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date: 27 January 2023

# Quantum Quench and Universal Scaling

• Sumit R. DasSumit R. DasDepartment of Physics and Astronomy, University of Kentucky

### Summary

A quantum quench is a process in which a parameter of a many-body system or quantum field theory is changed in time, taking an initial stationary state into a complicated excited state. Traditionally “quench” refers to a process where this time dependence is fast compared to all scales in the problem. However in recent years the terminology has been generalized to include smooth changes that are slow compared to initial scales in the problem, but become fast compared to the physical scales at some later time, leading to a breakdown of adiabatic evolution. Quantum quench has been recently used as a theoretical tool to study many aspects of nonequilibrium physics like thermalization and universal aspects of critical dynamics. Relatively recent experiments in cold atom systems have implemented such quench protocols, which explore dynamical passages through critical points, and study in detail the process of relaxation to a steady state. On the other hand, quenches which remain adiabatic have been explored as a useful technique in quantum computation.

### Subjects

• Condensed Matter and Materials Physics
• Cosmology and Astrophysics
• Particles and Fields

### 1. Introduction

In many-body physics or quantum field theory, quantum quench refers to a process where a system, initially in some nice state like the vacuum state, is subject to external driving by a time-dependent coupling. We will discuss processes where the coupling asymptotes to a constant at early and late times, so that the time dependence is appreciable over some timescale $δt$. Sometimes “quantum quench” refers specifically to an instantaneous change (i.e. $δt→0$). This article will, however, use the more general definition involving a finite $δt$. The external driving excites the system which then relaxes: this makes quantum quench a useful tool to investigate many issues in nonequilibrium physics.

One set of issues relates to questions of thermalization. Generic interacting systems are expected to thermalize. When the initial state is a pure state, the final state cannot be a mixed thermal state—in this case thermalization means that expectation values of a large class of physically interesting observables approach thermal expectation values. This question is important for the validity of statistical mechanics. These observables include correlation functions, as well as nonlocal quantities like the entanglement entropy of a subregion. A related issue is that of quantum recurrence for finite-sized systems, where the state at some late time has $O(1)$ overlap with the initial state, thus causing a revival.

A second issue relates to quenches that start in a gapped phase and either cross or approach a critical point or surface. It turns out that quantities display universal scaling behavior as functions of the quench rate, with exponents related to the equilibrium-critical exponents.

This article concentrates on theoretical aspects of the second issue, though some salient features of thermalization will also be mentioned.

### 2. The Problem

Consider a many-body system with a Hamiltonian given by

$Display mathematics$ (2.1)

$H0$ has no explicit time dependence, while the time-dependent part $H1(t)$ can be generally expressed as

$Display mathematics$ (2.2)

where $Oi$ consist of a set of operators, and their space- and time-dependent couplings are $λi(x→,t)$. We have used a continuum notation appropriate for a quantum field theory. For a lattice system, the integral over the $(d−1)$ space directions is replaced by a sum over lattice points and the spatial location $x→$ is replaced by a set of discrete variables which specify a point on the lattice. $λi(x→,t)$ asymptote to constant values at early and late times: $λi(x→,t)→λi,in(x→)$ as $t→−∞$, while $λi(x→,t)→λi,out(x→)$ as $t→+∞$.

Suppose we start with a stationary initial state, e.g., the ground state of the initial Hamiltonian, or a thermal density matrix. Sometimes the latter case is called a thermal quench and the phrase quantum quench is reserved for the case when the initial state is a pure state. The time-dependent coupling then excites the system. The aim is to describe the time evolution of the system. This may involve an understanding of the system at late times, long after the Hamiltonian has been tuned to a new time-independent Hamiltonian described by the set of couplings $λi,out$. This may also involve studying properties soon after the quench has ended, that is, when the couplings have basically attained their asymptotic values $λi,out$, but the system is still evolving nontrivially in time. The latter is often referred to as an early-time response.

A special case involves couplings which are homogeneous in space, i.e. $λi(t)$: this is called a global quench, while a more general $λi(x→,t)$ of this type is called a local quench. A given set of functions $λi(x→,t)$ is referred to as a quench protocol. In this article we will consider only global quenches and restrict to the simple case where only one of the couplings is time dependent.

When the timescale over which the coupling changes, $δt$, approaches zero, the process is called an instantaneous, sudden, or abrupt quench. Physically this means that the timescale $δt$ is small compared to all other timescales in the problem. In this situation a time-independent Hamiltonian $Hin$ suddenly changes to another time-independent Hamiltonian $Hout$. The Schrodinger picture states immediately before and after the quench are identical, since the system does not get enough time to react. For example, when the initial state is the ground state of the initial Hamiltonian, $|0〉in$ the state right after the quench remains $|0〉in$. However this is not the ground state of $Hout$. Such an abrupt quench therefore studies the time evolution of a rather nontrivial excited state of the final Hamiltonian.

At the other extreme is a slow quench. Here $δt$ is large compared to all timescales of the initial Hamiltonian. This is possible if the initial Hamiltonian has a gapped spectrum. In this case, the early-time evolution is adiabatic. Following standard treatments of the adiabatic expansion in quantum mechanics it is useful to define an instantaneous Hamiltonian. This is simply $H(t)$ where $t$ is now regarded as a parameter rather than time. The eigenvalues of the instantaneous Hamiltonian are the instantaneous energy levels$En(t)$. Suppose the system is prepared at an early time $t=−T$ in the eigenstate labeled by $n$. To leading order in the adiabatic expansion, the time dependence of the wave functional is given by $ψn(t)∼exp[−i∫tEn(t′)dt′]$. The leading correction is given by (Gritsev & Polkovnikov, 2010; Schiff, 1968)

$Display mathematics$ (2.3)

where $ψk,t$ denotes the wave function of an instantaneous eigenstate which is denoted by $|k〉t$, that is, an eigenstate of the Hamiltonian at the time $t$. This shows that when the rate of change of the instantaneous energy levels is small compared to the gap in the instantaneous spectrum, the correction is small and the response remains approximately adiabatic. For simple systems whose energy spectrum is controlled by a single scale, the energy gap $Eg$, the condition that the correction (2.3) is small then becomes the Landau criterion

$Display mathematics$ (2.4)

where $Eg(t)$ is the gap in the spectrum of the Hamiltonian. Adiabaticity fails when the left-hand side of (2.4) becomes order one. An interesting case when this happens is a critical quench. For such a protocol, the time-dependent coupling goes through or approaches values at which the instantaneous Hamiltonian is gapless, so that the left-hand side of (2.4) becomes infinitely large. Such a gapless Hamiltonian describes a quantum critical point. A set of interesting questions for critical quenches relate to the exploration of universal features of the response. The earliest known universal behavior is Kibble–Zurek scaling, which will be discussed in the following section (see “Kibble–Zurek Scaling”).

