Exotic Symmetry Breaking Properties of Self-Dual Fracton Spin Models

TL;DR

This paper investigates the unique symmetry-breaking properties of two self-dual fracton spin models, revealing their unconventional phase transitions and order parameters, which enhance understanding of fracton codes for quantum error correction.

Contribution

It introduces and analyzes the exotic symmetry-breaking behaviors of the tetrahedral and fractal Ising models, linking them to fracton codes and their error-correction capabilities.

Findings

Both models exhibit strong first-order phase transitions with anomalous scaling.

The tetrahedral model shows extended semi-local order, while the fractal model has a fractal order parameter.

Numerical and analytical methods confirm unconventional symmetry-breaking phenomena.

Abstract

Fracton codes host unconventional topological states of matter and are promising for fault-tolerant quantum computation due to their large coding space and strong resilience against decoherence and noise. In this work, we investigate the ground-state properties and phase transitions of two prototypical self-dual fracton spin models -- the tetrahedral Ising model and the fractal Ising model -- which correspond to error-correction procedures for the representative fracton codes of type-I and type-II, the checkerboard code and the Haah's code, respectively, in the error-free limit. They are endowed with exotic symmetry-breaking properties that contrast sharply with the spontaneous breaking of global symmetries and deconfinement transition of gauge theories. To show these unconventional behaviors, which are associated with sub-dimensional symmetries, we construct and analyze the order parameters, correlators, and symmetry generators for both models. Notably, the tetrahedral Ising model acquires an extended semi-local ordering moment, while the fractal Ising model fits into a polynomial ring representation and leads to a fractal order parameter. Numerical studies combined with analytical tools show that both models experience a strong first-order phase transition with an anomalous $L^{-(D-1)}$ scaling, despite the fractal symmetry of the latter. Our work provides new understanding of sub-dimensional symmetry breaking and makes an important step for studying quantum-error-correction properties of the checkerboard and Haah's codes.