The Magic Barrier before Thermalization

Abstract

We investigate the time dependence of anti-flatness in the entanglement spectrum, a measure for non-stabilizerness and lower bound for non-local quantum magic, on a subsystem of a linear SU(2) plaquette chain during thermalization. Tracing the time evolution of a large number of initial states, we find that the anti-flatness exhibits a barrier-like maximum during the time period when the entanglement entropy of the subsystem grows rapidly from the initial value to the microcanonical entropy. The location of the peak is strongly correlated with the time when the entanglement exhibits the strongest growth. This behavior is found for generic highly excited initial computational basis states and persists for coupling constants across the ergodic regime, revealing a universal structure of the entanglement spectrum during thermalization. We conclude that quantitative simulations of thermalization for nonabelian gauge theories require quantum computing. We speculate that this property generalizes to other quantum chaotic systems.

Figures (3)

• Figure 1: (a) Real-time evolution of entanglement entropy $S_A$ (blue solid line, left scale) and anti-flatness $\mathcal{F}_A$ (red solid line, right scale) of a small two-plaquette subsystem in the middle of an aperiodic seven-plaquette chain with $j_\mathrm{max}=1$, ergodic coupling $g^2=1$ and asymmetric boundary conditions $\{j_{\rm ext}\}=\{0,0,0,1\}$ for a randomly chosen, highly excited initial electric basis state with energy $E-E_0\approx 19.17$. The blue dashed line shows the entanglement entropy growth rate. $t_{\rm MB}$ and $t_{\rm EE}$, indicated by gray lines, correspond to the time of the magic barrier and maximum entanglement growth rate, respectively. (b) Entanglement spectra of the above state at different times during the thermalization process. $\lambda_k$ denotes the $k$-th eigenvalue of the entanglement Hamiltonian $H_A=-\log\rho_A$. The flat-spectrum limit is given by $|\lambda_k|= \log({\rm dim}(\mathcal{H}_A)) \approx 4.727$.
• Figure 2: (a) Real-time evolution of $S_A$ and $\mathcal{F}_A$ for a small two-plaquette subsystem in the middle of an aperiodic seven-plaquette chain with $j_\mathrm{max}=1$ and asymmetric boundary conditions $\{j_{\rm ext}\}=\{0,0,0,1\}$ at ergodic coupling $g^2=1$ for all electric basis states within the highly excited energy window $E-E_0\in [19.71, 20.21]$. The energy window contains 2728 states. (b) Rescaled time evolution of the same system and state ensemble. The real time $t$ is replaced by $\kappa\log(t/t_0)$ with state-dependent fit parameters $\kappa$ and $t_0$, such that the thermalization of different states is synchronized. The solid lines and bands in (a) and (b) are the ensemble means and $1\sigma$-bands. The left and right scales correspond to $S_A$ and $\mathcal{F}_A$, respectively. (c) 2D histogram showing the joint distribution of magic barrier time $t_{\rm MB}$ and time of maximum entanglement entropy growth $t_{\rm EE}$ on logarithmic scales for all 18389 physical electric basis states of the above system with energies below the spectrum mean. Color intensity corresponds to bin counts. The red line represents a linear fit with slope 1.331 and intercept 0.530.
• Figure 3: Coupling dependence of entanglement dynamics. Real-time evolution of (a) $S_A$ and (b) $\mathcal{F}_A$ for a small two-plaquette subsystem in the middle of an aperiodic seven-plaquette chain with $j_\mathrm{max}=1$ and asymmetric boundary conditions $\{j_{\rm ext}\}=\{0,0,0,1\}$ at different ergodic couplings $g^2\in\{1.0, 0.8, 0.6\}$. The ensemble of electric basis states is chosen such that at $g^2=0.6$ they lie in the highly excited energy window $E-E_0\in [26.48, 26.98]$. This ensemble contains 2995 states. The solid lines and bands in (a) and (b) are the ensemble means and $1\sigma$-bands. (c) Relations between the ensemble means $\overline{S}_A$ an $\overline{{\mathcal{F}}}_A$ for different ergodic couplings, which exhibit very mild coupling dependence.