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Entanglement entropy in (1+1)D CFTs with multiple local excitations

Wu-zhong Guo, Song He, Zhu-Xi Luo

TL;DR

This work analyzes Rényi and entanglement entropies for locally excited states in (1+1)D CFTs with multiple operator insertions, employing both the replica trick and Schmidt/OPE methods. A central result is that, at late times, the entanglement computed from the product state S_L matches that from the OPE-based state S_R, yielding a nontrivial identity between fusion data, F-symbols, and quantum dimensions in rational CFTs. The authors derive a general lambda_p = d_p/d_O^2 constraint, prove it within a tensor-category framework, and validate it through concrete RCFT examples (free boson, Ising, and minimal models). They further connect these entanglement constraints to bulk-edge correspondence and discuss implications for large-c CFTs and holography. Overall, the paper provides a coherent bridge between quantum information measures and the intrinsic algebraic structure of (1+1)D CFTs, supported by explicit examples and a formal categorical proof.

Abstract

In this paper, we use the replica approach to study the Rényi entropy $S_L$ of generic locally excited states in (1+1)D CFTs, which are constructed from the insertion of multiple product of local primary operators on vacuum. Alternatively, one can calculate the Rényi entropy $S_R$ corresponding to the same states using Schmidt decomposition and operator product expansion, which reduces the multiple product of local primary operators to linear combination of operators. The equivalence $S_L=S_R$ translates into an identity in terms of the $F$ symbols and quantum dimensions for rational CFT, and the latter can be proved algebraically. This, along with a series of papers, gives a complete picture of how the quantum information quantities and the intrinsic structure of (1+1)D CFTs are consistently related.

Entanglement entropy in (1+1)D CFTs with multiple local excitations

TL;DR

This work analyzes Rényi and entanglement entropies for locally excited states in (1+1)D CFTs with multiple operator insertions, employing both the replica trick and Schmidt/OPE methods. A central result is that, at late times, the entanglement computed from the product state S_L matches that from the OPE-based state S_R, yielding a nontrivial identity between fusion data, F-symbols, and quantum dimensions in rational CFTs. The authors derive a general lambda_p = d_p/d_O^2 constraint, prove it within a tensor-category framework, and validate it through concrete RCFT examples (free boson, Ising, and minimal models). They further connect these entanglement constraints to bulk-edge correspondence and discuss implications for large-c CFTs and holography. Overall, the paper provides a coherent bridge between quantum information measures and the intrinsic algebraic structure of (1+1)D CFTs, supported by explicit examples and a formal categorical proof.

Abstract

In this paper, we use the replica approach to study the Rényi entropy of generic locally excited states in (1+1)D CFTs, which are constructed from the insertion of multiple product of local primary operators on vacuum. Alternatively, one can calculate the Rényi entropy corresponding to the same states using Schmidt decomposition and operator product expansion, which reduces the multiple product of local primary operators to linear combination of operators. The equivalence translates into an identity in terms of the symbols and quantum dimensions for rational CFT, and the latter can be proved algebraically. This, along with a series of papers, gives a complete picture of how the quantum information quantities and the intrinsic structure of (1+1)D CFTs are consistently related.

Paper Structure

This paper contains 19 sections, 110 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Entanglement of locally excited states
  3. Product of primary operator
  4. Linear combination of operators
  5. Identity from the constraint
  6. General discussion
  7. The states
  8. Normalization
  9. Rényi entropy of the state $\ket{p}$
  10. The Induced Equality from Entanglement Entropy
  11. More general cases
  12. Some examples
  13. Free massless scalar field
  14. Ising model or Minimal model $\mathcal{M}(p=4,p'=3)$
  15. Operator $\phi_{(2,1)} \phi_{(r,s)}$ in Minimal Model
  16. ...and 4 more sections

Figures (4)

  • Figure 1: Graphical representation of fusion and splitting.
  • Figure 2: Graphical representation of $F$-move.
  • Figure 3: Resolution of identity.
  • Figure 4: One identifies $F_{0p}$ with $F^{-1}_{0p}$ from the above figure.