Quantum Quench and Scaling of Entanglement Entropy

TL;DR

This paper investigates how entanglement entropy in a harmonic chain scales during a mass quench crossing critical points, revealing size independence at large subsystem sizes and connecting to known quench scaling laws.

Contribution

It provides an exactly solvable model demonstrating entanglement entropy scaling behavior during a mass quench crossing critical points.

Findings

Entanglement entropy becomes size-independent for large subsystems.

Results align with Kibble-Zurek scaling for slow quenches.

Supports 'fast quench scaling' in certain regimes.

Abstract

Global quantum quench with a finite quench rate which crosses critical points is known to lead to universal scaling of correlation functions as functions of the quench rate. In this work, we explore scaling properties of the entanglement entropy of a subsystem in a harmonic chain during a mass quench which asymptotes to finite constant values at early and late times and for which the dynamics is exactly solvable. When the initial state is the ground state, we find that for large enough subsystem sizes the entanglement entropy becomes independent of size. This is consistent with Kibble-Zurek scaling for slow quenches, and with recently discussed "fast quench scaling" for quenches fast compared to physical scales, but slow compared to UV cutoff scales.

Equations

$$⟨\mathcal{O}(t)⟩$$

$$⟨\mathcal{O}(\vec{x},t)\mathcal{O}({\vec{x}}^{\prime },t^{\prime })⟩$$

$$⟨\mathcal{O}(t)⟩\sim \delta t^{d-2\Delta }$$

$$H=\frac{1}{2}\sum _{n=-\infty }^{\infty }\left[P_n^2+(X_{n+1}-X_n{)}^2+m^2(t)X_n^2\right]$$

$$X_n(t)={\int }_{-\pi }^{\pi }\frac{dk}{2\pi }{X}_k(t)e^{ikn}{P}_n(t)={\int }_{-\pi }^{\pi }\frac{dk}{2\pi }{P}_k(t)e^{ikn}$$

$$\frac{d^2{X}_k}{d{t}^2}+[4{\sin }^2(k/2)+m^2(t)]X_k=0$$

$${\omega }_0^2=4{\sin }^2(k/2)+m_0^2.$$

$$X_k(t)=f_k(t)a_k+f_k^{\star }(t)a_{-k}^{†}$$

$${\Psi }_0(X_k,t)=\prod _k\frac{1}{[\sqrt{2\pi }{f}_k^{\star }(t){]}^{1/2}}\exp \left[\frac{1}{2}{\left(\frac{\dot{f_k(t)}}{f_k(t)}\right)}^{\star }{X}_k{X}_{-k}\right]$$

$$m^2(t)=m_0^2{\tanh }^2(t/\delta t)$$

$$f_k=\frac{1}{\sqrt{2{\omega }_0}}\frac{(2{)}^{i{\omega }_0\delta t}{\cosh }^{2\alpha }(t/\delta t)}{E_{1/2}^{\prime }{\tilde{E}}_{3/2}-E_{1/2}{\tilde{E}}_{3/2}^{\prime }}\times$$

$$[{\tilde{E}}_{3/2}^{\prime }{}_2{F}_1(\tilde{a},\tilde{b},\frac{1}{2};-{\sinh }^2(t/\delta t))$$

$$+E_{1/2}^{\prime }\sinh (t/\delta t){}_2{F}_1(\tilde{a}+\frac{1}{2},\tilde{b}+\frac{1}{2},\frac{3}{2};-{\sinh }^2(t/\delta t))]$$

$${\omega }_0^2(k)$$

$$\alpha$$

$$\tilde{a}$$

$$\tilde{b}$$

$$E_{1/2}$$

$${\tilde{E}}_{3/2}$$

$$E_c^{\prime }$$

$$C=\left[\begin{array}{cc}{X}_{mn}(t)& D_{mn}(t)\\ D_{mn}(t)& P_{mn}(t)\end{array}\right]J=\left[\begin{array}{cc}0& I_{l\times l}\\ -I_{l\times l}& 0\end{array}\right]$$

$$X_{mn}(t)$$

$$D_{mn}(t)$$

$$S=\sum _{n=1}^l\{[{\gamma }_n(t)+\frac{1}{2}]\log [{\gamma }_n(t)+\frac{1}{2}]-[{\gamma }_n(t)-\frac{1}{2}]\log [{\gamma }_n(t)-\frac{1}{2}]\}$$

$$X_{mn}(t)$$

$$P_{mn}(t)$$

$$D_{mn}(t)$$

$$X_{nn}\sim (\mathrm{const})+(\delta t{)}^2{P}_{nn}\sim (\mathrm{const})+(\delta t{)}^0{D}_{nn}\sim \delta t$$

