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.
Highlights
Recent numerical studies [1, 2] of the spin S = 1/2 antiferromagnet on the triangular lattice have presented convincing evidence for a spin liquid ground state in the presence of a next-nearest neighbor exchange interaction (J2)
We postulate here that there is a universal spectrum at the critical point between the Z2 spin liquid and an ordered antiferromagnet which is described by the O(2N) critical theory in Eq (4) subject to the boundary condition in Eq (6) and the constraint in Eq (7)
Following the notation of Ref. 11, we will call this the O(2N)∗ critical theory, while the theory obeying the boundary condition in Eq (5) is the conventional O(2N) theory. It was previously pointed out [25] that the O(2N)∗ critical theory has a distinct entanglement entropy from the O(2N) theory; our results show that the distinction applies to the finite-size spectrum on a torus
Summary
A conventional LGW description of a transition from this magnetically ordered state to a paramagnetic state would begin with an effective action for the fluctuations of the vectors n1,2 This phase transition would drive the system into a trivial gapped paramagnetic state with a non-degenerate ground state, which cannot occur in a system with an odd number of half-integer spins per unit cell such as the triangular antiferromagnet [23]. We postulate here that there is a universal spectrum at the critical point between the Z2 spin liquid and an ordered antiferromagnet which is described by the O(2N) critical theory in Eq (4) subject to the boundary condition in Eq (6) and the constraint in Eq (7).
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