Modular commutator as a robust topological invariant and approximate Markovianity

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}.

Quantum Physics (quant-ph); Strongly Correlated Electrons (cond-mat.str-el)

arXiv:2609.09019 [quant-ph]

(or arXiv:2609.09019v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.09019 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.