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Local quantum quenches in critical one-dimensional systems: entanglement, the Loschmidt echo, and light-cone effects

Published Aug 17, 2016• 101 citations
OpenAlex

Abstract

Abstract. We study a particular type of local quench in a generic quantum critical one-dimensional system, using conformal field theory (CFT) techniques, and providing numerical checks of the results in free fermion systems. The system is initially cut into two subsystems A and B which are glued together at time t = 0. We study the entanglement entropy (EE) between the two parts A and B, using previous results by Calabrese and Cardy [1, 2], and further extending them. We also study in detail the (logarithmic) Loschmidt echo (LLE). For finite-size systems both quantities turn out to be (almost) periodic in the scaling limit, and exhibit striking light-cone effects. While these two quantities behave similarly immediately after the quench—namely as c/3 log t for the EE and c/4 log t for the LLE—, we observe some discrepancy once the excitations emitted by the quench bounce on the boundary and evolve within the same subsystem A (or B). The decay of the EE is then non-universal, as noticed by Eisler and Peschel [3]. In contrast, we find that the evolution of the LLE is less sensitive than the EE to non-universal details of the model, and is still accurately described by our CFT prediction. To further probe these light-cone effects, we also introduce a variant of the Loschmidt echo specifically constructed to detect the excitations emitted just after the quench.

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Local quantum quenches in critical one-dimensional systems: entanglement, the Loschmidt echo, and light-cone effects

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We study a particular type of local quench in a generic quantum critical one-dimensional system, using conformal field theory (CFT) techniques, and providing numerical checks of the results in free fermion systems. The system is initially cut into two subsystems A and B which are glued together at time t = 0. We study the entropy of entanglement (EE) between the two parts A and B , using previous results obtained by Calabrese and Cardy (2004 J. Stat. Mech. P06002; 2007 J. Stat. Mech. P10004), and further extending them. We also study in detail the (logarithmic) Loschmidt echo (LLE). For finite size systems both quantities turn out to be (almost) periodic in the scaling limit, and exhibit striking light-cone effects. While these two quantities behave similarly immediately after the quench—namely as c /3log t for the EE and c /4log t for the LLE—we observe some discrepancy once the excitations emitted by the quench bounce on the boundary and evolve within the same subsystem A (or B ). The decay of the EE is then non-universal, as noticed by Eisler and Peschel (2007 J. Stat. Mech. P06005). In contrast, we find that the evolution of the LLE is less sensitive than the EE to non-universal details of the model, and is still accurately described by our CFT prediction. To further probe these light-cone effects, we also introduce a variant of the Loschmidt echo specifically constructed to detect the excitations emitted just after the quench.

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Universal corrections to entanglement entropy of local quantum quenches

Justin R. David, Surbhi Khetrapal, S. Prem Kumar

We study the time evolution of single interval Rényi and entanglement entropies following local quantum quenches in two dimensional conformal field theories at finite temperature for which the locally excited states have a finite temporal width ϵ. We show that, for local quenches produced by the action of a conformal primary field, the time dependence of Rényi and entanglement entropies at order ϵ2 is universal. It is determined by the expectation value of the stress tensor in the replica geometry and proportional to the conformal dimension of the primary field generating the local excitation. We also show that in CFTs with a gravity dual, the ϵ2 correction to the holographic entanglement entropy following a local quench precisely agrees with the CFT prediction. We then consider CFTs admitting a higher spin symmetry and turn on a higher spin chemical potential μ. We calculate the time dependence of the order ϵ2 correction to the entanglement entropy for small μ, and show that the contribution at order μ 2 is universal. We verify our arguments against exact results for minimal models and the free fermion theory.

