Quenched vs Annealed: Glassiness from SK to SYK

TL;DR

This paper demonstrates that SYK-like models with local degrees of freedom differ fundamentally from fermionic models by failing to be described by an annealed free energy at low temperatures, linking them to spin glasses.

Contribution

It proves that models with local interactions and tensor product Hilbert spaces cannot be described by annealed free energy at low temperatures, revealing a fundamental difference from fermionic SYK models.

Findings

Annealed free energy diverges at low temperature in tensor product models.

Phase transition from annealed to non-annealed phase as temperature decreases.

Non-bosonic particles are essential for SYK physics and its connection to glassiness.

Abstract

We show that any SYK-like model with finite-body interactions among \textit{local} degrees of freedom, e.g., bosons or spins, has a fundamental difference from the standard fermionic model: the former fails to be described by an annealed free energy at low temperature. In this respect, such models more closely resemble spin glasses. We demonstrate this by two means: first, a general theorem proving that the annealed free energy is divergent at low temperature in any model with a tensor product Hilbert space; and second, a replica treatment of two prominent examples which exhibit phase transitions from an "annealed" phase to a "non-annealed" phase as a function of temperature. We further show that this effect appears only at $O(N)$'th order in a $1/N$ expansion, even though lower-order terms misleadingly seem to converge. Our results prove that the non-bosonic nature of the particles in SYK is an essential ingredient for its physics, highlight connections between local models and spin glasses, and raise important questions as to the role of fermions and/or glassiness in holography.