Physics Colloquium: Hidden Order in Topological Phases—Entangled Pauli Principles

The search for hidden order is the unifying theme underlying the study of condensed matter. Its success is what allows us to organize the myriad forms of matter that surround us into “phases of matter.” Traditional organization principles are largely due to Landau, and rest at least in part on Hamiltonian descriptions: We simplify the interactions in a manner that does not drastically alter the physics, but allows for understanding of universal properties. Fascinating examples include superconductivity, a problem that is considered solved since Bardeen, Cooper, and Schrieffer presented a simple model interaction that explains the physics.

In many ways, the more recent paradigm of “topological phases” represents a radical departure from this thinking. In the fundamental description of these phases, Hamiltonian principles are often treated as an afterthought, while other organization principles are sought in intricate properties that are directly encoded in the many-body ground state wave functions, regardless of the interactions that stabilize them: Such properties include both entanglement, as well as topological invariants that lend a certain built-in robustness to the fascinating physical behaviors of these phases.

In this talk, I will introduce an organization principle for a large class of topological phases: Entangled Pauli principles. These principles are intimately connected to model-Hamiltonians of the traditional kind, allowing construction of model-Hamiltonians where none were known before. More importantly, they can be understood as the “DNA” of the phases they describe. In particular, they allow for a simple, efficient, and totally quantum-field-theory-free way to understand and visualize the properties that may make fault-tolerant topological quantum computing possible.