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Entanglement Entropy and Quantum Field Theory

Pasquale Calabrese, John Cardy

TL;DR

This work provides a unified framework to compute entanglement entropy in relativistic quantum field theory across critical and noncritical regimes. Employing the replica trick and conformal field theory, it derives exact expressions for S_A in 1+1D systems for single and multiple intervals, finite systems, and finite temperatures, and shows how off-criticality introduces universal logarithmic scaling with the correlation length, modulated by the number of boundary points. The authors defend these results with checks against free massive fields and integrable lattice models, and extend the analysis to higher dimensions where an area law with universal subleading terms is anticipated. A key contribution is the proposed finite-size scaling form that bridges critical and off-critical behavior, supported by Gaussian-model calculations and CTM techniques. The work also clarifies limitations and future directions, including caveats about multi-interval generalizations and their dependence on full CFT data.

Abstract

We carry out a systematic study of entanglement entropy in relativistic quantum field theory. This is defined as the von Neumann entropy S_A=-Tr rho_A log rho_A corresponding to the reduced density matrix rho_A of a subsystem A. For the case of a 1+1-dimensional critical system, whose continuum limit is a conformal field theory with central charge c, we re-derive the result S_A\sim(c/3) log(l) of Holzhey et al. when A is a finite interval of length l in an infinite system, and extend it to many other cases: finite systems,finite temperatures, and when A consists of an arbitrary number of disjoint intervals. For such a system away from its critical point, when the correlation length ξis large but finite, we show that S_A\sim{\cal A}(c/6)\logξ, where \cal A is the number of boundary points of A. These results are verified for a free massive field theory, which is also used to confirm a scaling ansatz for the case of finite-size off-critical systems, and for integrable lattice models, such as the Ising and XXZ models, which are solvable by corner transfer matrix methods. Finally the free-field results are extended to higher dimensions, and used to motivate a scaling form for the singular part of the entanglement entropy near a quantum phase transition.

Entanglement Entropy and Quantum Field Theory

TL;DR

This work provides a unified framework to compute entanglement entropy in relativistic quantum field theory across critical and noncritical regimes. Employing the replica trick and conformal field theory, it derives exact expressions for S_A in 1+1D systems for single and multiple intervals, finite systems, and finite temperatures, and shows how off-criticality introduces universal logarithmic scaling with the correlation length, modulated by the number of boundary points. The authors defend these results with checks against free massive fields and integrable lattice models, and extend the analysis to higher dimensions where an area law with universal subleading terms is anticipated. A key contribution is the proposed finite-size scaling form that bridges critical and off-critical behavior, supported by Gaussian-model calculations and CTM techniques. The work also clarifies limitations and future directions, including caveats about multi-interval generalizations and their dependence on full CFT data.

Abstract

We carry out a systematic study of entanglement entropy in relativistic quantum field theory. This is defined as the von Neumann entropy S_A=-Tr rho_A log rho_A corresponding to the reduced density matrix rho_A of a subsystem A. For the case of a 1+1-dimensional critical system, whose continuum limit is a conformal field theory with central charge c, we re-derive the result S_A\sim(c/3) log(l) of Holzhey et al. when A is a finite interval of length l in an infinite system, and extend it to many other cases: finite systems,finite temperatures, and when A consists of an arbitrary number of disjoint intervals. For such a system away from its critical point, when the correlation length ξis large but finite, we show that S_A\sim{\cal A}(c/6)\logξ, where \cal A is the number of boundary points of A. These results are verified for a free massive field theory, which is also used to confirm a scaling ansatz for the case of finite-size off-critical systems, and for integrable lattice models, such as the Ising and XXZ models, which are solvable by corner transfer matrix methods. Finally the free-field results are extended to higher dimensions, and used to motivate a scaling form for the singular part of the entanglement entropy near a quantum phase transition.

Paper Structure

This paper contains 15 sections, 111 equations, 2 figures.

Table of Contents

  1. Introduction.
  2. von Neumann Entropy and Riemann surfaces.
  3. Entanglement entropy in 2d conformal field theory.
  4. Single interval
  5. Finite system with a boundary.
  6. General case.
  7. Entropy in non-critical 1+1-dimensional models
  8. Massive field theory - general case
  9. Free bosonic field theory
  10. Integrable models and the corner transfer matrix
  11. Finite-size effects
  12. FSS in the Gaussian model
  13. Higher dimensions: Scaling of entropy and area law
  14. Note added.
  15. Two-point function in a $n$-sheeted Riemann geometry: Infinite and finite volume results

Figures (2)

  • Figure 1: Entanglement entropy for the 1D Ising chain as function of $\lambda$. The dashed line is the limit for $\lambda\rightarrow\infty$, i.e. $\log 2$.
  • Figure 2: $s_{\rm FSS}^{(1)}(x)$: Exact expression obtained as numerical sums over the first 1000 zeroes of the Bessel functions compared with small (quadratic, quartic and sextic) and large $x$ ($\log x+s_0^\infty$) approximations. Even the use of quadratic and large $x$ approximation may reproduce the right formula over all the range. Inset: Comparison of several small $x$ approximants for $x<2$.