An abrupt quench to a critical point can also exihibit universal behavior. More recently it has been found that there is an intermediate regime of quench rates where universal behavior also holds. This is the fast-quench regime. Here the timescale $δt$ is large compared to the ultraviolet scale of the system, but small compared to the infrared scale. For a system on a lattice this means that $δt$ is large compared to the lattice spacing, but small compared to an appropriate power of the correlation length which controls the long-distance behavior of the initial system. In this regime, expectation values again scale as a power of $δt$ with exponents that are determined in terms of the equilibrium-critical exponents.

### 3. Kibble–Zurek Scaling

Universal scaling in slow thermal quenches first appeared in a discussion of defect formation in cosmology. An expanding universe undergoes various kinds of phase transitions as it cools. In particular there are symmetry-breaking transitions where an order parameter condenses. These are phase transitions caused by a thermal quench, since the expansion renders the temperature (and other couplings) time dependent. When the order parameter is for example a complex scalar, its orientation in internal space would in general be different in different causally separated spatial regions, leading to topological defects. Roughly, there would be $O(1)$ defects in a spatial volume of size equal to the correlation length at a given time. In the 1970s Kibble (1976) proposed a mechanism for such defect formation, which was generalized to transitions in condensed-matter systems by Zurek (1985). This mechanism led to a prediction for the number density of such defects that scales as a function of the quench rate with universal exponents. In more recent years, this scaling and defect-formation mechanism has been extended to quantum phase transitions (for reviews and more references see Dziarmaga, 2010; Gogolin & Eisert, 2016; Gritsev & Polkovnikov, 2010; Polkovnikov, Sengupta, Silva, & Vengalattore, 2011).

Consider a Hamiltonian of the form (2.1) with a single time-dependent coupling $λ(t)$ which is of the form

$Display mathematics$ (3.1)

The Hamiltonian $H0$ is a critical Hamiltonian, that is, its spectrum is gapless. When $f=0$ there is a scaling symmetry $x→→αx,t→αzt$. The number $z$ is the dynamical critical exponent, $z=1$ for relativistic theories. The operator has dimension $Δ$, that is, under this scaling $O→α−ΔO$, and $λ0$ is a quantity with dimension $d+(z−1)−Δ$. The function $f(x)$ above is of the form

$Display mathematics$ (3.2)

where the constant is $O(1)$.

We will consider three kinds of quench protocols. The first is a trans-critical protocol (TCP). Here the function $f(x)$ monotonically interpolates between two constant values, crossing a zero at some value of $x$ which we can choose to be $x=0$. One example of a TCP is $f(x)=tanh(x)$. The second is a cis-critical protocol (CCP). Here the function changes from some constant value $f0$ to zero at $x=0$ and then turns back, eventually approaching some other constant value $f1$. One example of a CCP is $f(x)=tanh2x$. Finally we have an end-critical protocol (ECP) where the function $f(x)$ goes monotonically to zero asymptotically as $x→∞$. An example is $f(x)=1−tanhx$. These profiles are shown in Figure 1.

In all these three protocols, the theory has an energy gap at early times. Consider the system whose Heisenberg-picture state is the ground state of the initial Hamiltonian which has a gap $Eg,in$ Therefore for slow quenches where

$Display mathematics$ (3.3)

the initial time evolution is expected to be adiabatic. Adiabatic evolution holds as long as the Landau criterion is satisfied, and it fails at a time $t=tKZ$ where the Landau criterion is saturated. For the TCP and CCP protocols described above, this holds before $t=0$. In the critical region near $t=0$, standard scaling in equilibrium-critical phenomena implies

$Display mathematics$ (3.4)

where we have assumed that $λ(t)∼λ0(t/δt)r$ as $t→0$. In (3.4), $ν$ is the equilibrium correlation length critical exponent, that is, the correlation length $ξ$ at a (time-independent) coupling $λ$ close to the critical coupling $λ=0$ is given by $ξ∼λ−ν$. In the explicit TCP and CCP protocols described above, $r=1,2$ respectively. When the Landau criterion is saturated in the critical region, the time when adiabaticity fails, $tKZ$, can be obtained by substituting (3.4) in the left-hand side of (2.4) and setting the right-hand side to 1,

$Display mathematics$ (3.5)

The instantaneous energy gap at this time, $EKZ$ is, using (3.4)

$Display mathematics$ (3.6)

while the instantaneous correlation length at this time, $ξKZ$ is

$Display mathematics$ (3.7)

For quench protocols that are roughly constant for $t<−δt$, the above considerations are valid when the Kibble–Zurek time $|tKZ|$ is much smaller than $δt$. Then (3.5) leads to the condition (3.3). However, the Kibble–Zurek energy $EKZ$ should also be much smaller than any microscopic scale $ΛUV$. Using (3.6), this leads to the condition

$Display mathematics$ (3.8)

When $E0≪ΛUV$, the condition (3.3) implies (3.8). However in lattice models where $ΛUV=1/a$ where $a$ is the lattice spacing, one may investigate quenches where $E0∼ΛUV$. In that case, the condition (3.8) is stronger.

Kibble and Zurek assumed that:

Once adiabaticity breaks, the system becomes diabatic, that is, it remains frozen in time. This freezing continues till the Landau criterion is satisfied again.

During this freezout region, the only length scale that controls the physics is the instantateous correlation length at the Kibble–Zurek time, $ξKZ$.

These regimes are shown in Figure 2 for a TCP.

The first assumption is a drastic simplification of the real process. The second assumption is inspired by equilibrium-critical behavior and boldly extended to the nonequilibrium situation. These assumptions make predictions for the behavior of observables right after the quench.

The first assumption implies that observables such as correlation functions and entanglement entropies in the critical region are the same as at the Kibble–Zurek time, while the second assumption means that their behavior should be governed by dimensional analysis with $ξKZ$ as the only scale.

An operator $O$ whose scaling dimension is $Δ$ will therefore behave as

$Display mathematics$ (3.9)

A special case is the number density of defects for an order–disorder transition. It is natural to expect that there are $O(1)$ defects in a volume whose size is the correlation length. Thus the density of defects should be proportional to $ξKZ−d$ in $d$ space dimensions, behaving as an operator with conformal dimension $Δ=d$.

