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arXiv:2609.09019

Modular commutator as a robust topological invariant and approximate Markovianity

Tai Hsuan Yang

quant-phcond-mat.str-elSubmitted Sep 8, 2026• Announced Sep 9, 2026• 0 citations
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Abstract

The modular commutator provides a bulk, local, single-wave-function probe of the chiral central charge $c_-$ for gapped ground states. Its invariance under deformations was previously established under a local quantum Markov condition---namely, the conditional mutual information $I(A:C|B)$ is zero for all tripartitions of a disk into three consecutive strips A, B, and C \cite{Modular-commutator-Gapped}. However, the local quantum Markov condition also forces the probe to vanish, leaving open whether modular commutator remains robust in physically relevant states where the Markov property holds only approximately. In this paper, we resolve this tension for finite-dimensional Hilbert spaces: the approximate local quantum Markov property implies the change of the modular commutator under topology-preserving deformations vanishes asymptotically. We next prove trace-norm continuity of the modular commutator. Combining deformation invariance with trace-norm continuity, we establish that the modular commutator remains asymptotically invariant within gapped quantum phases connected by quasi-local unitary paths that preserve the approximate local quantum Markov condition. Conversely, by analyzing finite-time dynamics generated by modular Hamiltonians, we derive a quantitative lower bound on the conditional mutual information required for a non-zero modular commutator: $I(A:C|B)$ across such strip tripartitions cannot decay faster than exponentially with the width of B given that the state satisfies entanglement area law, extending the exact no-go theorem of \cite{strict-J-2024} to a quantitative finite bound . Finally, we demonstrate that those conclusions apply equally to the Hall conductance estimator \cite{FanSahayVishwanath2023}.

Comments
42 pages of main text, 23 pages of appendices; 5 main-text figures and 3 appendix figures

