Constructing symmetric topological phases of bosons in three dimensions via fermionic projective construction and dyon condensation

Abstract

Recently, there is a considerable study on gapped symmetric phases of bosons that do not break any symmetry. Even without symmetry breaking, the bosons can still be in many exotic new states of matter, such as symmetry-protected trivial (SPT) phases which are short-range entangled and symmetry-enriched topological (SET) phases which are long-range entangled. It is well-known that non-interacting fermionic topological insulators are SPT states protected by time-reversal symmetry and U(1) fermion number conservation symmetry. In this paper, we construct three-dimensional exotic phases of bosons with time-reversal symmetry and boson number conservation U(1) symmetry by means of fermionic projective construction. We first construct an algebraic bosonic insulator which is a symmetric bosonic state with an emergent U(1) gapless gauge field. We then obtain many gapped bosonic states that do not break the time-reversal symmetry and boson number conservation via proper dyon condensations. We identify the constructed states by calculating the allowed electric and magnetic charges of their excitations, as well as the statistics and the symmetric transformation properties of those excitations. This allows us to show that our constructed states can be trivial SPT states (i.e. trivial Mott insulators of bosons with symmetry), non-trivial SPT states (i.e. bosonic topological insulators) and SET states (i.e. fractional bosonic topological insulators). In non-trivial SPT states, the elementary monopole (carrying zero electric charge but unit magnetic charge) and elementary dyon (carrying both unit electric charge and unit magnetic charge) are fermionic and bosonic, respectively. In SET states, intrinsic excitations may carry fractional charge.

Model Realization and Numerical Studies of a Three-Dimensional Bosonic Topological Insulator and Symmetry-Enriched Topological Phases

“…This measurement, predicted in both the fermionic [29] and bosonic [16][17][18][19] TIs, is called a Witten effect, and is the tendency of an external magnetic monopole in a TI to bind to charge. In our system with Uð1Þ × Uð1Þ symmetry, the Witten effect corresponds to monopoles of one of the symmetries acquiring one-half charge of the other symmetry.…”

“…One measurement that we can make is called the Witten effect [16][17][18][19]. This is the binding of one-half of a boson charge to external monopoles in the spin sector.…”

Symmetry enrichment in three-dimensional topological phases

“…Mathematically, this conclusion can be achieved from Eqs. (19,20) by simply setting N 1 = 2, N 2 = 1. Physically, it means that there is only one Z 2 GT which is described by the BF term with level-2: i 2 2π´b ∧ da.…”

“…Before imposing symmetry, there are already four topologically distinct GTs labeled by (q,q) = (0 mod 4, 0 mod 4), (0 mod 4, 2 mod 4), (2 mod 4, 0 mod 4), and (2 mod 4, 2 mod 4), which can be derived from Eqs. (19,20) by setting N 1 = N 2 = 2. Under this circumstances, Step-5 in Sec.…”

Topological paramagnetism in frustrated spin-1 Mott insulators

“…SPT phases in three dimensions have been accessed previously through confinement of emergent U(1) gauge fields [37]. However, in these previous studies the confinement was achieved in a highly nontrivial way involving the condensation of dyons (bound states of magnetic and electric charges).…”