UC Santa Barbara

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.