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Generalized entanglement entropies in two-dimensional conformal field theory

Sara Murciano, Pasquale Calabrese, Robert M. Konik

TL;DR

The paper introduces generalized mixed state Rényi entropies (GMSREs) for two-dimensional CFTs and develops a path-integral framework that reduces the second generalized Rényi entropy to a four-point function on a two-sheet Riemann surface. Focusing on the free boson (c=1), the authors construct an efficient method to compute GMSREs for arbitrary bosonic states, including descendants and vertex operators, deriving explicit results for simple cases and a general Hafnian-based approach for broader content. They extend the formalism to include vertex insertions, providing a comprehensive recipe (Eq. vv) that maps GMSREs to sums over contractions, and validate their predictions against XX spin-chain numerics. The work enables systematic computation of non-diagonal entanglement measures, with potential applications to time evolution, relative entropy, and symmetry-resolved entanglement via truncated conformal space approaches across various CFTs.

Abstract

We introduce and study generalized Rényi entropies defined through the traces of products of ${\rm Tr}_B (|Ψ_i\rangle\langle Ψ_j|)$ where $|Ψ_i\rangle$ are eigenstates of a two-dimensional conformal field theory (CFT). When $|Ψ_i\rangle=|Ψ_j\rangle$ these objects reduce to the standard Rényi entropies of the eigenstates of the CFT. Exploiting the path integral formalism, we show that the second generalized Rényi entropies are equivalent to four-point correlators. We then focus on a free bosonic theory for which the mode expansion of the fields allows us to develop an efficient strategy to compute the second generalized Rényi entropy for all eigenstates. As a byproduct, our approach also leads to new results for the standard Rényi and relative entropies involving arbitrary descendent states of the bosonic CFT.

Generalized entanglement entropies in two-dimensional conformal field theory

TL;DR

The paper introduces generalized mixed state Rényi entropies (GMSREs) for two-dimensional CFTs and develops a path-integral framework that reduces the second generalized Rényi entropy to a four-point function on a two-sheet Riemann surface. Focusing on the free boson (c=1), the authors construct an efficient method to compute GMSREs for arbitrary bosonic states, including descendants and vertex operators, deriving explicit results for simple cases and a general Hafnian-based approach for broader content. They extend the formalism to include vertex insertions, providing a comprehensive recipe (Eq. vv) that maps GMSREs to sums over contractions, and validate their predictions against XX spin-chain numerics. The work enables systematic computation of non-diagonal entanglement measures, with potential applications to time evolution, relative entropy, and symmetry-resolved entanglement via truncated conformal space approaches across various CFTs.

Abstract

We introduce and study generalized Rényi entropies defined through the traces of products of where are eigenstates of a two-dimensional conformal field theory (CFT). When these objects reduce to the standard Rényi entropies of the eigenstates of the CFT. Exploiting the path integral formalism, we show that the second generalized Rényi entropies are equivalent to four-point correlators. We then focus on a free bosonic theory for which the mode expansion of the fields allows us to develop an efficient strategy to compute the second generalized Rényi entropy for all eigenstates. As a byproduct, our approach also leads to new results for the standard Rényi and relative entropies involving arbitrary descendent states of the bosonic CFT.
Paper Structure (17 sections, 108 equations, 4 figures)

This paper contains 17 sections, 108 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Path integral for the second generalized Rényi entropy
  3. Bosonic theory: a CFT approach beyond primary fields
  4. A warmup: zero-momentum states with a pair of chiral and anti-chiral modes
  5. An example involving vertex operators
  6. A more complicated case: two chiral modes
  7. A more complicated case: two chiral modes
  8. GMSREs involving level 2 chiral states
  9. GMSREs for arbitrary bosonic states
  10. Generalized mixed state Rényi entropies without vertex operators
  11. Generalized mixed state Rényi entropies with vertex operators
  12. Conclusions
  13. Short-interval expansions
  14. Numerical tools
  15. Correlation matrices of excited states in the XX spin chain
  16. ...and 2 more sections

Figures (4)

  • Figure 1: Left: Path integral representation of matrix elements of $\ket{\Psi_i}\bra{\Psi_j}$, cf. Eq. \ref{['eq:fig1']}. Right: Path integral representation of $\mathrm{Tr}_B(\ket{\Psi_i}\bra{\Psi_j})$, as in Eq. \ref{['eq:off-rdm']}.
  • Figure 2: Representation of the spacetime geometry involving a 2-sheeted Riemann surface, $\mathcal{R}_2$, that appears in computing the second generalized entropy $R_{i,j;i',j'}$ in Eq. \ref{["eq:rii'"]}.
  • Figure 3: Tests of the CFT results for the generalized Rényi entropies in the XX spin chain. We consider spin chains made of $N=128$ spins. Left panel: The quantity $\frac{R_{0,0;1,1}}{R_{\mathbbm{1},\mathbbm{1},\mathbbm{1},\mathbbm{1}}}$ as a function of $r=\ell/N$. The symbols are the numerical data, while the continuous line is the square root of Eq. \ref{['eq:results']}. Right panel: $\frac{R_{1,1;1,1}}{R_{\mathbbm{1},\mathbbm{1},\mathbbm{1},\mathbbm{1}}}$ as a function of $r=\ell/N$. The analytical prediction is the square root of Eq. \ref{['eq:sierra']}.
  • Figure 4: Further tests of the generalized Rényi entropies in the XX spin chains with $N=128$ spins. Left panel: The quantity $\frac{R_{1,1;V_{1},V_{-1}}}{R_{\mathbbm{1},\mathbbm{1};\mathbbm{1},\mathbbm{1}}}$ as a function of $r=\ell/N$. The continuous line describes the analytical prediction in Eq. \ref{['eq:test2']} while the symbols refer to the numerical data. Right panel: We test Eq. \ref{['eq:test3']}, i.e. the excess of entropy for descendent states at level 2, as a function of $r=\ell/N$.