Abstract We study aspects of chaos and thermodynamics at strong coupling in a scalar model using LCT numerical methods. We find that our eigenstate spectrum satisfies Wigner-Dyson statistics and that the coefficients describing eigenstates in our basis satisfy Random Matrix Theory (RMT) statistics. At weak coupling, though the bulk of states satisfy RMT statistics, we find several scar states as well. We then use these chaotic states to compute the equation of state of the model, obtaining results consistent with Conformal Field Theory (CFT) expectations at temperatures above the scale of relevant interactions. We also test the Eigenstate Thermalization Hypothesis by computing the expectation value of local operators in eigenstates, and check that their behavior is consistent with thermal CFT values at high temperatures. Finally, we compute the Spectral Form Factor (SFF), which has the expected behavior associated with the equation of state at short times and chaos at long times. We also propose a new technique for extracting the connected part of the SFF without the need of disorder averaging by using different symmetry sectors.
Abstract We study aspects of chaos and thermodynamics at strong coupling in a scalar model using LCT numerical methods. We find that our eigenstate spectrum satisfies Wigner-Dyson statistics and that the coefficients describing eigenstates in our basis satisfy Random Matrix Theory (RMT) statistics. At weak coupling, though the bulk of states satisfy RMT statistics, we find several scar states as well. We then use these chaotic states to compute the equation of state of the model, obtaining results consistent with Conformal Field Theory (CFT) expectations at temperatures above the scale of relevant interactions. We also test the Eigenstate Thermalization Hypothesis by computing the expectation value of local operators in eigenstates, and check that their behavior is consistent with thermal CFT values at high temperatures. Finally, we compute the Spectral Form Factor (SFF), which has the expected behavior associated with the equation of state at short times and chaos at long times. We also propose a new technique for extracting the connected part of the SFF without the need of disorder averaging by using different symmetry sectors.
Quantum chaos and the eigenstate thermalization hypothesis are based on the assumption of the validity of random matrix theory description on the spectrum and eigenstates. They provide the foundation and descriptions for the typical dynamics and thermalization in generic closed quantum systems. In this thesis, we investigate situations where the systems show atypical dynamics or anomalous thermalization, conflicting with the usual expectations from quantum chaos and eigenstate thermalization hypothesis. We first examine weak thermalization in a nonintegrable spin chain. The system shows long-lived strong oscillations and relaxes to the thermal equilibrium weakly. We identify the dynamics describable by quasiparticles and recognize the oscillation frequency to be the quasiparticle mass gap. We also estimate the damping time for the oscillations. Next, we study prethermalization, a phenomenon where a system relaxes to an intermediate almost-equilibrium stage before reaching the true thermal equilibrium. We study a nonintegrable spin chain in the strong coupling limit, where an almost-conserved quantity emerges and gives rise to the prethermalization. We also study a newly proposed diagnostic for quantum chaos: out-of-time-ordered correlators. Contrasting to the chaotic systems, we inspect their behaviors in various noninteracting integrable models. Finally, we dig into the quantum many-body scar states in the PXP model which describes a Rydberg atom chain. These special states do not satisfy the random matrix theory description nor the eigenstate thermalization hypothesis, therefore defying quantum chaos.
<!--HTML-->Nonintegrable QFTs are expected to thermalize and exhibit emergence of hydrodynamics and chaos. In weakly coupled QFTs, kinetic theory captures local thermalization; such a versatile tool is absent away from the perturbative regime. I will present analytical and numerical results using nonperturbative methods to study thermalization at strong coupling. I will show how requiring causality in the thermal state leads to strong analytic constraints on the thermodynamics and out of equilibrium properties of any relativistic 1+1d QFT. I will then discuss Lightcone Conformal Truncation (LCT) as a powerful numerical tool to study thermalization of QFTs. Applied to ϕ4 theory in 1+1d, LCT reveals eigenstate thermalization and onset of random matrix universality at any nonzero coupling. Finally, I will discuss prospects for observing the emergence of hydrodynamics in QFTs using Hamiltonian truncation.
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.
This review gives a pedagogical introduction to the eigenstate thermalization hypothesis (ETH), its basis, and its implications to statistical mechanics and thermodynamics. In the first part, ETH is introduced as a natural extension of ideas from quantum chaos and random matrix theory (RMT). To this end, we present a brief overview of classical and quantum chaos, as well as RMT and some of its most important predictions. The latter include the statistics of energy levels, eigenstate components, and matrix elements of observables. Building on these, we introduce the ETH and show that it allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath. We examine numerical evidence of eigenstate thermalization from studies of many-body lattice systems. We also introduce the concept of a quench as a means of taking isolated systems out of equilibrium, and discuss results of numerical experiments on quantum quenches. The second part of the review explores the implications of quantum chaos and ETH to thermodynamics. Basic thermodynamic relations are derived, including the second law of thermodynamics, the fundamental thermodynamic relation, fluctuation theorems, the fluctuation–dissipation relation, and the Einstein and Onsager relations. In particular, it is shown that quantum chaos allows one to prove these relations for individual Hamiltonian eigenstates and thus extend them to arbitrary stationary statistical ensembles. In some cases, it is possible to extend their regimes of applicability beyond the standard thermal equilibrium domain. We then show how one can use these relations to obtain nontrivial universal energy distributions in continuously driven systems. At the end of the review, we briefly discuss the relaxation dynamics and description after relaxation of integrable quantum systems, for which ETH is violated. We present results from numerical experiments and analytical studies of quantum quenches at integrability. We introduce the concept of the generalized Gibbs ensemble and discuss its connection with ideas of prethermalization in weakly interacting systems.
The emergence of statistical mechanics from quantum dynamics is conventionally understood through quantum chaos, the Eigenstate Thermalization Hypothesis (ETH), and canonical typicality, which together explain the thermal behavior of local observables in quantum many-body systems. We provide a distinct mechanism in which, for a quantum many-body system under strong random perturbation, the chaotic behavior of the system's eigenvectors and a constraint on the total energy directly generate a canonical ensemble for the entire system. The canonical ensemble occurs as the first moment of the Porter-Thomas distribution of the system's chaotic eigenstates, and the constraint on the total energy sets the temperature in the ensemble. This yields canonically distributed expectation values without invoking ETH, a microcanonical ensemble, or a subsystem-bath partition for local and global observables. We numerically demonstrate the mechanism for the 1D transverse-field Ising model by showing canonically distributed eigenstate projectors and spin-spin correlators with distinctive temperature dependence of the correlation length. An implementation of this thermalization approach on a quantum computer can be utilized for thermal state preparation.
We use field-theoretic methods to explore the statistics of eigenfunctions of the Floquet operator for a large family of Floquet random quantum circuits. The correlation function of the quasienergy eigenstates is calculated and shown to exhibit random matrix circular unitary ensemble statistics, which is consistent with the analogue of Berry's conjecture for quantum circuits. This quantity determines all key metrics of quantum chaos, such as the spectral form factor and thermalizing time-dependence of the expectation value of an arbitrary observable. It also allows us to explicitly show that the matrix elements of local operators satisfy the eigenstate thermalization hypothesis (ETH); i.e., the variance of the off-diagonal matrix elements of such operators is exponentially small in the system size. These results represent a proof of ETH for the family of Floquet random quantum circuits at a physical level of rigor. An outstanding open question for this and most of other sigma-model calculations is a mathematically rigorous proof of the validity of the saddle-point approximation in the large-N limit.