Anyon Condensation: Coherent states, Symmetry Enriched Topological Phases, Goldstone Theorem, and Dynamical Rearrangement of Symmetry

TL;DR

This paper establishes the physical foundation of anyon condensation by constructing an effective Hamiltonian and Hilbert space, demonstrating the emergence of symmetry enriched topological phases through explicit coherent state constructions and generalized Goldstone theorem.

Contribution

It provides a concrete Hamiltonian framework for anyon condensation and generalizes the Goldstone theorem to topological phases, linking symmetry breaking to topological order.

Findings

Constructed the effective Hamiltonian and Hilbert space for anyon condensation.

Explicitly constructed the vacuum as coherent states of condensed anyons.

Proved the condensed phase is a symmetry enriched topological phase.

Abstract

Although the mathematics of anyon condensation in topological phases has been studied intensively in recent years, a proof of its physical existence is tantamount to constructing an effective Hamiltonian theory. In this paper, we concretely establish the physical foundation of anyon condensation by building the effective Hamiltonian and the Hilbert space, in which we explicitly construct the vacuum of the condensed phase as the coherent states that are the eigenstates of the creation operators that create the condensate anyons. Along with this construction, which is analogous to Laughlin's construction of wavefunctions of fractional quantum hall states, we generalize the Goldstone theorem in the usual spontaneous symmetry breaking paradigm to the case of anyon condensation. We then prove that the condensed phase is a symmetry enriched (protected) topological phase by directly constructing the corresponding symmetry transformations, which can be considered as a generalization of the Bogoliubov transformation.