Chiral Central Charge from Real-Space Twist Operator Correlator

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 $Z_{g,h}=⟨{\psi }^{(N)}\mid {\mathcal{T}}_A(g){\mathcal{T}}_B(h){\mathcal{T}}_C(g){\mathcal{T}}_C(h)\mid {\psi }^{(N)}⟩$, where $\mid {\psi }^{(N)}⟩=\mid \psi {⟩}^{\otimes N}$ 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 $Z_{g,h}\mathrm{/}\mid Z_{g,h}\mid =({\theta }_g{\theta }_h\mathrm{/}{\theta }_{gh}{)}^{{\mathfrak{c}}_{-}}$, where ${\theta }_g^{{\mathfrak{c}}_{-}}$ 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.