Landau-Ginzburg Paradigm of Topological Phases

TL;DR

This paper demonstrates that topological phases, traditionally seen as beyond Landau-Ginzburg paradigms, can be described within this framework by reformulating string-net models as lattice gauge theories with matter, capturing phase transitions via Higgs-like mechanisms.

Contribution

The authors reformulate string-net models as lattice gauge theories with matter, enabling Landau-Ginzburg descriptions of topological phase transitions and phenomena.

Findings

Topological phases can be described by Landau-Ginzburg theory with modifications.

Anyon condensation corresponds to Higgs mechanism in gauge theories.

Exact formulation of topological phase transitions with order parameters.

Abstract

Topologically ordered matter phases have been regarded as beyond the Landau-Ginzburg symmetry breaking paradigm of matter phases. Recent studies of anyon condensation in topological phases, however, may fit topological phases back in the Landau-Ginzburg paradigm. To truly do so, we realized that the string-net model of topological phases is in fact an effective lattice gauge theory coupled with anyonic matter once two modifications are made: (1) We reinterpret anyons as matter fields coupled to lattice gauge fields, thus extending the HGW model to a genuine Hamiltonian lattice gauge theory. (2) By explicitly incorporating the internal degrees of freedom of anyons, we construct an enlarged Hilbert space that supports well-defined gauge transformations and covariant coupling, restoring the analogy with conventional lattice gauge field theory. In this modified string-net model, topological phase transitions induced by anyon condensation and their consequent phenomena, such as order parameter fields, coherent states, Goldstone modes, and gapping gauge degrees of freedom, can be formulated exactly as Landau's effective theory of the Higgs mechanism. To facilitate the understanding, we also compare anyon condensation to/with the Higgs boson condensation in the electroweak theory and the Cooper pair condensation.