Abstract We show that for a system governed by a random-matrix Hamiltonian (a member of the time-reversal invariant Gaussian Orthogonal Ensemble (GOE) of random matrices of dimension N), all functions Tr ( A ρ ( t ) ) in the ensemble thermalize: For N → ∞ every such function tends to the value Tr ( A ρ eq ( ∞ ) ) + Tr ( A ρ ( 0 ) ) g 2 ( t ) . Here ρ ( t ) is the time-dependent density matrix of the system, A is a Hermitean operator standing for an observable, and ρ eq ( ∞ ) is the equilibrium density matrix at infinite temperature. The oscillatory function g(t) is the Fourier transform of the average GOE level density and falls off as 1 / | t | 3 / 2 for large t. With g ( t ) = g ( − t ) , thermalization is symmetric in time. Analogous results, including the symmetry in time of thermalization, are derived for the time-reversal non-invariant Gaussian Unitary Ensemble of random matrices. Comparison with the ‘eigenstate thermalization hypothesis’ of (Srednicki 1999 J. Phys. A: Math. Gen. 32 1163) shows overall agreement but raises significant questions.
Abstract We show that for a system governed by a random-matrix Hamiltonian (a member of the time-reversal invariant Gaussian Orthogonal Ensemble (GOE) of random matrices of dimension N), all functions Tr ( A ρ ( t ) ) in the ensemble thermalize: For N → ∞ every such function tends to the value Tr ( A ρ eq ( ∞ ) ) + Tr ( A ρ ( 0 ) ) g 2 ( t ) . Here ρ ( t ) is the time-dependent density matrix of the system, A is a Hermitean operator standing for an observable, and ρ eq ( ∞ ) is the equilibrium density matrix at infinite temperature. The oscillatory function g(t) is the Fourier transform of the average GOE level density and falls off as 1 / | t | 3 / 2 for large t. With g ( t ) = g ( − t ) , thermalization is symmetric in time. Analogous results, including the symmetry in time of thermalization, are derived for the time-reversal non-invariant Gaussian Unitary Ensemble of random matrices. Comparison with the ‘eigenstate thermalization hypothesis’ of (Srednicki 1999 J. Phys. A: Math. Gen. 32 1163) shows overall agreement but raises significant questions.
If we prepare an isolated, interacting quantum system in an eigenstate and perturb a local observable at an initial time, its expectation value will relax towards a thermal expectation value, even though the time evolution of the system is deterministic. The eigenstate thermalization hypothesis (ETH) of Deutsch and Srednicki suggests that this is possible because each eigenstate of the full quantum system acts as a thermal bath to its subsystems, such that the reduced density matrices of the subsystems resemble thermal density matrices. Here, we use the observation that the eigenvalue distribution of interacting quantum systems is a Gaussian under very general circumstances, and Dyson Brownian motion random matrix theory, to derive the ETH and thereby elevate it from hypothesis to theory. Our analysis provides a derivation of statistical mechanics which neither requires the concepts of ergodicity or typicality, nor that of entropy. Thermodynamic equilibrium follows solely from the applicability of quantum mechanics to large systems and the absence of integrability.
We analyze thermalization and eigenstate thermalization in the random free fermion model, a quadratic Hamiltonian with a Gaussian Orthogonal Ensemble hopping matrix, by an exact random-matrix calculation. For a single-particle observable A we compute the ensemble average and fluctuations of Tr(Aρ(t)): we derive the relaxation envelope g2(t/τλ) with τλ=ℏ/(2ηN) and its oscillatory ∣t∣−3 tail, both properties of the semicircular limiting density, and we show that the variance of the fluctuations is O(1/N), so that the relaxation profile self-averages at fixed scaled time. Incorporating particle-number conservation, we prove a factorization theorem for the energy-shell statistics of the many-body eigenstates: the energy-independent ETH structure follows from the statistical independence of the eigenvector matrix and the spectrum, and at fixed realization the energy constraint enters only through a covariance subtraction of relative order 1/N. A controlled eigenvector--eigenvalue correlation deformation H(ε)=H0+εH1 then generates, at first order in ε, the energy dependence of the diagonal energy-resolved slope and of the off-diagonal spectral function, a single-particle diagnostic of the ETH structure in the allowed particle-hole sector. Correlation-function fluctuations and the single-particle spectral form factor finally identify the subleading O(1/N) late-time fingerprints of two-level spectral correlations. The model thus separates eigenbasis chaos from spectral chaos and identifies the statistical origin of energy-independent versus energy-dependent ETH.
