A Line of Critical Points in 2+1 Dimensions: Quantum Critical Loop Gases and Non-Abelian Gauge Theory

TL;DR

This paper constructs and analyzes a family of lattice models with exact ground states, revealing a critical line related to non-Abelian gauge theories and loop gases, with implications for quantum criticality in 2+1 dimensions.

Contribution

It introduces a new family of exactly solvable lattice models linked to non-Abelian gauge theories and critical loop gases, expanding understanding of quantum critical lines.

Findings

Models are critical for $d\, extless=\\,\sqrt{2}$.

Special values relate to doubled level-$k$ SU(2) Chern-Simons theory.

One-loop $eta$-function vanishes, indicating a critical line.

Abstract

We (1) construct a one-parameter family of lattice models of interacting spins; (2) obtain their exact ground states; (3) derive a statistical-mechanical analogy which relates their ground states to O(n) loop gases; (4) show that the models are critical for $d\leq \sqrt{2}$, where $d$ parametrizes the models; (5) note that for the special values $d=2\cos(\pi/(k+2))$, they are related to doubled level-$k$ SU(2) Chern-Simons theory; (6) conjecture that they are in the universality class of a non-relativistic SU(2) gauge theory; and (7) show that its one-loop $\beta$-function vanishes for all values of the coupling constant, implying that it is also on a critical line.