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

TL;DR

This work introduces a family of exactly solvable interacting Majorana fermion models in three or more dimensions that realize a new type of topological order with extensive ground-state degeneracy. By representing commuting Majorana Hamiltonians via stabilizer maps encoded as vectors of GF(2) Laurent polynomials, the authors provide a geometric-algebraic framework where the ground state structure and excitations are captured by algebraic varieties and quotient rings. They uncover a hierarchy of excitations, including immobile fractons and dimension-n anyons, and establish mobility constraints through polynomial conditions, supplemented by a detailed Majorana cubic model and a dimension-1 hopping model. The results extend topological order beyond TQFT, offering a constructive path to 3D fracton-like physics with exact solvability and a computable degeneracy scaling, potentially impacting fault-tolerant quantum information architectures and the study of 3D topological phases.

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.

Figures (5)

• Figure 1: Majorana Cubic Model: The Majorana cubic model is defined on a cubic lattice, as in (a), with a single Majorana fermion per lattice site (colored red). The operator $\mathcal{O}_{n}$ is the product of the $8$ Majorana fermions at the vertices of a cube. The Hamiltonian is a sum of these local operators over every other cube (colored blue) in a checkerboard pattern. As any pair of operators either share exactly one edge or none, all operators mutually commute. We choose to label the cubic operators $A$, $B$, $C$, and $D$ as shown in (b). Acting with a single Majorana operator $\gamma_{j}$ creates these four excitations.
• Figure 2: Dimension-1 Particle: Excitations (colored) may be created by acting with Wilson line operators. In (a), a straight Wilson line creates pairs of dimension-1 particles at the endpoints. The dimension-1 particle may hop freely in the direction of the Wilson line, by acting with Majorana bilinear terms. Remarkably, the dimension-1 particle cannot hop in any other direction without creating additional excitations. Introducing a "corner" in the Wilson line, as in (b), creates an additional topological excitation localized at the corner.
• Figure 3: Dimension-2 Anyon: Acting with two adjacent Wilson line operators $\hat{W} _1$ and $\hat{W}_2$ creates pairs of excitations at the endpoints of the same type ($AA$, $BB$, $CC$ or $DD$). These two-fracton excitations are free to move in a two-dimensional plane orthogonal to the shortest line segment connecting the pair of Wilson lines. Furthermore, in (b) we may detect a fracton (colored blue) by braiding a dimension-2 anyon around a closed loop enclosing the fracton. As the braiding operator, a pair of closed Wilson line operators $\hat{W}_{1}\hat{W}_{2}$, is equal to the product of the enclosed cube operators as shown above. Therefore, the braiding produces an overall minus sign if an odd number of fractons are enclosed.
• Figure 4: Membrane Operator & Fracton Excitations: Acting with a product of Majorana operators on a surface $\Sigma$ creates localized excitations at the corners of the boundary $\partial\Sigma$ as shown above.
• Figure 5: The Majorana plaquette model, as studied in vijay. Consider a honeycomb lattice with a single Majorana fermion on each lattice site. We define an operator $\mathcal{O}_{p}$ as the product of the six Majorana fermions on the vertices of a hexagonal plaquette $p$, as shown in (a). The colored plaquettes in (b) correspond to the three distinct bosonic excitations ($A, B,$ or $C$) that may each be created in pairs by acting with Wilson line operators.