Z 2 symmetry is ubiquitous in quantum mechanics, where it drives various phase transitions and emergent physics. The role of Z 2 symmetry in the thermalization of a local observable in a disordered system can be understood using random matrix theory. To do so, we consider random symmetric centrosymmetric (SC) matrix as a toy model where a Z 2 symmetry, namely, the exchange symmetry, is conserved. Such a conservation law splits the Hilbert space into decoupled subspaces such that the energy spectrum of an SC matrix is a superposition of two pure spectra. After discussing the known results on the correlations of such mixed spectra, we consider different initial states and analytically compute the time evolution of their survival probability and associated timescales. We show that there exist certain low-energy initial states which do not decay over a very long timescales such that a measure zero fraction of random SC matrices exhibit spontaneous symmetry breaking. Later, we look at the equilibrium values of local observables like the density-density correlation, kinetic energy operator, and compare them against the average values from the microcanonical and canonical ensembles. We find that when the observable violates (respects) the global symmetry of the Hamiltonian, the equilibrium value is independent (dependent) of the symmetry of the initial state. However, irrespective of such symmetry constraints, the fluctuations of the diagonal terms of the observables within microcanonical shells decay with system size such that the ansatz of the eigenstate thermalization hypothesis remains valid. We show that the equilibrium value converges to the canonical average for all the observables and initial states, indicating that thermalization occurs despite the presence of a global symmetry.
In this work, we consider a model of a subsystem interacting with a reservoir and study dynamics of entanglement assuming that the overall time-evolution is governed by non-integrable Hamiltonians. We also compare to an ensemble of Integrable Hamiltonians. To do this, we make use of unitary invariant ensembles of random matrices with either Wigner-Dyson or Poissonian distributions of energy. Using the theory of Weingarten functions, we derive universal average time evolution of the reduced density matrix and the purity and compare these results with several physical Hamiltonians: randomized versions of the transverse field Ising and XXZ models, Spin Glass and, Central Spin and SYK model. The theory excels at describing the latter two. Along the way, we find general expressions for exponential n-point correlation functions in the gas of GUE eigenvalues.
We derive the eigenstate thermalization hypothesis (ETH) from a random matrix Hamiltonian by extending the model introduced by Deutsch (1991 Phys. Rev. A 43 2046). We approximate the coupling between a subsystem and a many-body environment by means of a random Gaussian matrix. We show that a common assumption in the analysis of quantum chaotic systems, namely the treatment of eigenstates as independent random vectors, leads to inconsistent results. However, a consistent approach to the ETH can be developed by introducing an interaction between random wave-functions that arises as a result of the orthonormality condition. This approach leads to a consistent form for off-diagonal matrix elements of observables. From there we obtain the scaling of time-averaged fluctuations of generic observables with system size for which we calculate an analytic form in terms of the inverse participation ratio. The analytic results are compared to exact diagonalizations of a quantum spin chain for different physical observables in multiple parameter regimes.
We derive the eigenstate thermalization hypothesis (ETH) from a random matrix Hamiltonian by extending the model introduced by Deutsch (1991 Phys. Rev. A 43 2046). We approximate the coupling between a subsystem and a many-body environment by means of a random Gaussian matrix. We show that a common assumption in the analysis of quantum chaotic systems, namely the treatment of eigenstates as independent random vectors, leads to inconsistent results. However, a consistent approach to the ETH can be developed by introducing an interaction between random wave-functions that arises as a result of the orthonormality condition. This approach leads to a consistent form for off-diagonal matrix elements of observables. From there we obtain the scaling of time-averaged fluctuations of generic observables with system size for which we calculate an analytic form in terms of the inverse participation ratio. The analytic results are compared to exact diagonalizations of a quantum spin chain for different physical observables in multiple parameter regimes.