String-Net Condensation

Definition

String-net condensation (Levin–Wen, 2005) is a microscopic mechanism that produces topologically ordered phases and, with them, emergent gauge fields and their charges. The ground state is a coherent superposition — a “condensate” — of fluctuating networks of strings, whose amplitudes are fixed so that the state is invariant under local deformations.

A string-net model is specified by combinatorial data, presented operationally without category theory:

• String types and fusion rules $N_{ab}^c$ (which strings can merge at a vertex),
• $F$-symbols $F_d^{abc}$ (the amplitude for local recoupling moves), constrained by the pentagon identity.

The Hamiltonian enforces the branching rules (vertex terms) and $F$-move invariance (plaquette terms), and is a sum of commuting projectors like the toric-code. The simplest case — two string types with trivial $F$ — is the toric code (the ${\mathbb{Z}}_2$ string-net). Richer data (Fibonacci, doubled Ising) yields non-abelian anyons.

Role in Research

String-net condensation is Wen’s answer to “where do gauge fields come from”: a deconfined gauge theory is a condensate of its own electric strings, so the gauge field and its charges emerge together from a bosonic spin system. In the condensed-matter-connections line and the generalized-symmetries course it closes the loop opened by Kogut–Susskind — the electric flux strings of lattice-gauge-theory are the strings that condense — and provides the general framework in which the toric-code is just the smallest example.

• It realizes all non-chiral (doubled) topological orders from local, exactly solvable Hamiltonians.
• The $F$-symbol data is the same fusion-category input that governs the fusion of anyons and of condensation-defects.

Relations

• toric-code — the ${\mathbb{Z}}_2$ string-net; the simplest condensate
• topological-order — string-net condensation is the general mechanism producing it
• higher-form-symmetries — the condensed strings are the charged objects of the emergent gauge symmetry
• lattice-gauge-theory — Kogut–Susskind electric strings are what condense

Papers

Levin–Wen, “String-net condensation,” Phys. Rev. B 71 (2005) 045110 [cond-mat/0404617]; Wen, Quantum Field Theory of Many-Body Systems (2004). See the course bibliography.

Notes

• “Condensation” here is condensation of extended objects (strings), the higher-form analogue of ordinary Bose condensation of particles — the same generalization that underlies condensation-defects and the julia-toulouse-mechanism.
• String-net models produce only non-chiral topological orders (the doubled theories); genuinely chiral phases like the Laughlin state need extra input (a chiral edge), which is why the framework naturally yields “doubled Ising” rather than Ising.