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.

• The paper develops a QMC method that accurately computes symmetry-resolved entanglement in interacting many-body quantum systems.
• It leverages disorder operator measurements on replica manifolds to extract sector-specific entanglement scaling.
• Benchmarking with TFIM and Heisenberg models confirms universal scaling laws and the method’s scalability in various dimensions.

Detecting Symmetry-Resolved Entanglement via Quantum Monte Carlo: An Expert Analysis

Introduction and Motivation

The entanglement structure of many-body quantum states encodes nonlocal correlations that reveal far more than is accessible via local order parameters. Recent developments in symmetry-resolved entanglement entropies (SREEs) have enabled a sector-wise decomposition of entanglement according to conserved charges, providing both enhanced discrimination between quantum phases and more granular insight into criticality, topological order, and dynamics. However, calculating SREEs—especially in higher dimensions for generic, interacting systems—has been severely limited by the lack of scalable, unbiased numerical approaches.

The paper “Detecting Symmetry-Resolved Entanglement: A Quantum Monte Carlo Approach” (2604.02307) addresses this gap by developing a QMC methodology for computing SREEs, circumventing the scaling limitations of ED and tensor-network methods and leveraging the formalism of disorder (symmetry-twisted) operators on replica manifolds. The work provides both a general algorithmic framework and extensive benchmarking in canonical models, positioning the approach to enable future high-precision investigations in strongly correlated systems.

Framework for Symmetry-Resolved Entanglement

Formalism

If a system possesses a global symmetry, the reduced density matrix $\rho_A$ for a spatial region $A$ can be block-diagonalized according to the eigenvalues $q$ of a conserved charge $Q_A$. SREEs then quantify entanglement within each symmetry sector: $$S_n(q) = \frac{1}{1-n} \ln\!\left[\operatorname{Tr}\left(\rho_A(q)^n\right)\right],$$ where $\rho_A(q)$ is the normalized RDM block for eigenvalue $q$.

The method links the $n$th charged moment

$$Z_n(\alpha) = \operatorname{Tr}\left(\rho_A^n e^{i\alpha Q_A}\right)$$

to sector-projected traces by Fourier transformation. Thus, $Z_n(q) = \operatorname{Tr}(\Pi_q \rho_A^n)$. The SREE can be reconstructed as

$A$0

Figure 1: (a) Block-diagonal structure of $A$1 in the $A$2 eigenbasis; (b) Associated $A$3 distribution; (c) Geometry for measurement of disorder operators on single- and two-replica manifolds.

Methodological Implementation

The key technical insight is that $A$4 is accessible as a QMC expectation value of a nonlocal disorder operator $A$5 on an $A$6-replica (replica-glued) manifold. In practice, it suffices to perform QMC sampling in both the ordinary and the replica manifolds to extract the necessary expectation values. By scanning $A$7 (or, for discrete symmetry, evaluating the requisite points), one can reconstruct the full set of sector-resolved moments.

For abelian discrete symmetries such as $A$8, only two points need to be evaluated ($A$9), whereas for continuous symmetries ($q$0), a discrete Fourier transform is used over the spectrum of $q$1. This enables an efficient extraction of SREEs for arbitrary sector $q$2 in a single tour of large-scale QMC simulation.

Numerical Results and Benchmarking

1D and 2D Transverse Field Ising Model (TFIM)

The TFIM is examined at criticality (1D: $q$3; 2D: $q$4), focusing on the even and odd parity sectors of $q$5 symmetry. For 1D, the scaling of the disorder operator $q$6 in both single and double replica ensembles is predicted by CFT to have logarithmic corrections: $q$7 The prefactor extracted numerically, $q$8 ($q$9) and $Q_A$0 ($Q_A$1), matches the exact predictions to high precision.

Figure 2: Scaling of $Q_A$2 in single- and two-replica ensembles for 1D and 2D TFIM, showing both area law and logarithmic corrections.

