A duality web in 2+1 dimensions and condensed matter physics

Abstract

Building on earlier work in the high energy and condensed matter communities, we present a web of dualities in 2 + 1 dimensions that generalize the known particle/vortex duality.Some of the dualities relate theories of fermions to theories of bosons. Others relate different theories of fermions. For example, the long distance behavior of the 2 + 1-dimensional analog of QED with a single Dirac fermion (a theory known as U (1) 1 2 ) is identified with the O(2) Wilson-Fisher fixed point. The gauged version of that fixed point with a ChernSimons coupling at level one is identified as a free Dirac fermion. The latter theory also has a dual version as a fermion interacting with some gauge fields. Assuming some of these dualities, other dualities can be derived. Our analysis resolves a number of confusing issues in the literature including how time reversal is realized in these theories. It also has many applications in condensed matter physics like the theory of topological insulators (and their gapped boundary states) and the problem of electrons in the lowest Landau level at half filling. (Our techniques also clarify some points in the fractional Hall effect and its description using flux attachment.) In addition to presenting several consistency checks, we also present plausible (but not rigorous) derivations of the dualities and relate them to 3 + 1-dimensional S-duality.

A master bosonization duality

“…These dualities have been the subject of recent attention from a variety of viewpoints. For the special case N = k = N f = 1, these dualities are related to the surface states of time-reversal invariant topological insulators and the fractional quantum Hall effect at half filling [10][11][12][13], lead to a web of dualities [14][15][16][17], and can even be proven on the lattice [18]. They are crucial actors in mapping out the phase diagram of QCD 3 as well as some of its cousins.…”

“…In this way the U(k) N − Ns 2 theory becomes 15) which is the right side of the U/U duality (4.4). So the SU/U duality (1.4) implies the U/U duality (4.4) as promised.…”

“…The first is that this proposal reduces to Aharony's when N s = 0 or N f = 0. The second concerns recent works which use the basic dualities as "seed dualities" to generate new ones by gauging global symmetries on both sides of the seed [14][15][16][17]22]. In the context of quiver gauge theories, one of the results of [22] is that one can dualize node-by-node: given a quiver with a SU or U gauge group factor coupled to only bosons or fermions, one can generate a dual quiver by replacing a node and the matter attached to it with its dual according to (1.2) and (1.3).…”

Three-dimensional topological insulators and bosonization

“…Recently, several authors have suggested that flux attachment also holds for relativistic theories [74][75][76][77][78][79][80]. For example, a map exists between the Dirac fermion and complex scalar theories, both including self-interactions, with the boson coupled to the Chern-Simons field for changing statistics [79,80].…”

Scaling dimensions in QED3 from the ϵ-expansion

“…Namely, refs. [33][34][35] argued, based on 3d bosonization dualities [36][37][38][39], that for N f = 1 the SU(2) × U(1) symmetry is in fact enhanced to O(4) (this is related to the self-duality present in this theory [40]). Also, a recent lattice study [41] found no evidence for a symmetry-breaking condensate (for previous lattice studies see refs.…”