Transition from the $\mathbb{Z}_2$ spin liquid to antiferromagnetic order: spectrum on the torus

TL;DR

This paper analyzes the finite-size spectrum near a quantum critical point between a $ ext{Z}_2$ spin liquid and an antiferromagnet, revealing universal features that distinguish fractionalized quantum criticality from conventional transitions.

Contribution

It introduces a theoretical framework relating the spectrum of the fractionalized O(4)^ ext{*} model to the Wilson-Fisher fixed point, providing a universal signature of proximate topological and magnetic phases.

Findings

Spectrum evolves from topological degeneracy to tower-of-states

Spectrum computed at large N, revealing universal features

Provides a basis for numerical comparison in quantum antiferromagnets

Abstract

We describe the finite-size spectrum in the vicinity of the quantum critical point between a $\mathbb{Z}_2$ spin liquid and a coplanar antiferromagnet on the torus. We obtain the universal evolution of all low-lying states in an antiferromagnet with global SU(2) spin rotation symmetry, as it moves from the 4-fold topological degeneracy in a gapped $\mathbb{Z}_2$ spin liquid to the Anderson "tower-of-states" in the ordered antiferromagnet. Due to the existence of nontrivial order on either side of this transition, this critical point cannot be described in a conventional Landau-Ginzburg-Wilson framework. Instead it is described by a theory involving fractionalized degrees of freedom known as the O$(4)^\ast$ model, whose spectrum is altered in a significant way by its proximity to a topologically ordered phase. We compute the spectrum by relating it to the spectrum of the O$(4)$ Wilson-Fisher fixed point on the torus, modified with a selection rule on the states, and with nontrivial boundary conditions corresponding to topological sectors in the spin liquid. The spectrum of the critical O($2N$) model is calculated directly at $N=\infty$, which then allows a reconstruction of the full spectrum of the O($2N)^\ast$ model at leading order in 1/N. This spectrum is a unique characteristic of the vicinity of a fractionalized quantum critical point, as well as a universal signature of the existence of proximate $\mathbb{Z}_2$ topological and antiferromagnetically-ordered phases, and can be compared with numerical computations on quantum antiferromagnets on two dimensional lattices.