$$\Delta S_A=S(t=0)-S(t=-\infty )$$

$$\Delta S=c{\Gamma }_Q^2,$$

$$\Delta S_A=(\mathrm{constant})+\frac{1}{6}\log ({\Gamma }_Q)$$

Full text

††preprint: EFI-17-4,YITP-17-16,UK/17-02

1. Introduction: The behavior of entanglement of a many-body system that undergoes a quantum quench has been a subject of great interest in recent times. When the quench is instantaneous (i.e. a sudden change of the hamiltonian), several results are known. Perhaps the best known result pertains to the entanglement entropy (EE) of a region of size $l$ in a $1+1$ dimensional conformal field theory following a global instantaneous quench, $S_{EE}(l)$. As shown in cc1 , $S_{EE}(l)$ grows linearly in time till $t\approx l/2$ and then saturates to a constant value typical of a thermal state - a feature which has been studied extensively in both field theory and in holography. Generalisations of this result to conserved charges and higher dimensions have been discussed more recently Caputa:2013eka ; tsunami ; mezei . The emphasis of these studies is to probe the time evolution of the entanglement entropy.

In physical situations, quantum quench has a finite rate, characterized by a time scale $\delta t$, that can vary from very small to very large. When the quench involves a critical point, universal scaling behavior has been found for correlation functions at early times. The most famous scaling appears for a global quench which starts from a massive phase with an initial gap $m_{g}$, crosses a critical point (chosen to be e.g. at time $t=0$) and ends in another massive phase. For slow quenches (large $\delta t$), it has been conjectured that quantities obey Kibble-Zurek scaling kz : evidence for this has been found in several solvable models and in numerical simulations kz2 ; dziarmaga . Such scaling follows from two assumptions. First, it is assumed that as soon as the initial adiabatic evolution breaks down at some time $-t_{KZ}$ (the Kibble-Zurek time) the system becomes roughly diabatic. Secondly, one assumes that the only length scale in the critical region is the instantaneous correlation length $\xi_{KZ}$ at the time $t=-t_{KZ}$. This implies that, for example, one point functions scale as $\langle{\cal O}(t)\rangle\sim\xi_{KZ}^{-\Delta}$, where $\Delta$ denotes the conformal dimension of the operator ${\cal O}$ at the critical point. An improved conjecture involves scaling functions. For example, one and two point correlation functions are expected to be of the form Zurek96 ; Cincio ; Deng ; DamskiZurek ; DziarmagaRams ; Kolodrubetz ; Chandran

[TABLE]

Some time ago, studies of slow quenches in AdS/CFT models have led to some insight into the origin of such scaling without making these assumptions holokz .

For protocols in relativistic theories which asymptote to constant values at early times, one finds a different scaling behavior in the regime $\Lambda_{UV}^{-1}\ll\delta t\ll m_{phys}^{-1}$, where $\Lambda_{UV}$ is the UV cutoff scale, and $m_{phys}$ denotes any physical mass scale in the problem. For example,

[TABLE]

where $d$ is the space-time dimension. This ”fast quench scaling” behavior was first found in holographic studies blmv and subsequently shown to be a completely general result in any relativistic quantum field theory dgm . The result follows from causality, and the fact that in this regime linear response becomes a good approximation. Finally, in the limit of an instantaneous quench, suitable quantities saturate as a function of the rate : for quench to a critical theory a rich variety of universal results are known in $1+1$ dimensions cc2 .

