Adiabatic corrections to holographic entanglement in thermofield doubles and confining ground states

TL;DR

This paper investigates how slow variations in the geometry of holographic spacetimes affect entanglement entropy and phase transitions in thermofield double and confining ground states, revealing dimension-dependent effects of geometric gradients.

Contribution

It introduces adiabatic corrections to holographic entanglement entropy in geometries with slowly varying metrics, highlighting novel dimension-dependent effects on phase transition scales.

Findings

Gradients in the geometry can either increase or decrease the critical length scale depending on the dimension.

In thermofield double states, the effect of gradients on the phase transition scale changes sign at dimension four.

In confining ground states, gradients consistently decrease the critical length scale, especially in higher dimensions.

Abstract

We study entanglement in states of holographic CFTs defined by Euclidean path integrals over geometries with slowly varying metrics. In particular, our CFT spacetimes have $S^1$ fibers whose size $b$ varies along one direction ($x$) of an ${\mathbb R}^{d-1}$ base. Such examples respect an ${\mathbb R}^{d-2}$ Euclidean symmetry. Treating the $S^1$ direction as time leads to a thermofield double state on a spacetime with adiabatically varying redshift, while treating another direction as time leads to a confining ground state with slowly varying confinement scale. In both contexts the entropy of slab-shaped regions defined by $|x - x_0| \le L$ exhibits well-known phase transitions at length scales $L= L_{crit}$ characterizing the CFT entanglements. For the thermofield double, the numerical coefficients governing the effect of variations in $b(x)$ on the transition are surprisingly small and exhibit an interesting change of sign: gradients reduce $L_{crit}$ for $d \le 3$ but increase $L_{crit}$ for $d\ge4$. This means that, while for general $L > L_{crit}$ they significantly increase the mutual information of opposing slabs as one would expect, for $d\ge 4$ gradients cause a small decrease near the phase transition. In contrast, for the confining ground states gradients always decrease $L_{crit}$, with the effect becoming more pronounced in higher dimensions.