Topological order in an exactly solvable 3D spin model

Sergey Bravyi, Bernhard Leemhuis, Barbara M. Terhal

Research output: Contribution to journalArticleScientificpeer-review

95 Citations (Scopus)


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 Ω(R2) 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.

Original languageEnglish
Pages (from-to)839-866
Number of pages28
JournalAnnals of Physics
Issue number4
Publication statusPublished - 2011
Externally publishedYes


  • Quantum error correcting code
  • Topological quantum order


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