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

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 fourfold 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 $\mathrm{O}{(4)}^{*}$ 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 $\mathrm{O}(4)$ 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 $\mathrm{O}(2N$) model is calculated directly at $N=\ensuremath{\infty}$, which then allows a reconstruction of the full spectrum of the $\mathrm{O}{(2N)}^{*}$ 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.

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