For time-independent excited states in conformal field theories, the entanglement entropy of small subsystems satisfies a ‘first law’-like relation, in which the change in entanglement is proportional to the energy within the entangling region. Such a law holds for time-dependent scenarios as long as the state is perturbatively close to the vacuum, but is not expected otherwise. In this paper we use holography to investigate the spread of entanglement entropy for unitary evolutions of special physical interest, the so-called global quenches. We model these using AdS-Vaidya geometries. We find that the first law of entanglement is replaced by a linear response relation, in which the energy density takes the role of the source and is integrated against a time-dependent kernel with compact support. For adiabatic quenches the standard first law is recovered, while for rapid quenches the linear response includes an extra term that encodes the process of thermalization. This extra term has properties that resemble a time-dependent ‘relative entropy’. We propose that this quantity serves as a useful order parameter to characterize far-from-equilibrium excited states. We illustrate our findings with concrete examples, including generic power-law and periodically driven quenches.
For time-independent excited states in conformal field theories, the entanglement entropy of small subsystems satisfies a ‘first law’-like relation, in which the change in entanglement is proportional to the energy within the entangling region. Such a law holds for time-dependent scenarios as long as the state is perturbatively close to the vacuum, but is not expected otherwise. In this paper we use holography to investigate the spread of entanglement entropy for unitary evolutions of special physical interest, the so-called global quenches. We model these using AdS-Vaidya geometries. We find that the first law of entanglement is replaced by a linear response relation, in which the energy density takes the role of the source and is integrated against a time-dependent kernel with compact support. For adiabatic quenches the standard first law is recovered, while for rapid quenches the linear response includes an extra term that encodes the process of thermalization. This extra term has properties that resemble a time-dependent ‘relative entropy’. We propose that this quantity serves as a useful order parameter to characterize far-from-equilibrium excited states. We illustrate our findings with concrete examples, including generic power-law and periodically driven quenches.
We use a simple holographic toy model to study global quantum quenches in strongly coupled, hyperscaling-violating-Lifshitz quantum field theories using entanglement entropy as a probe. Generalizing our conformal field theory results, we show that the holographic entanglement entropy of small subsystems can be written as a simple linear response relation. We use this relation to derive a time-dependent first law of entanglement entropy. In general, this law has a time-dependent term resembling relative entropy which we propose as a good order parameter to characterize out-of-equilibrium states in the post-quench evolution. We use these tools to study a broad class of quantum quenches in detail: instantaneous, power law, and periodic.
Growth of entanglement entropy in time-dependent states formed due to a global quench in holographic conformal field theories which admit an Einstein-Gauss-Bonnet dual gravity description is studied. The global quench in the bulk is modelled by an AdS-Vaidya solution with an electric charge. It is observed that the Gauss-Bonnet correction parameter leads to faster thermalization, and lower saturation entropy. The rate of growth also depends crucially on the correction, and may exceed 1 in general spacetime dimensions. Nevertheless, the growth still follows the universal pattern expected for relativistic CFTs. Additionally, a time-dependent analogue of relative entropy introduced in [1] is generalized to include correction from the Gauss-Bonnet parameter. We demonstrate our findings through concrete examples, including instantaneous, linear, and periodically driven quenches. We also briefly mention the evolution of mutual information.
For a perturbation of the state of a conformal field theory (CFT), the response of the entanglement entropy is governed by the so-called ``first law'' of entanglement entropy, in which the change in entanglement entropy is proportional to the change in energy. Whether such a first law holds for other types of perturbations, such as a change to the CFT Lagrangian, remains an open question. We use holography to study the evolution in time t of entanglement entropy for a CFT driven by a t-linear source for a conserved U(1) current or marginal scalar operator. We find that although the usual first law of entanglement entropy may be violated, a first law for the rates of change of entanglement entropy and energy still holds. More generally, we prove that this first law for rates holds in holography for any asymptotically (d+1)-dimensional anti--de Sitter metric perturbation whose t dependence first appears at order zd in the Fefferman-Graham expansion about the boundary at z=0.
Time-dependent entanglement entropy (EE) is computed for a single interval in two-dimensional conformal theories from a quenched initial state in the presence of spatial boundaries. The EE is found to be periodic in time with periodicity equal to the system size L. For large enough L, the EE shows a rise to a thermal value (characterized by a temperature 1/β determined by the initial state), followed by periodic returns to the original value. This works irrespective of whether the conformal field theory (CFT) is rational or irrational. For conformal field theories with a holographic dual, the large c limit plays an essential role in ensuring that the EE computed from the CFT is universal (independent of the details of the CFT and of boundary conditions) and is exactly matched by the holographic EE. The dual geometry is computed and it interpolates between a BTZ black hole at large L and global AdS at large β.
In this thesis we study the time dependence of correlators such as Green function, entanglement (Renyi) entropy and mutual information following quantum quench in Conformal Field Theories (CFTs) and their holographic duals (AdS).
We develop a new recursive method to analytically study the thermalization of retarded Green function in CFTs holographically dual to thin shell AdS Vaidya space times. Using this method, analytic results for short time thermalization of the Green function are obtained; its long time behaviour is obtained by implementing this method numerically. This method is also used to study the late time behaviour of the minimally coupled scalar in AdS3 Vaidya and the shear vector mode in AdS5 Vaidya. Studying the latter, we obtain the universal ratio of the shear viscosity to entropy density from a time dependent process.
We study the time evolution of single interval Renyi and entanglement entropies following local quantum quenches in CFTs at finite temperature for which the locally excited states have a finite temporal width. We show that the time dependence of Renyi and entanglement entropies, at second order in width, is universal and is determined by the expectation value of the stress tensor in the replica geometry.
We study local quenches in two dimensional CFT at large central charge by operators carrying higher spin charge. The holographic description is used to obtain the single interval entanglement entropy, mutual information and scrambling time following the quench. We find that the change in entanglement entropy is finite (and real) only if the spin-three charge, q, is bounded by the energy of the perturbation, E, as |q|/c < E^2/c^2. For larger values of the spin-three charge, the scrambling time is shorter than for pure gravity and controlled by the spin-three Lyapunov exponent.
We use holographic duality to study the entanglement entropy (EE) of Conformal Field Theories (CFTs) in various spacetime dimensions d, in the presence of various deformations: a relevant Lorentz scalar operator with constant source, a temperature T , a chemical potential μ, a marginal Lorentz scalar operator with source linear in a spatial coordinate, and a circle-compactified spatial direction. We consider EE between a strip or sphere sub-region and the rest of the system, and define the “entanglement density” (ED) as the change in EE due to the deformation, divided by the sub-region’s volume. Using the deformed CFTs above, we show how the ED’s dependence on the strip width or sphere radius, L, is useful for characterizing states of matter. For example, the ED’s small-L behavior is determined either by the dimension of the perturbing operator or by the first law of EE. For Lorentz-invariant renormalization group (RG) flows between CFTs, the “area theorem” states that the coefficient of the EE’s area law term must be larger in the UV than in the IR. In these cases the ED must therefore approach zero from below as L→∞. However, when Lorentz symmetry is broken and the IR fixed point has different scaling from the UV, we find that the ED often approaches the thermal entropy density from above, indicating area theorem violation.