Ramp from Replica Trick

TL;DR

This paper analytically computes the spectral form factor of the modular Hamiltonian for Haar random states using the replica trick, revealing a ramp behavior and connecting it to gravitational geometries.

Contribution

It introduces a novel replica trick approach to derive the spectral form factor's ramp for modular Hamiltonians, linking permutation sums to gravitational double trumpet geometries.

Findings

Demonstrated the existence of a spectral ramp for the modular Hamiltonian.

Derived an explicit expression for the ramp's slope.

Established a connection between permutation sums and gravitational geometries.

Abstract

We compute the spectral form factor of the modular Hamiltonian $K=-\ln\rho_A$ associated to the reduced density matrix of a Haar random state. A ramp is demonstrated and we find an analytic expression for its slope. Our method involves an application of the replica trick, where we first calculate the correlator $<\text{tr}\rho_A^n\;\text{tr}\rho_A^m>$ at large bond dimension and then analytically continue the indices $n,m$ from integers to arbitrary complex numbers. We use steepest descent methods at large modular times to extract the ramp. The large bond dimension limit of the replicated partition function is dominated by a sum over \emph{annular non-crossing permutations}. We explored the similarity between our results and calculations of the spectral form factor in low dimensional gravitational theories where the ramp is determined by the double trumpet geometry. We find there is an underlying resemblance in the two calculations, when we interpret the annular non-crossing permutations as representing a discretized version of the double trumpet. Similar results are found for an equilibrated pure state in place of the Haar random state.