The result (3.9) is an early-time response, valid for times just after the quench has ended. However for some quantities, for example the excess energy produced, this is also the result at late times, since after $t∼δt$, the Hamiltonian is roughly constant, so that the energy remains the same. The defect density is another example of such a quantity.

The first assumption cannot be entirely correct: the quantity $O$ must have some slow time dependence in the critical region. This can be taken into account by postulating a scaling form

$Display mathematics$ (3.10)

where $f(x)$ is a slowly varying function. There is a similar scaling hypothesis for correlation functions (Chandran, Erez, Gubser, & Sondhi, 2012).

$Display mathematics$ (3.11)

In $1+1$-dimensional solvable systems, universal scaling also appears to hold for nonlocal quantities like the entanglement entropy of a region. For such a system in its ground state close to the critical point, the entanglement entropy of a region $A$ of length $L$ is given by Calabrese and Cardy (2004)

$Display mathematics$ (3.12)

where $c$ is the central charge of the critical theory and $nA$ is the number of boundaries of the region. In the Kibble–Zurek scenario, the system is adiabatic till time $t=tKZ$ and becomes diabatic after that time till the quench is over. In this critical region, the system should roughly behave like an equilibrium system with correlation length $ξKZ$. Therefore the entanglement entropy should be given by an expression (3.12) where $ξ$ is replaced by $ξKZ$. This result seems to hold in explicit calculations in models, for example the Ising model with a time-dependent transverse field (Cincio, Dziarmaga, Rams, & Zurek, 2007; Francuz, Dziarmaga, Gardas, & Zurek, 2016) and a harmonic chain (Caputa, Das, Nozaki, & Tomiya, 2017).

While the assumptions underlying Kibble–Zurek scaling are rather drastic, the result holds in many exactly solvable models, such as the Ising chain in a time-dependent transverse field, the Bose–Hubbard model in one spatial dimension, and the Kitaev Honeycomb model in two spatial dimensions. In the fermionic formulation of these models, a generic linear passage through a critical point becomes a set of Landau–Zener transitions for the decoupled momentum modes of fermions (Dziarmaga, 2010; Gogolin & Eisert, 2016; Gritsev & Polkovnikov, 2010; Polkovnikov et al., 2011).

There has been some experimental evidence for defect formation for low-temperature quenches following the Kibble–Zurek mechanism in a variety of systems: for example, liquid crystals, superfluid helium, Bose–Einstein condensates, trapped ions. It has been generally difficult to experimentally confirm scaling and determine the exponents. However there has recently been impressive progress in trapped-ion experiments (for references see Cui et al., 2016; see also Alba & Calabrese, 2018, and references therein).

### 4. Abrupt Quench

At the other end of the scale are abrupt quenches, where the Hamiltonian changes suddenly from an initial time-independent Hamiltonian $H0$ to another time-independent Hamiltonian $H1$ at some time which we can choose to be $t=0$. The quench then simply provides an initial condition for the time evolution of an observable for $t>0$. The Heisenberg-picture state of the system is the ground state of $H0$, which we denote by $|0〉in$. The operator $O$ at $t=0$ is determined from its early-time value by unitary time evolution with the constant Hamiltonian $H0$. For $t>0$, the time evolution is determined by the new Hamiltonian $H1$ so that

$Display mathematics$ (4.1)

Note that the time dependence for $t>0$ is highly nontrivial, since the state is not the ground state of $H1$.

When the final Hamiltonian is critical, one might expect that there would be universal behavior. The subsequent time evolution is difficult to calculate in general. However some exact results are known for relativistic theories in $1+1$ dimensions, when $H0$ is a gapped Hamiltonian and $H1=HCFT$ is the Hamiltonian of a conformal field theory.

For $1+1$ dimensional critical quenches, Calabrese and Cardy (2006) argued that, for purposes of computing long-distance quantities, the state $|0〉in$ can be approximated by the state $|CC〉$ which is given by

$Display mathematics$ (4.2)

where $|B〉$ is a conformally invariant boundary state of the final conformal field theory, and $τ0$ is a length scale which is essentially the correlation length associated with $H0$. The precise connection depends on the details of the Hamiltonian $H0$.

The nature of the approximation involved in (4.2) can be understood by considering a simple example: a $1+1$-dimensional theory of a free bosonic field $ϕ(t,x)$ whose mass is time dependent, $m(t)$, and goes from a value $m0$ at $t→−∞$ to zero at $t→∞$ over a timescale $δt$. In this case, the final conformal field theory is simply the theory of a massless scalar

$Display mathematics$ (4.3)

The field can be expanded as usual in terms of a set of modes $ϕk(t)eikx$ which solve the equations of motion. Since $m(t)$ becomes a constant at $t→−∞$, the solution for a momentum mode $ϕk(t)$ will approach some linear combination of $eiωin(k)t$ and $e−iωin(k)t$, where $ωin=k2+m02$. Similarly to $t→∞$ the solution will be some linear combination of $ei|k|t$ and $e−i|k|t$. Let $uin(k,t)$ be the solutions that asymptote to $e−iωin(k)t$ as $t→−∞$ and $uout(k,t)$ be the normalized solutions that asymptote to $e−i|k|t$ as $t→∞$. The field operator can then be expanded either in terms of $uin$ or in tems of $uout$,

$Display mathematics$ (4.4)

The annihilation and creation operators $ain(k),ain†(k),aout(k),aout†(k)$ satisfy the commutation relations

$Display mathematics$ (4.5)

These operators are related by Bogoliubov transformations

$Display mathematics$ (4.6)

The Bogoliubov coefficients $α(k)$ and $β(k)$ depend on the details of the quench protocol $m(t)$. The system is prepared in the ground state at early times; that is the Heisenberg-picture state is $|0〉in$ defined by

$Display mathematics$ (4.7)

This state can be expressed in terms of the “out” operators in the following form (Mandal, Paranjape, & Sorokhaibam, 2018)

$Display mathematics$ (4.8)

where

$Display mathematics$ (4.9)

and the Dirichlet state $|D〉$ is a boundary state of the final massless theory. What this means is the following: for the massless theory defined over the time domain $0≤t≤∞$, this state implements a Dirichlet boundary condition on the scalar field on the spacelike surface $t=0$. The expression for $|D〉$ is

$Display mathematics$ (4.10)

and the state $|0〉out$ satisfies $aout(k)|0〉out=0∀k$. The function $κ(k)$ admits an expansion in powers of $|k|/m0$ which turns out to be

$Display mathematics$ (4.11)

so that we can write

$Display mathematics$ (4.12)

where $W2l$ are the infinite set of conserved charges in the theory

$Display mathematics$ (4.13)

In particular, $W2$ is the final Hamiltonian; that is, the Conformal Field Theory (CFT) Hamiltonian. Thus in this case the state $|0〉in$ is a generalization of the state $|CC〉$ in (4.2). The preceding discussion holds for arbitrary $m(t)$; for profiles for which the equations of motion are analytically solvable, one can obtain the coefficients $κ2l$. In particular they hold for a sudden quench. In this case explicit calculations yield $κ2=1,κ4=−5/160$, and so on (Mandal et al., 2018).

