Transition from the Z2 spin liquid to antiferromagnetic order: spectrum on the torus

We study the finite-size spectrum of the quantum critical point between a $\mathbb{Z}_2$ spin liquid and a coplanar antiferromagnet on the torus. Due to the existence of nontrivial order on either side of this transition, this critical point cannot be described in a conventional Landau-Ginzburg framework. Instead it is described by a theory involving fractionalized degrees of freedom known as the $O(4)^*$ 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, along with a selection rule on the states, and with nontrivial boundary conditions corresponding to topological sectors in the spin liquid. The spectrum of the Wilson-Fisher fixed points is then calculated directly from the $\epsilon$- and large-$N$ expansions, which allows a reconstruction of the full spectrum of the $O(4)^*$ model. This spectrum is a unique characteristic of a fractionalized quantum critical point as well as a universal signature of the existence of a proximate $\mathbb{Z}_2$ topological phase which can be compared with numerical computations.