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Register a new userMeasuring effective temperatures in a generalized Gibbs ensemble
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The local physical properties of an isolated quantum statistical system in the stationary state reached long after a quench are generically described by the Gibbs ensemble, which involves only its Hamiltonian and the temperature as a parameter. If the system is instead integrable, additional quantities conserved by the dynamics intervene in the description of the stationary state. The resulting generalized Gibbs ensemble involves a number of temperature-like parameters, the determination of which is practically difficult. Here we argue that in a number of simple models these parameters can be effectively determined by using fluctuation-dissipation relationships between response and correlation functions of natural observables, quantities which are accessible in experiments.
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In this work, we show that the dissipation in a many-body system under an arbitrary non-equilibrium process is related to the R{e}nyi divergences between two states along the forward and reversed dynamics under very general family of initial conditions. This relation generalizes the links between dissipated work and Renyi divergences to quantum systems with conserved quantities whose equilibrium state is described by the generalized Gibbs ensemble. The relation is applicable for quantum systems with conserved quantities and can be applied to protocols driving the system between integrable and chaotic regimes. We demonstrate our ideas by considering the one-dimensional transverse quantum Ising model which is driven out of equilibrium by the instantaneous switching of the transverse magnetic field.
The generalized Gibbs ensemble (GGE), which involves multiple conserved quantities other than the Hamiltonian, has served as the statistical-mechanical description of the long-time behavior for several isolated integrable quantum systems. The GGE may involve a noncommutative set of conserved quantities in view of the maximum entropy principle, and show that the GGE thus generalized (noncommutative GGE, NCGGE) gives a more qualitatively accurate description of the long-time behaviors than that of the conventional GGE. Providing a clear understanding of why the (NC)GGE well describes the long-time behaviors, we construct, for noninteracting models, the exact NCGGE that describes the long-time behaviors without an error even at finite system size. It is noteworthy that the NCGGE involves nonlocal conserved quantities, which can be necessary for describing long-time behaviors of local observables. We also give some extensions of the NCGGE and demonstrate how accurately they describe the long-time behaviors of few-body observables.
We numerically study the unitary time evolution of a nonintegrable model of hard-core bosons with an extensive number of local Z2 symmetries. We find that the expectation values of local observables in the stationary state are described better by the generalized Gibbs ensemble (GGE) than by the canonical ensemble. We also find that the eigenstate thermalization hypothesis fails for the entire spectrum, but holds true within each symmetry sector, which justifies the GGE. In contrast, if the model has only one global Z2 symmetry or a size-independent number of local Z2 symmetries, we find that the stationary state is described by the canonical ensemble. Thus, the GGE is necessary to describe the stationary state even in a nonintegrable system if it has an extensive number of local symmetries.
In a recent paper, Dunkel and Hilbert [Nature Physics 10, 67-72 (2014)] use an entropy definition due to Gibbs to provide a consistent thermostatistics which forbids negative absolute temperatures. Here we argue that the Gibbs entropy fails to satisfy a basic requirement of thermodynamics, namely that when two bodies are in thermal equilibrium, they should be at the same temperature. The entropy definition due to Boltzmann does meet this test, and moreover in the thermodynamic limit can be shown to satisfy Dunkel and Hilberts consistency criterion. Thus, far from being forbidden, negative temperatures are inevitable, in systems with bounded energy spectra.
We study a classical integrable (Neumann) model describing the motion of a particle on the sphere, subject to harmonic forces. We tackle the problem in the infinite dimensional limit by introducing a soft version in which the spherical constraint is imposed only on average over initial conditions. We show that the Generalized Gibbs Ensemble captures the long-time averages of the soft model. We reveal the full dynamic phase diagram with extended, quasi-condensed, coordinate-, and coordinate and momentum-condensed phases. The scaling properties of the fluctuations allow us to establish in which cases the strict and soft spherical constraints are equivalent, confirming the validity of the GGE hypothesis for the Neumann model on a large portion of the dynamic phase diagram.
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