Since the terms involving the $W2l$ involve higher and higher powers of $|k|/m0$ one might expect that the long-distance correlators are insensitive to the values of $κ2l$ for $l>1$: this would mean that for these quantities, $|CC〉$ is a good approximation. While this is indeed true for correlators of some operators (e.g. the field itself; Calabrese & Cardy, 2007), this is not correct for other operators, in particular those involving derivatives (Mandal et al., 2018).

In the free-field theory example, the higher $W2l$ charges are nonzero in the initial ground state $|0〉in$. However, one can fine-tune the initial state such that only a few of these charges are nonzero (Mandal et al., 2018). In particular, there are squeezed states where only the energy is nonzero. In such a situation, the state $|CC〉$ is a good approximation to the quench state.

The above considerations in fact hold for arbitrary $1+1$-dimensional CFTs (Cardy, 2016; Mandal, Sinha, & Sorokhaibam, 2015). A general quench state is a modification of $|CC〉$ by irrelevant operators. Thus the state $|CC〉$ should really be replaced by

$Display mathematics$ (4.14)

where $Qn$ denote the conserved charges and $κn$ are coefficients which depend on the microscopic details of the theory. In fact, general $1+1$-dimensional conformal field theories do have an infinite number of conserved charges, known as KdV charges. If the Heisenberg-picture state has nonvanishing values of these charges, one needs a generalization of the state $|CC〉$.

In states like $|CC〉Q$, the calculation of observables can be reduced to calculations in a conformal field theory. Then powerful methods of $1+1$-dimensional conformal field theory can be used to obtain several results for the time dependence of observables. For example, when $|CC〉$ is a good approximation, the one-point function of an operator of dimension $Δ$ that is not conserved is given by

$Display mathematics$ (4.15)

at late times $t≫τ0$. The decay is characteristic of an approach to a state that resembles a thermal state with a relaxation time given by

$Display mathematics$ (4.16)

This is not universal, since $τ0$ depends on the details of the initial Hamiltonian $H0$. However the ratio of the relaxation times of two different operators $O1$ and $O2$ with conformal dimensions $Δ1$ and $Δ2$ respectively is universal:

$Display mathematics$ (4.17)

Another important universal result for critical quench in $1+1$ dimensions concerns the time dependence of the entanglement entropy of a spatial interval in the field theory. Using the approximation (4.2), and the replica formalism, the calculation of the entanglement entropy can be reduced to a calculation of correlations of twist operators located at the end points of the interval. In the limit of a vanishingly small correlation length of the initial theory, the result for an interval of length $L$ is (Calabrese & Cardy, 2006, 2007):

$Display mathematics$ (4.18)

Since this result uses the Calabrese–Cardy approximation to a state after the quench, such a sharp change of behavior in time is valid when the initial correlation length is small compared to $L$. In this limit, the result has an intuitive explanation. For $t<0$, the system is in the ground state of a gapped system, so that $SEE∼log(L/ξ)$ which is a small quantity. Assuming that the system gets excited at the time of the quench $(t=0)$ by production of a pair of quasiparticles, one moving to the left and the other moving to the right, only quasiparticles that are produced at the same point are entangled with each other since the correlation length is small. These quasiparticles then move at the speed of light (chosen to be unity). At any given time $t$, the contribution to the entanglement entropy of a region of size $L$ comes from the quasiparticles that are produced at a point such that either the left-mover or the right-mover, but not both, has reached this region. It is then straightforward to see that, for $t≤L/2$, these must have originated from a region whose size is $4t$, so $SEE(L,t)$ should be proportional to $4t$. The size of this region therefore increases and reaches $2L$ at $t=L/2$. For $t>L/2$ both the left- and right-moving quasiparticles produced at any point outside this region of size $2L$ enter the region A and do not contribute anything more to $SEE$. Therefore $SEE$ saturates beyond this time. As discussed later, this is a signature that the system is thermalizing, and the saturation value is the thermal entropy. For a finite initial correlation length, the saturation happens smoothly.

A similar spread of entanglement entropy holds for quenches to field theories that are not relativistic, as well as in theories on a lattice (Alba & Calabrese, 2018). It has been shown that this kind of linear behavior holds in free-field theories in higher dimensions (Cotler, Hertzberg, Mezei, & Mueller, 2016) and it is expected that this is in fact a general result.

The quasiparticle picture of entanglement spread holds only for integrable systems: this does not account for the dynamics of entanglement spread for generic systems. In particular it does not work for chaotic systems. However, in a certain scaling regime a different effective model called the “membrane model” is conjectured to hold (Jonay, Huse, & Nahum, 2018). Consider a subregion $A$ with a typical size $R$. We are interested in calculating the entanglement of this subregion at time $t$. This scaling regime is then given by

$Display mathematics$ (4.19)

Here, $tth$ is the timescale for local thermalization. The membrane model can be described as follows. In a space-time picture, consider a codimension-one surface which extends in the time direction (backward in time) between the time slice at time $t$ and time $t=0$, such that it is anchored on the subregion $A$. The surface ends at the time slice $t=0$ and is perpendicular to this slice. If the number of spatial dimensions of the theory is $d$ this is also a $d$-dimensional surface, and one of the directions along this surface is time. This is called a membrane. Let $v$ denote the magnitude of the local velocity of a point on the membrane. According to the membrane model, the entanglement entropy $S(A,t)$ of the subregion $A$ is given by the minimum value of the quantity

$Display mathematics$ (4.20)

where $ε(v)$ is a velocity-dependent membrane tension that depends on the theory, $seq$ is the equilibrium entropy density, and $γ$ is the determinant of the induced metric on the membrane. This conjecture is motivated by results in random circuits, and has been shown to hold for chaotic one-dimensional spin chains. As discussed in section 8, “Quantum Quench and Holography,” this membrane model can be derived for theories with dual-gravity descriptions.

