We investigate the quench dynamics of the transverse field Ising model on a finite fully connected lattice as a prime example of non-equilibrium mean field dynamics. Using a rate function approach we compute the leading order corrections to the mean field behavior analytically. Our focus is threefold: i) We analyze the validity of the mean field approximation and observe that deviations can occur quickly even for large systems. ii) We study the variance of the order parameter and identify four qualitative different regions that cannot be distinguished by solely looking at its mean. iii) We derive the complete entanglement Hamiltonian for a bipartition of the lattice, which remarkably turns out to be a time-dependent harmonic oscillator within the validity of the mean field analysis.
We investigate the quench dynamics of the transverse field Ising model on a finite fully connected lattice as a prime example of non-equilibrium mean field dynamics. Using a rate function approach we compute the leading order corrections to the mean field behavior analytically. Our focus is threefold: i) We analyze the validity of the mean field approximation and observe that deviations can occur quickly even for large systems. ii) We study the variance of the order parameter and identify four qualitative different regions that cannot be distinguished by solely looking at its mean. iii) We derive the complete entanglement Hamiltonian for a bipartition of the lattice, which remarkably turns out to be a time-dependent harmonic oscillator within the validity of the mean field analysis.
The transverse field Ising model of N fully connected spins provides a mathematical accessible, yet non-trivial, system to investigate the non-thermal dynamics after a sudden quantum quench. The focus in this thesis is three-fold. First, in a 1/N expansion the dynamics of the mean magnetization within its variance is obtained analytically in the large N limit. The variance constitutes a leading order correction to the mean field limit at infinite N. While mean field theories are one of the most accessible approximations to deal with the quantum complexity, its validity in time out of equilibrium has to be checked on a case by case basis. By studying the dynamics of the variance, we see that the mean field approximation can break down at surprisingly early times scaling with the square root of N. The underlying mechanism in the fully connected Ising model is identified as a dephasing effect between nearby effective orbits. The dynamics of the magnetization variance discriminates four qualitatively different regions in the dynamical phase diagram that are indistinguishable by just looking at the mean magnetization. Second, bipartite entanglement between disjoint groups of the spins in the fully connected Ising model is studied. In the large N limit the entanglement Hamiltonian is shown to be a harmonic oscillator. A quantitative relation between the angular frequency of the oscillator and a squeezing measure of the spin state is derived. Explicit expressions for the Renyi entanglement entropies are given. In contrast to the variance, which vanishes in the thermodynamic limit, the entanglement entropy saturate to an N-independent positive value. The dynamics of the entanglement Hamiltonian and the entanglement entropies is discussed. Linear growth as a function of time is linked to the unstable fixed point and the homoclinic orbit of a pitchfork bifurcation, while logarithmic increase is explained due to a dephasing mechanism of nearby effective orbits. A generalization to other models that permit a semiclassical effective description (on a possibly higher dimensional phase space) is given. As a byproduct the connection between entangled Gaussian states and the symplectic eigenvalues of its reduced covariance is derived. An upper bound on the von Neumann entanglement entropy of any bipartite pure state is given in terms of the symplectic properties of its reduced covariance. Third, the existence of dynamical phase transitions (DPTs) in the fully connected Ising model is argued by two different approaches. DPTs are defined as non-analyticities in the time-dependence of the return probability rate function in the thermodynamic limit. Like entanglement entropy, the behavior of the return rate function is an observable-independent quantity used to characterize non-equilibrium states. A numerically based connection between the DPT and the analytic property of the infinite time averaged order parameter as a function of quench strength is discussed. The concept of Feynman history states is used to devise the notion of history entanglement. An information theoretic interpretation of the history entanglement entropy as the precision of a quantum clock follows from Holevo’s bound. The dependence of this entanglement entropy on the observation time constitutes an operator-independent characterization of the quantum dynamics in closed systems. For observation times comparable to the Heisenberg time, the history entanglement becomes sensitive to the statistics of the energy gaps, and thus distinguishes integrable and non-integrable dynamics. This is demonstrated for two examples, a random matrix ensemble, and a one-dimensional hardcore boson model with integrability breaking next nearest neighbor interaction.
