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arXiv:1709.08654 · Journal of High Energy Physics

Universality of fast quenches from the conformal perturbation theory

Anatoly Y. Dymarsky, Michael Smolkin

Published Jan 1, 2018• 10 citations
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Abstract

We consider global quantum quenches, a protocol when a continuous field theoretic system in the ground state is driven by a homogeneous time-dependent external interaction. When the typical inverse time scale of the interaction is much larger than all relevant scales except for the UV-cutoff the system’s response exhibits universal scaling behavior. We provide both qualitative and quantitative explanations of this universality and argue that physics of the response during and shortly after the quench is governed by the conformal perturbation theory around the UV fixed point. We proceed to calculate the response of one and two-point correlation functions confirming and generalizing universal scalings found previously. Finally, we discuss late time behavior after the quench and argue that all local quantities will equilibrate to their thermal values specified by an excess energy acquired by the system during the quench.

DOI
10.1007/jhep01(2018)112

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We consider global quantum quenches, a protocol when a continuous field theoretic system in the ground state is driven by a homogeneous time-dependent external interaction. When the typical inverse time scale of the interaction is much larger than all relevant scales except for the UV-cutoff the system’s response exhibits universal scaling behavior. We provide both qualitative and quantitative explanations of this universality and argue that physics of the response during and shortly after the quench is governed by the conformal perturbation theory around the UV fixed point. We proceed to calculate the response of one and two-point correlation functions confirming and generalizing universal scalings found previously. Finally, we discuss late time behavior after the quench and argue that all local quantities will equilibrate to their thermal values specified by an excess energy acquired by the system during the quench.
We expand on the investigation of the universal scaling properties in the early time behaviour of fast but smooth quantum quenches in a general d-dimensional conformal field theory deformed by a relevant operator of dimension Δ with a time-dependent coupling. The quench consists of changing the coupling from an initial constant value λ 1 by an amount of the order of δλ to some other final value λ 2, over a time scale δt. In the fast quench limit where δt is smaller than all other length scales in the problem, δt ≪ λ 1 1/() , λ 2 1/() , δλ 1/(Δ − d), the energy (density) injected into the system scales as δℰ ∼ (δλ)2(δt) d− 2Δ . Similarly, the change in the expectation value of the quenched operator at times earlier than the endpoint of the quench scales as ⟨OΔ⟩∼δλ(δt)d−2Δ \left\langle {\mathcal{O}}_{\varDelta}\right\rangle \sim \delta \lambda {{\left(\delta t\right)}^d}^{-2\varDelta } ⟨OΔ​⟩∼δλ(δt)d−2Δ , with further logarithmic enhancements in certain cases. While these results were first found in holographic studies, we recently demonstrated that precisely the same scaling appears in fast mass quenches of free scalar and free fermionic field theories. As we describe in detail, the universal scaling refers to renormalized quantities, in which the UV divergent pieces are consistently renormalized away by subtracting counterterms derived with an adiabatic expansion. We argue that this scaling law is a property of the conformal field theory at the UV fixed point, valid for arbitrary relevant deformations and insensitive to the details of the quench protocol. Our results highlight the difference between smooth fast quenches and instantaneous quenches where the Hamiltonian abruptly changes at some time.
We examine in detail the relationship between smooth fast quantum quenches, characterized by a time scale δt, and instantaneous quenches, within the framework of exactly solvable mass quenches in free scalar field theory. Our earlier studies [1, 2] highlighted that the two protocols remain distinct in the limit δt → 0 because of the relation of the quench rate to the UV cut-off, i.e., 1/δt ≪ Λ always holds in the fast smooth quenches while 1/δt ∼ Λ for instantaneous quenches. Here we study UV finite quantities like correlators at finite spatial distances and the excess energy produced above the final ground state energy. We show that at late times and large distances (compared to the quench time scale) the smooth quench correlator approaches that for the instantaneous quench. At early times, we find that for small spatial separation and small δt, the correlator scales universally with δt, exactly as in the scaling of renormalized one point functions found in earlier work. At larger separation, the dependence on δt drops out. The excess energy density is finite (for finite mδt) and scales in a universal fashion for all d. However, the scaling behaviour produces a divergent result in the limit mδt → 0 for d ≥ 4, just as in an instantaneous quench, where it is UV divergent for d ≥ 4. We argue that similar results hold for arbitrary interacting theories: the excess energy density produced is expected to diverge for scaling dimensions Δ > d/2.
We study the time evolution of a conformal field theory deformed by a relevant operator under a smooth but fast quantum quench which brings it to the conformal point. We argue that when the quench time scale δt is small compared to the scale set by the relevant coupling, the expectation value of the quenched operator scales universally as δλ/δt(2Δ-d), where δλ is the quench amplitude. This growth is further enhanced by a logarithmic factor in even dimensions. We present explicit results for free scalar and fermionic field theories, supported by an analytic understanding of the leading contribution for fast quenches. Our results suggest that this scaling result, first found in holography, is in fact quite general. Our considerations also show that this limit of fast smooth quenches is quite different from an instantaneous quench from one time-independent Hamiltonian to another, where the state at the time of the quench serves as an initial condition for subsequent evolution with the final Hamiltonian.
We study the time evolution of a conformal field theory deformed by a relevant operator under a smooth but fast quantum quench which brings it to the conformal point. We argue that when the quench time scale δt\delta tδt is small compared to the scale set by the relevant coupling, the expectation value of the quenched operator scales universally as δg/δt2Δ−d\delta g/ \delta t ^{2\Delta-d}δg/δt2Δ−d where δg\delta gδg is the quench amplitude. This growth is further enhanced by a logarithmic factor in even dimensions. We present explicit results for free scalar and fermionic field theories, supported by an analytic understanding of the leading contribution for fast quenches. Results from this Letter suggest that this scaling result, first found in holography, is in fact universal to quantum quenches. Our considerations also show that this limit of fast smooth quenches is quite different from an instantaneous quench from one time-independent Hamiltonian to another, where the Schrodinger picture state at the time of the quench simply serves as an initial condition for subsequent evolution with the final Hamiltonian.
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.