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Geometric Entropy and Hagedorn/Deconfinement Transition

Mitsutoshi Fujita, Tatsuma Nishioka, Tadashi Takayanagi

TL;DR

The paper introduces geometric entropy, defined as a double Wick rotation of entanglement entropy, and investigates its behavior as an order parameter for confinement/deconfinement. It computes S_G holographically from AdS/CFT, showing a discontinuous jump at the Hagedorn transition, and corroborates this with a free ${ m N}=4$ SYM matrix-model calculation that also exhibits a jump at the transition. The results align qualitatively across gravity and field theory, with quantitative differences understood in terms of coupling and sector choices, and the authors extend the concept to topological field theories and 2D YM where S_G captures phase-structure signals. Overall, geometric entropy emerges as a robust, universal probe of phase transitions in diverse quantum field theories and their holographic duals.

Abstract

It has recently been proposed that the entanglement entropy can be an order parameter of confinement/deconfinement transitions. To find a clear evidence, we introduce a new quantity called the geometric entropy, which is related to the entanglement entropy via a double Wick rotation. We analyze the geometric entropy and manifestly show that its value becomes discontinuous at the Hagedorn temperature both in the free N =4 super Yang-Mills and in its supergravity dual.

Geometric Entropy and Hagedorn/Deconfinement Transition

TL;DR

The paper introduces geometric entropy, defined as a double Wick rotation of entanglement entropy, and investigates its behavior as an order parameter for confinement/deconfinement. It computes S_G holographically from AdS/CFT, showing a discontinuous jump at the Hagedorn transition, and corroborates this with a free SYM matrix-model calculation that also exhibits a jump at the transition. The results align qualitatively across gravity and field theory, with quantitative differences understood in terms of coupling and sector choices, and the authors extend the concept to topological field theories and 2D YM where S_G captures phase-structure signals. Overall, geometric entropy emerges as a robust, universal probe of phase transitions in diverse quantum field theories and their holographic duals.

Abstract

It has recently been proposed that the entanglement entropy can be an order parameter of confinement/deconfinement transitions. To find a clear evidence, we introduce a new quantity called the geometric entropy, which is related to the entanglement entropy via a double Wick rotation. We analyze the geometric entropy and manifestly show that its value becomes discontinuous at the Hagedorn temperature both in the free N =4 super Yang-Mills and in its supergravity dual.

Paper Structure

This paper contains 10 sections, 46 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Geometric Entropy in ${\cal N} =4$ SYM and $AdS/CFT$
  3. Definition of Geometric Entropy
  4. Holographic Calculation of $S_G$
  5. Geometric Entropy in Free ${\cal N} =4$ super Yang-Mills
  6. Analysis Near the Transition
  7. Analysis of High Temperature Limit
  8. Geometric Entropy in TQFT and 2D YM
  9. Conclusion
  10. Computation of the single particle partition functions

Figures (2)

  • Figure 1: The behavior of $\Delta S_G/N^2$ for free Yang-Mills (a) and IIB supergravity (b). The horizontal axis corresponds to the temperature $T$. The temperature at which the phase transition occurs is $T_H = -1/\ln (7-4\sqrt3)= 0.379$ in the Yang-Mills theory and $T_H = 3 /2\pi = 0.477$ (dashed line) in the dual gravity. Notice that the line starts at $T = \sqrt2 /\pi = 0.450$ in (b) above which the AdS black hole exists.
  • Figure 2: The behavior of $\Delta S_G/N^2$ for free Yang-Mills with the specified values of the chemical potential $(\mu_1,\mu_2,\mu_3)=(\mu,0,0)$ of the $R$-charges.