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Abstract. We study a particular type of local quench in a generic quantum critical one-dimensional system, using conformal field theory (CFT) techniques, and providing numerical checks of the results in free fermion systems. The system is initially cut into two subsystems A and B which are glued together at time t = 0. We study the entanglement entropy (EE) between the two parts A and B, using previous results by Calabrese and Cardy [1, 2], and further extending them. We also study in detail the (logarithmic) Loschmidt echo (LLE). For finite-size systems both quantities turn out to be (almost) periodic in the scaling limit, and exhibit striking light-cone effects. While these two quantities behave similarly immediately after the quench—namely as c/3 log t for the EE and c/4 log t for the LLE—, we observe some discrepancy once the excitations emitted by the quench bounce on the boundary and evolve within the same subsystem A (or B). The decay of the EE is then non-universal, as noticed by Eisler and Peschel [3]. In contrast, we find that the evolution of the LLE is less sensitive than the EE to non-universal details of the model, and is still accurately described by our CFT prediction. To further probe these light-cone effects, we also introduce a variant of the Loschmidt echo specifically constructed to detect the excitations emitted just after the quench.
We study a particular type of local quench in a generic quantum critical one-dimensional system, using conformal field theory (CFT) techniques, and providing numerical checks of the results in free fermion systems. The system is initially cut into two subsystems A and B which are glued together at time t = 0. We study the entropy of entanglement (EE) between the two parts A and B , using previous results obtained by Calabrese and Cardy (2004 J. Stat. Mech. P06002; 2007 J. Stat. Mech. P10004), and further extending them. We also study in detail the (logarithmic) Loschmidt echo (LLE). For finite size systems both quantities turn out to be (almost) periodic in the scaling limit, and exhibit striking light-cone effects. While these two quantities behave similarly immediately after the quench—namely as c /3log t for the EE and c /4log t for the LLE—we observe some discrepancy once the excitations emitted by the quench bounce on the boundary and evolve within the same subsystem A (or B ). The decay of the EE is then non-universal, as noticed by Eisler and Peschel (2007 J. Stat. Mech. P06005). In contrast, we find that the evolution of the LLE is less sensitive than the EE to non-universal details of the model, and is still accurately described by our CFT prediction. To further probe these light-cone effects, we also introduce a variant of the Loschmidt echo specifically constructed to detect the excitations emitted just after the quench.
Abstract We study the effect of two simultaneous local quenches on the evolution of the Loschmidt echo (LE) and entanglement entropy (EE) of a one dimensional transverse Ising model. In this work, one of the local quenches involves the connection of two spin-1/2 chains at a certain time and the other corresponds to a sudden change in the magnitude of the transverse field at a given site in one of the spin chains. We numerically calculate the dynamics associated with the LE and the EE as a result of such double quenches, and discuss the various timescales involved in this problem using the picture of quasiparticles (QPs) generated as a result of such quenches. We perform a detailed analysis of the probability of QPs produced at the two sites and the nature of the QPs in various phases, and obtain interesting results. More specifically, we find partial reflection of these QPs at the defect center or the site of h -quench, resulting in new timescales which have never been reported before.
Quantum quench is a non-equilibrium process where the Hamiltonian is suddenly changed during the quantum evolution. The change can be made by spatially local perturbations (local quench) or globally switching to a completely different Hamiltonian (global quench). This thesis investigates the post-quench non-equilibrium dynamics with an emphasis on the time dependence of the quantum entanglement. We inspect the scaling of entanglement entropy (EE) to learn how correlation and entanglement built up in a quench. We begin with two local quench examples. In Chap. 2, we apply a local operator to the groundstate of the quantum Lifshitz model and monitor the change of the EE. We find that the entanglement grows according to the dynamical exponent z = 2 and then saturates to the scaling dimension of the perturbing operator -- a value representing its strength. In Chap. 3, we study the evolution after connecting two different one-dimensional critical chains at their ends. The Loschmidt echo which measures the similarity between the evolved state and the initial one decays with a power law, whose exponent is the scaling dimension of the defect (junction). Among other conclusions, we see that the local quench dynamics contain universal information of the (critical) theory. In the global quench scenario, the change of the Hamiltonian affects all parts of the system. In this thesis, we focus on the global chaotic quench driven by generic non-integrable Hamiltonians. In Chap. 4, we propose to use the operator entanglement entropy of the unitary operator as a probe. Its fast linear entanglement production is sharply contrasted to the slow logarithmic spreading of the many-body localized system. The entanglement saturation suggests that the evolution operator in the long time can be modeled by a random unitary matrix. In Chap. 5, we construct a random tensor network which consists of random unitary matrices connected locally to model chaotic evolution with local interactions. We find that the entanglement dynamics is mapped to the statistical mechanics of interacting random walks. This appealing emergent picture allows us to understand the universal linear growth as well as the fluctuations of entanglement in a chaotic quench.
Abstract: We discuss the behaviour of holographic entanglement entropy following a local quench in 2+1 dimensional strongly coupled CFTs. The entanglement generated by the quench propagates along an emergent light-cone, reminiscent of the Lieb-Robinson light-cone propagation of correlations in non-relativistic systems. We find the speed of propagation is bounded from below by the entanglement tsunami velocity obtained earlier for global quenches in holographic systems, and from above by the speed of light. The former is realized for sufficiently broad quenches, while the latter pertains for well localized quenches. The non-universal behavior in the intermediate regime appears to stem from finite-size effects. We also note that the entanglement entropy of subsystems reverts to the equilibrium value exponentially fast, in contrast to a much slower equilibration seen in certain spin models.
We investigate a class of local quantum quenches in which the conformal boundary condition of a (1+1)-dimensional conformal field theory is abruptly changed. We derive a remarkably simple and universal expression for the time evolution of one-point functions on the half-line. This result provides a direct description of the propagation of the disturbance generated by the quench and, in turn, allows us to determine the dynamics of bipartite entanglement for subsystems adjacent to the boundary. We show that, once the subsystem becomes fully causally connected to the quench event, the entanglement entropy undergoes a sharp finite jump whose magnitude is universally given by the logarithm of the ratio of the boundary g-factors associated with the initial and final boundary conditions. We benchmark these analytical predictions against Matrix Product State simulations of the critical Ising spin chain, finding excellent agreement. The numerical analysis further allows us to investigate the time evolution of the spin-flip entanglement asymmetry, revealing how the symmetry-breaking perturbation emitted from the boundary propagates through the system. Our results uncover universal dynamical signatures of boundary quenches and establish a direct connection between nonequilibrium entanglement dynamics and boundary critical phenomena.
We study the time evolution of single interval Rényi and entanglement entropies following local quantum quenches in two dimensional conformal field theories at finite temperature for which the locally excited states have a finite temporal width ϵ. We show that, for local quenches produced by the action of a conformal primary field, the time dependence of Rényi and entanglement entropies at order ϵ2 is universal. It is determined by the expectation value of the stress tensor in the replica geometry and proportional to the conformal dimension of the primary field generating the local excitation. We also show that in CFTs with a gravity dual, the ϵ2 correction to the holographic entanglement entropy following a local quench precisely agrees with the CFT prediction. We then consider CFTs admitting a higher spin symmetry and turn on a higher spin chemical potential μ. We calculate the time dependence of the order ϵ2 correction to the entanglement entropy for small μ, and show that the contribution at order μ 2 is universal. We verify our arguments against exact results for minimal models and the free fermion theory.