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.
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 \ensuremathρ\ifmmode^\elseˆ\fiA 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.
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.
Under the framework of unified time scale, boundary time geometry, Matrix Universe THE--MATRIX, and Quantum Cellular Automaton (QCA) Universe, we construct a structural unified theory for ``Quantum Chaos and Eigenstate Thermalization''. The Bohigas--Giannoni--Schmit (BGS) conjecture suggests that spectral statistics of quantum chaotic systems follow Random Matrix Theory (RMT), while the Eigenstate Thermalization Hypothesis (ETH) explains thermalization of isolated quantum systems from the perspective of matrix elements of local observables. Based on the unified time scale mother formula equation \kappa(\omega) =\varphi'(\omega){\pi} =\rho_{rel}(\omega) =1{2\pi}tr Q(\omega), equation this paper interprets quantum chaos and thermalization as: ergodicity of scattering phase flow in the Matrix Universe and operator mixing in the QCA Universe; and further regards spectral rigidity and RMT universality as statistical inevitable results of unified time scale density under maximum entropy principle. Specifically: 1. In the Matrix Universe, defining the ``scattering phase flow'' dynamical system on the energy shell, we prove that when the scattering matrix S(\omega) exhibits chaotic behavior (positive Lyapunov exponent or mixing), the spectral correlation function of the unified time scale \kappa(\omega) asymptotically approaches the RMT (GOE/GUE) kernel function equation R_2(\omega_1,\omega_2) \simeq 1-\Bigl(\sin[\pi\bar\kappa(\omega_1-\omega_2)]{\pi\bar\kappa(\omega_1-\omega_2)}\Bigr)^2, equation thereby providing a scattering--spectral unified proof for the BGS conjecture. 2. In the QCA Universe, considering the operator algebra \mathcal A_{qloc} of local cell observables and the Heisenberg evolution generated by the unitary update U, we prove that for generic chaotic QCA (satisfying algebraic mixing condition), local operators spread to global operators under time evolution, and their matrix elements in energy eigenstates satisfy the ETH ansatz equation \langle E_a|\mathcal O|E_b\rangle \simeq \mathcal O(E)\delta_{ab} +e^{-S(E)/2}f_{\mathcal O}(E,\Delta E)R_{ab}, equation where R_{ab} is a random variable, establishing the microscopic mechanism of thermalization in the discrete universe. 3. Investigating the growth of entanglement entropy S_{ent}(t) in QCA time evolution, we find that under unified time scale control, the entanglement growth rate is bounded by the Lieb--Robinson velocity and related to the ``boundary time area law''. For chaotic systems, linear growth S_{ent}(t)\propto t is consistent with the entropy production rate of the scattering matrix. 4. Under unified time scale constraints, writing the spectral form factor (SFF) K(t) as the Fourier transform of the two-point correlation of \kappa(\omega), we demonstrate the ``slope--dip--plateau'' structure of SFF in the Matrix--QCA universe, interpreting the linear ramp region as a direct manifestation of the spectral rigidity of the unified time scale. 5. The appendix systematically organizes: standard definitions of quantum chaos, RMT spectral statistics and SFF; operator spreading and out-of-time-order correlator (OTOC) description in QCA; rigorous formulation of ETH; and the maximum entropy derivation of unified time scale spectral correlations. Results indicate a purely structural unified picture: quantum chaos and eigenstate thermalization can be viewed as the statistical behavior of scattering phase flow in the Matrix Universe and operator mixing in the QCA Universe under the unified time scale, thereby answering the structural version of ``why quantum chaos and thermalization occur'' within the unified universe mother structure.
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 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.
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
The spectral form factor (SFF) captures universal spectral fluctuations as signatures of quantum chaos, and has been instrumental in advancing multiple frontiers of physics including the studies of black holes and quantum many-body systems. The measurement of the SFF in many-body systems is however challenging due to the difficulty in resolving level spacings that become exponentially small with increasing system size. Here, we utilize the random measurement toolbox to perform a direct experimental measurement of the SFF, and hence probe the presence or absence of chaos in quantum many-body systems on superconducting quantum processors. For a Floquet chaotic system, we observe signatures of both short- and long-range spectral correlations in the SFF given by the ramp-plateau behavior. Furthermore, for a Hamiltonian system we utilize the SFF to distinguish a quantum many-body chaotic phase and the prethermal many-body localization. We observe the dip-ramp-plateau behavior of random matrix theory in the chaotic phase and contrast the scaling of the plateau time in system size between the many-body chaotic and localized phases. Finally, we probe the eigenstate statistics by measuring a generalization of the SFF, known as the partial SFF, and observe distinct behaviors in the purity of the reduced density matrices in these two phases. This work unveils a new experimental way of extracting the universal signatures of many-body quantum chaos in quantum devices by probing short- and long-range correlations in the energy spectrum and eigenstates.