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We describe the finite-size spectrum in the vicinity of the quantum criticalpoint between a $\mathbb{Z}_2$ spin liquid and a coplanar antiferromagnet onthe torus. We obtain the universal evolution of all low-lying states in anantiferromagnet with global SU(2) spin rotation symmetry, as it moves from the4-fold topological degeneracy in a gapped $\mathbb{Z}_2$ spin liquid to theAnderson "tower-of-states" in the ordered antiferromagnet. Due to the existenceof nontrivial order on either side of this transition, this critical pointcannot be described in a conventional Landau-Ginzburg-Wilson framework. Insteadit is described by a theory involving fractionalized degrees of freedom knownas the O$(4)^\ast$ model, whose spectrum is altered in a significant way by itsproximity to a topologically ordered phase. We compute the spectrum by relatingit 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 boundaryconditions corresponding to topological sectors in the spin liquid. Thespectrum 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 thevicinity of a fractionalized quantum critical point, as well as a universalsignature of the existence of proximate $\mathbb{Z}_2$ topological andantiferromagnetically-ordered phases, and can be compared with numericalcomputations on quantum antiferromagnets on two dimensional lattices.

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