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

TL;DR

The paper studies a line of quantum critical points in 2+1 dimensions via a Kitaev-inspired honeycomb spin model Hd-iso, whose ground-state manifold is exactly known and maps to an $O(n)$ loop model with $n=d^2$. The authors show that the ground state is a $d$-weighted superposition of multi-loop configurations, and that the system is critical for $1<d<\sqrt{2}$ with gapless, quadratic low-energy excitations; they also propose a non-relativistic SU(2) gauge theory without an $E^2$ term as an effective field theory, for which the one-loop beta function vanishes ($\frac{dg}{d\ell}=0$) and the dynamical exponent remains $z=2$, suggesting a second fixed line controlling the infrared behavior. The two viewpoints converge on a picture of a topological-critical phase described by fluctuating unoriented loops and a non-relativistic non-Abelian gauge theory, with nonlocal Wilson-loop correlators displaying power-law scaling and local correlations remaining short-ranged. The work opens avenues for understanding quantum critical loop gases and their connections to non-Abelian gauge theories, with potential implications for topological quantum computation and the design of robust quantum phases.

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 $d=2\cos(π/(k+2))$, 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 $β$-function vanishes for all values of the coupling constant, implying that it is also on a critical line.

Figures (1)

• Figure 1: The configuration space of the system has the shape of a dumbbell with two regions $X$, $Y$ which are not directly connected by the Hamiltonian. Consequently, a trial wavefunction which has value $-\Psi_0$ on $Y$ and $\Psi_0$ on $X$ has vanishing energy in the thermodynamic limit.