A priori characterization of exact entanglement equipartition and reconstruction of the entanglement Hamiltonian from symmetry-resolved entanglement in gapped quantum systems

For a half-infinite subsystem of an infinite gapped chain, Baxter's corner transfer matrix (CTM) yields exact symmetry-resolved entanglement entropies at finite correlation length. Applying this to two U(1)-symmetric models, the source work found a puzzling dichotomy: the interacting XXZ spin chain with anisotropy $\Delta\geq1$ shows exact equipartition of entanglement (sector-resolved entropies independent of the charge sector, for all $\Delta\geq1$ and all Renyi indices), while the free complex harmonic chain shows none, with effective equipartition recovered only in special limits; the harmonic chain's charged moments also contain a linear-in-$|\alpha|$ term and an $\alpha\to0$ non-uniformity tied to the non-compact continuum limit. The work asked which structural properties guarantee exact equipartition and how they relate to integrability, whether such statements survive beyond integrability and in higher dimensions, and whether symmetry-resolved data determine the entanglement (modular) Hamiltonian. Later literature resolved two adjacent offshoots — the field-theory recovery of the charged-moment asymptotics (linear term reproduced, zero mode identified as the origin of the $\alpha\to0$ non-uniformity) and the computation of symmetry-resolved entanglement in massive two-dimensional free and integrable field theories — and showed that leading-order equipartition is generic rather than integrability-specific. The surviving open problem is the a priori characterization of exact (all-orders) equipartition, whether it survives beyond integrability and in higher dimensions, and the question of how much of the entanglement Hamiltonian is encoded in symmetry-resolved entanglement data.

Background

Entanglement entropy is a standard probe of quantum many-body ground states: for a subsystem $A$ with reduced density matrix $\rho_A$, the Renyi entropies are $S_n=(1-n)^{-1}\log\mathrm{Tr}\,\rho_A^n$, whose $n\to1$ limit is the von Neumann entropy. When the system conserves a U(1) charge $Q$ (particle number, transverse magnetisation, electric charge), $\rho_A$ commutes with the subsystem charge $Q_A$ and block-diagonalises into charge sectors $\rho_A(q)$. The symmetry-resolved entanglement entropies $S_n(q)$ quantify the entanglement carried by each sector; they are extracted from the charged moments $Z_n(\alpha)=\mathrm{Tr}(\rho_A^n e^{i\alpha Q_A})$ by a Fourier transform that projects onto fixed charge. This decomposition was introduced as a systematic object by Goldstein and Sela and by Xavier, Alcaraz and Sierra (2018), who observed the regularity now called equipartition of entanglement: in many one-dimensional systems $S_n(q)$ is independent of $q$ at leading order, with computable subleading corrections.

For a half-infinite subsystem of an infinite one-dimensional gapped chain, $\rho_A$ is accessible in closed form through Baxter's corner transfer matrix (CTM), the object generating the partition function of a quadrant in the associated two-dimensional classical model: one has $\rho_A\propto e^{-H_{\mathrm{CTM}}}$, so the entanglement Hamiltonian $H_E=-\log\rho_A$ (also called the modular Hamiltonian) is proportional to the CTM and the entanglement spectrum — its eigenvalues and, crucially, their degeneracies — is known exactly. Gapped integrable chains are therefore rare grounds where symmetry-resolved entanglement is exactly computable at finite correlation length rather than asymptotically.

