Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2206.02823 · Physical Review Letters

From Entanglement Generated Dynamics to the Gravitational Anomaly and Chiral Central Charge

Ruihua Fan

Published Dec 22, 2022• 25 citations
arXivHTMLPDF

Abstract

We apply modular flow-entanglement generated dynamics-to characterize quantum orders of ground state wave functions. In particular, we study the linear response of the entanglement entropy of a simply connected region with respect to modular flow. First, we apply it to (1+1)D conformal field theories and demonstrate its relationship to the chiral central charge-or, equivalently, the perturbative gravitational anomaly-which is shown to vanish. Next, we apply it to (2+1)D gapped ground states where it reduces to a recently proposed formula by Kim et al. that is conjectured to compute the edge chiral central charge. Modular flow provides an intuitive picture for this conjecture based on bulk-edge correspondence. We also provide numerics on free-fermion models that lend support to our picture.

DOI
10.1103/physrevlett.129.260403

Related papers

Entanglement Measure Response to Modular Flow and Chiral Topological Phases

Yunlong Zang

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, $\langleρ_{AB}^α\mathrm{e}^{λ{Q}_{AB}}\mathrm{e}^{μ{Q}_{BC}}ρ_{BC}^β\rangle$. 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.

arXiv:2603.09717•Mar 10, 2026•0 citations

Entanglement entropy and anomaly inflow

Taylor L. Hughes, Robert G. Leigh, Onkar Parrikar +1 more

We study entanglement entropy for parity-violating (time-reversal breaking) quantum field theories on ${\mathbb{R}}^{1,2}$ in the presence of a domain wall between two distinct parity-odd phases. The domain wall hosts a $1+1$-dimensional conformal field theory (CFT) with nontrivial chiral central charge. Such a CFT possesses gravitational anomalies. It has been shown recently that, as a consequence, its intrinsic entanglement entropy is sensitive to Lorentz boosts around the entangling surface. Here, we show using various methods that the entanglement entropy of the three-dimensional bulk theory is also sensitive to such boosts owing to parity-violating effects, and that the bulk response to a Lorentz boost precisely cancels the contribution coming from the domain wall CFT. We argue that this can naturally be interpreted as entanglement inflow (i.e., inflow of entanglement entropy analogous to the familiar Callan-Harvey effect) between the bulk and the domain-wall, mediated by the low-lying states in the entanglement spectrum. These results can be generally applied to $2+1$-d topological phases of matter that have edge theories with gravitational anomalies, and provide a precise connection between the gravitational anomaly of the physical edge theory and the low-lying spectrum of the entanglement Hamiltonian.

arXiv:1509.04969•Physical review. D/Physical review. D.•Mar 30, 2016•19 citations

Identifying chiral topological order in microscopic spin models by modular commutator

Avijit Maity, Aman Kumar, Vikram Tripathi

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.

arXiv:2510.06086•Oct 7, 2025•0 citations

Rényi-like entanglement probe of the chiral central charge

Julian Gass, Michael Levin

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 $ω_{α,β}$, 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 $ω_{α,β}$ for gapped ground states of non-interacting fermion Hamiltonians as well as ground states of string-net models. In both cases, we find that $ω_{α,β}$ takes a universal value related to the chiral central charge. For integer values of $α$ and $β$, our quantity $ω_{α,β}$ can be expressed as an expectation value of permutation operators acting on an appropriate replica system, providing a natural route to measuring $ω_{α,β}$ in numerical simulations and potentially, experiments.

cond-mat.str-elCross-listedarXiv:2512.20608•Physical review. B./Physical review. B•Dec 24, 2025•1 citation

Modular Commutators in Conformal Field Theory

Yijian Zou, Bowen Shi, Jonathan Sorce +2 more

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.

arXiv:2206.00027•Physical Review Letters•Dec 22, 2022•23 citations

Modular commutator in gapped quantum many-body systems

Isaac H. Kim, Bowen Shi, Kohtaro Kato +1 more

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 $0.7%$.

arXiv:2110.10400•Physical review. B./Physical review. B•Aug 26, 2022•45 citations
We apply modular flow-entanglement generated dynamics-to characterize quantum orders of ground state wave functions. In particular, we study the linear response of the entanglement entropy of a simply connected region with respect to modular flow. First, we apply it to (1+1)D conformal field theories and demonstrate its relationship to the chiral central charge-or, equivalently, the perturbative gravitational anomaly-which is shown to vanish. Next, we apply it to (2+1)D gapped ground states where it reduces to a recently proposed formula by Kim et al. that is conjectured to compute the edge chiral central charge. Modular flow provides an intuitive picture for this conjecture based on bulk-edge correspondence. We also provide numerics on free-fermion models that lend support to our picture.
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.
We study entanglement entropy for parity-violating (time-reversal breaking) quantum field theories on R1,2{\mathbb{R}}^{1,2}R1,2 in the presence of a domain wall between two distinct parity-odd phases. The domain wall hosts a 1+11+11+1-dimensional conformal field theory (CFT) with nontrivial chiral central charge. Such a CFT possesses gravitational anomalies. It has been shown recently that, as a consequence, its intrinsic entanglement entropy is sensitive to Lorentz boosts around the entangling surface. Here, we show using various methods that the entanglement entropy of the three-dimensional bulk theory is also sensitive to such boosts owing to parity-violating effects, and that the bulk response to a Lorentz boost precisely cancels the contribution coming from the domain wall CFT. We argue that this can naturally be interpreted as entanglement inflow (i.e., inflow of entanglement entropy analogous to the familiar Callan-Harvey effect) between the bulk and the domain-wall, mediated by the low-lying states in the entanglement spectrum. These results can be generally applied to 2+12+12+1-d topological phases of matter that have edge theories with gravitational anomalies, and provide a precise connection between the gravitational anomaly of the physical edge theory and the low-lying spectrum of the entanglement Hamiltonian.
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.
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,βω_{α,β}ωα,β​, is parameterized by two positive real numbers α,βα, βα,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,βω_{α,β}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,βω_{α,β}ωα,β​ takes a universal value related to the chiral central charge. For integer values of ααα and βββ, our quantity ωα,βω_{α,β}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,βω_{α,β}ωα,β​ in numerical simulations and potentially, experiments.
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.
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.