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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.

Studying the evolution of quantum field theories (QFTs) after generic time-dependent perturbations is an important problem. If

Study of global quenches was initiated in [

Entanglement entropy of a subregion

In such cases, the AdS/CFT correspondence [

In 2006, Ryu and Takayanagi [

Using the HRT proposal, [_{3}. If these operators have conformal dimensions much smaller than the central charge, the bulk fields are light and then the quench is described by a 3D geometry with a thin, uniform shell of infalling matter that forms a black hole. This geometry is well-studied and known as the_{4}. They also observed a linear behavior for the growth of entanglement entropy, but noticed some novel phenomena, such as a discontinuity in the time derivative of entanglement entropy near the time when it is about to saturate. Motivated by these examples, [

However, these works focused on the limit of large subsystem sizes. Reference [

In this paper, we would like to study global quantum quenches when the CFT is in an excited state

The idea of investigating holography for theories without conformal symmetry is not new. See [

In this paper, we will study a class of excited states in the CFTs that is further qualified as follows. The IR geometry describing these states has an extra critical exponent, called the

Hyperscaling violating metrics can be obtained from the action [

The plan of the paper is as follows. In Section

From now on, we will work in an asymptotically

In this subsection, we will discuss holographic renormalization for asymptotically

For a review of holographic renormalization in the case of CFTs, see [

When we do a global quantum quench in the field theory, this will in general change the geometry by a finite amount but because the subregion

But the area of the HRT surface will be divergent, reflecting the fact that entanglement entropy of

In the previous subsection, we have made precise the class of universal corrections we will aim to capture by calculating entanglement entropy holographically. In this subsection, we briefly review the results of [

Let us start with the

We will denote the duration of the quench by

where we have normalized the value of

In this case, we keep on perturbing the system indefinitely in time and as a result keep on inserting energy. One can still formally expand in a small parameter

In this paper, we will mainly study quenches of finite duration.

In [

Equation (

As we argue in equation (

For

If a source is a linear combination of two independent sources

A convolution is left invariant by translating it

If

If

There always exists a First Law of Entanglement Entropy for small subregions. For the kind of quenches we are studying, the reduced density matrix for the subregion

In this subsection, we will show that there exists such a first law even in the time-dependent case for the small subregions. In (

We now generalize this law to the case of adiabatic sources. In this case, the function

Naively, this gives us the full expression for the time-dependent change in entanglement entropy. However, there is still a choice of the integration constant in the definition of the function

In time-independent cases, relative entropy between two density matrices

We can in fact show that, for

Apart from providing an order parameter to understand out-of-equilibrium states, the time-dependent relative entropy also helps us to organize the post-quench time evolution of the field theory. To see how this is done, let us first calculate the time derivative of the relative entropy using the differentiation rule of convolution

For

For

The distance in the Hilbert space between the vacuum and the excited state thus keeps decreasing with time in this regime.

In this section, we will study the growth of entanglement for small subsystems for some explicit quenches. As we will see, they cover a wide range of time-dependent perturbations to the field theory.

As a first example, we will study the instantaneous global quantum quench defined by

When

When

For

In terms of this parameter, the time evolution of the entanglement entropy is

where

Having described the regimes, we now observe that the problem of studying the evolution of entanglement entropy simplifies to the problem of studying the behavior of

Entanglement entropy (a) and relative entropy (b) after an instantaneous quench in

We now study in detail the time-dependent regime. To organize our study, we observe that the time-dependent regime can be further classified into three different subregimes. The existence of these subregimes can also be seen from Figure

Entanglement velocity as a function of time for different values of

To understand this instantaneous velocity better, we study it in some limits. First we set

Time dependence of entanglement velocity for different values of

The maximum entanglement velocity in our case can be found using

(a) Maximum and (b) average entanglement velocities as a function of

It is also interesting to consider the maximum entanglement velocity as a function of

(a) Maximum and (b) average entanglement velocities as a function of

We now consider global quench that is a power law with respect to time. We will denote the power by

In Figure

Schematic representation of the convolution integral for a power-law quench with

The pre-quench and post-saturation regimes are in equilibrium. In particular

We now derive analytic expressions for entanglement growth. To write them succinctly, we define the following indefinite integral that depends on the subsystem:

Now, using the binomial series

Entanglement entropy, relative entropy, and entanglement velocity for different powers after a power-law quench. Colors blue, green, orange, and red represent

It is interesting to ask how does the entanglement grow in different dimensions. As shown in Figure

Entanglement entropy, relative entropy, and entanglement velocity for different dimensions. Colors blue, green, orange, and red represent

Finally, in Figures

Entanglement entropy, relative entropy, and entanglement velocity for different values of

Entanglement entropy, relative entropy, and entanglement velocity for different values of

In this subsection we will comment on the case of a linear quench

Time periodic forces are commonly used in laboratory situations. The study of differential equations with a periodic function in the differential operator is called Floquet Theory. Adopting this name, we refer to a quench that is periodic in time as Floquet quench. Thus, the time-dependent function

From the convolution equation (

Expanding the sin function in a Taylor series, the exact expression for the growth of entanglement entropy becomes

In Figure

Entanglement entropy (solid line) as a function of time after a Floquet quench, plotted against the periodic source (dashed line). The colors blue and red denote frequencies

As the dashed red line in Figure

In Figure

Entanglement entropy after a Floquet quench as a function of (a) dimension and (b) frequency. The colors blue, green, orange, and red denote

We see that, at higher dimensions, the amplitude of the entanglement entropy is higher but it decreases as the frequency increases.

In Figure

Entanglement entropy after a Floquet quench as a function of (a)

Amplitude

In Figure

Relative phase

In this paper, we studied global quantum quenches to holographic hyperscaling-violating-Lifshitz (

In Section

Using the perturbative expansion in

In Section

For the case of the power law quench, we studied the entanglement entropy in detail in Section

Finally, in Section

In this paper, we could not study the problem of holographic renormalization in detail. That is an obvious direction to extend our work in. Further, notwithstanding the expectation that our simple holographic toy model captures gross universal features of thermalization, it is not very realistic. It would be interesting to have a more realistic holographic model for studying quenches in

In this appendix, we use the so-called Minimal Approach to find the stress-energy tensor for an asymptotically

We could now consider making the bulk solution time-dependent in an adiabatic way, such that the blackening function is given by

We may now consider a time-dependent bulk geometry that is not a small perturbation of (

The data used to support the findings of this study are included within the article.

The author declares that they have no conflicts of interest.

I would like to thank Juan F. Pedraza for several useful discussions. I am grateful to Juan F. Pedraza, Jay Armas, Rik van Bruekelen, and Gerben W.J. Oling, for suggestions to improve the draft. This work is supported by the Netherlands Organisation for Scientific Research (NWO-I), which is funded by the Dutch Ministry of Education, Culture and Science (OCW).