Exotic symmetry breaking properties of self-dual fracton spin models

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 paper, 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 subdimensional symmetries, we construct and analyze the order parameters, correlators, and symmetry generators for both models. Notably, the tetrahedral Ising model acquires an extended semilocal 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 paper provides a unique understanding of subdimensional symmetry breaking and makes an important step for studying quantum -error -correction properties of the checkerboard and Haah's codes.