# Universality Classes of Stabilizer Code Hamiltonians

## TL;DR

This paper analyzes the finite temperature behavior of stabilizer code Hamiltonians by solving their partition functions using duality, revealing their universality classes and effective dimensions, with specific results on the 4D Toric Code and Haah's Code.

## Contribution

It introduces a generic duality-based method to determine the universality class and effective dimension of stabilizer code Hamiltonians at finite temperature.

## Key findings

- 4D Toric Code belongs to 4D Ising universality class
- Haah's Code exhibits dimensional reduction to 1D Ising class
- Method provides insights into robustness and dynamics of quantum memories

## Abstract

Stabilizer code quantum Hamiltonians have been introduced with the intention of physically realizing a quantum memory because of their resilience to decoherence. In order to analyze their finite temperature thermodynamics, we show how to generically solve their partition function using duality techniques. By unveiling each model's universality class and effective dimension, insights may be gained on their finite temperature dynamics and robustness. Our technique is demonstrated in particular on the 4D Toric Code and Haah's Code -- we find that the former falls into the 4D Ising universality class, whereas Haah's Code exhibits dimensional reduction and falls into the 1D Ising universality class.