Topological order in an exactly solvable 3D spin model

TL;DR

This work analyzes Chamon's exactly solvable 3D spin model, a generalization of the toric code that lives on an FCC-derived lattice with six-qubit stabilizers $S_u$ and Hamiltonian $H=-\sum_{u\in \Lambda_{odd}} S_u$. It shows that elementary excitations, monopoles, are corners of rectangular membranes and cannot be created by string-like operators, with creation requiring operators of weight $\Omega(R^2)$, while dipoles and quadrupoles arise as end-points of rigid and flexible strings, respectively. The ground space on a 3-torus encodes $k=4g$ logical qubits ($g=\gcd(p_x,p_y,p_z)$), and a subsystem encoding using closed rigid strings and membranes is constructed to manage logical and gauge degrees of freedom; closed strings and membranes are used to realize logical operators, with a complete topological-charge classification of excitations. The model highlights unique 3D topological features distinct from the 4D toric code, discusses potential glassy dynamics and thermal stability issues, and poses open questions about the optimal distance scaling $d_{\mathcal{G}}$ and practical fault-tolerance implementations in 3D topological codes.

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.

Figures (15)

• Figure 1: The generator $S_u$ and two possible ways for a pair of generators to have overlapping supports. Black dots indicate qubit locations. Generators $S_u$ are centered at sites indicated by open dots.
• Figure 2: (Color Online) Qubits (black dots) live on the FCC lattice $\Lambda_{even}$. The stabilizer generators (red) $S_u$ are centered on the open dots $u$ in $\Lambda_{odd}$. The double (blue) lines are examples of the six face-diagonals and the two dashed (green) lines are examples of the four body-diagonals.
• Figure 3: (a) The restriction of a generator $S_u$ onto a horizontal plane. (b) The correspondence between even sites lying in the chosen plane and links of the toric code lattice. (c) The change of basis $X\to Y$, $Y\to X$, $Z\to -Z$ applied to every horizontal link maps the in-plane generators $S_u$ to the plaquette ($X$-type) and star ($Y$-type) operators of the toric code. A string-like operator comprised only of Pauli $Z$'s can commute with both plaquette and star operators only if it follows the diagonal of the lattice (the pink shaded region). Strings that contain $X$ or $Y$ Paulis fail to commute with the generators $S_u$ located in the adjacent horizontal planes. Hence extending the toric code to 3D leads to rigidity of strings. Double circles indicate excitations created near the end-point of the string.
• Figure 4: Example of a rigid string $\gamma$ lying in the $[001]$-plane. The string operator $W(\gamma)$ acts by $Z$ on the qubits in the shaded region. Double circles near the end-points of the string indicate locations of excitations created by the string operator $W(\gamma)$. A pair of excitations located near each end-point of the string is called a dipole.
• Figure 5: (Color Online) Construction of a closed rigid string-net or tetrahedron operator. The six edges of the tetrahedron $T$ are rigid strings of the six possible types lying on the faces of the cube. The string-net operator $W(T)$ is the product of the six corresponding string operators. Note that $W(T)$ acts trivially at the qubits located at vertices of $T$ since the triple of strings incident to any vertex of $T$ cancel each other through the identity $XYZ\sim I$. The open dot with two incident blue lines indicates the location of excitations created by a pair of string operators. The two excitations cancel each other.

Theorems & Definitions (17)

• Theorem 1
• Theorem 2
• Definition 1
• Definition 2: Flexible bilayer strings
• Lemma 1
• Lemma 2
• proof
• Definition 3
• Lemma 3
• proof