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The modular commutator provides a bulk, local, single-wave-function probe of the chiral central charge c−c_-c−​ for gapped ground states. Its invariance under deformations was previously established under a local quantum Markov condition---namely, the conditional mutual information I(A:C∣B)I(A:C|B)I(A:C∣B) is zero for all tripartitions of a disk into three consecutive strips A, B, and C \cite{Modular-commutator-Gapped}. However, the local quantum Markov condition also forces the probe to vanish, leaving open whether modular commutator remains robust in physically relevant states where the Markov property holds only approximately. In this paper, we resolve this tension for finite-dimensional Hilbert spaces: the approximate local quantum Markov property implies the change of the modular commutator under topology-preserving deformations vanishes asymptotically. We next prove trace-norm continuity of the modular commutator. Combining deformation invariance with trace-norm continuity, we establish that the modular commutator remains asymptotically invariant within gapped quantum phases connected by quasi-local unitary paths that preserve the approximate local quantum Markov condition. Conversely, by analyzing finite-time dynamics generated by modular Hamiltonians, we derive a quantitative lower bound on the conditional mutual information required for a non-zero modular commutator: I(A:C∣B)I(A:C|B)I(A:C∣B) across such strip tripartitions cannot decay faster than exponentially with the width of B given that the state satisfies entanglement area law, extending the exact no-go theorem of \cite{strict-J-2024} to a quantitative finite bound . Finally, we demonstrate that those conclusions apply equally to the Hall conductance estimator \cite{FanSahayVishwanath2023}.
In Phys. Rev. Lett. 128, 176402 (2022), we argued that the chiral central charge---a topologically protected quantity characterizing the edge theory of a gapped (2+1)(2+1)(2+1)-dimensional system---can be extracted from the bulk by using an order parameter called the modular commutator. In this paper, we reveal general properties of the modular commutator and strengthen its relationship with the chiral central charge. First, we identify connections between the modular commutator and conditional mutual information, time reversal, and modular flow. Second, we prove, within the framework of the entanglement bootstrap program, that two topologically ordered media connected by a gapped domain wall must have the same modular commutator in their respective bulk. Third, we numerically calculate the value of the modular commutator for a bosonic lattice Laughlin state for finite sizes and extrapolate to the infinite-volume limit. The result of this extrapolation is consistent with the proposed formula up to an error of about 0.70.7%0.7.
The modular commutator J(A,B,C)=i⟨[KAB,KBC]⟩J(A,B,C) = i\langle[K_{AB},K_{BC}]\rangleJ(A,B,C)=i⟨[KAB​,KBC​]⟩ extracts the chiral central charge c−c_-c−​ from a single bulk wavefunction of a \emph{gapped} 2d state, where 3J/π=c−3J/π=c_-3J/π=c−​. Inspired by the recent developments in the field of gapless symmetry-protected topological phases, we ask: what does the modular commutator measure, if it is well-defined at all, when the 2d bulk becomes \textit{gapless}? Several interesting new insights can already be obtained using the simple Haldane honeycomb model. For the critical point hosting an isolated Dirac node we find that JJJ remains sharp: it converges to a \textit{half-quantized} value, with corrections that decay as a power law in the subsystem size rather than exponentially, mirroring the power-law correlations in gapless systems. We prove the half-quantization using an emergent reflection symmetry of the massless Dirac cone, and show that the half-quantized contribution comes from the other gapped cone (the massive partner of the massless one). This massive partner can be interpreted as the physical incarnation of the Pauli-Villars regulator, which is the origin of the parity-breaking level-12\frac{1}{2}21​ Chern-Simons term (with half-quantized Hall conductance) and the parity anomaly. When protected chiral edge modes coexist with a bulk Dirac node we obtain 3J/π=c−+123J/π= c_-+\frac{1}{2}3J/π=c−​+21​. The half-quantization is also shown to be robust against tripartition deformation, tuning Dirac velocity and Dirac cone anisotropy. We further investigate other types of gaplessness---quadratic nodes (in contrast to linear Dirac) and the case with Fermi surface---and show that the robust half-quantization of JJJ is lost in such non-Dirac cases. These results generalize the modular commutator beyond gapped phases, and at the same time provide an information-theoretic measurement of the parity anomaly.
A defining feature of chirality is the distinction between a system and its mirror image. Despite extensive experimental observations of chiral phases and theoretical advances, a quantum-information theoretic characterization of chirality based solely on the entanglement structure of many-body quantum states remains elusive. Here, we introduce the notion of many-body chirality by formulating it as an obstruction to transforming a quantum state into its complex conjugate through finite-depth local operations. We rigorously establish many-body chirality for stabilizer realizations of Zd(k)\mathbb{Z}_d^{(k)}Zd(k)​ anyon theories, proving that complex conjugation can be implemented by local quantum channels if and only if the underlying anyon data are mirror invariant. This reveals forms of chirality that evade conventional diagnostics, including examples with vanishing modular commutator, vanishing chiral central charge, and commuting-projector realizations. We further show that this obstruction is intrinsically four-partite, while invisible to tripartite entanglement structure. Finally, we prove that Zd(k)\mathbb{Z}_d^{(k)}Zd(k)​ states with d>2d>2d>2 possess intrinsic many-body imaginarity: their complex phase structure cannot be removed by finite-depth local unitaries. Remarkably, this includes states that are not many-body chiral.
Chirality is a property of a gapped phase of matter in two spatial dimensions that can be manifested through nonzero thermal or electrical Hall conductance. In this Letter, we prove two no-go theorems that forbid such chirality for a quantum state in a finite dimensional local Hilbert space with strict area law entanglement entropies. We also show that the finite dimensional local Hilbert space condition can be relaxed to the condition that the state has finite local entanglement entropies. As a crucial ingredient in the proofs, we introduce a new quantum information-theoretic primitive called instantaneous modular flow, which has many other potential applications.
The chiral central charge c−c_-c−​ is a key topological invariant of the edge characterizing the bulk two-dimensional chiral topological order, but its direct evaluation in microscopic spin models has long been a challenge, especially for non-abelian topological order. Building on the recently developed modular commutator formalism, we numerically obtain c−c_-c−​ directly from single ground-state wave functions of two-dimensional interacting spin models that have chiral topological order. This provides a geometry-independent and bulk diagnostic of chirality. We study two nonintegrable systems -- the Zeeman-Kitaev honeycomb model and the kagome antiferromagnet -- both subjected to scalar spin chirality perturbations. We find that the modular commutator yields results consistent with the expected topological quantum field theories. We also compute the topological entanglement entropy which provides an independent diagnostic of the topological orders. Our work establishes modular commutators as a powerful numerical probe of chiral topological order in strongly correlated quantum magnets.
Recent years have witnessed significant progress in the entanglement-based characterization of quantum phases of matter. The primary objects of interest are the reduced density matrix and its associated entanglement Hamiltonian. As intrinsic properties of a quantum state, these quantities theoretically determine all experimentally accessible local observables. In this work, we investigate the response of two entanglement measures to the real-time dynamics driven by the entanglement Hamiltonian--a process known as modular flow. We demonstrate that our results can be unified into a single generating function, ⟨ρABαeλQABeμQBCρBCβ⟩\langleρ_{AB}^α\mathrm{e}^{λ{Q}_{AB}}\mathrm{e}^{μ{Q}_{BC}}ρ_{BC}^β\rangle⟨ρABα​eλQAB​eμQBC​ρBCβ​⟩. This function is of independent interest as it represents a generalization of the recently proposed Rényi modular commutator. In appropriate limits, this function yields the response of Rényi entropy and its charged version, which we find to be uniquely determined by chiral topological invariants, specifically the chiral central charge and the Hall conductance. Our analytical findings are validated through two independent approaches: (i) free fermion systems using the real-space Chern number formula, and (ii) an effective field theory treatment that regularizes the entanglement cut via chiral conformal field theory. Both methods yield consistent results.