Modular Hamiltonian of Excited States in Conformal Field Theory

TL;DR

This work introduces a replica trick that breaks the $\mathbb{Z}_n$ symmetry to access the modular Hamiltonian and relative entropy of excited states in conformal field theories. Relative entropy and diagonal modular-operator elements are expressed via analytic continuations of correlation functions on replica geometries, enabling UV-finite, gauge-ambiguity-free quantities. For near-vacuum states, the quantum Fisher information reduces to universal two-point functions on the replica manifold, yielding a universal vacuum QFI that depends only on energy and subsystem size. The authors demonstrate concrete 2D CFT results for a free boson, including explicit expressions for $S(\alpha\|\beta)$, the diagonal modular Hamiltonian, and the mutual information for multiple intervals, and discuss limitations and holographic connections.

Abstract

We present a novel replica trick that computes the relative entropy of two arbitrary states in conformal field theory. Our replica trick is based on the analytic continuation of partition functions that break the replica Z_n symmetry. It provides a method for computing arbitrary matrix elements of the modular Hamiltonian corresponding to excited states in terms of correlation functions. We show that the quantum Fisher information in vacuum can be expressed in terms of two-point functions on the replica geometry. We perform sample calculations in two-dimensional conformal field theories.

Figures (5)

• Figure 1: ($a$) Operator-state correspondence in radial quantization of conformal field theories. ($b$) Reduced density matrix corresponds to a path-integral with two operator insertions and a cut on the subsystem.
• Figure 2: ($a$) Entanglement entropy replica trick: the Euclidean path-integral on the $n$-sheeted manifold corresponding to the partition function $tr(\psi^n)$. ($b$) The $Z_n$-breaking partition $tr(\phi\psi^{n-1})$ that appears in our relative entropy replica trick.
• Figure 3: The type of two-point functions on the replica manifold whose analytic continuation determine quantum Fisher information in vacuum.
• Figure 4: ($a$) The $n$-sheeted manifold corresponding to the partition function $Z_n^{AB}$. ($b$) The $Z_n$-breaking partition $Z_n^{A,B}=tr(\sigma_{AB} (\sigma_A\otimes \sigma_B)^{\otimes n-1})$.
• Figure 5: Inserting the resolution of identity in $Z_n^{AB}$ we observe that at each $K$ we multiply sphere one-point functions that are zero unless $\Phi_K$ is identity.