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Entanglement Entropy of 3-d Conformal Gauge Theories with Many Flavors

Igor R. Klebanov, Silviu S. Pufu, Subir Sachdev, Benjamin R. Safdi

TL;DR

The paper analyzes the finite part F of the entanglement entropy for 3d conformal gauge theories with many flavors, using the relation F = −log|Z| on S^3. It develops a 1/N_F expansion for non-supersymmetric CS-matter theories and performs exact localization for N=2 SUSY theories, finding agreement and a characteristic subleading log N term. By studying RG flows in SUSY models, the authors provide strong evidence for the F-theorem, showing F decreases along flows between fixed points. The results connect gauge dynamics, entanglement, and RG structure in a broad class of 3d CFTs, with implications for conformal windows and dualities.

Abstract

Three-dimensional conformal field theories (CFTs) of deconfined gauge fields coupled to gapless flavors of fermionic and bosonic matter describe quantum critical points of condensed matter systems in two spatial dimensions. An important characteristic of these CFTs is the finite part of the entanglement entropy across a circle. The negative of this quantity is equal to the finite part of the free energy of the Euclidean CFT on the three-sphere, and it has been proposed to satisfy the so called F-theorem, which states that it decreases under RG flow and is stationary at RG fixed points. We calculate the three-sphere free energy of non-supersymmetric gauge theory with a large number N_F of bosonic and/or fermionic flavors to the first subleading order in 1/N_F. We also calculate the exact free energies of the analogous chiral and non-chiral {\cal N} = 2 supersymmetric theories using localization, and find agreement with the 1/N_F expansion. We analyze some RG flows of supersymmetric theories, providing further evidence for the F-theorem.

Entanglement Entropy of 3-d Conformal Gauge Theories with Many Flavors

TL;DR

The paper analyzes the finite part F of the entanglement entropy for 3d conformal gauge theories with many flavors, using the relation F = −log|Z| on S^3. It develops a 1/N_F expansion for non-supersymmetric CS-matter theories and performs exact localization for N=2 SUSY theories, finding agreement and a characteristic subleading log N term. By studying RG flows in SUSY models, the authors provide strong evidence for the F-theorem, showing F decreases along flows between fixed points. The results connect gauge dynamics, entanglement, and RG structure in a broad class of 3d CFTs, with implications for conformal windows and dualities.

Abstract

Three-dimensional conformal field theories (CFTs) of deconfined gauge fields coupled to gapless flavors of fermionic and bosonic matter describe quantum critical points of condensed matter systems in two spatial dimensions. An important characteristic of these CFTs is the finite part of the entanglement entropy across a circle. The negative of this quantity is equal to the finite part of the free energy of the Euclidean CFT on the three-sphere, and it has been proposed to satisfy the so called F-theorem, which states that it decreases under RG flow and is stationary at RG fixed points. We calculate the three-sphere free energy of non-supersymmetric gauge theory with a large number N_F of bosonic and/or fermionic flavors to the first subleading order in 1/N_F. We also calculate the exact free energies of the analogous chiral and non-chiral {\cal N} = 2 supersymmetric theories using localization, and find agreement with the 1/N_F expansion. We analyze some RG flows of supersymmetric theories, providing further evidence for the F-theorem.

Paper Structure

This paper contains 16 sections, 98 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mapping to $S^3$ and large $N_F$ expansion
  3. Gauge field contribution to the free energy
  4. Performing the Gaussian integrals
  5. Diagonalizing the kernel $K^{\mu \nu}(r,r')$
  6. Warm-up: Diagonalization on $\mathbb{R}^3$
  7. Diagonalization on $S^3$
  8. Contribution to the free energy
  9. Generalization to $U(N_c)$ theory
  10. SUSY gauge theory with flavors
  11. ${\cal N}=4$ theory
  12. ${\cal N}=3$ theory
  13. Non-chiral ${\cal N}=2$ theory
  14. Chiral ${\cal N} = 2$ theory
  15. Discussion
  16. ...and 1 more sections

Figures (2)

  • Figure 1: The exact free energy of the ${\cal N} =4$ theory obtained from eq. \ref{['ZN4']} (solid orange) and the analytical approximation \ref{['FN4']} (dashed black).
  • Figure 2: The R-charge $\Delta$ plotted as a function of $N$ for $\kappa = 0, 4/\pi, 8/\pi$, with darker plots corresponding to larger $\kappa$. The solid lines are calculated using the approximation in eq. \ref{['DeltaN2']}. The circles are computed by numerically maximizing the free energy with respect to $\Delta$. Note that the two computations match well even for small $N$.