### 5. Fast Quench

An abrupt quench involves a change of the Hamiltonian over a timescale $δt$ which is small compared to all scales in the theory, including the ultraviolet scale $ΛUV−1$ (e.g. a lattice spacing). It turns out that there is another universal scaling regime in between slow and abrupt quenches: this is called the fast-quench regime (Das, Galante, & Myers, 2014, 2015). In this regime

$Display mathematics$ (5.1)

where $Lphys$ denotes any physical infrared length scale in the problem.

Consider first renormalized, relativistic, continuum quantum field theory where $ΛUV$ has been taken to infinity so that the first inequality is automatic. Consider the time-dependent Hamiltonian (2.1) and (2.2) with a single nonzero $λi$ which depends only on time.

The Hamiltonian $H0$ is a quantum field theory which flows out of a UV fixed point, which has space- and time-translation invariance and has a mass gap $m0$. The operator $O$ has a dimension $Δ$ in this UV fixed point. The time-dependent coupling $λ(t)$ starts from zero at $t=0$ and smoothly turns on. The subsequent time dependence lasts for a time $δt$, and can be fairly arbitrary, which may be written in the form

$Display mathematics$ (5.2)

and we consider functions $F(x)$ whose magnitude ranges from zero to of order one. The Heisenberg-picture state of the system is the ground state of $H0$, which we denote by $|ψ0〉$.

The fast-quench regime is then

$Display mathematics$ (5.3)

since the coupling $λ0$ has dimension $(d−Δ)$.

Calculating the response using perturbation theory, where $H1$ is regarded as small (the sense in which this is small will become clear soon) to the lowest nontrivial order gives

$Display mathematics$ (5.4)

Here $GR$ is a retarded Green’s function evaluated in the theory whose Hamiltonian is $H0$:

$Display mathematics$ (5.5)

We have used the fact that $H0$ is space- and time-translation invariant. The second term on the right-hand side of (5.4) is the linear response term. The higher-order terms in (5.4) involve higher-order correlators of $O$. This coupling will be denoted by $λ$ and the corresponding operator by $O$.

The integral over $x→′$ in (5.4) is over all space. However, $GR$ is a retarded Green’s function: since the time extent is from $t′=0$ to $t′=t$, causality of the relativistic field theory with Hamiltonian $H0$ implies that $GR(t−t′,x→−x→′)$ is nonzero only for $|x→−x→′|≤t$.1

Consider now the response at early times, $t∼δt$. Then the Green’s function which appears in the linear response term needs to be evaluated for space and time intervals less than $δt$. In the fast-quench regime (5.3), these intervals are much smaller than the correlation length of the $H0$ theory. However, in a relativistic theory, the correlators at space-time distances much smaller than the correlation length approach correlators of the massless conformal field theory at the UV fixed point. Therefore in this limit, the early-time response only involves retarded Green’s functions of this conformal field theory. This means that the integral in (5.4) involves only one length scale, which is $cδt$ (Figure 3). Since $〈ψ0|O|ψ0〉$ must have dimension $Δ$, dimensional analysis implies that

$Display mathematics$ (5.6)

where $c0$ is a pure number.

The higher-order terms show that the perturbation theory is in fact an expansion in the dimensionless parameter $g=λ0(δt)d−Δ$, so that one may write

$Display mathematics$ (5.7)

However, the fast-quench regime (5.3) is precisely the regime where $g≪1$, so the first term in the expansion dominates, leading to a universal scaling

$Display mathematics$ (5.8)

There are similar scaling relations for the excess energy produced $〈ΔE〉$

$Display mathematics$ (5.9)

which is consistent with the Ward identity

$Display mathematics$ (5.10)

All expectation values in the above discussion need to be renormalized. The necessary counterterms typically involve terms with time derivatives of the coupling in addition to the usual counterterms that appear for actions with constant couplings. There are similar universal results for expectation values of operators other than the one which is quenched. The contributions for such operators come from higher-order terms in the conformal perturbation expansion (Dymarsky & Smolkin, 2018).

From the above discussion, it is clear that this scaling happens because the early-time response in this regime of quench rates is controlled entirely by the UV fixed point. Once the quench is over, the coupling is basically constant and the observables evolve according to the final Hamiltonian. However, quantities like the excess energy remain constant after the quench, so that the scaling law (5.9) continues to hold at late times.

Fast-quench scaling has been shown to hold in model systems where the time evolution can be exactly solved, for example free bosonic and fermionic field theories with time-dependent masses (Das et al., 2014, 2015). For these systems in $1+1$ dimensions, there is some evidence that the early-time entanglement entropy also scales (Caputa et al., 2017).

### 6. Joining the Quench Regimes

While the above discussion has relied on renormalized continuum field theories, one expects that there is a regime in theories with a finite UV cutoff where these relationships hold. This is simply because the long-distance properties of a cutoff theory are captured by a renormalized continuum theory—which is why one expects the regime of validity of fast-quench scaling to be given by (5.1).

This has been explicitly shown for solvable lattice spin systems like the transverse-field Ising chain and the Kitaev model on a two (spatial) dimensional honeycomb lattice with time-dependent couplings (Das, Das, Galante, Myers, & Sengupta, 2017). Consider the case of an infinite Ising chain with a time-dependent transverse field. In terms of the spin variables, the Hamiltonian of the system is

$Display mathematics$ (6.1)

where $τ(i)$ denote the Pauli spin operators, and $n$ denotes the site indices of the one-dimensional chain. The time-dependent coupling $h(t)$ denotes the transverse magnetic field and $J$ is the interaction strength between the nearest-neighbor spins. It is well known that there is a Jordan–Wigner transformation which maps this to a theory of Majorana fermions (for a review see e.g. Kogut, 1979). In terms of a two-component momentum-space fermion field $χ(k,t)$, the Hamiltonian becomes

$Display mathematics$ (6.2)

where $σi$ denote Pauli matrices in the particle-hole space of fermions and we have introduced the (dimensionless) coupling $g(t)=h(t)/J$. For a given momentum $k$, the instantaneous energy eigenvalues are given by

$Display mathematics$ (6.3)

It is clear from this dispersion relation that $g=1$ corresponds to a critical point where the gapless mode is $k=0$. We are interested in quench profiles where $g(t)$ asymptotes to constant values different from $1$ and crosses $g=1$ at some time $t=0$ over a timescale $δt$. It turns out that it is possible to find such physically interesting quench profiles such that the quantum dynamics can be solved exactly (Das et al., 2017). The system starts off at $t→−∞$ in the ground state of the instantaneous Hamiltonian. The response $〈χ†σ3χ〉(t)$ in this Heisenberg-picture state can then be calculated analytically for all values of $δt$.