Much less is known about the behavior of entanglement and Renyi entropies as functions of the quench rate - a key ingredient of universality. This has been studied for the 1d Ising model (and generalizations) with a transverse field which depends linearly on time, $g(t)=1-\frac{t}{\tau_{Q}}$ moore -Cincio ; Francuz . The system is prepared in the instantaneous ground state at some initial time, crossing criticality at $t=0$. The emphasis of moore and Cincio ; Francuz is on the slow regime, which means $\tau_{Q}\gg a$ where $a$ is the lattice spacing, while canovi also studies smaller values of $\tau_{Q}$. In particular, moore and canovi studied the EE for half of a finite chain and found that the answer approaches $S_{EE}\sim\frac{1}{12}\log\xi_{KZ}$ after sufficiently slow quenches. This is consistent with the standard assumptions which lead to Kibble-Zurek scaling mentioned above. According to these assumptions, the system evolves adiabatically till $t=-t_{KZ}$ and enters a phase of diabatic evolution soon afterwards. Thus the state of the system at $t=0$ is not far from the ground state of the instantaneous hamiltonian at $t=-t_{KZ}$. Furthermore when $\tau_{Q}\gg 1$ in lattice units, $\xi_{KZ}$ is large, and the instantaneous state is close to criticality. In such a state, the entanglement entropy of a subregion of a large chain with $N_{A}$ boundary points should obey an ”area law” $\frac{c}{6}N_{A}\log(\xi_{KZ})$, where $c$ is the central charge. When the subsystem is half space $N_{A}=1$ and for the Ising model the central charge is $c=1/2$. Similarly, Cincio ; Francuz studied the EE of a subsystem of finite size $l$ in an infinite 1d Ising model, with a transverse field linear in time, starting with the ground state at $t=-\infty$. The EE close to the critical point for $l\gg\xi_{KZ}$ was found to saturate to $S_{EE}=(\rm{constant})+\frac{1}{6}\log(\kappa(t)\xi_{KZ})$. The factor $\kappa(t)$ depends mildly on the time of measurement and $\kappa(-t_{KZ})\approx 1$. Once again, this result is roughly that of a stationary system with correlation length $\xi_{KZ}$, as would be expected from Kibble-Zurek considerations. The factor $\kappa(t)$ is a correction to the extreme adiabatic-diabatic assumption. The paper canovi investigates an intermediate regime of fast quench (as described above). While this paper investigates scaling of $S_{EE}$ as a function of quench rate in the slow regime, there is no similar analysis in the fast regime.

In this letter, we study entanglement entropy for a simple system: an infinite harmonic chain (i.e. a 1+1 dimensional bosonic theory on a lattice) with a time dependent mass term which asymptotes to constant finite values at early and late times. We choose a mass function for which the quantum dynamics can be solved exactly. The use of such a protocol allows us to explore the whole range of quench rates, where the speed of quench is measured in units of the initial gap rather than the lattice scale. We compute the entanglement entropy for a subsystem of size $l$ (in lattice units) in the middle of the quench and find that it scales in interesting ways as we change the quench rate. The dimensionless quantity which measures the quench timescale is $\Gamma_{Q}=m_{0}\delta t$ where $m_{0}$ is the initial gap.

2. Our setup and quench protocols: The hamiltonian of the harmonic chain is given by

[TABLE]

where $(X_{n},P_{n})$ are the usual canonically conjugate scalar field variables on an one dimensional lattice whose sites are labelled by the integer $n$. The mass term $m(t)$ is time dependent. All quantities are in lattice units. In terms of momentum variables $X_{k},P_{k}$

[TABLE]

the equation of motion is given by

[TABLE]

We are interested in functions $m(t)$ which asymptote to constant values $m_{0}$ at $t\rightarrow\pm\infty$, and pass through zero at $t=0$. Let $f_{k}(t)$ be a solution of (5) which asymptotes to a purely positive frequency solution $\sim e^{-i\omega_{0}t}/\sqrt{2\omega_{0}}$ at $t\rightarrow-\infty$, where

[TABLE]

A mode decomposition

[TABLE]

with $[a_{k},a^{\dagger}_{-k^{\prime}}]=2\pi\delta(k-k^{\prime})$ can be then used to define the ”in” vacuum by $a_{k}|0\rangle=0$ for all $k$. The solutions $f_{k}(t)$ are chosen to satisfy the Wronskian condition $f_{k}({\dot{f_{k}}})^{\star}-({\dot{f_{k}}})f_{k}^{\star}=i$. The state $|0\rangle$ then denotes the Heisenberg picture ground state of the initial Hamiltonian. The normalized wavefunctional for the ”in” vacuum state is given by

[TABLE]

We will choose a quench protocol for a mass function for which the mode functions $f_{k}(t)$ can be solved exactly. The particular mass function we use is

[TABLE]

The corresponding mode functions are given by

[TABLE]

where we have defined

[TABLE]

3. Entanglement Entropy: Our aim is to calculate the entanglement entropy for a subregion of the infinite chain consisting of $l$ lattice points. This is most conveniently calculated by considering the $2l\times 2l$ matrix of correlators, and the symplectic matrix correntangle ; mezei

[TABLE]

where $(m,n)$ denote lattice sites inside the subsystem and

[TABLE]

The eigenvalues of the matrix $iJC$ then occur in pairs $\pm\gamma_{n}(t)~{}~{}n=1,\cdots l$ where $\gamma_{n}(t)>0$. The entanglement entropy $S$ is given by

[TABLE]

In terms of the mode functions (10), the correlators (11) are given by the expressions

[TABLE]

The correlator $D_{mn}(t)$ is particularly important in the calculations which follow. This is because this quantity is exactly zero when the mass is time independent. For the same reason, this quantity vanishes in the leading order of the adiabatic expansion (i.e. when observables are replaced by their static answers with the instantaneous value of the mass), i.e. $f^{adia}_{k}=\frac{1}{\sqrt{2\omega_{k}(t)}}e^{-i\omega_{k}(t)t}$ with $\omega_{k}(t)^{2}=4\sin^{2}(k/2)+m^{2}(t)$.

