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arXiv:2411.08186

Two types of quantum chaos: testing the limits of the Bohigas-Giannoni-Schmit conjecture

Javier M. Magán, Qingyue Wu

Published Nov 12, 2024• 1 citation
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Abstract

There are two types of quantum chaos: eigenbasis chaos and spectral chaos. The first type controls the early-time physics, e.g. the thermal relaxation and the sensitivity of the system to initial conditions. It can be traced back to the Eigenstate Thermalization Hypothesis (ETH), a statistical hypothesis about the eigenvectors of the Hamiltonian. The second type concerns very late-time physics, e.g. the ramp of the Spectral Form Factor. It can be traced back to Random Matrix Universality (RMU), a statistical hypothesis about the eigenvalues of the Hamiltonian. The Bohigas-Giannoni-Schmit (BGS) conjecture asserts a direct relationship between the two types of chaos for quantum systems with a chaotic semiclassical limit. The BGS conjecture is challenged by the Poissonian Hamiltonian ensembles, which can be used to model any quantum system displaying RMU. In this paper, we start by analyzing further aspects of such ensembles. On general and numerical grounds, we argue that these ensembles can have chaotic semiclassical limits. We then study the Poissonian ensemble associated with the Sachdev-Ye-Kitaev (SYK) model. While the distribution of couplings peaks around the original SYK model, the Poissonian ensemble is not $k$-local. This suggests that the link between ETH and RMU requires of physical $k$-locality as an assumption. We test this hypothesis by modifying the couplings of the SYK Hamiltonian via the Metropolis algorithm, rewarding directions in the space of couplings that do not display RMU. The numerics converge to a $k$-local Hamiltonian with eigenbasis chaos but without spectral chaos. We finally comment on ways out and corollaries of our results.

DOI
10.48550/arxiv.2411.08186

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There are two types of quantum chaos: eigenbasis chaos and spectral chaos. The first type controls the early-time physics, e.g. the thermal relaxation and the sensitivity of the system to initial conditions. It can be traced back to the Eigenstate Thermalization Hypothesis (ETH), a statistical hypothesis about the eigenvectors of the Hamiltonian. The second type concerns very late-time physics, e.g. the ramp of the Spectral Form Factor. It can be traced back to Random Matrix Universality (RMU), a statistical hypothesis about the eigenvalues of the Hamiltonian. The Bohigas-Giannoni-Schmit (BGS) conjecture asserts a direct relationship between the two types of chaos for quantum systems with a chaotic semiclassical limit. The BGS conjecture is challenged by the Poissonian Hamiltonian ensembles, which can be used to model any quantum system displaying RMU. In this paper, we start by analyzing further aspects of such ensembles. On general and numerical grounds, we argue that these ensembles can have chaotic semiclassical limits. We then study the Poissonian ensemble associated with the Sachdev-Ye-Kitaev (SYK) model. While the distribution of couplings peaks around the original SYK model, the Poissonian ensemble is not kkk-local. This suggests that the link between ETH and RMU requires of physical kkk-locality as an assumption. We test this hypothesis by modifying the couplings of the SYK Hamiltonian via the Metropolis algorithm, rewarding directions in the space of couplings that do not display RMU. The numerics converge to a kkk-local Hamiltonian with eigenbasis chaos but without spectral chaos. We finally comment on ways out and corollaries of our results.
Abstract Thermalization of a closed chaotic quantum system is commonly addressed in terms of the ‘eigenstate thermalization hypothesis’ (ETH). An alternative approach uses the Bohigas–Giannoni–Schmit (BGS) conjecture. The comparison shows that the two approaches differ significantly. In contrast to ETH, BGS fully uses the statistical properies of the chaotic Hamiltonian. In both approaches, the criterion for thermalization is similar. BGS goes beyond ETH in predicting quantitatively the time dependence of thermalization.
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.
Abstract The Bohigas-Giannoni-Schmit (BGS) conjecture states that the Hamiltonian of a microscopic analogue of a classical chaotic system can be modeled by a random matrix from a Gaussian ensemble. Here, this conjecture is considered in the context of a recently discovered geometric relationship between classical and quantum mechanics. Motivated by BGS, we conjecture that the Hamiltonian of a system whose classical counterpart performs a random walk can be modeled by a family of independent random matrices from the Gaussian unitary ensemble. By accepting this conjecture, we find a relationship between the process of observation in classical and quantum physics, derive irreversibility of observation and describe the boundary between the micro and macro worlds.
Abstract We investigate thermalization of a closed chaotic many-body quantum system by combining the Hartree–Fock approach with the Bohigas–Giannoni–Schmit conjecture. The conjecture implies that locally, the residual interaction causes the statistics of eigenvalues and eigenfunctions of the full Hamiltonian to agree with random-matrix predictions. The agreement is confined to an interval Δ (the correlation width). The results are used to calculate Tr ( A ρ ( t ) ) . Here ρ ( t ) is the time-dependent density matrix of the system, and A represents an observable. In the semiclassical regime, the average ⟨ Tr ( A ρ ( t ) ) ⟩ decays on the time scale ℏ / Δ toward an asymptotic value. If the energy spread of the system is of order Δ, that value is given by Tr ( A ρ eq ) where ρ eq is the density matrix of statistical equilibrium. The correlation width Δ is the central parameter of our approach. We argue that Δ occurs generically in chaotic quantum systems and plays the same central role.
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.
Quantum chaos remains a challenging concept because classical signatures of chaos do not directly transfer to quantum systems. This thesis investigates which indicators of chaos are shared between classical and quantum mechanics and how they can be used to characterise non-integrable many-body systems. We examine ergodicity and thermalisation by means of the Eigenstate Thermalisation Hypothesis (ETH) as signatures of non-integrability, and study the Out-of-Time-Ordered Correlator (OTOC) as a probe of information scrambling and as a measure of sensitivity to initial conditions. We briefly discuss the appearance of chaotic wavefunctions and observations such as quantum scars and quantum ergodicity in individual eigenmodes of the Bunimovich stadium potential. Using the OTOC, we discuss extensions of the Lyapunov exponent to quantum systems and apply these ideas to the Sachdev–Ye–Kitaev (SYK) model. In the large-N limit, and using the AdS/CFT correspondence, we find that the model might potentially be the dual of a black hole and that it is maximally chaotic. We conclude by discussing additional proposed signatures of quantum chaos and prospects for experimental realisations of the SYK model.