Universality of equilibration dynamics after quantum quenches

TL;DR

The paper demonstrates a universal structure in the lower part of the entanglement spectrum after global quantum quenches in 1D integrable systems. By linking the spectrum to the large-$\alpha$ expansion of Rényi entropies and reconstructing $P(\lambda)$ from the moments $M_\alpha$, the authors derive a universal distribution near the largest eigenvalue, parameterized by $a_0,a_1$ and the scaling variable $\xi$. Depending on the regime and model, the lower spectrum exhibits either a CFT-like continuous form or a staircase of degenerate levels (delta peaks), with precise signatures in models such as the rule 54 chain, TFIC, XX, and XXZ via Bethe Ansatz and free-fermion techniques. Numerical benchmarks across these models confirm the predicted universality, highlighting a deep connection between equilibration dynamics and universal entanglement spectra that may extend to higher dimensions and other quench protocols.

Abstract

We investigate the distribution of the eigenvalues of the reduced density matrix (entanglement spectrum) after a global quantum quench. We show that in an appropriate scaling limit the lower part of the entanglement spectrum exhibits ``universality''. In the scaling limit and at asymptotically long times the distribution of the entanglement spectrum depends on two parameters that can be determined from the Rényi entropies. We show that two typical scenarios occur. In the first one, the distribution of the entanglement spectrum levels is similar to the one describing the ground-state entanglement spectrum in Conformal Field Theories. In the second scenario, the lower levels of the entanglement spectrum are highly degenerate and their distribution is given by a series of Dirac deltas. We benchmark our analytical results in free-fermion chains, such as the transverse field Ising chain and the XX chain, in the rule 54 chain, and in Bethe ansatz solvable spin models.

Figures (11)

• Figure 1: Cartoon of the setup that employed in this work. A subsystem $A$ of length $\ell$ is embedded in an infinite chain. The whole system undergoes unitary dynamics with an Hamiltonian $H$. We are interested in structure of the eigenvalues of the reduced density matrix $\rho_A$ (entanglement spectrum) in the hydrodynamic limit $t,\ell\to\infty$. We consider global quenches, i.e., quench protocols giving rise to linear entanglement growth at short times $t/\ell\ll1$ (short-time regime) and volume-law entanglement in the long-time regime at $t/\ell\gg1$.
• Figure 2: Cumulative distribution function $n(\lambda)$ of the entanglement spectrum plotted versus the scaling variable $\xi=[-\ln(\lambda_\mathrm{m})\ln(\lambda_\mathrm{m}/\lambda)]^{1/2}$, with $\lambda_\mathrm{m}$ the largest eigenvalue of the reduced density matrix. We employ a logarithmic scale on the $y$-axis. The continuous curve is the theoretical result $n(\lambda)=I_0(2r_1\xi)$ (cf. \ref{['eq:n-fin-fin']}) in the limit $t\to\infty$. Here $r_1=\sqrt{-a_1/a_0}$ and we choose $a_0=-0.2, a_1=0.3$. The dashed and dashed dotted lines are the theoretical results at times $t=10, 40$ obtained by including the subleading contributions with $a_3=0.5$ in the expansion \ref{['eq:Fa']}. At long times one recovers the continuous line. The dotted line is the CFT result, which corresponds to $r_1=1$.
• Figure 3: Quantum quenches in the XXZ chain. The coefficient $a_0$ (cf. \ref{['eq:Fa']}) plotted as a function of the chain anisotropy $\Delta$. The full line is the exact result for the Néel quench, whereas the dotted line is for the dimer quench. The red lines are obtained from \ref{['eq:large-alpha-sol']} and \ref{['eq:large-alpha-sol-dimer']} which are valid for $\Delta>\Delta_c$.
• Figure 4: Large $\alpha$ behavior of $F_\alpha$ (cf. \ref{['eq:Fa']}) after the Néel quench in the XXZ chain. We plot $-\ln M_\alpha-a_0\alpha$, with $M_\alpha=\mathrm{Tr}\rho_A^\alpha$ and $a_0$ as obtained analytically by solving \ref{['eq:large-alpha-c']}. On the y-axis we employ a logarithmic scale. The exponential decay at large $\alpha$ is clearly visible for any $\Delta$.
• Figure 5: Dimer quench in the XXZ chain. We plot $-\ln M_\alpha-a_0\alpha$ versus $\alpha$. The data are for the dimer quench in the XXZ with $\Delta=4$. In contrast with the Néel quench (see Fig. \ref{['fig:a1-xxz-neel']}) the large $\alpha$ behavior is power-law, as we show in the inset by using a double logarithmic scale. The dashed line is the analytical prediction for $a_1$ obtained by using \ref{['eq:a1-dimer']}.