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Physics at the entangling surface

Kantaro Ohmori, Yuji Tachikawa

TL;DR

This paper argues that defining entanglement entropy in quantum field theories necessitates a boundary condition at the entangling surface, effectively inserting a thin physical boundary whose type is captured by Cardy states. Using a 2d CFT framework, it derives the leading universal entanglement term for a single interval as $S_n \sim (1+1/n)\frac{c_{\text{eff}}}{6}\log\frac{L}{\epsilon}$ with $c_{\text{eff}}=c-12\Delta_0$, plus a boundary-entropy dependent constant and a subleading power-law correction controlled by the lowest-dimension operator coupling to both boundaries; in the $n\to\infty$ limit, a conformal cylinder partition function $Z^{\text{conf}}(q;a_1^{(\infty)},a_2^{(\infty)})$ emerges, encoding semi-universal data. The critical Ising model is then analyzed as a concrete example: the UV cutting procedure reads off Cardy states at the entangling point, with the pure boundary state $|\sigma\rangle$ arising in the free-boundary case and other Cardy states $|1\rangle$, $|\varepsilon\rangle$ corresponding to specific fixed-boundary cuttings, yielding explicit relations between entanglement data and minimal-model characters. Together, these results illuminate how entanglement in QFT is shaped by boundary conditions at the entangling surface and connect boundary CFT data to universal features of entanglement, with implications for holography and foundational questions in quantum gravity.

Abstract

To consider the entanglement between the spatial region $A$ and its complement in a QFT, we need to assign a Hilbert space $\mathcal{H}_A$ to the region, by making a certain choice on the boundary $\partial A$. We argue that a small physical boundary is implicitly inserted at the entangling surface. We investigate these issues in the context of 2d CFTs, and show that we can indeed read off the Cardy states of the $c=1/2$ minimal model from the entanglement entropy of the critical Ising chain.

Physics at the entangling surface

TL;DR

This paper argues that defining entanglement entropy in quantum field theories necessitates a boundary condition at the entangling surface, effectively inserting a thin physical boundary whose type is captured by Cardy states. Using a 2d CFT framework, it derives the leading universal entanglement term for a single interval as with , plus a boundary-entropy dependent constant and a subleading power-law correction controlled by the lowest-dimension operator coupling to both boundaries; in the limit, a conformal cylinder partition function emerges, encoding semi-universal data. The critical Ising model is then analyzed as a concrete example: the UV cutting procedure reads off Cardy states at the entangling point, with the pure boundary state arising in the free-boundary case and other Cardy states , corresponding to specific fixed-boundary cuttings, yielding explicit relations between entanglement data and minimal-model characters. Together, these results illuminate how entanglement in QFT is shaped by boundary conditions at the entangling surface and connect boundary CFT data to universal features of entanglement, with implications for holography and foundational questions in quantum gravity.

Abstract

To consider the entanglement between the spatial region and its complement in a QFT, we need to assign a Hilbert space to the region, by making a certain choice on the boundary . We argue that a small physical boundary is implicitly inserted at the entangling surface. We investigate these issues in the context of 2d CFTs, and show that we can indeed read off the Cardy states of the minimal model from the entanglement entropy of the critical Ising chain.

Paper Structure

This paper contains 11 sections, 36 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Comments on our description of the entanglement entropy:
  3. Organization:
  4. 2d CFT analysis
  5. Setup
  6. Leading universal piece
  7. Subleading corrections
  8. Critical Ising model
  9. Free boundary condition
  10. Fixed boundary condition
  11. Future directions

Figures (2)

  • Figure 1: The cutting operation is given by a linear map $\iota :\mathcal{H}\to\mathcal{H}_{A,a} \otimes \mathcal{H}_{B,a}$. The wiggly line represents "thickened" entangling surface with a boundary condition $a$ specified.
  • Figure 2: The path-integral expression for the reduced density matrix $\rho_A$. For two-dimensional QFTs, the entangling surface consists of two points, where different conditions $a_1$, $a_2$ can be specified.