  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.
