String-net condensation: A physical mechanism for topological phases

2004/04/26 by Michael Levin, Michael A. Levin, Xiao-Gang Wen · 1 voice · 6 citations
Physics and Astronomy · #Physics of Superconductivity and Magnetism #Quantum and electron transport phenomena #Quantum many-body systems #cond-mat.mes-hall #cond-mat.str-el #hep-th

paper · pdf · doi:10.1103/physrevb.71.045110

arxiv published 2004/04/26 · arxiv updated 2004/04/27 · openalex publication_date 2005/01/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

We show that quantum systems of extended objects naturally give rise to a large class of exotic phases---namely topological phases. These phases occur when extended objects, called ``string-nets,'' become highly fluctuating and condense. We construct a large class of exactly soluble 2D spin Hamiltonians whose ground states are string-net condensed. Each ground state corresponds to a different parity invariant topological phase. The models reveal the mathematical framework underlying topological phases: tensor category theory. One of the Hamiltonians---a spin-1∕2 system on the honeycomb lattice---is a simple theoretical realization of a universal fault tolerant quantum computer. The higher dimensional case also yields an interesting result: we find that 3D string-net condensation naturally gives rise to both emergent gauge bosons and emergent fermions. Thus, string-net condensation provides a mechanism for unifying gauge bosons and fermions in 3 and higher dimensions.

Citations

• Field theories with « Superconductor » solutions
• Equivalence of the resonating-valence-bond and fractional quantum Hall states
• An introduction to lattice gauge theory and spin systems
• Fault-tolerant quantum computation by anyons
• Hamiltonian formulation of Wilson's lattice gauge theories
• Classical and quantum conformal field theory
• Theory of Superconductivity

Cited by

• The commuting local Hamiltonian on locally-expanding graphs is in NP
• Emergent Symmetry and Phase Transitions on the Domain Wall of ℤ₂ Topological Orders
• Fibre bundle framework for unitary quantum fault tolerance
• Generalized topological state-sum constructions and their universality
• Models for Gapped Boundaries and Domain Walls
• Fusion rules from entanglement

Discussions

• An important open mathematical problem in the theoretical foundation of topological order is to prove that the universal properties (braiding and fusion) of point-like excitations in any gapped phase [stackexchange, 3 points]