Strongly Coupled Gauge Theories: High and Low Temperature Behavior of Non-local Observables

TL;DR

The paper develops systematic finite-temperature expansions for non-local observables in strongly coupled gauge theories with AdS duals, focusing on equal-time two-point functions, spatial Wilson loops, and entanglement entropy. By separating near-boundary and near-horizon contributions, it derives high- and low-temperature behaviors across relativistic and non-relativistic (hyperscaling-violating) backgrounds. Key results include exponential decay of two-point functions and area- or volume-law leading terms for entanglement and Wilson loops at high temperature, with subleading corrections encoding full bulk geometry and quantum entanglement. The analyses are concretely applied to N=4 SYM in 3+1 dimensions, illustrating quantitative HV generalizations and setting the stage for studying thermalization and mutual information in holographic settings.

Abstract

We explore the high and low temperature behavior of non-local observables in strongly coupled gauge theories that are dual to AdS. We develop a systematic expansion for equal time two-point correlation, spatial Wilson loops and entanglement entropy at finite temperature using the AdS/CFT correspondence, leading to analytic expressions for these observables at high and low temperature limits. This approach enables the identification of the contributions of different regions of the bulk geometry to these gauge theory observables.

Figures (5)

• Figure 1: At high temperature ($r_H l\gg 1$), the actual geodesic (solid blue line) can be approximated by the dashed red line curve that consists of $x=-l/2, r=r_H, x=l/2$.
• Figure 2: (a) The total system can be divided into two subsystems A and B; the entanglement entropy $S_A$ measures the amount of information loss because of smearing out in region B. (b) A schematic diagram of the rectangular geometry and the corresponding extremal surface used for the calculations of spatial Wilson loops and the entanglement entropy.
• Figure 3: Variation of $f(x,y)\equiv \frac{1}{\Delta}\ln \langle {\cal O} _{\Delta}(t,x){\cal O} _{\Delta}(t,y)\rangle$ with $T|x-y|$ for the 4-dimensional ${\cal N} =4$ SYM theory. The solid black line represents the exact numerical result. Blue and red lines represent the two-point functions computed using low and high temperature approximations, respectively.
• Figure 4: Variation of $S_{NG;ren}$ (in the units of $\frac{\sqrt{\lambda} L}{2\pi}$) of the spatial rectangular (infinite) Wilson loop with $T l$ for the 4-dimensional ${\cal N} =4$ SYM theory. The expectation value of the Wilson loop is given by $\langle W({\cal C} )\rangle=e^{- S_{NG;ren}}$. The solid black line represents the exact numerical result. Blue and red lines represent the two-point functions computed using low and high temperature approximations, respectively.
• Figure 5: Variation of $S_{finite}$, the finite part of the entanglement entropy (in the units of $\frac{N^2 L^2}{2\pi}$) of an infinite rectangular strip with $T l$ for the 4-dimensional ${\cal N} =4$ SYM theory. The solid black line represents the exact numerical result. Blue and red lines represent the two-point functions computed using low and high temperature approximations, respectively.