Universality in fast quantum quenches

TL;DR

The paper establishes a universal early-time scaling for fast but smooth quantum quenches in any $d$-dimensional CFT deformed by a relevant operator of dimension $\Delta$, with a time-dependent coupling. By deriving renormalization counterterms from an adiabatic expansion and solving exactly for free scalar and fermion quenches, it shows that the injected energy and the quenched operator obey $\delta \mathcal{E} \sim (\delta\lambda)^2 (\delta t)^{d-2\Delta}$ and $\langle \mathcal{O}_\Delta\rangle \sim \delta\lambda (\delta t)^{d-2\Delta}$ (with logarithmic enhancements in certain cases), independent of the quench protocol details. It further demonstrates that these scalings persist in CFT-to-CFT quenches and extend to general initial/final masses, and extends the analysis to higher-spin currents in free theories, suggesting broad universality. The work highlights the distinctive nature of smooth fast quenches versus instantaneous quenches, and provides a framework for renormalized observables in dynamic settings, with implications for holography and interacting theories near a UV fixed point.

Abstract

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 \ll λ_1^{1/(Δ-d)}, λ_2^{1/(Δ-d)}, δλ^{1/(Δ-d)}$, the energy (density) injected into the system scales as $δ{\cal E} \sim (δλ)^2 (δt)^{d-2Δ}$. Similarly, the change in the expectation value of the quenched operator at times earlier than the endpoint of the quench scales as $<{\cal O}_Δ> \sim δλ(δt)^{d-2Δ}$, 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.

Figures (21)

• Figure 1: (Colour online) Renormalized expectation values $\langle \phi^2\rangle_{ren}$ as a function of time $t/\delta t$, for $d=3$ and $4$. In each plot, the different curves correspond to different quench rates: $\delta t = 1/1,1/2, \cdots, 1/10$ where the curves exhibiting higher peaks (in absolute value) correspond to smaller values of $\delta t$. Note that the expectation value is multiplied by the numerical constant $\sigma_s=\frac{2(2\pi)^{d-1}}{\Omega_{d-2}}$. Further, at each time, the expectation value for an 'adiabatic' quench is subtracted.
• Figure 2: (Colour online) Renormalized expectation values $\langle \phi^2\rangle_{ren}$ as a function of time $t/\delta t$, for $d=5$, 6 and $7$. In each plot, the different curves correspond to different quench rates: $\delta t = 1/1,1/2, \cdots, 1/10$ where the curves exhibiting higher peaks (in absolute value) correspond to smaller values of $\delta t$. Note that the expectation value is multiplied by the numerical constant $\sigma_s=\frac{2(2\pi)^{d-1}}{\Omega_{d-2}}$. Further, at each time, the expectation value for an 'adiabatic' quench is subtracted.
• Figure 3: (Colour online) Renormalized expectation values $\langle \phi^2\rangle_{ren}$ as a function of time $t/\delta t$, for $d=8$ and $9$. In each plot, the different curves correspond to different quench rates: $\delta t=1/20, 1/21, \cdots, 1/30$ where the curves exhibiting higher peaks (in absolute value) correspond to smaller values of $\delta t$. As in the previous figure, the expectation value is multiplied by the numerical constant $\sigma_s=\frac{2(2\pi)^{d-1}}{\Omega_{d-2}}$. Further, at each time, the expectation value for an 'adiabatic' quench is subtracted.
• Figure 4: (Colour online) Expectation value $\langle\phi^2\rangle_{ren}(t=0)$ as a function of the quench times $\delta t$ for spacetime dimensions from $d=3$ to 9. Note that in the plot, the expectation values are multiplied by the numerical factor: $\sigma_s=\frac{2(2\pi)^{d-1}}{\Omega_{d-2}}$. The slope of the linear fit in each case is shown in the brackets beside the labels. The results support the power law scaling $\langle\phi^2\rangle_{ren} \sim \delta t^{4-d}$.
• Figure 5: (Colour online) Expectation value $\langle\phi^2\rangle_{ren}(t=\delta t/2)$ as a function of $\delta t$ for spacetime dimensions $d=6$ and $d=8$ --- the lower curve corresponds to $d=6$. As in previous plots, the expectation values are multiplied by $\sigma_s=\frac{2(2\pi)^{d-1}}{\Omega_{d-2}}$. We show in a blue solid curve the best fit by a function $f(\delta t) = \delta t^{-\alpha} (a \log \delta t +b)$, where we get $\alpha = 1.9995$ for $d=6$ and $\alpha = 4.0097$ for $d=8$. The purple curve is the best fit for a function $f(\delta t) = a \delta t^{4-d}$. The plots clearly show that there is an extra logarithmic divergence in expectation values. The results support the scaling $\langle\phi^2\rangle_{ren} \propto \delta t^{4-d} \log (\delta t)$ for even $d$.