Symmetry Resolved Entanglement Entropy: Equipartition under Driven and Non-unitary Evolution in a Compact Boson CFT

Abstract

We study the evolution of symmetry-resolved entanglement entropy in bulk-driven Floquet conformal field theories (CFTs). Focusing on the two-dimensional free compact boson CFT, we analyze how symmetry-resolved Rényi entropies approach or depart from equipartition among charge sectors. We show that the existence of an $\mathfrak{sl}^{(k)}(2,\mathbb{R})$ subalgebra of the Virasoro algebra introduces a free parameter, the label $k$, which allows us to control the breakdown of equipartition. We argue that this effect originates from an explicit coupling between low- and high-frequency modes. Based on a general oscillator representation of the Virasoro algebra, we expect this mechanism to persist beyond the free boson CFT. Finally, we discuss how the real-time dynamics of fine-grained symmetry-resolved entropies of a boundary state are modified under non-unitary evolution, which can be associated with post-selected weak measurements.

Figures (4)

• Figure 1: Evolution of the symmetry-resolved entanglement entropy, $S_A(q;N)$, as a function of the number of drive cycles $N$ in the heating phase (top-left), at the phase transition (top-right), and in the non-heating phase (bottom) of the driving protocol \ref{['eq:drive-two-step-drive-protocol']}. We take $k=3$, $q=2$, $L=1000$, and the driving parameter $\theta=0.1$. As a benchmark, we also show the evolution of the total von Neumann entanglement entropy, $S_A(N)$, as a function of the number of drive cycles $N$.
• Figure 2: The top-left panel shows the $q$-dependent contribution to the SREE, shown as a function of charge $q$ for a representative moderate width ($B_1=5$, blue dots) and a large width ($B_1=5\times10^{4}$, orange squares). The top-right, bottom-left, and bottom-right panels display stroboscopic evolution of SREE for different charge sectors $q$ in the heating, phase-transition, and non-heating phases of the two-step driving protocol \ref{['eq:drive-two-step-drive-protocol']}, respectively. For these three figures, we set $L=1200$, $k=120$, and the driving parameter $\theta=0.1$, such that the effective scale $\ell=L/k$ is sufficiently small to make the breakdown of charge equipartition clearly visible.
• Figure 3: Evolution of the number (Shannon) entropy $S_{\rm num}(N)$ as a function of the number of drive cycles $N$ in the heating, non-heating, and phase-transition regimes of the two-step driving protocol \ref{['eq:drive-two-step-drive-protocol']}. Here we fix $k=3$ and $\theta=0.1$.
• Figure 4: Euclidean path-integral representation of the reduced density matrix of the state \ref{['eq:state_tau12']} for the half-line subsystem $A$ and its conformal mapping to the cylinder. The cylinder length $W(t)$ after the analytic continuation \ref{['eq:anal_cont']} is given by Eq. \ref{['eq:quench-annulus-width-Wt']}. The red line represents the line defect that implements the insertion $e^{i\alpha Q_A}$ in the charged moments $Z_n(\alpha; t)= \mathrm{Tr}\left(\rho_A(t)^{n}\,e^{i\alpha Q_A}\right)$.