Detecting Symmetry-Resolved Entanglement: A Quantum Monte Carlo Approach

Abstract

Symmetry and entanglement are two fundamental concepts in quantum many-body physics. Their interplay is captured by symmetry-resolved entanglement, which decomposes the total entanglement into contributions from different symmetry sectors. Computing symmetry-resolved entanglement in strongly interacting higher-dimensional quantum systems remains challenging. Here, we introduce a quantum Monte Carlo (QMC) approach for computing symmetry-resolved Rényi entropies (SRRE) in large-scale interacting systems by measuring disorder (symmetry-twisted) operators on replica manifolds and reconstructing SRRE from the corresponding charged moments. We apply this method to the transverse-field Ising model (TFIM) in one and two dimensions. In one dimension, we recover the conformal-field-theory prediction for the logarithmic scaling of the disorder operator and observe the expected approach to entanglement equipartition. In two dimensions, our data provide numerical evidence consistent with entanglement equipartition at the (2+1)D Ising critical point. We further apply the framework to the 1D Heisenberg chain and obtain results consistent with the expected asymptotic scaling and finite-size corrections in the U(1)-resolved sectors. Our work establishes a practical numerical route to symmetry-resolved entanglement in interacting lattice models and provides a useful framework for future studies beyond one dimension.

Figures (5)

• Figure 1: (a) Block-diagonal structure of $\rho_A$ in the eigenbasis of the conserved charge $Q_A$. (b) Typical probability distribution $P(q)=\mathrm{Tr}[\tilde{\rho}_A(q)]$ in the 1D Heisenberg model, with $q=S_A^z$. (c) Single-replica (upper panel) and two-replica (lower panel) configurations corresponding to the ordinary partition function $Z$ and the replica partition function $Z_A^{(2)}$, respectively. The geometry of $Z_A^{(2)}$ can be viewed as two replicas of the single-replica configuration, with region $A$ glued together along the imaginary-time direction. The disorder operator $e^{i\alpha Q_A}$ is measured in the entangling region $A$.
• Figure 2: Scaling of the disorder operator $\langle X \rangle$ in the single- and two-replica ensembles at the quantum critical point of the TFIM. (a) 1D chain of length $L$ with a half-chain entangling region ($L_A=L/2$). The data are fitted by $-\ln \langle X \rangle_Z = 0.251(1)\ln L + 0.149(5)$ and $-\ln \langle X \rangle_{Z_A^{(2)}} = 0.1242(4)\ln L - 0.074(2)$. (b) 2D $L\times L$ square lattice with an entangling region of size $L/2\times L$ and a smooth boundary. The single-replica data are fitted by $-\ln \langle X \rangle_Z = 0.0762(1)L + 0.066(2)$, while the two-replica data are well described by $-\ln \langle X \rangle_{Z_A^{(2)}} = 0.00793(2)L + 0.0343(3)\ln L - 0.0347(5)$. The inset shows $-\ln \langle X \rangle - aL$ versus $\ln L$ for both ensembles, where $a$ denotes the fitted coefficient of the leading linear term.
• Figure 3: Asymptotic scaling of the subtracted symmetry-resolved Rényi entropy $S_2(q)-S_A^{(2)}$ in the 1D (panel (a)) and 2D (panel (b)) TFIM at their respective QCPs. Symbols denote the numerical data, while dashed lines are obtained by inserting the fitted scaling forms from Fig. \ref{['fig:fig2']} into Eq. \ref{['eq:subSREE']}. The inset in panel (a) shows the subtracted symmetry-resolved Rényi entropy in the large-$L$ limit.
• Figure 4: Analysis of symmetry-resolved entanglement for the 1D Heisenberg model. (a) Disorder operator $\langle e^{i\alpha S^{z}_{A}} \rangle$ as a function of $\alpha$ in the single-replica and two-replica ensembles at $L=768$. (b) Probability distribution $P(q)$ of the subsystem charge $q$ for $L=64$ and $L=768$. (c) Scaling of the charge variance $\mathrm{Var}_{q}$ versus $\ln L$. The red line represents a linear fit, yielding $\mathrm{Var}_{q}(L)=0.0547(1)\ln L+0.1334(8)$. (d) Scaling of the number entropy $S^{\mathrm{num}}$ versus $\ln(\ln L)$. The red line is a linear fit, yielding $S^{\mathrm{num}}(L)=0.3483(9)\ln(\ln L)+0.409(2)$. (e) Finite-size scaling of the subtracted symmetry-resolved Rényi entropy, $S_2(q)-S_{A}^{(2)}+\frac{1}{2}\ln(\ln L)$, plotted as a function of $1/\ln L$ for the charge sectors $q=0$ and $q=1$. The dashed lines are linear extrapolations to the thermodynamic limit, $1/\ln L \to 0$.
• Figure 5: Comparison of ED and QMC results for the disorder operator in the 1D Heisenberg model.