A New Kind of Topological Quantum Order: A Dimensional Hierarchy of Quasiparticles Built from Stationary Excitations

TL;DR

This paper presents exactly solvable models in three or more dimensions that exhibit a novel topological quantum order with a hierarchy of quasiparticles, including some that are completely stationary, using an algebraic geometric framework.

Contribution

It introduces a new class of topological models with hierarchical quasiparticles and a polynomial-based algebraic approach to describe their properties.

Findings

Models exhibit extensive ground-state degeneracy

Existence of stationary topological excitations

Topological properties encoded in algebraic varieties

Abstract

We introduce exactly solvable models of interacting (Majorana) fermions in $d \ge 3$ spatial dimensions that realize a new kind of topological quantum order, building on a model presented in ref. [1]. These models have extensive topological ground-state degeneracy and a hierarchy of point-like, topological excitations that are only free to move within sub-manifolds of the lattice. In particular, one of our models has fundamental excitations that are completely stationary. To demonstrate these results, we introduce a powerful polynomial representation of commuting Majorana Hamiltonians. Remarkably, the physical properties of the topologically-ordered state are encoded in an algebraic variety, defined by the common zeros of a set of polynomials over a finite field. This provides a "geometric" framework for the emergence of topological order.