The source work applied this to two prototypical gapped models. The complex harmonic chain is a free bosonic lattice model (two coupled real harmonic chains) whose continuum limit is the massive complex Klein-Gordon field, a non-compact boson (its compactification radius is infinite, so the field's zero mode is not quantised); the conserved U(1) charge counts particles minus antiparticles. The XXZ spin-1/2 chain with anisotropy $\Delta\geq1$ is a genuinely interacting model that is integrable, i.e. exactly solvable by the Bethe ansatz (an exact method for one-dimensional quantum systems), in its gapped antiferromagnetic regime; the conserved U(1) charge is the transverse magnetisation, and the degeneracy structure of its entanglement spectrum was determined by Alba, Haque and Läuchli (2012). The results contrast sharply. For the harmonic chain, $S_n(q)$ depends explicitly on $q$ (no exact equipartition), with effective equipartition recovered only (i) for sector charge $q$ exceeding the logarithm of the correlation length $\xi$ and (ii) in the critical regime for $n\epsilon\ll1$, where $\epsilon\ll1$ parametrises proximity to criticality ($\log\xi\simeq\pi^2/\epsilon$); its charged moments contain a term linear in $|\alpha|$ (whereas for the compactified boson conformal field theory the analogous term is proportional to $\alpha^2$), and the limits $\alpha\to0$ and $\epsilon\to0$ do not commute. For the XXZ chain, $S_n(q)$ is exactly independent of $q$ for every $\Delta\geq1$ and every $n$, a property rooted in the equispaced, highly degenerate CTM spectrum, whose physical origin the authors could not identify.

Two of the directions the source work left open — recovering the charged-moment asymptotics of the non-compact massive boson directly in field theory, and computing symmetry-resolved entanglement in massive two-dimensional quantum field theories, free and integrable — were subsequently carried out by the authors' own follow-up and by the form-factor-bootstrap program (an exact method for computing matrix elements of fields in integrable quantum field theories). What remains is the interpretive core of the source work's agenda: understanding, a priori rather than model by model, what forces the charge-sector decomposition of entanglement to take the form it takes in gapped systems, and what that decomposition reveals about the underlying entanglement Hamiltonian.

Problem Statement

Consider the ground state of a gapped quantum many-body system with a conserved U(1) charge $Q$. For a subsystem $A$ with reduced density matrix $\rho_A$ and subsystem charge $Q_A$, the charged moments are $Z_n(\alpha)=\mathrm{Tr}(\rho_A^n e^{i\alpha Q_A})$ and the symmetry-resolved Renyi entropies $S_n(q)$ are the Renyi entropies of the charge-$q$ block of $\rho_A$, obtained from $Z_n(\alpha)$ by Fourier transform in $\alpha$. The problem is the following linked set of questions, whose common objective is to replace model-by-model exact computation with an a priori structural understanding of the charge-sector decomposition of entanglement in gapped systems — both its cause and its information content.

(1) Exact equipartition. In the ground state of the gapped XXZ chain, $S_n(q)$ is exactly independent of $q$ for every anisotropy $\Delta\geq1$ and every Renyi index (for the half-infinite subsystem treated by the corner transfer matrix, and for finite blocks of even length), whereas in the complex harmonic chain $S_n(q)$ depends on $q$ and equipartition is recovered only effectively (for sector charge $q$ exceeding the logarithm of the correlation length, and in the critical regime for $n\epsilon\ll1$). Identify the structural properties — integrability (Bethe-ansatz structure), the degeneracy structure of the corner-transfer-matrix/entanglement spectrum, spectrum degeneracies, additional symmetries, or other features — that are necessary and/or sufficient for exact, all-orders equipartition at finite correlation length, and explain the mechanism by which they make the sector-resolved entropies of the two models differ in this way.

(2) Persistence beyond integrability and in higher dimensions. Determine to what extent general statements about equipartition or effective-equipartition regimes survive when integrability is removed (genuinely interacting, non-integrable gapped models) and in spatial dimensions higher than one: do non-integrable or higher-dimensional gapped systems with exact equipartition exist, or is only leading-order/effective equipartition generic?

