Detecting Symmetry-Resolved Entanglement: A Quantum Monte Carlo Approach

The gist: This work formulates and implements an estimator-based quantum Monte Carlo (QMC) framework for computing symmetry-resolved Renyi entropies in sign-problem-free interacting lattice systems by measuring disorder (symmetry-twisted) operators in ordinary and replica ensembles.

Symmetry-Resolved Entanglement Entropy

Entanglement entropy is a powerful probe of quantum many-body states, offering insights that transcend the paradigm of conventional order parameters <ref:2604.02307#pg2>. Symmetry-resolved entanglement decomposes the total entanglement into contributions from different symmetry sectors <ref:2604.02307#pg2>. A central result is the equipartition of entanglement, where the leading asymptotic contribution to the symmetry-resolved entropy becomes independent of the sector label <ref:2604.02307#pg2>. The symmetry-resolved Renyi entropy (SRRE) in each charge sector is defined as 1/Sn(q) = 1/(1−n ln Tr[ρA(q)n] <ref:2604.02307#pg2>. This decomposition relation only holds for the Renyi entropy if equipartition of entanglement holds in the system under consideration <ref:2604.02307#pg2>.

How it works

The core of the method involves relating charged moments to subsystem disorder operators measured in ordinary and glued replica ensembles and reconstructing the symmetry-resolved moments by Fourier transformation <ref:2604.02307#pg2>. The essential extension relative to the earlier disorder-operator QMC program is the measurement on replica manifolds, where the symmetry-twisted observables directly provide the charged moments required for symmetry resolution <ref:2604.02307#pg2>. A practical advantage is that one simulation in the ordinary ensemble and one in the replica ensemble provide all required counting fields and multiple symmetry sectors, rather than requiring an independent simulation for every counting field or charge sector <ref:2604.02307#pg2>.

The general method starts with the charged moment defined as Zn(α) = Tr(ρn A e i α Q A) <ref:2604.02307#pg2>. Treating α as a continuous variable in (−π, π), the Fourier transform of Zn(α) yields Zn(q) = Z π −π dα 2π e −iαqZn(α) = Tr(Πqρn A) <ref:2604.02307#pg2>. This leads to the expression for the n-th SRRE: Sn(q) = 1/(1−n ln Zn(q)/Z1(q) n <ref:2604.02307#pg2>.

For n = 1, the charged moment reduces to Z1(α) = Tr(ρAe i α Q A) = ⟨e i α Q A ⟩Z, which is the ordinary subsystem disorder operator illustrated in Fig. 1(c) <ref:2604.02307#pg2>. For n ≥ 2, we have Zn(α) = Tr(ρ n Ae i α Q A) / Tr(ρ n A) = ⟨e i α Q A ⟩Z (n)A <ref:2604.02307#pg2>.

Models and Validation

The framework is validated in two controlled one-dimensional settings: the transverse-field Ising model (TFIM), for which exact conformal-field-theory predictions are available, and the interacting Heisenberg chain, which tests the U(1) symmetry-sector reconstruction and its finite-size behavior <ref:2604.02307#pg2>. The two-dimensional TFIM illustrates the main intended application in an interacting system beyond one dimension <ref:2604.02307#pg2>.

For the Z2 TFIM, the global Z2 spin-flip symmetry splits the Hilbert space of subsystem A into even (q = 0) and odd (q = 1) parity sectors <ref:2604.02307#pg2>. From the two key observables, ⟨X⟩Z and ⟨X⟩Z(2)A, we can reconstruct the symmetry-resolved moments for the charge sectors q = 0, 1 <ref:2604.02307#pg2>. Substituting these expressions into Eq. (12) yields the closed-form result: S2(q) − S(2)A = − ln" 1 + (−1)q⟨X⟩Z(2)A 1 + (−1)q⟨X⟩Z 2 / - ln 2 <ref:2604.02307#pg2>.

Results and Equipartition

For the Z2 TFIM, the asymptotic criterion for Z2 equipartition is explicit: if the nontrivial charged moments ⟨X⟩Z and ⟨X⟩Z(2)A vanish in the thermodynamic limit, then P(0) = P(1) → 1/2 and S2(q) = S(2)A − ln 2 + o(1), q = 0, 1 <ref:2604.02307#pg2>. The second model considered is the spin-1/2 antiferromagnetic Heisenberg chain, whose low-energy physics is described by a (1+1)D conformal field theory <ref:2604.02307#pg2>. In (1 + 1)D CFT with a global U(1) symmetry, the SRRE follows a universal scaling behavior given by Sn(q) = S(n)A − 1/2 ln(lnL) + cn + O(1/ lnL), highlighting a characteristic subleading double-logarithmic term <ref:2604.02307#pg2>.