The difference of this quantity from the initial value in the middle of the quench ($t=0$) is plotted as a function of $Jδt$ in Figure 4 for the profile $g(t)=1−εtanh(t/δt)$(Das et al., 2017). The two sets of data points are for two different quench amplitudes. Note that $J$ is essentially the inverse of the lattice spacing. For slow quenches $Jδt>>1$ the data fits the Kibble–Zurek prediction, which is shown by the red solid lines. In this case Kibble–Zurek scaling predicts a behavior $〈χ†σ3χ〉∼δt−1/2$.

For $Jδt≪1$, the behavior saturates as a function of $Jδt$. The reason for this saturation is the following. The expression (6.3) shows that, for the quench protocol chosen above, the Landau criterion for validity of adiabatic evolution at time $t=0$ is given by

$Display mathematics$ (6.4)

Clearly modes around $k=0$ are nonadiabatic. However, as $Jδt$ becomes smaller, more and more modes become nonadiabatic until all the modes inside the Brillioun zone are excited. For smaller $Jδt$, there are no more modes to excite—which is the reason why the response saturates. In fact the saturation value agrees well with the result for an abrupt quench obtained many years ago in Barouch and McCoy (1970).

In between this abrupt regime and the Kibble–Zurek regime there is another scaling regime. The orange solid curve has a behavior $∼log(δt)$ which is the fast-quench scaling prediction for this model obtained in the continuum limit.

The results for the more complex and interesting Kitaev model are similar. These results indicate that in physical systems that have a finite lattice spacing, there is an intermediate regime of universal scaling which agrees with the fast-quench scaling predicted by the continuum quantum field theory.

### 7. Thermalization

At late times, the state should resemble a thermal state for many local observables; that is, the expectation values of these observables are close to thermal averages in a density matrix characterized by some temperature. Of course, a pure state remains pure under unitary evolution. One way to characterize this approximate thermality is to consider the entanglement entropy of a region $A$. Thermality would then mean that the reduced density matrix of this region is close to a thermal density matrix; that is,

$Display mathematics$ (7.1)

where $B$ denotes the complement of $A$. The rough idea is that in this case the complement of this region acts like a heat bath. The effective temperature is $T=1/β$.

We have seen above that after a sudden quench to a critical theory, the entanglement entropy increases with time, and saturates after a certain time to a value proportional to the size of the subsystem. This dependence on the subsystem size is characteristic of a thermal entropy. For smooth quenches at finite rate, it is natural to expect that late-time behavior is not very different from an abrupt quench—this means that this is the fate for such quenches as well. This behavior is indicative of the fact that the system is thermalizing.

In a higher-dimensional relativistic theory, consider a region $Σ$ of size $R$, the boundary of which has an area $AΣ$. Studies in holography (see section 8, “Quantum Quench and Holography”) reveal that there is a generalization of the linearly growing entanglement entropy shown to hold in $1+1$-dimensional critical systems. This would mean that after local equilibration, the change of entropy of $Σ$ from that at the time of quench has a universal form (Liu & Suh, 2014a, 2014b),

$Display mathematics$ (7.2)

where $vE$ is called the entanglement velocity and $seq$ is the equilibrium entropy density. When $t∼R$, the growth of entanglement saturates. Here $teq$, is the time required to reach equilibrium, which is expected to be of the order of $1T$, where $T$ is the final equilibrium temperature. When the size is of order $teq$, the saturation value of the entanglement entropy is $Ssaturation=seqVA$ where $VA=RAA$ is the volume of the region. This is the thermal value. It has been shown that $vE$ is always smaller than the speed of light (Casini, Liu, & Mezei, 2016).

While this behavior is expected in general nonintegrable systems, it cannot hold for integrable systems that possess an infinite number of conserved charges. It turns out, however, that in many such systems which approach a steady state at late times, the final state is described well by a Generalized Gibbs Ensemble (GGE; for a review and references see Essler, Mussardo, & Panfil, 2015), whose density matrix has the form

$Display mathematics$ (7.3)

where the conserved charges are $Qn$ and their corresponding chemical potentials are $μn$. Under the assumption that the quench state is given by (4.14), there is evidence for this in quenches to general $1+1$-dimensional CFTs as well as in lattice models.

There are of course systems that do not thermalize. An important class of examples are systems that display many-body localization—and quantum quenches are not expected to lead to thermal or GGE states for such systems (Abanin, Altman, Bloch, & Serbyn, 2019).

### 8. Quantum Quench and Holography

Universal scaling in quantum quench is difficult to establish in generic field theories. In equilibrium, universality in critical phenomena is understood in the framework of the renormalization group (RG). However, RG for time-dependent Hamiltonians is not well understood. This has motivated the application of the anti-de-Sitter (AdS)/CFT correspondence (for a discussion of applied AdS/CFT see Ammon & Erdmenger, 2015) to the problem of quantum quench (for a recent review of applications of AdS/CFT to problems out of equilibrium, see Liu & Sonner, 2018).

In its weak form, the AdS/CFT correspondence is a duality between a class of strongly coupled field theories without dynamical gravity and gravitational theories in a higher number of dimensions. When the field theory is a CFT, the gravitational theory lives in an asymptotically AdS space-time—the field theory is defined on the boundary of this space-time. When a $d$ dimensional CFT is in its ground state, the bulk space-time is exactly $AdSd+1$, whose metric can be chosen to be

$Display mathematics$ (8.1)

This is the Poincaré patch of AdS. The coordinate $z$ is referred to as the AdS radial coordinate: the boundary is at $z=0$. The field theory is defined on the $d$-dimensional flat space-time $(t,x→)$ with a metric

$Display mathematics$ (8.2)

For each field of the gravitational theory in the bulk of AdS, there is an operator $𝒜$ of the CFT. Consider, for example, a bulk scalar $ϕ(z,t,x→)$ of mass $m$ whose linearized equation of motion is the Klein–Gordon equation. The solution of the wave equation near the boundary is of the form

$Display mathematics$ (8.3)

The coefficients at higher orders in $z$ are determined in terms of the two functions $A(t,x→)$ and $λ(t,x→)$ via the equations of motion. The number $Δ$ is given by

$Display mathematics$ (8.4)

When $m2>0$, the function $A(t,x→)$ is a normalizable mode, while the function $λ(t,x→)$ is a nonnormalizable mode.

