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Entanglement entropy through conformal interfaces in the 2D Ising model

Enrico M. Brehm, Ilka Brunner

TL;DR

This work analyzes entanglement entropy between two halves of a 2D Ising CFT separated by a conformal defect. By applying the replica trick, the authors derive a universal logarithmic scaling of the vacuum EE, with a prefactor $\sigma(\mathcal{T})$ that depends on the defect transmission $\mathcal{T}$, and they show how topological and ground-state data contribute to subleading constants. The calculation employs a detailed free-fermion formulation, including NS/R sectors, GSO projection, and a careful evaluation of the defect-augmented torus partition function via theta and eta functions; neutral and charged defects are treated, revealing how $\mathcal{T}$ controls the leading term while constants encode topological information. In the supersymmetric extension, oscillator contributions cancel, leaving a simplified leading behavior that highlights the role of $\mathcal{T}$ and the ground-state index. These results illuminate how conformal defects encode entanglement structure and suggest generalizations to higher-dimensional tori and RG-domain walls, linking defect data to universal EE features.

Abstract

We consider the entanglement entropy for the 2D Ising model at the conformal fixed point in the presence of interfaces. More precisely, we investigate the situation where the two subsystems are separated by a defect line that preserves conformal invariance. Using the replica trick, we compute the entanglement entropy between the two subsystems. We observe that the entropy, just like in the case without defects, shows a logarithmic scaling behavior with respect to the size of the system. Here, the prefactor of the logarithm depends on the strength of the defect encoded in the transmission coefficient. We also comment on the supersymmetric case.

Entanglement entropy through conformal interfaces in the 2D Ising model

TL;DR

This work analyzes entanglement entropy between two halves of a 2D Ising CFT separated by a conformal defect. By applying the replica trick, the authors derive a universal logarithmic scaling of the vacuum EE, with a prefactor that depends on the defect transmission , and they show how topological and ground-state data contribute to subleading constants. The calculation employs a detailed free-fermion formulation, including NS/R sectors, GSO projection, and a careful evaluation of the defect-augmented torus partition function via theta and eta functions; neutral and charged defects are treated, revealing how controls the leading term while constants encode topological information. In the supersymmetric extension, oscillator contributions cancel, leaving a simplified leading behavior that highlights the role of and the ground-state index. These results illuminate how conformal defects encode entanglement structure and suggest generalizations to higher-dimensional tori and RG-domain walls, linking defect data to universal EE features.

Abstract

We consider the entanglement entropy for the 2D Ising model at the conformal fixed point in the presence of interfaces. More precisely, we investigate the situation where the two subsystems are separated by a defect line that preserves conformal invariance. Using the replica trick, we compute the entanglement entropy between the two subsystems. We observe that the entropy, just like in the case without defects, shows a logarithmic scaling behavior with respect to the size of the system. Here, the prefactor of the logarithm depends on the strength of the defect encoded in the transmission coefficient. We also comment on the supersymmetric case.

Paper Structure

This paper contains 22 sections, 88 equations, 3 figures.

Table of Contents

  1. Introduction and summary
  2. Conformal interfaces of the Ising model
  3. Interfaces and boundary conditions
  4. Free fermion description
  5. How to derive the entanglement entropy
  6. Derivation of the partition function
  7. The partition function for a single NS operator I
  8. Explicit calculation
  9. The partition function for a single R-operator I
  10. Explicit calculation
  11. The partition function for the neutral interface operator
  12. The partition function for the charged interface operator
  13. Derivation of the entanglement entropy
  14. Explicit derivation of S
  15. Comments on the result and special cases
  16. ...and 7 more sections

Figures (3)

  • Figure 1: The folding trick transforms interfaces (red) of the critical Ising model to boundary conditions of the $c=1$$\mathbb{Z}_2$-orbifold theory.
  • Figure 2: Sketch of the $K$-sheet Riemann surface $\mathcal{R}_K$ that one needs for the replica trick. After imposing a IR citoff $\epsilon$, an UV cutoff $L$, and suitable variable transformation, $\mathcal{R}_K$ looks like on the right.
  • Figure 3: The prefactor $\sigma$ as a function of the transmission coefficient $\mathcal{T}$. As expected, it vanishes for totally reflecting interfaces, where $\mathcal{T} = 0$, and reproduces the familiar result $\sigma(\mathcal{T}=1) = 1/6 = c/3$ for topological interfaces in the Ising model.