(3) Information content for the entanglement Hamiltonian. Establish whether, and to what extent, symmetry-resolved entanglement data (sector-resolved entropies $S_n(q)$ and charged moments $Z_n(\alpha)$) determine or constrain the full entanglement (modular) Hamiltonian $H_E=-\log\rho_A$ of the subsystem — for instance through an explicit reconstruction scheme valid on a solvable class, or through a proof of underdetermination by exhibiting two inequivalent entanglement Hamiltonians arising from U(1)-symmetric gapped ground states of the same class but with identical symmetry-resolved data.

Two computational offshoots raised alongside these questions in the source work — the field-theory recovery of the charged-moment asymptotics of the non-compact massive boson (including the linear-in-$|\alpha|$ term and the zero-mode origin of the $\alpha\to0$ non-uniformity) and the computation of symmetry-resolved entanglement in massive two-dimensional free and integrable field theories — have since been carried out and are not part of this problem.

References

1. Sara Murciano, Giuseppe Di Giulio, Pasquale Calabrese, "Symmetry resolved entanglement in gapped integrable systems: a corner transfer matrix approach", SciPost Phys. 8, 046 (2020), DOI: 10.21468/SciPostPhys.8.3.046, arXiv:1911.09588, https://arxiv.org/abs/1911.09588 (source paper)
2. Sara Murciano, Giuseppe Di Giulio, Pasquale Calabrese, "Entanglement and symmetry resolution in two dimensional free quantum field theories", JHEP 08 (2020) 073, DOI: 10.1007/JHEP08(2020)073, arXiv:2006.09069, https://arxiv.org/abs/2006.09069
3. Pasquale Calabrese, Mario Collura, Giuseppe Di Giulio, Sara Murciano, "Full counting statistics in the gapped XXZ spin chain", EPL 129, 60007 (2020), DOI: 10.1209/0295-5075/129/60007, arXiv:2002.04367, https://arxiv.org/abs/2002.04367
4. Dávid X. Horváth, Pasquale Calabrese, "Symmetry resolved entanglement in integrable field theories via form factor bootstrap", JHEP 11 (2020) 131, DOI: 10.1007/JHEP11(2020)131, arXiv:2008.08553, https://arxiv.org/abs/2008.08553
5. Dávid X. Horváth, Luca Capizzi, Pasquale Calabrese, "U(1) symmetry resolved entanglement in free 1+1 dimensional field theories via form factor bootstrap", JHEP 05 (2021) 197, DOI: 10.1007/JHEP05(2021)197, arXiv:2103.03197, https://arxiv.org/abs/2103.03197
6. David X. Horvath, Pasquale Calabrese, Olalla A. Castro-Alvaredo, "Branch Point Twist Field Form Factors in the sine-Gordon Model II: Composite Twist Fields and Symmetry Resolved Entanglement", SciPost Phys. 12, 088 (2022), DOI: 10.21468/SciPostPhys.12.3.088, arXiv:2105.13982, https://arxiv.org/abs/2105.13982
7. Xhek Turkeshi, Paola Ruggiero, Vincenzo Alba, Pasquale Calabrese, "Entanglement equipartition in critical random spin chains", Phys. Rev. B 102, 014455 (2020), DOI: 10.1103/PhysRevB.102.014455, arXiv:2005.03331, https://arxiv.org/abs/2005.03331
8. Benoit Estienne, Yacine Ikhlef, Alexi Morin-Duchesne, "Finite-size corrections in critical symmetry-resolved entanglement", SciPost Phys. 10, 054 (2021), DOI: 10.21468/SciPostPhys.10.3.054, arXiv:2010.10515, https://arxiv.org/abs/2010.10515