Discussion

The one-dimensional calculations serve as controlled benchmarks of complementary parts of the framework <ref:2604.02307#pg2>. The 1D TFIM validates the Z2 replica estimator against an exact CFT prediction, while the interacting Heisenberg chain validates the U(1) symmetry-sector reconstruction and its finite-size scaling <ref:2604.02307#pg2>. The two-dimensional TFIM illustrates the main intended application beyond one dimension <ref:2604.02307#pg2>. Across the models studied here, the resolved sectors exhibit a common asymptotic entanglement scaling <ref:2604.02307#pg2>. In the Z2 TFIM, vanishing nontrivial charged moments imply the common offset S2(q) − S(2)A → − ln 2 for the two parity sectors <ref:2604.02307#pg2>. In the U(1) Heisenberg chain, the charge sectors instead share the same double-logarithmic term and asymptotic intercept, while their differences survive in O(1/ lnL) corrections <ref:2604.02307#pg2>. Access to charged moments on replica manifolds opens several concrete directions <ref:2604.02307#pg2>.

Future Directions

A natural next step is to generalize the method from Abelian symmetries to non-Abelian ones, where symmetry resolution is organized by irreducible thermodynamic limit representations rather than a single conserved charge <ref:2604.02307#pg2>. Another important direction is to apply the method to additional (2 + 1)D critical systems in order to clarify the structure of subleading corrections to symmetry-resolved entanglement and to test the robustness of entanglement equipartition beyond the Ising universality class <ref:2604.02307#pg2>. It would also be interesting to use this approach in sign-problem-free models with more exotic ground states, such as interacting topological phases or deconfined quantum critical points, where symmetry-resolved entanglement may provide information that is not accessible from conventional entropy measures alone <ref:2604.02307#pg2>. The method provides a practical numerical tool for studying the interplay between symmetry and entanglement in strongly correlated quantum manybody systems <ref:2604.02307#pg2>.

Benchmark of QMC Results

To verify the validity and accuracy of our stochastic series expansion QMC implementations, we first compare the results of ⟨X⟩Z and ⟨X⟩Z(2)A obtained from QMC simulations at extremely low temperatures with those derived from exact diagonalization (ED) at zero temperature <ref:2604.02307#pg2>. For the 1D Ising model (hc = 1), we compare the ED results with our QMC results at β = 256 and L = 16, where subsystem A is defined as half of the periodic chain <ref:2604.02307#pg2>. For the 2D Ising model (hc ≈ 3.044), we compare the ED results with the QMC results at Lx = Ly = 4 and β = 128, where subsystem A is chosen as an (Lx/2) × Ly cylinder on an Lx × Ly torus <ref:2604.02307#pg2>.

Numerical Implementation

In both models, the ordinary and two-replica ensembles use the same model-specific update machinery <ref:2604.02307#pg2>.

1. No longer reliant on conventional order parameters for phase classification, this system can diagnose more exotic quantum phases by measuring symmetry-resolved entanglement (SRRE). This allows for a refined probe of entanglement structure beyond simple area law scaling, as it decomposes total entanglement into contributions from different symmetry sectors.

2. The improved AI system will be able to perform sign-problem-free interacting lattice simulations using an estimator-based Quantum Monte Carlo (QMC) framework. It can compute symmetry-resolved Renyi entropies (SRRE) by measuring disorder (symmetry-twisted) operators in ordinary and replica ensembles and reconstructing them from corresponding charged moments.

3. The system can achieve direct operational connection between fieldtheoretic formulations and lattice simulations, as the method allows for the measurement of observables on glued replica manifolds, which provides a practical advantage by requiring only one simulation in the ordinary ensemble and one in the replica ensemble rather than independent simulations for every counting field.

4. The AI can rigorously test hypotheses regarding entanglement equipartition by comparing numerical results with theoretical predictions. For instance, it can show that for the Z2 TFIM, vanishing nontrivial charged moments imply the common offset S2(q) − S(2)A → - ln 2 for the two parity sectors.

5. The system can provide quantitative finite-size scaling information to distinguish between different asymptotic behaviors. It can fit data to phenomenological forms like − ln⟨X⟩Z(2)A = aL + b lnL + c to characterize the approach to the thermodynamic limit, noting that for the 2D TFIM, this suggests an approach on length scales of order L ∼ 500.

6. The AI can perform complex reconstruction of entanglement spectra from measured moments using a direct linear cosine transform. This enables the determination of subtracted SRRE by calculating it as S2(q) − S(2)A = − ln R2(q)/P(q) squared, even for non-Abelian symmetries where resolution is organized by irreducible representations.

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 formulate and implement an estimator-based quantum Monte Carlo (QMC) framework for computing symmetry-resolved Rényi entropies (SRRE) in sign-problem-free interacting lattice systems by measuring disorder (symmetry-twisted) operators in ordinary and replica ensembles and reconstructing SRRE from the corresponding charged moments. We validate the framework in two controlled one-dimensional settings: the transverse-field Ising model (TFIM), for which exact conformal-field-theory predictions are available, and the interacting Heisenberg chain, which tests the U(1) symmetry-sector reconstruction and its finite-size behavior. We then apply the method to the two-dimensional TFIM. Within the accessible system sizes and a phenomenological finite-size extrapolation, our data provide numerical evidence consistent with entanglement equipartition at the (2+1) D Ising critical point. Our work establishes a practical numerical route to symmetry-resolved entanglement in interacting lattice models and provides a framework for future studies beyond one dimension.