Boson condensation in topologically ordered quantum liquids

Abstract

Boson condensation in topological quantum field theories (TQFT) has been previously investigated through the formalism of Frobenius algebras and the use of vertex lifting coefficients. While general, this formalism is physically opaque and computationally arduous: analyses of TQFT condensation are practically performed on a case by case basis and for very simple theories only, mostly not using the Frobenius algebra formalism. In this paper we provide a new way of treating boson condensation that is computationally efficient. With a minimal set of physical assumptions, such as commutativity of lifting and the definition of confined particles, we can prove a number of theorems linking Boson condensation in TQFT with chiral algebra extensions, and with the factorization of completely positive matrices over Z+. We present numerically efficient ways of obtaining a condensed theory fusion algebra and S matrices; and we then use our formalism to prove several theorems for the S and T matrices of simple current condensation and of theories which upon condensation result in a low number of confined particles. We also show that our formalism easily reproduces results existent in the mathematical literature such as the noncondensability of 5 and 10 layers of the Fibonacci TQFT.

Surfaces and slabs of fractional topological insulator heterostructures

“…This anyon condensation [42][43][44] procedure effectively glues the two FTI slabs together along the TR symmetric surfaces (see Fig. 1).…”

Fermion condensation and gapped domain walls in topological orders

“…The self-bosons that condense in B transform into the trivial boson 1 ∈ B ′ , when they cross the GWD. The connection between GDWs in bTOs and self-boson condensation has been thoroughly studied recently [16,17,[24][25][26][27][28][29][30]. It has been found that GDWs between bosonic topological orders can be classified by anyon condensation [16,17].…”

Gapped spin liquid with Z2 topological order for the kagome Heisenberg model

“…For a TNS that describes a topologically ordered wave function in 2D, the gauge symmetry of local tensors is necessary [29]; i.e., each local tensor should be invariant under local symmetry transformations on virtual legs. For a gauge-symmetric TNS, we are able to compute modular matrices [20][21][22] which can detect topological phase transitions [30][31][32].…”