The AdS/CFT dictionary then states that this solution is dual to a field theory obtained by deforming a conformal field theory with action $SCFT$,

$Display mathematics$ (8.5)

while the expectation value of the operator in this state is given by

$Display mathematics$ (8.6)

$Δ$ is the conformal dimension of the operator in the CFT. Equation (8.5) shows that the function $λ(t,x→)$ is a source for the operator $𝒜(t,x→)$.

There is a similar dictionary for other bulk fields. For example, a vector gage field in the bulk is dual to a conserved current. The graviton in the bulk is dual to the energy momentum tensor of the field theory. In particular, if the nonnormalizable mode of the bulk metric is nonvanishing, the quantum field theory is defined on a space-time with a nontrivial metric.

This dictionary then implies that a space-time-dependent coupling of the field theory is a space-time-dependent nonnormalizable mode of the dual bulk field, while the normalizable mode is the response. Therefore a quantum quench is simply a space-time-dependent boundary condition of the dual field: a global quench corrresponds to a boundary condition that depends only on time. The idea is to solve the bulk equations of motion with this boundary condition. The normalizable mode $A(x→,t)$ can be then read out of the solution, and the dictionary (8.6) then provides the response in the boundary field theory. In most cases, this involves a numerical calculation.

As an example, consider a situation where the field theory is a CFT until time $t=0$, at which time a time-dependent boundary condition for a massless field is turned on for some time $δt$. From the bulk point of view, this causes an injection of energy into the system, resulting in a wave that enters the bulk from the boundary at the speed of light. Typically this leads to a collapse into a black hole. This process has been studied numerically (Chesler & Yaffe, 2009) and analytically for small amplitudes (Bhattacharyya & Minwalla, 2009). The result is that the resulting space-time is well approximated for small enough $δt$ by an AdS–Vaidya metric

$Display mathematics$ (8.7)

where retarded null coordinates $v$ have been used. Setting $m(v)=0$ corresponds to pure $AdS$ and the coordinate $v$ is related to the usual time coordinate $t$ which appears in (8.1) by $v=t−z$. The function $m(v)$ is a function that rises from zero to some constant value $M$ over a timescale $δt$. When $m(v)$ becomes a constant, the metric of (8.7) is that of a AdS–Schwarzschild black hole with mass $M$. In the limit $δt→0$ this becomes the metric of a collapsing null thin shell, $m(v)=Mθ(v)$ (Danielsson, Keski-Vakkuri, & Kruczenski, 1999). In that case, the Penrose diagram is shown in Figure 5..

Just as pure AdS is dual to the ground state of a CFT, a black hole in AdS is dual to a finite temperature state in the CFT. Therefore, such a quantum quench leads to a black hole at late times. A black hole has a temperature, the Hawking temperature $TH$. A consequence of this is that all correlation functions outside a black-hole horizon are thermal correlation functions at this temperature. AdS/CFT then implies that the correlation functions of the dual field theory on the boundary are also thermal. Therefore the process of black-hole formation caused by a quantum quench is the dual description of thermalization of the field theory.

Normally one might expect that the timescale for thermalization is set by $1/T$, where $T$ is the temperature of the final equilibrium state. One important result of holographic studies is that local observables thermalize in a time which is much smaller than $1/TH$. In fact, for an infinitely thin shell, this happens instantly. Nonlocal observables like correlation functions and entanglement entropies take longer periods of time to settle down to thermal values.

When the initial background is a global AdS, so that the boundary theory is defined on a compact sphere of radius $Rb$, an immediate collapse happens only when the energy injected in the quench exceeds the energy gap in the field theory which is of order $1/Rb$. For lower energies, the gravitational excitations oscillate inside the bulk, getting reflected from the boundary. However, it is generally believed that after many such reflections there will be an eventual collapse. This belief is based on numerical relativity studies of the dynamics of small perturbations of AdS (these are excitations of normalizable modes), where it was found that there is a collapse regardless of the strength of the perturbation (Bizon & Rostworowski, 2011). However, there are classes of initial conditions for which this does not happen (Buchel, Liebling, & Lehner, 2013).

The state $|CC〉$ in (4.2) resulting from an abrupt quench in $1+1$-dimensional CFT also has a dual gravitational description in terms of a $2+1$-dimensional geometry (Hartman & Maldacena, 2013). On the other hand, a sudden injection of energy is expected to be dual to a Vaidya-like geometry. Indeed, in a $1+1$-dimensional conformal field theory with a large central charge (so that the bulk space-time is classical), the emergence of a Vaidya geometry that is dual to a sudden injection of energy can be explicitly seen in a direct field-theory calculation (Anous, Hartman, Rovai, & Sonner, 2016). Such a quench is implemented by insertion of operators. Using conformal-field-theory techniques, it is possible to calculate correlation functions and other observables at later times, and compare with the corresponding bulk calculations in a Vaidya geometry, which gives perfect agreement.

Under certain circumstances, it is possible for a sector of the theory to thermalize. In the context of thermalization of the quark-gluon fluid modeled by a large-N gage theory, it is possible for the quark sector to thermalize separately from the gluon sector. In AdS/CFT the quark sector is described by introducing probe branes in AdS which do not back-react on the geometry in leading order of large N. Probe brane thermalization happens by the formation of horizons for the induced metric on the probe brane (Das, Nishioka, & Takayanagi, 2010). Since the induced metric is nondynamical, such a horizon need not enclose a singularity.

The AdS/CFT correspondence offers a way to calculate the entanglement entropy of a subregion in a strongly coupled field theory that has a classical gravitational dual. This is given by the Ryu–Takayanagi (RT) formula (Ryu & Takayanagi, 2006). Consider first a stationary state in a $d$-dimensional CFT. The dual gravity is a time-independent asymptotically $AdSd+1$ geometry. For a region on the boundary $Σ$ bounded by a surface $∂Σ$, the RT prescription is that the entanglement entropy of the region $Σ$ in the field theory is given by

$Display mathematics$ (8.8)

where $AreaΣ$ is the area of the codimension-two surface in the $d+1$-dimensional bulk with the boundary $∂Σ$ which has the minimal area, and $GN$ is Newton’s gravitational constant. Figure 6 shows the construction. For time-dependent geometries, as would appear for quantum quenches, this prescription has to be modified: instead of a minimal-area surface in the bulk, one needs to find the extremal-area surface called a HRT surface (Hubeny, Rangamani, & Takayanagi, 2007). The RT prescription has been essentially derived in Lewkowycz and Maldacena (2013) while the time-dependent version has been derived in Dong, Lewkowycz, and Rangamani (2016).

