Two-Dimensional Homological Order of Non-Local Quantum Matter from qLDPC Codes

In recent years, the assumption of geometric locality has become increasingly artificial in emerging quantum technologies, where it is becoming possible to entangle arbitrary degrees of freedom in a system irrespective of their distance. When geometric locality is relaxed, even basic organizing principles in many-body physics such as a system’s “dimension” can be ambiguous— e.g., a 2D system with non-local couplings can be turned into a 3D system. In this talk, we introduce a notion of dimension, inspired from the theory of qLDPC codes, that is homological rather than geometric and use it to define the concept of a “two-dimensional homological order”. These orders form a superset of 2D topological orders and 3D fracton phases and crucially can arise in settings where a naive notion of the spatial dimension is arbitrary or even infinite. Within this framework, we first show that two-dimensional homological orders host isolated quasi-particles, and that these quasi-particles can bind into dyons when the complex possesses a generalized Poincaré duality. Subsequently, we rigorously prove that every such order admits a self-exchange move that defines a notion of self-statistics for these quasi-particles. We conclude by developing twisted variants of these orders analogous to the double semion model.