Abstract We study quantum quenches and subsequent non-equilibrium dynamics of free Dirac fermions in 1 + 1 spacetime dimensions using time dependent mass. The final state is a normalized boundary state which is called generalized Calabrese-Cardy (gCC) state and the system thermalizes to a generalized Gibb’s Ensemble(GGE). We can also tune the initial states so that the final states are exact Calabrese-Cardy (CC) state and special gCC states. The system in the CC state thermalizes to a Gibb’s ensemble. We derive closed-form analytic expressions for the growth of entanglement entropy of subsystems consisting of arbitrary number of disjoint intervals in CC state. We show that the entanglement entropy of a single interval grows monotonically before saturation. In case of certain gCC states, for particular charges, the entanglement entropy of a single interval grows non-monotonically when the effective chemical potential is increased beyond a critical value. We argue that the non-monotonic growth of entanglement entropy is a boundary effect which arises due to increase in long range correlation and decrease in short range correlation at early times.
Abstract We study quantum quenches and subsequent non-equilibrium dynamics of free Dirac fermions in 1 + 1 spacetime dimensions using time dependent mass. The final state is a normalized boundary state which is called generalized Calabrese-Cardy (gCC) state and the system thermalizes to a generalized Gibb’s Ensemble(GGE). We can also tune the initial states so that the final states are exact Calabrese-Cardy (CC) state and special gCC states. The system in the CC state thermalizes to a Gibb’s ensemble. We derive closed-form analytic expressions for the growth of entanglement entropy of subsystems consisting of arbitrary number of disjoint intervals in CC state. We show that the entanglement entropy of a single interval grows monotonically before saturation. In case of certain gCC states, for particular charges, the entanglement entropy of a single interval grows non-monotonically when the effective chemical potential is increased beyond a critical value. We argue that the non-monotonic growth of entanglement entropy is a boundary effect which arises due to increase in long range correlation and decrease in short range correlation at early times.
We investigate the crossover of the entanglement entropy toward its thermal value in nearly integrable systems. We employ equations-of-motion techniques to study the entanglement dynamics in a lattice model of weakly interacting spinless fermions after a quantum quench. For weak enough interactions we observe a two-step relaxation of the entanglement entropies of finite subsystems. Initially, the entropies follow a nearly integrable evolution, approaching the value predicted by the generalized Gibbs ensemble (GGE) of the unperturbed model. Then, they start a slow drift toward the thermal stationary value described by a standard Gibbs ensemble (GE). While the initial relaxation to the GGE is independent of the interaction, the slow drift from GGE to GE values happens on timescales proportional to the inverse interaction squared. For asymptotically large times and subsystem sizes the dynamics of the entropies can be predicted using a modified quasiparticle picture that keeps track of the evolution of the fermionic occupations caused by the integrability breaking. This picture gives a quantitative description of the results as long as the integrability-breaking timescale is much larger than the one associated with the (quasi)saturation to the GGE. In the opposite limit, the quasiparticle picture still provides the correct late-time behavior, but it underestimates the initial slope of the entanglement entropy.
We consider quantum quenches in models of free scalars and fermions with a generic time-dependent mass m(t) that goes from m0 to zero. We prove that, as anticipated in MSS \cite{Mandal:2015jla}, the post-quench dynamics can be described in terms of a state of the generalized Calabrese-Cardy form ∣ψ⟩= exp[−κ2H−∑n>2∞κnWn]∣Bd⟩. The Wn (n=2,3,..., W2=H) here represent the conserved W∞ charges and ∣Bd⟩ represents a conformal boundary state. Our result holds irrespective of whether the pre-quench state is a ground state or a squeezed state, and is proved without recourse to perturbation expansion in the κn's as in MSS. We compute exact time-dependent correlators for some specific quench protocols m(t). The correlators explicitly show thermalization to a generalized Gibbs ensemble (GGE), with inverse temperature β=4κ2, and chemical potentials μn=4κn. In case the pre-quench state is a ground state, it is possible to retrieve the exact quench protocol m(t) from the final GGE, by an application of inverse scattering techniques. Another notable result, which we interpret as a UV/IR mixing, is that the long distance and long time (IR) behaviour of some correlators depends crucially on all κn's, although they are highly irrelevant couplings in the usual RG parlance. This indicates subtleties in RG arguments when applied to non-equilibrium dynamics.
