We show that in classical spin systems, the precise nature of the late-time hydrodynamic tails of the autocorrelation functions of a generic observable is determined by (i) the dynamical critical exponent and (ii) the equilibrium thermodynamic properties of the corresponding observable. We provide numerical results for one- and two-dimensional systems and present theoretical considerations that only rely on the notion of ergodicity. Our result extends to the classical framework the relaxation-overlap inequality, first introduced in [Phys. Rev. X 15, 011059 (2025)2160-330810.1103/PhysRevX.15.011059] for quantum many-body systems satisfying the eigenstate thermalization hypothesis.
We show that in classical spin systems, the precise nature of the late-time hydrodynamic tails of the autocorrelation functions of a generic observable is determined by (i) the dynamical critical exponent and (ii) the equilibrium thermodynamic properties of the corresponding observable. We provide numerical results for one- and two-dimensional systems and present theoretical considerations that only rely on the notion of ergodicity. Our result extends to the classical framework the relaxation-overlap inequality, first introduced in [Phys. Rev. X 15, 011059 (2025)2160-330810.1103/PhysRevX.15.011059] for quantum many-body systems satisfying the eigenstate thermalization hypothesis.
Boltzmann's ergodic hypothesis furnishes a possible explanation for the emergence of statistical mechanics in the framework of classical physics. In quantum mechanics, the Eigenstate Thermalization Hypothesis (ETH) is instead generally considered as a possible route to thermalization. This is because the notion of ergodicity itself is vague in the quantum world and it is often simply taken as a synonym for thermalization. Here we show, in an elementary way, that when quantum ergodicity is properly defined, it is, in fact, equivalent to ETH. In turn, ergodicity is equivalent to thermalization, thus implying the equivalence of thermalization and ETH. This result previously appeared in [De Palma et al., Phys. Rev. Lett. 115, 220401 (2015)], but becomes particularly clear in the present context. We also show that it is possible to define a classical analogue of ETH which is implicitly assumed to be satisfied when constructing classical statistical mechanics. Classical and quantum statistical mechanics are built according to the familiar standard prescription. This prescription, however, is ontologically justified only in the quantum world.
The eigenstate thermalization hypothesis (ETH) describes the properties of diagonal and off-diagonal matrix elements of local operators in the eigenenergy basis. In this work, we propose a relation between (i) the singular behavior of the off-diagonal part of ETH at small energy differences and (ii) the smooth profile of the diagonal part of ETH as a function of the energy density. We establish this connection from the decay of the autocorrelation functions of local operators, which is constrained by the presence of local conserved quantities whose evolution is described by hydrodynamics. We corroborate our predictions through numerical simulations of two distinct nonintegrable spin-1 Ising models, exhibiting diffusive and superdiffusive transport behaviors. The simulations are performed using dynamical quantum typicality up to 18 spins, for both infinite- and finite-temperature regimes.
Abstract Obtaining rigorous and general results about the non-equilibrium dynamics of extended many-body systems is a difficult task. In quantum lattice models with short-range interactions, the Lieb–Robinson bound tells us that the spatial extent of operators grows at most linearly in time. But what happens within this light-cone? We discuss rigorous results on ergodicity and the emergence of the Euler hydrodynamic scale in correlation functions, which establish fundamental principles at the root of non-equilibrium physics. One key idea of the present work is that general structures of Euler hydrodynamics, obtained under ballistic scaling, follow independently from the details of the microscopic dynamics, and in particular do not necessitate chaos; they are consequences of “extensivity”. Another crucial observation is that these apply at arbitrary frequencies and wavelengths. That is, long-time, persistent oscillations of correlation functions over ballistic regions of spacetime, which may be of microscopic frequencies and wavelengths, are predicted by a general Euler-hydrodynamic theory that takes the same form as that for smoothed-out correlation functions. This involves a natural extension of notions of conserved quantities and hydrodynamic projection and shows that the Euler hydrodynamic paradigm covers the full frequency-wavelength plane.
We investigate the relation between the classical ergodicity and the quantum eigenstate thermalization in the fully connected Ising ferromagnets. In the case of spin-1/2, an expectation value of an observable in a single-energy eigenstate coincides with the long-time average in the underlying classical dynamics, which is a consequence of the Wentzel-Kramers-Brillouin approximation. In the case of spin-1, the underlying classical dynamics is not necessarily ergodic. In that case, it turns out that, in the thermodynamic limit, the statistics of the expectation values of an observable in the energy eigenstates coincides with the statistics of the long-time averages in the underlying classical dynamics starting from random initial states sampled uniformly from the classical phase space. This feature seems to be a general property in semiclassical systems, and the result presented here is crucial in discussing equilibration, thermalization, and dynamical transitions of such systems.
We provide a pure state formulation for hydrodynamics of isolated quantum many-body systems. A pure state describing quantum systems in local thermal equilibrium is constructed, which we call a local thermal pure quantum ($\ensuremath{\ell}\mathrm{TPQ}$) state. We show that the thermodynamic functional and the expectation values of local operators (including a real-time correlation function) calculated from the $\ensuremath{\ell}\mathrm{TPQ}$ state converge to those from a local Gibbs ensemble in the large fluid-cell limit. As a numerical demonstration, we investigate a one-dimensional spin chain and observe the hydrodynamic relaxation obeying Fourier's law. We further prove the second law of thermodynamics and the quantum fluctuation theorem, which are also validated numerically. The $\ensuremath{\ell}\mathrm{TPQ}$ formulation gives a useful theoretical basis to describe the emergent hydrodynamic behavior of quantum many-body systems furnished with a numerical efficiency applicable to both the nonrelativistic and relativistic regimes.
Autocorrelation functions play a key role in the theoretical characterization of the dynamical properties of interacting many-body quantum systems. Recently, bringing together the eigenstate thermalization hypothesis and hydrodynamics, it was theoretically predicted that the relaxation of autocorrelators can be described by the \textit{relaxation-overlap inequality}, which, when saturated, predicts a hierarchy of relaxation exponents for a set of operators that are easily identified and constructed. Here, we employ a trapped-ion quantum simulator to experimentally demonstrate it in the diffusive regime; to do so, we have extended the theory of the overlap-relaxation inequality to systems with multiple conservation laws. Our study thus opens the path to a thorough characterization of the relaxation to equilibrium and appearance of hydrodynamic behavior in quantum matter.