The computation of correlation functions and the entanglement entropy then involves integrals over momenta $k$, which are computed numerically. As is well known, numerical integrals with oscillating functions in the integrand are tricky. Numerical methods reduce this to a summation of an alternating series, leading to slow convergence. To confirm our numerical integration, we check the convergence of the integration with varying precision. We perform all calculations with interval sizes $dk=0.00001,0.000001$.

4. Regimes of the quench rate: Before we examine these entropies, it is necessary to understand the various regimes of the dimensionless parameter $\Gamma_{Q}\equiv m_{0}\delta t$. This can be done by studying the scaling behavior of correlation functions. Since we are interested in the slow as well as the fast regime, we will stay close to the continuum limit where the dimensionless mass $m_{0}a$ is small. In the following we express everything in lattice units. Then, Kibble Zurek scaling is expected to hold for these quantities for $\Gamma_{Q}\gg 1$, while we get the regime of fast quench when $\Gamma_{Q}\ll 1$, but $\delta t>1$.

For slow quench, the usual Landau criterion leads to a Kibble Zurek time and the instantaneous correlation length at this time to be $t_{KZ}=\xi_{KZ}=\sqrt{\delta t/m_{0}}$, and we expect e.g. the following leading answers at $t=0$ : $X_{mn}(0)\sim\xi_{KZ}^{0}F_{X}(|m-n|/\xi_{KZ}),P_{mn}(0)\sim\xi_{KZ}^{-2}F_{P}(|m-n|/\xi_{KZ}),D_{mn}(0)\sim\xi_{KZ}^{-1}F_{D}(|m-n|/\xi_{KZ})$, where $F_{X},F_{P},F_{D}$ are smooth functions. We indeed find this behavior for $\Gamma_{Q}\geq 100$. For example, Figure (1) shows $\xi_{KZ}D_{mn}(0)$ as a function of $|m-n|$ for various values of $\Gamma_{Q}$. Clearly for large enough $\Gamma_{Q}$ the points fall on top of each other, showing that the above scaling behavior holds. The results for $X_{mn}(0)$ and $\xi_{KZ}^{2}P_{mn}(0)$ are also consistent with the above expectations.

In the fast quench regime $\Gamma_{Q}\ll 1$ the analysis of dgm leads to the following scaling relations for coincident correlators

[TABLE]

We found this behavior for small $\Gamma_{Q}<0.1$. For correlation functions $X_{mn},P_{mn},D_{mn}$, with the constant mass values subtracted, we expect these scalings to hold for large $|m-n|<\delta t<m_{0}^{-1}$, while for $\delta t<|m-n|<m_{0}^{-1}$ they should become independent of $\delta t$ dgm . We indeed see this behavior for sufficiently small $\Gamma_{Q}$. This is most clearly seen for $D_{mn}$ since the constant mass value of this quantity is exactly zero.

5. Scaling of entropies: Now that we have identified the slow and fast quench regimes, we go ahead and explore possible scaling properties of the entanglement entropy $S_{EE}$. This quantity is studied in detail at $t=0$. (which is the middle of the quench) for both fast and slow quenches.

First, we consider the fast quench regime, $\Gamma_{Q}\ll 1$. We have computed the difference of the entanglement entropy at $t=0$ from its value at $t=-\infty$,

[TABLE]

for various subsystem sizes $l$ as a function of $\Gamma_{Q}$. Figure (2) shows the results for $\xi_{i}=600$ as a function of $\Gamma_{Q}$. When $l$ is smaller than the initial correlation length $\xi_{i}=m_{0}^{-1}$ this quantity depends on $l$ for a given $\Gamma_{Q}$. However for $l$ sufficiently large compared to $\xi_{i}$ we find that the $l$-dependence saturates.

Namely, the data for large $l/\xi_{i}$ fits to the following result for small $\Gamma_{Q}$

[TABLE]

where the constant $c$ is independent of $l$ and $\xi_{i}$ and approximately equal to $0.67$.

