Subsystem symmetry has emerged as a powerful organizing principle for unconventional quantum phases of matter, most prominently fracton topological orders. Here, we focus on a special subclass of such symmetries, known as higher-form subsystem symmetries, which allow us to adapt tools from the study of conventional topological phases to the fracton setting. We demonstrate that certain transitions out of familiar fracton phases, including the X-cube model, can be understood in terms of the spontaneous breaking of higher-form subsystem symmetries. We find simple pictures for these seemingly complicated fracton topological phase transitions by relating them in an exact manner, via gauging, to spontaneous higher-form subsystem symmetry breaking phase transitions of decoupled stacks of lower-dimensional models. We harness this perspective to construct a sequence of unconventional subdimensional critical points in two and three spatial dimensions based on the stacking and gauging of canonical models with higher-form symmetry. Through numerous examples, we illustrate the ubiquity of coupled layer constructions in theories with higher-form subsystem symmetries.
We consider topological entanglement entropy (TEE) at finite temperature for CSS codes, which include some ordinary topological-ordered systems such as the toric code and some fracton models such as the Haah's code and the X-cube model. We find, under the assumption that there is no extended critical phase, the finite-temperature TEE is a piecewise constant function of the temperature, with possible discontinuities only at phase transitions. We then consider phase transitions of CSS codes. We claim that there must exist a phase transition at zero temperature for any CSS codes in 2D and 3D with topological order. This statement can be rigorous proved for some familiar examples, while for general models it can be argued based on the low-temperature expansion. This indicates the break down of topological orders at finite temperature. We also discuss possible connections with self-correcting quantum memory.
Understanding quantum phases and phase transitions in the presence of symmetries is a central objective of quantum many-body physics. A powerful modern paradigm for investigating this problem is topological holography, which relates symmetries in
k dimensions to "bulk" topological orders in
(k+1) dimensions. While conceptually profound, most existing bulk construction methods rely on sophisticated mathematical formalisms and can be difficult to apply to certain symmetry types. In this work, we propose a physically intuitive and versatile method, termed the layered gauging construction, to systematically generate
(k+1)-dimensional (liquid or fracton) topological orders from
k-dimensional generalized symmetries. Roughly speaking, the prescription is to stack many layers of
k-dimensional quantum systems with certain symmetries into a
(k+1)-dimensional pile, and then sequentially gauge a diagonal symmetry acting on each nearest-neighbor pair of layers. The detailed procedure depends on the specific symmetry types. We have successfully implemented the method in a number of examples in different spatial dimensions, with symmetries that are conventional, higher-form, subsystem, anomalous, nonabelian, or noninvertible. We hence conjecture the method to be very general. For example, from the subsystem symmetry of the
2d plaquette Ising model, we derive the X-cube model and also an anisotropic fracton topological order. Additionally, starting from an anomalous
Z2 symmetry in
1d, we construct a new square lattice model realizing the double semion topological order.
We introduce hybrid fracton orders: three-dimensional gapped quantum phases that exhibit the phenomenology of both conventional three-dimensional topological orders and fracton orders. Hybrid fracton orders host both (i) mobile topological quasiparticles and loop excitations, as well as (ii) pointlike topological excitations with restricted mobility, with nontrivial fusion rules and mutual braiding statistics between the two sets of excitations. Furthermore, hybrid fracton phases can realize either conventional three-dimensional topological orders or fracton orders after undergoing a phase transition driven by the condensation of certain gapped excitations. Therefore they serve as parent orders for both long-range-entangled quantum liquid and nonliquid phases. We study the detailed properties of hybrid fracton phases through exactly solvable models in which the resulting orders hybridize a three-dimensional
Z2 topological order with (i) the X-cube fracton order or (ii) Haah's code. The hybrid orders presented here can also be understood as the deconfined phase of a gauge theory whose gauge group is given by an Abelian global symmetry
G and subsystem symmetries of a normal subgroup
N along lower-dimensional subregions.