This prescription can therefore be used to find the time dependence of the entanglement entropy once the geometry dual to the quench process is found. One application of this is to the Vaidya-type geometries discussed above (Abajo-Arrastia, Aparicio, & Lopez, 2010); Balasubramanian et al., 2011).

The dynamics of entanglement entropy in a $|CC〉$ state have been reproduced holographically in (Hartman & Maldacena, 2013). Likewise, entanglement dynamics following a sudden injection of energy has been calculated for large central charge $1+1$-dimensional conformal field theories, in good agreement with the gravity results in a Vaidya geometry (Anous et al., 2016; Anous, Hartman, Rovai, & Sonner, 2017).

These holographic calculations have been extended to higher dimensions (Liu & Suh, 2014a, 2014b), which reveal several regimes of the time dependence of the entanglement entropy. Prior to local equilibrium, there is a quadratic growth followed by a universal linear growth and a final saturation. This suggests a picture of entanglement in which an “entanglement wave” proceeds from the boundary of the subregion to the interior where the region covered by the wave is entangled with the exterior. Saturation happens when the entire subregion has been filled up. This kind of entanglement spread has been called an “entanglement tsunami” (Liu & Suh, 2014a, 2014b) in analogy with tsunamis in oceans. The entanglement speed $vE$ which appears in (7.2) is called a “tsunami velocity.” One of the interesting features of these holographic calculations is that the extremal surfaces penetrate inside the black-hole horizon.

In fact, the holographic prescription yields a derivation of the membrane model for entanglement dynamics discussed in section 4, “Abrupt Quench.” This is possible because in the scaling limit (4.19) the value of the AdS radial coordinate of a point on the HRT surface can be determined by solving an algebraic equation. Upon substitution, the extremization problem for a codimension-two surface in the bulk reduces to a minimization of a codimension-one surface which resides entirely along the boundary directions (Mezei, 2018).

Investigation of universal scaling in quenches across critical points has led to some insight into the origins of Kibble–Zurek scaling. Here one starts with a holographic model for a critical point. The classic example is a holographic superfluid (Hartnoll, 2009). The simplest model involves a charged AdS black hole or an AdS soliton (Nishioka, Ryu, & Takayanagi, 2010). This corresponds to a dual field theory on the boundary which has a nonzero chemical potential $μ$ which is the value of the Maxwell scalar potential on the boundary. Consider a charged scalar field in this background. For small values of $μ$, the black hole or soliton background with a vanishing scalar field remains a stable solution of the bulk equations of motion. However, there exists a critical value $μc$ such that this solution becomes unstable. In this regime there is a new stable solution with a nonvanishing scalar field—this is a “hairy” black hole. Only the “normalizable” mode of the scalar field is excited; that is, the nonnormalizable mode is set to zero. The AdS/CFT dictionary then implies that there is a nonzero expectation value of a scalar operator in the boundary field theory in the absence of any source—that is, spontaneous symmetry-breaking. The broken phase is then a model for a superfluid. The transition to the broken phase is a continuous transition and $μ=μc$ is a critical point. In models of an AdS soliton (Nishioka et al., 2010), this scenario holds also at zero temperature, which is a quantum critical point.

In Basu and Das (2012) and Basu, Das, Das, and Nishioka (2013), a quantum quench across such holographic critical points was investigated by tuning the chemical potential to $μ=μc$ and adding a time-dependent nonnormalizable mode to the scalar that passes through zero. The response of this quench through the critical point is then calculated by solving the bulk equations of motion. In fact, an examination of these equations of motion reveals that in the slow quench limit, the dynamics are dominated by the zero mode. The zero-mode dynamics then become a version of the time-dependent Landau–Ginsburg equation that displays scaling solutions with Kibble–Zurek scaling. This analytic demonstration of scaling does not use the usual adiabatic–diabatic assumption. In addition, the zero-mode dominance provides an explanation of why only one scale dominates the dynamics in the critical regime. However this explanation is from the bulk point of view—an understanding of this mechanism in terms of the boundary field theory is still lacking.

Quenches across holographic critical points have also revealed new dynamical phases for thermal quenches. In Bhaseen, Gauntlett, Simons, Sonner, and Wiseman (2013), a time-dependent boundary condition for the scalar field was turned on, starting from a condensed phase of a holographic superfluid. The quench heats up the system. When the quench amplitude is small, the final state remains a symmetry-broken phase, while when the amplitude is large enough, the quench takes the system to a symmetry-restored phase where the order parameter vanishes. A numerical solution of the bulk equations of motion reveals that the approach to this final state can be very different, depending on the quench amplitude this shows three distinct dynamical phases. In fact, quantum quenches have been studied directly in Bardeen-Cooper-Schrieffer (BCS) theory (Barankov & Levitov, 2006; Barankov, Levitov, & Spivak, 2004). One set of protocols involves a sudden change of the pairing interaction, and the theoretical methods used in these studies are expected to hold for timescales shorter than the energy relaxation time. The holographic studies of Bhaseen et al. (2013) are in agreement with these results in the appropriate regime and yield new results at later times.

Holographic quenches with finite $δt$ in the fast regime were investigated in (Buchel, Lehner, Myers, & van Niekerk, 2013; Buchel, Myers, & van Niekerk, 2013). In fact, the universal fast-quench scaling discussed above was first discovered in these holographic studies. In these works, the initial geometry is pure AdS, so that the boundary conformal field theory is in the ground state. A time-dependent boundary condition is then imposed on a bulk scalar with some mass $m$. The solution of the bulk equations of motion then shows that in the fast-quench regime, the expectation value of the dual scalar operator obeys the scaling relation (5.8) and the energy obeys (5.9). These scaling forms were first seen in numerical calculations. Soon it was realized that in this fast-quench regime, the bulk solution is only needed near the boundary where the bulk equations simplify, allowing an analytic derivation of these scalings. Fast-quench scaling in quantum quenches provides an example of a phenomenon that was first discovered in holographic studies and subsequently shown to be a general result valid for all quantum field theories.

### Notes

• 1. This article uses natural units where the speed of light $c$ and the Planck constant $ħ$ have been set to unity.