A Physical Theory of Two-stage Thermalization

Abstract

One indication of thermalization time is subsystem entanglement reaching thermal values. Recent studies on local quantum circuits reveal two exponential stages with decay rates $r_1$ and $r_2$ of the purity before and after thermalization. We provide an entanglement membrane theory interpretation, with $r_1$ corresponding to the domain wall free energy. Circuit geometry can lead to $r_1 < r_2$, producing a ``phantom eigenvalue". Competition between the domain wall and magnon leads to $r_2 < r_1$ when the magnon prevails. However, when the domain wall wins, this mechanism provides a practical approach for measuring entanglement growth through local correlation functions.

Figures (8)

• Figure 1: Two-stage thermalization. (a): Schematics of 2-stage decay for staircase and brickwall geometry. Both insets show circuits of 4 time steps. Generally $r_1 < r_2$ for staircase geometry and $r_1 \ge r_2$ for brickwork geometry. (b) For the dual unitary $(1,1,a_z)$ circuits, $r_2$ depends on $a_z$ and the boundary condition (periodic or open).
• Figure 2: Domain wall configurations in brickwall ((a)(b)) and staircase ((c)(d)) geometries. (a) $t< t_{\rm sat}$, spin states at the lattice level (top) and domain wall random walk at the coarsed grained level (bottom). (b) $t> t_{\rm sat}$ for brickwall. (c) $t< t_{\rm sat}$, domain wall has shorter paths in the staircase. (d) Same as (b) for staircase.
• Figure 3: Magnon partition function and magnon decay rate. (a) heatmap of $Z_{\rm mag}(x, t) / \text{max}_x |Z_{\rm mag}(x, t)|$. The magnon mostly travels on the light cone. (b) Magnon decay rate for clean dual unitary circuits $(1,1,0.7)$, $(1,1,0.6)$ and clean non-dual unitary Floquet circuit $(0.9,0.8,0.5)$.
• Figure 4: Domain wall and magnon in staircase geometry. (a) The time spans of each parts for $t < t_{\rm sat}$. If the (red) domain wall (anti)-slope is $v$, then $t_2 = \frac{t}{1+v}$. For dual unitary circuit, (b) domain wall favors to have $v = 1$, reducing its rate to be $\frac{\mathcal{E}(0)}{2}$; (c) magnon propagates on the light cone, also reducing its rate to $\frac{1}{2}$ of its value computed from the brickwall geometry.
• Figure 5: $Z_{\rm mag}(x,t)$ for averaged dual unitary dynamics $(1,1,a_z)$, starting with a magnon mode $\ket{-+ \cdots +}$ from the top boundary. As $a_z$ increases, we observe the ballistic magnon mode becoming increasingly dominant in the dynamics, consistent with the symmetric unitarity $u_{sym}$ approaching a SWAP gate in the limit of $(1,1,1)$. For $(1,1,a_z \sim 0.2)$, the $b_-$ and $b_+$ moves have nearly equal magnitude but opposite signs, resulting in both a positive (red) and negative (blue) ballistic magnon.