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 Z2 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 Z2 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)∗ 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 = ∞, which then allows a reconstruction of the full spectrum of the 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 Z2 topological and antiferromagnetically-ordered phases, and can be compared with numerical computations on quantum antiferromagnets on two dimensional lattices.