Grand Canonical vs Canonical Krylov Complexity in Double-Scaled Complex SYK Model

TL;DR

This paper analyzes Krylov complexity in the double-scaled complex SYK model, deriving transfer matrices for grand canonical and canonical ensembles, and computes complexity analytically and numerically across different charge sectors.

Contribution

It introduces a transfer matrix approach for the grand canonical ensemble and relates it to canonical ensembles, providing analytical and numerical complexity calculations.

Findings

Krylov complexity in the grand canonical ensemble is a sum over charge sectors.

Analytical expressions for early and late time complexity limits.

Numerical results confirm the theoretical framework.

Abstract

We consider the complex SYK model in the double-scaling limit. We obtain the transfer matrix for the grand canonical ensemble and symmetrize it. In the (n,Q)- basis of chord states, the grand canonical transfer matrix is block diagonal, where each block is the canonical transfer matrix for the respective charge sector. We therefore conclude that the Krylov complexity for the grand canonical ensemble is given by the sum of the complexities in the charge sectors weighted by a probability function that depends on the chemical potential. Finally, we compute the Krylov complexity analytically in the limit of early and late time in the charge sector and numerically for both canonical and grand canonical ensemble.