In fact, for small $l/\xi_{i}$, $\Delta S_{A}$ does depend on $\xi_{i}$ for a given $\Gamma_{Q}$ and $l$, as shown in Figures (3) and (4). These are plots of $\Delta S_{A}$ as a function of $\Gamma_{Q}$ for a given $l$, and different values of $\xi_{i}$. Figure (3) has $l=100$ with $\frac{l}{\xi_{i}}$ ranging from $0.1$ to $0.2$ : the result has a clear dependence on $\xi_{i}$. Figure (4) has $l=2000$ with $\frac{l}{\xi_{i}}$ ranging from $2.0$ to $4.0$ - the data for various values of $\xi_{i}$ are right on top of each other.

This result (16) is in agreement with expectations based on dgm . In these papers it was argued that the fast quench scaling for one point functions follow from the fact that linear response becomes a good approximation in this regime. Even though the scaling behavior of the two point function is complicated (as mentioned above), one would expect that the $\delta t$ dependence of the matrix elements of $C$ would still be governed by perturbation theory. To lowest order this should be therefore proportional to $m_{0}^{2}$. In the regime of quench rates we are considering, the only other scale is $\delta t$. Since the entanglement entropy is dimensionless, we expect that the leading answer is proportional to $m_{0}^{2}\delta t^{2}$ which is $\Gamma_{Q}^{2}$. The fact that the result becomes independent of $l$ is also expected since the measurement is made at $t=0$.

It would be interesting to gain further insight by using the perturbation theory for entanglement entropy (as e.g. developed in smolkin ).

Let us now consider quench rates slow enough to ensure that the correlation functions discussed above display standard Kibble Zurek scaling. Figure (5) shows $S_{EE}$ as a function of $\Gamma_{Q}$ for a given $m_{0}$ various values of subsystem size $l$.

For large enough $l$ we therefore find that $\Delta S_{A}$ is independent of $l$ and fits the relation

[TABLE]

The constant in (17) is roughly $0.3$. This result is consistent with the expectations from Kibble Zurek scenario. More precisely, the state of the system at $t=0$ is fairly close to the instantaneous ground state at $t=-t_{KZ}$, i.e. the ground state of the system with a fixed (time independent) correlation length $\xi_{KZ}$. For such a system the entanglement entropy should behave as $\frac{1}{3}\log(\xi_{KZ})$, independent of $l$ for $l>\xi_{KZ}$. Since $\xi_{KZ}=\xi_{i}\sqrt{\Gamma_{Q}}$, this is $\frac{1}{3}\log(\xi_{i})+\frac{1}{6}\log(\Gamma_{Q})$. On the other hand, $S_{EE}(t=-\infty)$ is ground state entanglement entropy of the system at a fixed correlation length $\xi_{i}$ and therefore behaves as $\frac{1}{3}\log(\xi_{i})$ for $l\gg\xi_{i}$. Thus the KZ scenario predicts that when $\Delta S_{A}$ is expressed as a function of $\Gamma_{Q}$, the dependence on $\xi_{i}$ should cancel - which is exactly as we find.

In conclusion, we have calculated the entanglement entropy of a subsystem in a harmonic chain in the presence of an exactly solvable mass quench and examined the dependence of this quantity at the middle of the quench on the quench rate. Our nonlinear quench protocol asymptotes to constant values at initial and final times : in this respect it differs significantly from the kind of protocols most commonly studied in the literature, e.g. couplings which behave linearly in time. For quench rates which are fast compared to the initial mass (but slow compared to the lattice scale) our results show, for the first time, that the fast quench scaling established for correlation functions in dgm extend to non-local quantities like entanglement entropy. For quench rates slow compared to the initial mass, our results are consistent with the predictions of Kibble-Zurek scenario, and with earlier results in the 1d Ising model.

Note In an earlier version of this paper, we had reported that the entanglement entropies have a piece which is extensive in the subsystem size. This result is erroneous and stemmed from a mistake in our numerical code. This revised version corrects the error.

Acknowledgements. We thank Shinsei Ryu, Marek Rams, Krishnendu Sengupta and Tadashi Takayanagi for discussions and comments on the manuscript. The work of PC is supported by the Simons Foundation through the ”It from Qubit” collaboration. The work of S.R.D. is supported by a National Science Foundation grant NSF-PHY-1521045. The work of A.T. was supported in part by NSFC under grant number 11535012. A part of numerical computation in this work was carried out at the Yukawa Institute Computer Facility.