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 creating 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.
Abstract In a physical system undergoing a continuous quantum phase transition, spontaneous symmetry breaking occurs when certain symmetries of the Hamiltonian fail to be preserved in the ground state. In the traditional Landau theory, a symmetry group can break down to any subgroup. However, this no longer holds across a continuous phase transition driven by anyon condensation in symmetry enriched topological orders (SETOs). For a SETO described by aG-crossed braided extension
C⊆CG× C⊆CG× , we show that physical considerations require that a connected étale algebraA∈
C C admit aG-equivariant algebra structure for symmetry to be preserved under condensation ofA. Given any categorical actionG→EqBr(
C C ) such thatg(A) ≅Afor allg∈G, we show there is a short exact sequence whose splittings correspond toG-equivariant algebra structures. The non-splitting of this sequence forces spontaneous symmetry breaking under condensation ofA, while inequivalent splittings of the sequence correspond to different SETOs resulting from the anyon-condensation transition. Furthermore, we show that if symmetry is preserved, there is a canonically associated SETO of
CAloc CAloc , and gauging this symmetry commutes with anyon condensation.
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.
Although anyon condensation is a standard mechanism for relating topological orders, anyon condensation in symmetry-enriched topological (SET) phases is more intricate because the condensate must also be compatible with the global symmetry. We study symmetry-preserving anyon condensation in SET phases described by the enlarged Hu-Geer-Wu (HGW) string-net model with multifusion-category input data. We show that a
G-preserving condensation is characterized by a compatible grading of the input multifusion category, and that this grading constructs the multifusion-category input of the child SET phase. To make this construction concrete, we consider the case where the relevant data come from a finite group extension
E of
G by
N: an
E-graded fusion category induces a
G-SET input by passing to the quotient symmetry
G, and the resulting input naturally carries a compatible
N-grading that implements the further condensation inside the SET phase while preserving
G. We illustrate the construction using three quantum-double examples: the trivial extension
Z2×Z2, the non-Abelian semidirect product
S3, and the nontrivial central extension
Z4. The
Z4 example further shows that symmetry fractionalized anyons do not obstruct symmetry-preserving condensation once the condensate is treated as a physical coherent state.
In topological phases in
2+1 dimensions, anyons fall into representations of quantum group symmetries. As proposed in our work [Hung and Wan, Int. J. Mod. Phys. B 28, 1450172 (2014)], the physics of a symmetry enriched phase can be extracted by the mathematics of (hidden) quantum group symmetry breaking of a ``parent phase.'' This offers a unified framework and classification of the symmetry enriched (topological) phases, including symmetry protected trivial phases as well. In this paper, we extend our investigation to the case where the ``parent'' phases are non-Abelian topological phases. We show explicitly how one can obtain the topological data and symmetry transformations of the symmetry enriched phases from that of the ``parent'' non-Abelian phase. Two examples are computed: (1) the $\text{Ising}\ifmmode\times\else\texttimes\fi{}\overline{\text{Ising}}$ phase breaks into the
Z2 toric code with
Z2 global symmetry; (2) the
SU(2)8 phase breaks into the chiral Fibonacci $\ifmmode\times\else\texttimes\fi{}$ Fibonacci phase with a
Z2 symmetry, a first non-Abelian example of symmetry enriched topological phase beyond the gauge-theory construction.
We construct in the
K matrix formalism concrete examples of symmetry-enriched topological phases, namely intrinsically topological phases with global symmetries. We focus on the Abelian and nonchiral topological phases and demonstrate by our examples how the interplay between the global symmetry and the fusion algebra of the anyons of a topologically ordered system determines the existence of gapless edge modes protected by the symmetry and that a (quasi)group structure can be defined among these phases. Our examples include phases that display charge fractionalization and more exotic nonlocal anyon exchange under global symmetry that correspond to general group extensions of the global symmetry group.
Abstract A theory with anyon permuting symmetry is unchanged under permutation of the names of the particle types. The toric code is an example of a theory with such a symmetry. The toric code, and all theories with such symmetries can be viewed as arising as a condensation of anyons from some more complicated parent theory. Given that such a symmetry exists, we consider defects of this symmetry which generalize the mathematical structure of the TQFT to a so-called G-crossed extension. To clarify how this works we use a particularly tranparent example of the toric code where the symmetry of particle types is reduced to a lattice symmetry, and the defects correspond to dislocations in the lattice. It can be shown that these defects harbor Majorana zero modes. We discuss the general structure of a parent theory condensing via a G-crossed theory to a theory with permutation symmetry. Finally we mention theories with both on-site symmetries and anyons, which are known as symmetry enriched topological phases.