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arXiv:1403.3040 · Physical Review Letters

Thermalization and Revivals after a Quantum Quench in Conformal Field Theory

John Cardy

Published Jun 5, 2014• 136 citations
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Abstract

We consider a quantum quench in a finite system of length $L$ described by a $1+1$-dimensional conformal field theory (CFT), of central charge $c$, from a state with finite energy density corresponding to an inverse temperature $\ensuremath{\beta}\ensuremath{\ll}L$. For times $t$ such that $\ensuremath{\ell}/2<t<(L\ensuremath{-}\ensuremath{\ell})/2$ the reduced density matrix of a subsystem of length $\ensuremath{\ell}$ is exponentially close to a thermal density matrix. We compute exactly the overlap $\mathcal{F}$ of the state at time $t$ with the initial state and show that in general it is exponentially suppressed at large $L/\ensuremath{\beta}$. However, for minimal models with $c<1$ (more generally, rational CFTs), at times which are integer multiples of $L/2$ (for periodic boundary conditions, $L$ for open boundary conditions) there are (in general, partial) revivals at which $\mathcal{F}$ is $O(1)$, leading to an eventual complete revival with $\mathcal{F}=1$. There is also interesting structure at all rational values of $t/L$, related to properties of the CFT under modular transformations. At early times $t\ensuremath{\ll}(L\ensuremath{\beta}{)}^{1/2}$ there is a universal decay $\mathcal{F}\ensuremath{\sim}\mathrm{exp}(\ensuremath{-}(\ensuremath{\pi}c/3)L{t}^{2}/\ensuremath{\beta}({\ensuremath{\beta}}^{2}+4{t}^{2}))$. The effect of an irrelevant nonintegrable perturbation of the CFT is to progressively broaden each revival at $t=nL/2$ by an amount $O({n}^{1/2})$.