We construct the finite-temperature dynamical phase diagram of the fully connected transverse-field Ising model from the vantage point of two disparate concepts of dynamical criticality. An analytical derivation of the classical dynamics and exact diagonalization simulations are used to study the dynamics after a quantum quench in the system prepared in a thermal equilibrium state. The different dynamical phases characterized by the type of nonanalyticities that emerge in an appropriately defined Loschmidt-echo return rate directly correspond to the dynamical phases determined by the spontaneous breaking of Z2 symmetry in the long-time steady state. The dynamical phase diagram is qualitatively different depending on whether the initial thermal state is ferromagnetic or paramagnetic. Whereas the former leads to a dynamical phase diagram that can be directly related to its equilibrium counterpart, the latter gives rise to a divergent dynamical critical temperature at vanishing final transverse-field strength.
We study dynamical properties of the finite-size fully connected Ising model with a transverse field at zero temperature. In a quench dynamics, we study the time period and the first critical time, which play important roles in the dynamical phase transitions, based on a dynamical order parameter and the Loschmidt rate, respectively. When all the spins are initially polarized in the direction of their mutual interaction, we show that both the time period and critical time diverge logarithmically with the system size at the dynamical critical point. When all the spins are initially in the direction of transverse field, both the time period and critical time exhibit logarithmic or power-law divergences depending on the final field strength. In the case of convergence, we provide estimates for the finite-size scaling and converged value. We also investigate the equilibrium phase transition, presenting approximate ground and first excited states away from the criticality, and compare their energy gap and bipartite and multipartite entanglements with the exact eigenstates.
We study the nonequilibrium dynamics in the fermionic Hubbard model after a sudden change of the interaction strength. To this scope, we introduce a time-dependent variational approach in the spirit of the Gutzwiller ansatz. At the saddle-point approximation, we find at half filling a sharp transition between two different regimes of small and large coherent oscillations, separated by a critical line of quenches where the system is found to relax. Any finite doping washes out the transition, leaving aside just a sharp crossover. In order to investigate the role of quantum fluctuations, we map the model onto an auxiliary quantum Ising model in a transverse field coupled to free fermionic quasiparticles. Remarkably, the Gutzwiller approximation turns out to correspond to the mean-field decoupling of this model in the limit of infinite coordination lattices. The advantage is that we can go beyond mean field and include Gaussian fluctuations around the non-equilibrium mean-field dynamics. Unlike at equilibrium, we find that quantum fluctuations become massless and eventually unstable before the mean-field dynamical critical line, which suggests they could even alter qualitatively the mean-field scenario.
We investigate the non-equilibrium dynamics of the transverse field quantum Ising chain evolving from an inhomogeneous initial state given by joining two macroscopically different semi-infinite chains. We obtain integral expressions for all two-point correlation functions of the Jordan--Wigner Majorana fermions at any time and for any value of the transverse field. Using this result, we analytically compute the profiles of various physical observables in the space-time scaling limit and show that they can be obtained from a hydrodynamic picture based on ballistically propagating quasiparticles. Going beyond the hydrodynamic limit, we analyze the approach to the non-equilibrium steady state and find that the leading late time corrections display a lattice effect. We also study the fine structure of the propagating fronts which are found to be described by the Airy kernel and its derivatives. Near the front we observe the phenomenon of energy back-flow where the energy locally flows from the colder to the hotter region.
In the context of out of equilibrium dynamics of isolated interacting quantum many-body systems, we study the mean-field discretized and nonlinear Gross-Pitaevskii equation (GPE) of motion for a fully connected (or all-to-all coupled) Bose-Hubbard model describing a large population of bosons on a lattice of generic dimension V, with potential experimental applications ranging from ultra-cold atomic gases in optical traps to systems of Josephson junctions. Considering a quench on the system Hamiltonian, we describe the time evolution of a system initialized to the Mott insulating phase and taken in the strong superfluid regime, finding a sharp change in the dynamical behaviour when varying the quench intensity across a critical value, that is a dynamical phase transition. Our results generalize the study presented in some previous papers focusing on the particular cases of V = 2, V = 3 and 1D lattice, and place particular emphasis on the relaxation of some dynamical order parameters in the large V limit. On the side of the dynamical phase transition corresponding to the strong superfluid regime, we find such relaxation to consist of a π-synchronization of the mean field bosonic phases, that we find to compete against the eventual site-dependent disorder of the Hamiltonian in what, in analogy with the notorious Kuramoto model for nonlinear coupled oscillators, can be referred to as a synchronization transition. Additionally, in support of the validity of the mean field approximation far from the transition, we report a study beyond mean field based on the Bogolioubov-de Gennes method.