Duality between $(2+1)d$ Quantum Critical Points

TL;DR

This review surveys the burgeoning web of dualities in $(2+1)d$ quantum field theories, with emphasis on conformal fixed points governing quantum criticality in condensed matter. It organizes boson, fermion, and mixed fermion–boson dualities into an interconnected web generated by $S$ and $T$ operations, and relates boundary theories to $(3+1)d$ bulk electromagnetic duality. The discussion covers lattice realizations, coupled-wire constructions, large-$N$ limits, and numerical evidence, and connects these ideas to key condensed-matter problems such as the half-filled Landau level, TI surface states, and deconfined quantum critical points. The work highlights how dualities reveal hidden symmetries, unify seemingly disparate descriptions, and yield testable predictions for critical exponents, topological orders, and emergent symmetries, thereby providing a powerful framework for understanding beyond-Landau quantum phase transitions and topological phenomena.

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.

Figures (6)

• Figure 1: Charge-monopole lattice at $\theta = n\pi$ with $n$ even.
• Figure 2: Charge-monopole lattice at $\theta = n\pi$ with $n$ odd.
• Figure 3: Two different basis choices for the charge-monopole lattice at $\theta = n\pi$ with $n$ even. The blue arrow points to what - in that basis - is the electric charge and the red arrow points to the corresponding magnetic charge. The basis to the left is the standard one while that to the right is obtained by a 90 degree rotation, i.e by an $S$-transformation.
• Figure 4: Two other basis choices for the charge-monopole lattice at $\theta = n\pi$ with $n$ even. Note that time reversal does not keep the basis vectors fixed. Rather the basis in the left figure is transformed to the one in the right figure and vice versa.
• Figure 5: The basic logic of emergent symmetries from dualities.