On the Replica Problem in Supersymmetric SYK Models

We investigate the replica problem for Sachdev-Ye-Kitaev (SYK) models. First, we consider $n-$replicas of the non-supersymmetric SYK model, finding that this $n$-replica model is solvable only under specific conditions. We then introduce the $\mathcal{N}=1$ supersymmetry and utilize the superconformal symmetry to develop a ``multi-ordered trick" that covers the replica structure. By incorporating ordered off-diagonal couplings, we study the resulting thermal phase structure under higher-order interactions. The Lorentzian time dynamics is analyzed, and we plot the time evolution of the effective action. Furthermore, we investigate emergent superconformal symmetry in the low-energy limit of the replicated theory. In the superconformal limit, we propose an ordered super-Schwarzian action and derive reparameterization relations for the ordered coordinates. Corresponding constraints are derived for holographic matching to $\mathcal{N}=1$ super-Jackiw-Teitelboim(SJT) gravity. We numerically calculate the modular thermodynamics (modular entropy, the $n$-dependent relative entropy, and the entanglement capacity) using fully diagonal methods. Our result provide a framework for studying $n$-replica wormholes with supersymmetric SYK model.