An exact spacetime polymer gas for finite-temperature $\mathbb Z_N$ homological quantum code

TL;DR

This paper develops an exact spacetime model for finite-temperature $ ext{Z}_N$ homological quantum codes, revealing phase transitions and dualities through a polymer gas reformulation.

Contribution

It introduces an exact reformulation of the partition functions in terms of defect polymers and establishes conditions for exponential suppression of nontrivial polymers.

Findings

Bound the polymer gas with positive hard-core majorants for control.

Derive an exact higher-form Kramers-Wannier duality.

Identify a gauge-theory specialization linked to the plaquette random-cluster model.

Abstract

We study finite-temperature $P$-form $\mathbb Z_N$ homological codes via an exact finite-Trotter quantum-to-classical map to a $(d+1)$-dimensional spacetime model with electric and magnetic topological background charges. The resulting background-resolved partition functions admit an exact reformulation in terms of closed magnetic and electric defect polymers, with opposite-species interactions governed by linking phases. By bounding this complex polymer gas by positive same-species hard-core majorant gases, we obtain an explicit low-activity criterion under which all background-dependent partition functions are uniformly controlled and homologically nontrivial polymers are exponentially suppressed on the scale of the spacetime systole. We also derive an exact higher-form Kramers-Wannier duality exchanging electric and magnetic backgrounds, Wilson and 't Hooft operators, and $P$-form and $(d-P)$-form theories. Finally, for prime $N$, we identify an exact source-free gauge-theory specialization coupled to the plaquette random-cluster model, which imports sharp phase-transition results on special geometries into the spacetime framework.