DOI
10.1103/physrevlett.112.220401

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Abstract We investigate the behavior of the return amplitude  ( t ) = ∣ 〈 Ψ ( 0 ) ∣ Ψ ( t ) 〉 ∣ following a quantum quench in a conformal field theory (CFT) on a compact spatial manifold of dimension d − 1 and linear size O ( L ) , from a state ∣ Ψ ( 0 ) 〉 of extensive energy with short-range correlations. After an initial gaussian decay  ( t ) reaches a plateau value related to the density of available states at the initial energy. However for d = 3 , 4 this value is attained from below after a single oscillation. For a holographic CFT the plateau persists up to times at least O ( σ 1 / ( d − 1 ) L ) , where σ ≫ 1 is the dimensionless Stefan–Boltzmann constant. On the other hand for a free field theory on manifolds with high symmetry there are typically revivals at times t ∼ integer × L . In particular, on a sphere S d − 1 of circumference 2 π L , there is an action of the modular group on  ( t ) implying structure near all rational values of t / L , similar to what happens for rational CFTs in d = 2 .
Abstract. We describe several results concerning global quantum quenches from states with short-range correlations to quantum critical points whose low-energy properties are described by a 1 + 1-dimensional conformal field theory (CFT), extending the work of Calabrese and Cardy (2006 Phys. Rev. Lett . 96 136801 ): (a) for the special class of initial states discussed in that paper we show that, once a finite region falls inside the horizon, its reduced density matrix is exponentially close in L 2 norm to that of a thermal Gibbs state; (b) small deformations of this initial state in general lead to a (non-Abelian) generalized Gibbs distribution with, however, the possibility of parafermionic conserved charges; (c) small deformations of the CFT, corresponding to curvature of the dispersion relation and (non-integrable) left-right scattering, lead to a dependence of the speed of propagation on the initial state, as well as diffusive broadening of the horizon.
At thermal equilibrium, the concept of effective central charge for massive deformations of two-dimensional conformal field theories (CFT) is well understood, and can be defined by comparing the partition function of the massive to that of a CFT. This temperature-dependent effective charge interpolates monotonically between the central charge values corresponding to the IR and UV fixed points at low and high temperatures, respectively. We propose a non-equilibrium, time-dependent generalization of the effective central charge for integrable models after a quantum quench, ceff(t)c_{\rm eff}(t)ceff​(t), obtained by comparing the return amplitude to that of a CFT quench. We study this proposal for a large mass quench of a free boson, where the charge is seen to interpolate between ceff=0c_{\rm eff}=0ceff​=0 at t=0t=0t=0, and ceff∼1c_{\rm eff}\sim 1ceff​∼1 at t→∞t\to\inftyt→∞, as is expected. We use our effective charge to define an to quench protocol, where the charge evolves from ceff=1/2c_{\rm eff}=1/2ceff​=1/2 at t=0t=0t=0, to ceff=7/10c_{\rm eff}=7/10ceff​=7/10 at t→∞t\to\inftyt→∞, the corresponding values of the first two unitary minimal CFT models. We then argue that the inverse Tricritical Ising to quench is impossible with our methods. These conclusions can be generalized for quenches between any two adjacent unitary minimal CFT models. We finally study a large mass quench into the model (sinh-Gordon with a particular complex coupling). At short times after the quench, the effective central charge increases in a discrete staircase structure, where the values of the charge at the steps can be computed in terms of the central charges of unitary minimal CFT models. When the initial state is a pure state, one always finds that ceff(t→∞)≥ceff(t=0)c_{\rm eff}(t\to\infty)\geq c_{\rm eff}(t=0)ceff​(t→∞)≥ceff​(t=0), though ceff(t)c_{\rm eff}(t)ceff​(t), generally oscillates at finite times. We explore how this constraint may be related to RG flow irreversibility.
In a recent letter [J. Cardy, Phys. Rev. Lett. 112, 220401 (2014)], the author made a very interesting observation that complete revivals of quantum states after quantum quench can happen in a period that is a fraction of the system size. This is possible for critical systems that can be described by minimal conformal field theories with central charge c<1c<1c<1. In this paper, we show that these complete revivals are impossible in microscopic realizations of those minimal models. We will prove the absence of the mentioned complete revivals for the critical transverse field Ising chain analytically, and present numerical results for the critical line of the XY chain. In particular, for the considered initial states, we will show that criticality has no significant effect in partial revivals. We also comment on the applicability of quasiparticle picture to determine the period of the partial revivals qualitatively. In particular, we detect a regime in the phase diagram of the XY chain in which one can not determine the period of the partial revivals using the quasiparticle picture.
The eigenstate thermalization hypothesis (ETH) provides a way to understand how an isolated quantum mechanical system can be approximated by a thermal density matrix. We find a class of operators in (1+1)-dimensional conformal field theories, consisting of quasiprimaries of the identity module, which satisfy the hypothesis only at the leading order in large central charge. In the context of subsystem ETH, this plays a role in the deviation of the reduced density matrices, corresponding to a finite energy density eigenstate from its hypothesized thermal approximation. The universal deviation in terms of the square of the trace-square distance goes as the eighth power of the subsystem fraction and is suppressed by powers of inverse central charge (c). Furthermore, the nonuniversal deviations from subsystem ETH are found to be proportional to the heavy-light-heavy structure constants which are typically exponentially suppressed in sqrt[h/c], where h is the conformal scaling dimension of the finite energy density state. We also examine the effects of the leading finite-size corrections.
At thermal equilibrium, the concept of effective central charge for massive deformations of two-dimensional conformal field theories (CFT) is well understood, and can be defined by comparing the partition function of the massive model to that of a CFT. This temperature-dependent effective charge interpolates monotonically between the central charge values corresponding to the IR and UV fixed points at low and high temperatures, respectively. We propose a non-equilibrium, time-dependent generalization of the effective central charge for integrable models after a quantum quench, ceff(t)c_{\rm eff}(t)ceff​(t), obtained by comparing the return amplitude to that of a CFT quench. We study this proposal for a large mass quench of a free boson, where the charge is seen to interpolate between ceff=0c_{\rm eff}=0ceff​=0 at t=0t=0t=0, and ceff∼1c_{\rm eff}\sim 1ceff​∼1 at t→∞t\to\inftyt→∞, as is expected. We use our effective charge to define an "Ising to Tricritical Ising" quench protocol, where the charge evolves from ceff=1/2c_{\rm eff}=1/2ceff​=1/2 at t=0t=0t=0, to ceff=7/10c_{\rm eff}=7/10ceff​=7/10 at t→∞t\to\inftyt→∞, the corresponding values of the first two unitary minimal CFT models. We then argue that the inverse "Tricritical Ising to Ising" quench is impossible with our methods. These conclusions can be generalized for quenches between any two adjacent unitary minimal CFT models. We finally study a large mass quench into the "staircase model" (sinh-Gordon with a particular complex coupling). At short times after the quench, the effective central charge increases in a discrete "staircase" structure, where the values of the charge at the steps can be computed in terms of the central charges of unitary minimal CFT models. When the initial state is a pure state, one always finds that ceff(t→∞)≥ceff(t=0)c_{\rm eff}(t\to\infty)\geq c_{\rm eff}(t=0)ceff​(t→∞)≥ceff​(t=0), though ceff(t)c_{\rm eff}(t)ceff​(t), generally oscillates at finite times. We explore how this constraint may be related to RG flow irreversibility.