The modular commutator is a recently discovered entanglement quantity that quantifies the chirality of the underlying many-body quantum state. In this Letter, we derive a universal expression for the modular commutator in conformal field theories in 1+1 dimensions and discuss its salient features. We show that the modular commutator depends only on the chiral central charge and the conformal cross ratio. We test this formula for a gapped (2+1)-dimensional system with a chiral edge, i.e., the quantum Hall state, and observe excellent agreement with numerical simulations. Furthermore, we propose a geometric dual for the modular commutator in certain preferred states of the AdS/CFT correspondence. For these states, we argue that the modular commutator can be obtained from a set of crossing angles between intersecting Ryu-Takayanagi surfaces.
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}.
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)-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
0.7.
We propose a ground state entanglement probe for gapped, two-dimensional quantum many-body systems that involves taking powers of reduced density matrices in a particular geometric configuration. This quantity, which we denote by
ωα,β, is parameterized by two positive real numbers
α,β, and can be seen as a ``Rényi-like" generalization of the modular commutator -- another entanglement probe proposed as a way to compute the chiral central charge from a bulk wave function. We obtain analytic expressions for
ωα,β for gapped ground states of non-interacting fermion Hamiltonians as well as ground states of string-net models. In both cases, we find that
ωα,β takes a universal value related to the chiral central charge. For integer values of
α and
β, our quantity
ωα,β can be expressed as an expectation value of permutation operators acting on an appropriate replica system, providing a natural route to measuring
ωα,β in numerical simulations and potentially, experiments.
The chiral central charge
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− 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.
The modular commutator
J(A,B,C)=i⟨[KAB,KBC]⟩ extracts the chiral central charge
c− from a single bulk wavefunction of a \emph{gapped} 2d state, where
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
J 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-
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−+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
J 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.