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arXiv:2610.08788

Chiral Central Charge from Real-Space Twist Operator Correlator

Yunlong Zang

Published Oct 6, 2026• 0 citations
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Abstract

Recent years have witnessed substantial progress in the entanglement-based characterization of quantum phases of matter, with growing interest in real-space formulas due to their theoretical and experimental relevance. For two-dimensional gapped quantum systems, we propose a universal real-space formula for the chiral central charge based on the twist-operator correlator $Z_{g,h}=\langleψ^{(N)}\rvert\mathcal{T}_A(g)\mathcal{T}_B(h)\mathcal{T}_C(g)\mathcal{T}_C(h)\lvertψ^{(N)}\rangle$, where $\lvertψ^{(N)}\rangle=\lvertψ\rangle^{\otimes N}$ is the $N$-copy state and $g,h\in S_N$ are permutations. The twist operator $\mathcal{T}_R(g)$ permutes the $N$ replicas within region $R$ according to $g$. For pairs $(g,h)$ satisfying the spherical condition, we show that the phase of the correlator is determined by the chiral central charge $\mathfrak{c}_{-}$ through $Z_{g,h}/\left\lvert Z_{g,h}\right\rvert=(θ_gθ_h/θ_{gh})^{\mathfrak{c}_{-}}$, where $θ_g^{\mathfrak{c}_{-}}$ is the topological spin of the twist defect labeled by $g\in S_N$. The construction applies to both bosonic and fermionic systems. We derive the formula within a field-theoretic framework under suitable assumptions and further test it numerically in microscopic lattice models. Our results establish a universal connection between real-space entanglement structures and the chiral central charge, providing a systematic framework for characterizing chiral topological phases directly on the lattice.

DOI
10.48550/arxiv.2610.08788

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Recent years have witnessed substantial progress in the entanglement-based characterization of quantum phases of matter, with growing interest in real-space formulas due to their theoretical and experimental relevance. For two-dimensional gapped quantum systems, we propose a universal real-space formula for the chiral central charge based on the twist-operator correlator Zg,h=⟨ψ(N)∣TA(g)TB(h)TC(g)TC(h)∣ψ(N)⟩Z_{g,h}=\langleψ^{(N)}\rvert\mathcal{T}_A(g)\mathcal{T}_B(h)\mathcal{T}_C(g)\mathcal{T}_C(h)\lvertψ^{(N)}\rangleZg,h​=⟨ψ(N)∣TA​(g)TB​(h)TC​(g)TC​(h)∣ψ(N)⟩, where ∣ψ(N)⟩=∣ψ⟩⊗N\lvertψ^{(N)}\rangle=\lvertψ\rangle^{\otimes N}∣ψ(N)⟩=∣ψ⟩⊗N is the NNN-copy state and g,h∈SNg,h\in S_Ng,h∈SN​ are permutations. The twist operator TR(g)\mathcal{T}_R(g)TR​(g) permutes the NNN replicas within region RRR according to ggg. For pairs (g,h)(g,h)(g,h) satisfying the spherical condition, we show that the phase of the correlator is determined by the chiral central charge c−\mathfrak{c}_{-}c−​ through Zg,h/∣Zg,h∣=(θgθh/θgh)c−Z_{g,h}/\left\lvert Z_{g,h}\right\rvert=(θ_gθ_h/θ_{gh})^{\mathfrak{c}_{-}}Zg,h​/∣Zg,h​∣=(θg​θh​/θgh​)c−​, where θgc−θ_g^{\mathfrak{c}_{-}}θgc−​​ is the topological spin of the twist defect labeled by g∈SNg\in S_Ng∈SN​. The construction applies to both bosonic and fermionic systems. We derive the formula within a field-theoretic framework under suitable assumptions and further test it numerically in microscopic lattice models. Our results establish a universal connection between real-space entanglement structures and the chiral central charge, providing a systematic framework for characterizing chiral topological phases directly on the lattice.
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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.
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The hallmark of two-dimensional chiral topological phases is the existence of anomalous gapless modes at the spatial boundary. Yet, the manifestation of this edge anomaly within the bulk ground-state wavefunction itself remains only partially understood. In this work, we introduce a family of multipartite entanglement measures that probe chirality directly from the bulk wavefunction. Our construction involves applying different permutations between replicas of the ground state wavefunction in neighboring spatial regions, creating "permutation defects" at the boundaries between these regions. We provide general arguments for the robustness of these measures and develop a field-theoretical framework to compute them systematically. While the standard topological field theory prescription misses the chiral contribution, our method correctly identifies it as the chiral conformal field theory partition function on high-genus Riemann surfaces. This feature is a consequence of the bulk-edge correspondence, which dictates that any regularization of the theory at the permutation defects must introduce gapless boundary modes. We numerically verify our results with both free-fermion and strongly-interacting chiral topological states and find excellent agreement. Our results enable the extraction of the chiral central charge and the Hall conductance using a finite number of wavefunction replicas, making these quantities accessible to Monte-Carlo numerical techniques and noisy intermediate-scale quantum devices.