We study the dynamics of the entanglement in one-dimensional critical quantum systems after a local quench in which two independently thermalized semi-infinite halves are joined to form a homogeneous infinite system and left to evolve unitarily. We show that under certain conditions a nonequilibrium steady state (NESS) is reached instantaneously as soon as the entanglement interval is within the light cone emanating from the contact point. In this steady state, the exact expressions for the entanglement entropy and the logarithmic negativity are in agreement with the steady state density matrix being a boosted thermal state, as expected. We derive various general identities: relating the negativity after the quench with unequal left and right initial temperatures with that where the left and right temperatures are equal; and relating these with the negativity in equilibrium thermal states. In certain regimes the resulting expressions can be analytically evaluated. Immediately after the interval intersects the light cone, we find logarithmic growth. For a very long interval, we find that the negativity approaches a plateau after sufficiently long times, different from its NESS value. The NESS value is reached instantly as soon as the entire interval is contained in the light cone. This provides a theoretical framework explaining recently obtained numerical results.
We consider quantum quenches in models of free scalars and fermions with a generic time-dependent mass m(t) that goes from m0 to zero. We prove that, as anticipated in MSS [1], the post-quench dynamics can be described in terms of a state of the generalized Calabrese-Cardy form |ψ〉 = exp[−κ2H − ∑ >2 ∞ κ n W n ]|Bd〉. The W n (n = 2, 3, . . ., W2 = H) here represent the conserved W∞ charges and |Bd〉 represents a conformal boundary state. Our result holds irrespective of whether the pre-quench state is a ground state or a squeezed state, and is proved without recourse to perturbation expansion in the κ n ’s as in MSS. We compute exact time-dependent correlators for some specific quench protocols m(t). The correlators explicitly show thermalization to a generalized Gibbs ensemble (GGE), with inverse temperature β = 4κ2, and chemical potentials μ n = 4κ n . In case the pre-quench state is a ground state, it is possible to retrieve the exact quench protocol m(t) from the final GGE, by an application of inverse scattering techniques. Another notable result, which we interpret as a UV/IR mixing, is that the long distance and long time (IR) behaviour of some correlators depends crucially on all κ n ’s, although they are highly irrelevant couplings in the usual RG parlance. This indicates subtleties in RG arguments when applied to non-equilibrium dynamics.
We begin a systematic investigation of quench dynamics in higher-dimensional lattice systems considering the case of noninteracting fermions with conserved particle number. We prepare the system in a translational-invariant nonequilibrium initial state, the simplest example being a classical configuration with fermions at fixed positions on the lattice, and let it evolve in time. We characterize the system's dynamics by measuring the entanglement between a finite connected region and its complement. We observe the transmutation of entanglement entropy into thermodynamic entropy and investigate how this process depends on the shape and orientation of the region with respect to the underlying lattice. Interestingly, we find that irregular regions display a distinctive multislope entanglement growth, while the dependence on the orientation angle is generically fairly weak. This is particularly true for regions with a large (discrete) rotational symmetry group. The main tool of our analysis is the celebrated quasiparticle picture of Calabrese and Cardy, which we generalize to describe the case at hand. Specifically, we show that for generic initial configurations (even when restricting to classical ones) one has to allow for the production of multiplets involving n > 2 quasiparticles and carrying nondiagonal correlations. We obtain quantitatively accurate predictions, tested against exact numerics, and propose an efficient Monte Carlo based scheme to evaluate them for arbitrary connected regions of generic higher-dimensional lattices. Published by the American Physical Society 2024
Abstract In this series of works, we study exactly solvable non-unitary time evolutions in one-dimensional quantum critical systems ranging from quantum quenches to time-dependent drivings. In this part I, we are motivated by the recent works of Kontsevich and Segal (2021 arXiv: 2105.10161 ) and Witten (2021 arXiv: 2111.06514 ) on allowable complex spacetime metrics in quantum field theories. In general, such complex spacetime metrics will lead to non-unitary time evolutions. In this work, we study the universal features of such non-unitary time evolutions based on exactly solvable setups. Various physical quantities including the entanglement Hamiltonian and entanglement spectrum, entanglement entropy, and energy density at an arbitrary time can be exactly solved. Due to the damping effect introduced by the complex time, the excitations in the initial state are gradually damped out in time. The non-equilibrium dynamics exhibit universal features that are qualitatively different from the case of real-time evolutions. For instance, for an infinite system after a global quench, the entanglement entropy of the semi-infinite subsystem will grow logarithmically in time, in contrast to the linear growth in a real-time evolution. Moreover, we study numerically the time-dependent driven quantum critical systems with allowable complex spacetime metrics. It is found that the competition between driving and damping leads to a steady state with an interesting entanglement structure.