our strategy is to embed the Floquet quasienergy spectrum of a dual-unitary circuit into the energy spectrum of a static local Hamiltonian, and to prove that the latter's connected SFF inherits the BKP ramp within a symmetry sector. This is to our knowledge the first proof of a spectral ramp in a time-independent, geometrically local many-body system with finite local Hilbert space dimension.
Quantum chaos is commonly associated with the emergence of random-matrix statistics in the spectra of quantum systems. A useful diagnostic is provided by the spectral form factor (SFF), which for random matrix ensembles displays a universal linear-growth regime (ramp'). In the last decade, a landmark result by Bertini, Kos and Prosen (BKP) identified for the first time a class of geometrically local quantum dynamics of finite-dimensional particles where the SFF provably exhibits a random-matrix ramp: periodically driven (Floquet) qudit chains whose evolution is described by dual-unitary' circuits. Here, building on the BKP result and on a recently proposed variant of the Feynman-Kitaev clock construction, we obtain a spectral ramp in a class of static, geometrically local Hamiltonians. Our strategy is to embed the Floquet quasienergy spectrum of a dual-unitary circuit into the energy spectrum of a static local Hamiltonian, and to prove that the latter's connected SFF inherits the BKP ramp within a symmetry sector. This is to our knowledge the first proof of a spectral ramp in a time-independent, geometrically local many-body system with finite local Hilbert space dimension.
We investigate the appearance of quantum chaos in a single many-body wave function by analyzing the statistical properties of the eigenvalues of its reduced density matrix ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{A}$ of a spatial subsystem A. We find that (i): the spectrum of the density matrix is described by so-called Wishart random matrix theory, which (ii): exhibits besides level repulsion, spectral rigidity, and universal spectral correlations between eigenvalues separated by distances ranging from one up to many mean level spacings, which we investigate. We use these universal spectral characteristics of the reduced density matrix as a definition of chaos in the wave function. A simple and precise characterization of such universal correlations in a spectrum is a segment of strictly linear growth at sufficiently long times, recently called the ``ramp,'' of the spectral form factor which is the Fourier transform of the correlation function between a pair of eigenvalues. It turns out that Wishart and standard random matrix theory have the same universal ``ramp.'' Specifically, here numerical results for the spectral form factor of the density matrix of generic nonintegrable many-body systems, such as one-dimensional quantum Ising and Floquet spin models, are found to exhibit a universal ``ramp'' identical to that appearing for a ``random pure state'' (``Page state,'' or ``Haar state''). The density matrix of the latter is precisely the Wishart random matrix, the reduced density matrix of a completely random wave function. In addition, we study the development of chaos in the wave function by letting an initial direct product state evolve under the unitary time evolution. We find that the universal spectral correlations as manifested by the ``ramp'' set in as soon as the entanglement entropy begins to grow, and first develop for the eigenvalues at the top of the spectrum of the density matrix ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{A}$, subsequently spreading over the entire spectrum at later times. Finally, we study a prethermalized regime described by a generalized Gibbs ensemble, which develops in a rapidly driven Floquet model at intermediate times. We find that the prethermalized regime exhibits no chaos, as evidenced by the absence of a ``ramp'' in the spectral form factor of the density matrix, while the universal spectral correlations start to develop when the prethermalized regime finally relaxes at late times to the fully thermalized (infinite temperature) chaotic regime.
One manifestation of quantum chaos is a random-matrix-like fine-grained energy spectrum.Prior to the inverse level spacing time, random matrix theory predicts a "ramp" of increasing variance in the connected part of the spectral form factor.However, in realistic quantum chaotic systems, the finite-time dynamics of the spectral form factor is much richer, with the pure random matrix ramp appearing only at sufficiently late time.In this article, we present a hydrodynamic theory of the connected spectral form factor prior to the inverse level spacing time.We first derive a general formula for the spectral form factor of a system with almost-conserved sectors in terms of return probabilities and spectral form factors within each sector.Next we argue that the theory of fluctuating hydrodynamics can be adapted from the usual Schwinger-Keldysh contour to the periodic time setting needed for the spectral form factor, and we show explicitly that the general formula is recovered in the case of energy diffusion.We also initiate a study of interaction effects in this modified hydrodynamic framework and show how the Thouless time defined as the time required for the spectral form factor to approach the pure random matrix result is controlled by the slow hydrodynamics modes.We then extend the formalism to Floquet systems, where a ramp is expected but with a different coefficient, and we derive a crossover formula from the Hamiltonian ramp to the Floquet ramp when the Floquet drive is weak.Taken together, these results establish an effective field theory of chaotic spectral correlations which predicts the random matrix ramp at late time and computes corrections to it at earlier times.
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.
A bstract The fine grained energy spectrum of quantum chaotic systems is widely believed to be described by random matrix statistics. A basic scale in such a system is the energy range over which this behavior persists. We define the corresponding time scale by the time at which the linearly growing ramp region in the spectral form factor begins. We call this time t ramp . The purpose of this paper is to study this scale in many-body quantum systems that display strong chaos, sometimes called scrambling systems. We focus on randomly coupled qubit systems, both local and k -local (all-to-all interactions) and the Sachdev-Ye-Kitaev (SYK) model. Using numerical results, analytic estimates for random quantum circuits, and a heuristic analysis of Hamiltonian systems we find the following results. For geometrically local systems with a conservation law we find t ramp is determined by the diffusion time across the system, order N 2 for a 1D chain of N qubits. This is analogous to the behavior found for local one-body chaotic systems. For a k -local system like SYK the time is order log N but with a different prefactor and a different mechanism than the scrambling time. In the absence of any conservation laws, as in a generic random quantum circuit, we find t ramp ∼ log N , independent of connectivity.
While the notion of quantum chaos is tied to random matrix spectral correlations, also eigenstate properties in chaotic systems are often assumed to be described by random matrix theory. Analytic insights into eigenstate correlations can be obtained by the recently introduced partial spectral form factor. Here, we study the partial spectral form factor in chaotic dual-unitary quantum circuits in the thermodynamic limit. We compute the latter for a finite subsystem in a brickwork circuit coupled to an infinite complement. For initial times, shorter than the subsystem's size, spatial locality and (dual) unitarity implies a constant partial spectral form factor, clearly deviating from the linear ramp of the random matrix prediction. In contrast, for larger times we prove, that the partial spectral form factor follows the random matrix result up to exponentially suppressed corrections. We supplement our exact analytical results by semi-analytic computations performed in the thermodynamic limit as well as with numerics for finite-size systems.
We consider systems of fermions evolved by noninteracting unitary circuits with correlated on-site potentials. When these potentials are drawn from the eigenvalue distribution of a circular random matrix ensemble, the single-particle sector exhibits chaotic dynamics. The corresponding many-body spectral form factors (SFFs) can be computed exactly, revealing signatures of single-particle chaos in the many-body spectral statistics. Due to the absence of interactions the SFF grows exponentially in time, a result which we demonstrate through simple arguments, scaling collapses, and closed-form evaluation of the SFF. Upon introducing interactions, the SFF crosses over to a linear growth regime consistent with many-body random matrix universality. Our exact results for the SFF provide a baseline for future studies of the crossover between single-particle and many-body random matrix behavior.