Symmetry Resolved Entanglement with $U(1)$ Symmetry: Some Closed Formulae for Excited States

TL;DR

This work addresses the problem of computing symmetry-resolved entanglement entropies $S_n(q)$ for zero-density excited states in theories with a $U(1)$ symmetry in infinite volume. It introduces a qubit-based framework to express SREEs for states consisting of multiple subsets of excitations and derives closed formulas for charged moments $M_n^{\cdots}(r;\alpha)$ and symmetry-resolved partition functions $\mathcal{Z}_n^{\cdots}(r,q)$, including extensions to conformal field theories via Fourier transforms of the charged moments. A key finding is that, in the identical-charge qubit sector, the SREEs become independent of the region-to-system-size ratio $r$, while the configuration and number entropies retain $r$-dependence; for mixed-charge configurations, all entropies exhibit explicit $r$-dependence. The results connect combinatorial binomial structures to symmetry-resolved entanglement and provide exact closed-form expressions, including hypergeometric representations in mixed-charge cases, with implications for both theory and potential experimental measurements of SREEs. The paper also outlines natural extensions to non-Abelian symmetries and broader quantum-information contexts in critical and topological systems.

Abstract

In this work, we revisit a problem we addressed in previous publications with various collaborators, that is, the computation of the symmetry resolved entanglement entropies of zero-density excited states in infinite volume. The universal nature of the charged moments of these states has already been noted previously. Here, we investigate this problem further, by writing general formulae for the entropies of excited states consisting of an arbitrary number of subsets of identical excitations. When the initial state is written in terms of qubits with appropriate probabilistic coefficients, we find the final formulae to be of a combinatorial nature too. We analyse some of their features numerically and analytically and find that for qubit states consisting of particles of the same charge, the symmetry resolved entropies are independent of region size relative to system size, even if the number and configuration entropies are not.

Figures (3)

• Figure 1: The symmetry resolved Rényi entropies for charges $q=1 (\textrm{left}), 2 (\textrm{right})$ and various excited states consisting of five distinct groups of indistinguishable particles. In the first row the first four groups contain a single particle while in the second row they contain two. The fifth group has size $p=1 (\textrm{black}), 2(\textrm{green}), 3(\textrm{red}), 4(\textrm{dark blue}), 5(\textrm{orange})$ excitations. The entropy is highest when all groups consist of the same number of excitations and lowest for the largest value of $p$. Various values can be easily computed from the formulae (\ref{['39']}) and (\ref{['32']}). For instance, the black line in the top and bottom left figures stands at $\log 5$, which follows easily from the formulae (\ref{['39']}) and (\ref{['32']}). For the top right figure the black line stands at $\log 10$ which again follows from (\ref{['32']}). Finally, the top green curve in the bottom right figure is $n$-dependent but for $n$ large tends to the value $\log \frac{45}{4}$ which also follows from (\ref{['32']}). In these figures we have dropped the superindex '1' on the particle subgroups as they all have the same charge.
• Figure 2: The symmetry resolved von Neumann entropies for charges $q=1$ (top), $3$, and $5$ (bottom) and various excited states consisting of five distinct groups of indistinguishable particles. In this example three groups consist of one (left)/two (right) excitation(s). The fourth group contains $w=1 (\textrm{black}), 2(\textrm{green}), 3(\textrm{red}), 4(\textrm{dark blue}), 5(\textrm{orange})$ excitations and $p$ varies from 1 to 10 as shown. We observe that the maximum value of the entropy is typically reached when $p=q=w$. In these figures we have dropped the superindex '1' on the particle subgroups as they all have the same charge.
• Figure 3: The symmetry resolved Rényi entropies $S_n^{2^+ 2^+ 2^-}(r,q)$ for charges $q=-1,0,1,2$ of a state consisting of three groups of two identical particles each. Two of the groups have positive charge and one group has negative charge. The different colours correspond to different $n$. The values are $n=2$(black), $n=4$(green), $n=7$ (red) and $n=20$ (blue). The entropies exhibit the property $S_n^{2^+ 2^+ 2^-}(r,q)=S_n^{2^+ 2^+ 2^-}(1-r, 2-q)$ which is state-dependent.