Modular Berry Connection

Abstract

The Berry connection describes transformations induced by adiabatically varying Hamiltonians. We study how zero modes of the modular Hamiltonian are affected by varying the region that supplies the modular Hamiltonian. In the vacuum of a 2d CFT, global conformal symmetry singles out a unique modular Berry connection, which we compute directly and in the dual AdS$_3$ picture. In certain cases, Wilson loops of the modular Berry connection compute lengths of curves in AdS$_3$, reproducing the differential entropy formula. Modular Berry transformations can be measured by bulk observers moving with varying accelerations.

Figures (2)

• Figure 1: A causal diamond in 1+1 dimensions is stabilized by an $\rm{SO}{(1,1)} \times \overline{\rm{SO}{(1,1)}}$ conformal symmetry. Their symmetric combination is the modular Hamiltonian, which induces a flow from the bottom to the top of the diamond. The antisymmetric combination induces a flow from the left to the right endpoint. In the bulk of AdS$_3$ these symmetries generate, respectively, trajectories of accelerating observers and translations along spacelike geodesics.
• Figure 2: The bulk realization of the modular connection is determined by a map between neighboring geodesics. Given a fixed 1-parameter family of intersecting geodesics $[\lambda(\sigma)]$, a natural gauge is one where the connection leaves the point of intersection between $[\lambda(\sigma)]$ and $[\lambda(\sigma+d\sigma)$] fixed. The origin $s=0$ will then recede as it traverses the family of geodesics. The total precession is the length of the curve connecting all of the intersection points. We can think of the connection as a "rolling without slipping" condition, where the point on the geodesic tangent to this curve is always momentarily at rest.