Complexity as a novel probe of quantum quenches: universal scalings and purifications

TL;DR

The recently developed notion of complexity for field theory is applied to a quantum quench through a critical point in 1+1 dimensions, and it is demonstrated that complexity is capable of probing features to which the entanglement entropy is insensitive.

Abstract

We apply the recently developed notion of complexity for field theory to a quantum quench through a critical point in 1+1 dimensions. We begin with a toy model consisting of a quantum harmonic oscillator, and show that complexity exhibits universal scalings in both the slow and fast quench regimes. We then generalize our results to a 1-dimensional harmonic chain, and show that preservation of these scaling behaviours in free field theory depends on the choice of norm. Applying our set-up to the case of two oscillators, we quantify the complexity of purification associated to a subregion, and demonstrate that complexity is capable of probing features to which the entanglement entropy is insensitive. We find that the complexity of subregions is subadditive, and comment on potential implications for holography.

Figures (4)

• Figure 1: Log-log plot of complexity of the $(1\!+\!1)$-dimensional free field theory \ref{['eq:Cfield2']} at the critical point $t\!=\!0$ vs. the quench rate $\delta t$ (measured in units of the lattice spacing), with $\omega_0\!=\!0.005$. The straight-line fit (blue) reveals linear scaling in the fast regime.
• Figure 2: Single-mode contributions to the complexity \ref{['eq:Cfield2']} at the critical point $t\!=\!0$ for $\omega_0\!=\!0.005$ and $k\!=\!\{0.006, 0.111, 0.216, 0.320, 0.425 \}$ (resp. red, orange, yellow, green, blue). For large $\delta t$, the exact solutions (dotted) agree with the saturation values \ref{['eq:Csat']} predicted from KZ (solid).
• Figure 3: Zero-mode contribution to $\mathcal{C}(0)$\ref{['eq:Cfield2']} as a function of the quench rate $\omega_0\delta t$, with $\omega_0\!=\!0.005$. The complexity scales linearly in the fast regime ($\ln C/\ln \delta t=1$, blue), and smoothly transitions to a logarithmic scaling $\tfrac{1}{4}\log \delta t$ in the slow regime (red). The transition to KZ occurs at $\omega_0\delta t\!\sim\!1$ which in this case is $\delta t\!\sim\!200$ in lattice units.
• Figure 4: Comparison of the complexity \ref{['eq:Cfield']} as a function of time $t$ of the original target state (solid) and the optimum purification (dashed) for $\delta t=10$ (blue) and $\delta t=1$ (red), with $\omega_\mathrm{R}=0.5$ for both oscillators. Note that the latter never exceeds the former, and is always greater than $\mathcal{C}/2$; that is, the complexity of purification appears to satisfy superadditivity \ref{['eq:super']}. We have tested this conjecture numerically for $\sim\!70,\!000$ cases.