Starting from a product initial state, equal-time correlations in nonrelativistic quantum lattice models propagate within a lightcone-like causal region. The presence of entanglement in the initial state can modify this behavior, enhancing and accele
rating the growth of correlations. In this paper we give a quantitative description, in the form of Lieb-Robinson-type bounds on equal-time correlation functions, of the interplay of dynamics vs. initial entanglement in quantum lattice models out of equilibrium. We test the bounds against model calculations, and also discuss applications to quantum quenches, quantum channels, and Kondo physics.
The long-time asymptotic behavior is studied for a long-range variant of the Emch-Radin model of interacting spins. We derive upper and lower bounds on the expectation values of a class of observables. We prove analytically that the time scale at whi
ch the system relaxes to equilibrium diverges with the system size N, displaying quasistationary nonequilibrium behavior. This finding implies that, for large enough N, equilibration will not be observed in an experiment of finite duration.
The approach to equilibrium is studied for long-range quantum Ising models where the interaction strength decays like r^{-alpha} at large distances r with an exponent $alpha$ not exceeding the lattice dimension. For a large class of observables and i
nitial states, the time evolution of expectation values can be calculated. We prove analytically that, at a given instant of time t and for sufficiently large system size N, the expectation value of some observable <A>(t) will practically be unchanged from its initial value <A>(0). This finding implies that, for large enough N, equilibration effectively occurs on a time scale beyond the experimentally accessible one and will not be observed in practice.
Nonanalyticities of thermodynamic functions are studied by adopting an approach based on stationary points of the potential energy. For finite systems, each stationary point is found to cause a nonanalyticity in the microcanonical entropy, and the fu
nctional form of this nonanalytic term is derived explicitly. With increasing system size, the order of the nonanalytic term grows, leading to an increasing differentiability of the entropy. It is found that only asymptotically flat stationary points may cause a nonanalyticity that survives in the thermodynamic limit, and this property is used to derive an analytic criterion establishing the existence or absence of phase transitions. We sketch how this result can be employed to analytically compute transition energies of classical spin models.
The microcanonical entropy s(e,m) as a function of the energy e and the magnetization m is computed analytically for the anisotropic quantum Heisenberg model with Curie-Weiss-type interactions. The result shows a number of interesting properties whic
h are peculiar to long-range interacting systems, including nonequivalence of ensembles and partial equivalence. Furthermore, from the shape of the entropy it follows that the Curie-Weiss Heisenberg model is indistinguishable from the Curie-Weiss Ising model in canonical thermodynamics, although their microcanonical thermodynamics in general differs. The possibility of experimentally realizing quantum spin models with long-range interactions in a microcanonical setting by means of cold dipolar gases in optical lattices is discussed.
For the spherical model with nearest-neighbour interactions, the microcanonical entropy s(e,m) is computed analytically in the thermodynamic limit for all accessible values of the energy e and the magnetization m per spin. The entropy function is fou
nd to be concave (albeit not strictly concave), implying that the microcanonical and the canonical ensembles are equivalent, despite the long-range nature of the spherical constraint the spins have to obey. Two transition lines are identified in the (e,m)-plane, separating a paramagnetic phase from a ferromagnetic and an antiferromagnetic one. The resulting microcanonical phase diagram is compared to the more familiar canonical one.
