Charged moments and symmetry-resolved entanglement from Ballistic Fluctuation Theory

TL;DR

This work addresses symmetry-resolved entanglement in quantum many-body systems with a global $U(1)$ symmetry by introducing charged moments $Z_m(\alpha)$ and employing Ballistic Fluctuation Theory (BFT) together with the height-field twist-field formalism. The authors derive analytic expressions for charged Rényi entropies in both equilibrium generalized Gibbs ensembles (GGEs) and out-of-equilibrium settings after symmetry-preserving quenches for free fermions, including composite branch-point twist fields that incorporate a flux $\alpha$. The results connect the charged moments to full counting statistics and the dynamical large-deviation formalism, reproducing known quenches (e.g., Néel and Dimer) and supporting the quasiparticle interpretation of entanglement growth. This framework paves the way for extensions to bosonic systems, interacting integrable models, and more general quenches, offering a unified hydrodynamic perspective on symmetry-resolved entanglement via twist fields and large-deviation statistics.

Abstract

The charged moments of a reduced density matrix provide a natural starting point for deriving symmetry-resolved Rényi and entanglement entropies, which quantify how entanglement is distributed among symmetry sectors in the presence of a global internal symmetry in a quantum many-body system. In this work, we study charged moments within the framework of Ballistic Fluctuation Theory (BFT). This theory describes large-scale ballistic fluctuations of conserved charges and associated currents and, by exploiting the height-field formulation of twist fields, gives access to the asymptotic behaviour of their two-point correlation functions. In Del Vecchio Del Vecchio et al. $[1]$, this approach was applied to the special case of branch-point twist fields used to compute entanglement entropies within the replica approach. Here, we extend those results by applying BFT to composite branch-point twist fields, obtained by inserting an additional gauge field. Focusing on free fermions, we derive analytic expressions for charged Rényi entropies both at equilibrium, in generalized Gibbs ensembles, and out of equilibrium following a quantum quench from $U(1)$ preserving pair producing integrable initial states. In the latter case, our results agree with the conjecture arising from the quasiparticle picture.

Figures (1)

• Figure 1: The region $A$ is denoted by the red line, and is equivalent to the straight region from $(0,t)\to (x,t)$. (Left) In the regime $t \ll x$ there are long range correlations in $A$ due to the propagation of pairs from the initial state. The long range correlations cannot be described by a GGE, hence expectation values along $A$ are not equivalent to the expectation values in a GGE. One can avoid the long range correlation by integrating along $\tilde{A}$ which is defined by the blue line and still connects the point $(0,t)$ to $(x,t)$, but now via the path $(0,t)\to(0,0)\to(x,0)\to(x,t)$. (Right) In the regime $t\gg x$, the long range correlations exist within $\tilde{A}$ but not $A$, and this is what allows the expectation value to be mapped to a GGE. Note that pairs of particle propagate with momentum $k$ and $k+\pi$, the velocity associated to the particle with momentum $k+\pi$ is given by a sine function, hence it is equivalent to $-v_k$.