Filling in the Circle: Wormhole Partition Functions in Non-Holographic Quantum Systems

Abstract

Replica wormholes arise as natural contributions to the gravitational path integral when computing the nth Renyi entropies of density matrices for holographic systems. In this paper, we extend this notion to non-holographic systems by introducing an auxiliary bulk. The auxiliary bulk serves as a space over which we can integrate the symplectic form of the system in the path integral. This allows us to compute partition functions on wormhole topologies. We build on this concept by computing wormhole partition functions for the examples of a particle on a circle and a particle on a group. We also consider a class of geometric states that are obtained by slicing the topologies over which the partition functions are defined. These are the thermal density matrix (obtained by slicing the thermal circle), the TFD state (obtained by purifying the thermal density matrix), and the TMD state (obtained by slicing the partition functions on wormhole topologies). We also show how these wormhole partition functions show up as contributions to the overlaps of generalized TFD states, causing the Hilbert space spanned by them to become finite-dimensional. In an attempt to extend the topology obtained by considering wormholes and extract some notion of geometry from them, we also consider the concept of Krylov state complexity of TFD states \cite{spreadofstates}, which is conjectured to be the dual of the length of a wormhole connecting the left and right Hilbert spaces in holographic systems.