Topological order in an exactly solvable 3D spin model

TL;DR

This paper introduces an exactly solvable 3D spin model with topological order, analyzing its excitations, logical operators, and ground state properties, revealing novel features like monopole excitations and rigid string structures.

Contribution

It presents a new 3D topologically ordered spin model with detailed analysis of excitations, logical operators, and ground state encoding, extending the understanding of 3D topological phases.

Findings

Elementary excitations are monopoles with creation requiring operators on at least R^2 qubits.

Dipole and quadrupole excitations can be described as endpoints of rigid strings.

The ground space encodes 4g qubits with logical operators represented by closed strings and membranes.

Abstract

We study a 3D generalization of the toric code model introduced recently by Chamon. This is an exactly solvable spin model with six-qubit nearest neighbor interactions on an FCC lattice whose ground space exhibits topological quantum order. The elementary excitations of this model which we call monopoles can be geometrically described as the corners of rectangular-shaped membranes. We prove that the creation of an isolated monopole separated from other monopoles by a distance R requires an operator acting on at least R^2 qubits. Composite particles that consist of two monopoles (dipoles) and four monopoles (quadrupoles) can be described as end-points of strings. The peculiar feature of the model is that dipole-type strings are rigid, that is, such strings must be aligned with face-diagonals of the lattice. For periodic boundary conditions the ground space can encode 4g qubits where g is the greatest common divisor of the lattice dimensions. We describe a complete set of logical operators acting on the encoded qubits in terms of closed strings and closed membranes.