9. Pasquale Calabrese, Jérôme Dubail, Sara Murciano, "Symmetry-resolved entanglement entropy in Wess-Zumino-Witten models", JHEP 10 (2021) 067, DOI: 10.1007/JHEP10(2021)067, arXiv:2106.15946, https://arxiv.org/abs/2106.15946
10. Yuya Kusuki, Sara Murciano, Hirosi Ooguri, Sridip Pal, "Symmetry-resolved entanglement entropy, spectra & boundary conformal field theory", JHEP 11 (2023) 216, DOI: 10.1007/JHEP11(2023)216, arXiv:2309.03287, https://arxiv.org/abs/2309.03287
11. Giuseppe Di Giulio, Johanna Erdmenger, "Symmetry-resolved modular correlation functions in free fermionic theories", JHEP 07 (2023) 058, DOI: 10.1007/JHEP07(2023)058, arXiv:2305.02343, https://arxiv.org/abs/2305.02343
12. Giuseppe Di Giulio, Moritz Dorband, Johanna Erdmenger, Henri Scheppach, "Modular theory and symmetry resolution in hyperfinite von Neumann algebras", JHEP 05 (2026) 259, DOI: 10.1007/JHEP05(2026)259, arXiv:2510.02441, https://arxiv.org/abs/2510.02441
13. Sara Murciano, Paola Ruggiero, Pasquale Calabrese, "Symmetry resolved entanglement in two-dimensional systems via dimensional reduction", J. Stat. Mech. (2020) 083102, DOI: 10.1088/1742-5468/aba1e5, arXiv:2003.11453, https://arxiv.org/abs/2003.11453
14. Yuanzhu Huang, Yang Zhou, "Symmetry-Resolved Entanglement Entropy in Higher Dimensions", JHEP 10 (2025) 045, DOI: 10.1007/JHEP10(2025)045, arXiv:2503.09070, https://arxiv.org/abs/2503.09070
15. Kuangjie Chen, Weizhen Jia, Xiaopeng Li, René Meyer, Jiarui Zhao, "Detecting Symmetry-Resolved Entanglement: A Quantum Monte Carlo Approach", arXiv:2604.02307 (2026), https://arxiv.org/abs/2604.02307
16. Mark J. Arildsen, Valentin Crépel, Nicolas Regnault, Benoit Estienne, "Symmetry Resolved Entanglement Entropy in a Non-Abelian Fractional Quantum Hall State", Phys. Rev. B 113, 075108 (2026), DOI: 10.1103/h336-ysly, arXiv:2508.05494, https://arxiv.org/abs/2508.05494
17. Filiberto Ares, Jayashish Das, Arnab Kundu, "Symmetry Resolved Entanglement Entropy: Equipartition under Driven and Non-unitary Evolution in a Compact Boson CFT", arXiv:2603.28567 (2026), https://arxiv.org/abs/2603.28567
18. M. Reza Mohammadi Mozaffar, Ali Mollabashi, "Symmetry resolved entanglement in Lifshitz field theories", arXiv:2604.19082 (2026), https://arxiv.org/abs/2604.19082
19. Olalla A. Castro-Alvaredo, Lucía Santamaría-Sanz, "Symmetry-resolved measures in quantum field theory: A short review", Mod. Phys. Lett. B 39, 2430002 (2024), DOI: 10.1142/S0217984924300023, arXiv:2403.06652, https://arxiv.org/abs/2403.06652
20. J. C. Xavier, F. C. Alcaraz, G. Sierra, "Equipartition of the entanglement entropy", Phys. Rev. B 98, 041106 (2018), DOI: 10.1103/PhysRevB.98.041106, https://doi.org/10.1103/PhysRevB.98.041106
21. Moshe Goldstein, Eran Sela, "Symmetry-Resolved Entanglement in Many-Body Systems", Phys. Rev. Lett. 120, 200602 (2018), DOI: 10.1103/PhysRevLett.120.200602, https://doi.org/10.1103/PhysRevLett.120.200602
22. Vincenzo Alba, Masudul Haque, Andreas M. Läuchli, "Boundary-Locality and Perturbative Structure of Entanglement Spectra in Gapped Systems", Phys. Rev. Lett. 108, 227201 (2012), DOI: 10.1103/PhysRevLett.108.227201, https://doi.org/10.1103/PhysRevLett.108.227201