Duality between (2+1)d quantum critical points

Abstract

Duality refers to two equivalent descriptions of the same theory from different points of view. Recently there has been tremendous progress in formulating and understanding possible dualities of quantum many body theories in 2 + 1-spacetime dimensions. Of particular interest are dualities that describe conformally invariant quantum field theories in (2 + 1)d. These arise as descriptions of quantum critical points in condensed matter physics. The appreciation of the possible dual descriptions of such theories has greatly enhanced our understanding of some challenging questions about such quantum critical points. Perhaps surprisingly the same dualities also underlie recent progress in our understanding of other problems such as the half-filled Landau level and correlated surface states of topological insulators. Here we provide a pedagogical review of these recent developments from a point of view geared toward condensed matter physics.

Notes on 8 Majorana Fermions

“…In this case, the other theories in the triality orbit consist of eight Majorana fermions coupled to a Z 2 × Z 2 chiral gauge field. These gauge fields, in turn, couple to the background spin structure through the Arf invariant, a topological invariant which plays a crucial role in other 2d dualities [10][11][12][13][14][15]. We describe this in Section 3.…”

“…In this case, a new "quantum" Z 2 global symmetry emerges in the new theory allowing one to repeat the procedure. For simple theories, this leads to a duality web in d = 1 + 1 dimensions relating, for example, bosonization with Kramers-Wannier duality [10][11][12][13]15]. Aspects of this orbifolding procedure in supersymmetric dualities were discussed, for example, in [19,20].…”

Quantization of a self-dual conformal theory in (2 + 1) dimensions

“…In (2+1) dimensions, it is well-known that non-relativistic particles can change their statistics by coupling to a Chern-Simons gauge field. Recently, this mechanism was argued to hold for relativistic theories as well, leading to several conjectures that fit into a "web of dualities" [8]. For instance, the fermion-boson duality reads [19]:…”

“…• The boson-fermion correspondence, i.e. bosonization in (2 + 1) dimensions, is part of the web of duality relations that have been extensively analyzed in the recent years [7] [8]. The loop model provides a neat example of a massless theory that is covariant under duality transformations, corresponding to SL(2, Z) maps of the complex coupling τ = f + ig.…”

On duality between Cosserat elasticity and fractons

“…Recently there was a resurgence of interest in various dualities in a quantum field theory [1][2][3].…”