In 2D, the single-replica ensemble displays a pure area law, while the two-replica ensemble reveals a subleading logarithmic correction, indicating subtle contributions beyond what is captured by area-law scaling alone. The subtracted SREE $Q_A$3 converges to $Q_A$4, as expected from the equipartition theorem, but with significantly different rates in 1D versus 2D: whereas 2D reaches the asymptotic value at accessible $Q_A$5, the approach is logarithmically slow in 1D.

Figure 3: Subtracted SREE $Q_A$6 scaling in 1D and 2D TFIM across symmetry sectors, evidencing equipartition.

1D Heisenberg Chain and $Q_A$7 Symmetry

For the 1D spin-$Q_A$8 Heisenberg model, the $Q_A$9 symmetry sector $$S_n(q) = \frac{1}{1-n} \ln\!\left[\operatorname{Tr}\left(\rho_A(q)^n\right)\right],$$0 corresponds to the total $$S_n(q) = \frac{1}{1-n} \ln\!\left[\operatorname{Tr}\left(\rho_A(q)^n\right)\right],$$1 in the region $$S_n(q) = \frac{1}{1-n} \ln\!\left[\operatorname{Tr}\left(\rho_A(q)^n\right)\right],$$2. The disorder operator expectation values $$S_n(q) = \frac{1}{1-n} \ln\!\left[\operatorname{Tr}\left(\rho_A(q)^n\right)\right],$$3 are sampled, and discrete Fourier inversion yields the SREEs for each $$S_n(q) = \frac{1}{1-n} \ln\!\left[\operatorname{Tr}\left(\rho_A(q)^n\right)\right],$$4.

The scaling is governed by: $$S_n(q) = \frac{1}{1-n} \ln\!\left[\operatorname{Tr}\left(\rho_A(q)^n\right)\right],$$5 with the crucial subleading double-logarithmic term, in agreement with CFT predictions.

Figure 4: (a) Charged disorder operator measurements; (b) charge distribution $$S_n(q) = \frac{1}{1-n} \ln\!\left[\operatorname{Tr}\left(\rho_A(q)^n\right)\right],$$6; (c) logarithmic scaling of charge variance; (d) number entropy $$S_n(q) = \frac{1}{1-n} \ln\!\left[\operatorname{Tr}\left(\rho_A(q)^n\right)\right],$$7 vs. $$S_n(q) = \frac{1}{1-n} \ln\!\left[\operatorname{Tr}\left(\rho_A(q)^n\right)\right],$$8; (e) finite-size scaling collapse of corrected SREE for primary sectors.

Systematic QMC-ED benchmarking in both TFIM and Heisenberg models show essentially perfect agreement, affirming the absence of systematic bias.

Implications and Outlook

The methodology enables, for the first time, precise extraction of symmetry-resolved entanglement scaling in large-scale, interacting systems directly in the thermodynamic and continuum limits. This unlocks multiple directions:

• Non-Abelian Symmetries: Extension to irreducible representation-resolved entanglement in non-Abelian settings can clarify the distribution of entanglement contributions in, e.g., topologically ordered states or critical points with enhanced symmetry.
• Higher Dimensions & Exotic Phases: The approach is directly applicable to critical points with deconfined quantum criticality, topological order, or other non-Landau paradigms, to probe for sector selectivity and nontrivial SREE scaling.
• Experimental Probes: As symmetry-resolved entanglement becomes a measurable observable in quantum simulators, these advancements provide a direct bridge between field theory, numerics, and experiment.

Conclusion

By formulating SREE computation in terms of QMC-accessible disorder operators on replica manifolds and validating this method against analytic and ED results in canonical models, the paper "Detecting Symmetry-Resolved Entanglement: A Quantum Monte Carlo Approach" (2604.02307) establishes a new standard for unbiased, scalable SREE calculations in quantum many-body systems. The results confirm equipartition in 1D and 2D TFIM, reproduce universal subleading corrections in the 1D Heisenberg chain, and set the stage for comprehensive studies of symmetry-resolved entanglement in strongly correlated matter. The framework’s generality suggests it will play a pivotal role in elucidating the structure of entanglement across quantum phases and transitions in future theoretical and experimental explorations.