Transition from the ${\mathbb{Z}}_2$ 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 $\mathrm{O}{\left(4\right)}^{*}$ 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 $\mathrm{O}\left(4\right)$ 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 $N=\infty$, which then allows a reconstruction of the full spectrum of the $\mathrm{O}{\left(2N\right)}^{*}$ 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.

Figure 1

The geometry of the spatial torus, where the position is given by complex coordinates $w=x+iy$. We associate all points related by a lattice vector $n{\omega }_1+m{\omega }_2$ for $n,m\in \mathbb{Z}$, where the complex numbers ${\omega }_1$ and ${\omega }_2$ are the periods of the torus. We define the modular parameter $\tau ={\omega }_2/{\omega }_1$ and the length scale $L\equiv |{\omega }_1|=\sqrt{\mathcal{A}/\mathrm{Im}\left(\tau \right)}$.

Figure 2

The evolution of the spectrum $LE$ for the O(4) model as a function of the tuning parameter $L^{1/\nu }(s-s_c)$ on the square torus, $\tau =i$. Note that $\nu =1$ at leading order in $1/N$. The energy levels are defined so that $E=0$ at $s=s_c$ and $L=\infty$. We label the states by their behavior in the ordered region, distinguishing between the tower, the Goldstone modes, and the singlet states. Our choice of states is not not exhaustive, but they highlight the main features of each region.

Figure 3

(Left) The dimensionless ground-state energy density $L{E}_0/{\tau }_2=L^3{E}_0/\mathcal{A}$ for the O(4) model on an infinite cylinder with circumference $L$. This energy is defined so that $E/\mathcal{A}=0$ at $s=s_c$ and $L=\infty$. (Right) The energy gap above the ground state for the $\mathrm{O}(2N$) model at $N=\infty$ on the infinite cylinder as a function of $s-s_c$. For energies higher than the gap, the spectrum is continuous.

Figure 4

The evolution of the spectrum $LE$ for the $\mathrm{O}{\left(4\right)}^{*}$ model as a function of the tuning parameter $L^{1/\nu }(s-s_c)$ on the square torus, $\tau =i$. Note that $\nu =1$ to leading order in $1/N$. The energy levels are defined so that $E=0$ at $s=s_c$ and $L=\infty$. We label the states by their behavior in the ordered region, distinguishing between the tower, the Goldstone modes, and the singlet states. We also distinguish the four “ground states” of the different sectors $(a_1,a_2)$ according to Table 2, though the (A,P) and (P,A) sectors are degenerate for the square geometry. These states become degenerate in the topological phase, while they represent ${\mathbb{Z}}_2$ vortices in the magnetic phase. Our choice of states is not exhaustive, but highlights the main features of the proximate phases.

Figure 5

(Left) The splitting between the energy densities of the periodic and antiperiodic sectors for the $\text{O}{\left(4\right)}^{*}$ model on the infinite cylinder. The energy levels are defined so that $E_0/\mathcal{A}=0$ at $s=s_c$ and $L=\infty$. (Right) The energy gap above the ground state for the $\text{O}{\left(4\right)}^{*}$ model on the cylinder as a function of $s-s_c$. The spectrum above this gap is continuous.

Figure 6

The evolution of the spectrum $LE$ for the $\mathrm{O}{\left(4\right)}^{*}$ model as a function of the tuning parameter $L^{1/\nu }(s-s_c)$ on the triangular torus, $\tau =e^{i\pi /3}$. Note that $\nu =1$ to leading order in $1/N$. The energy levels are defined so that $E=0$ at $s=s_c$ and $L=\infty$. We label the states by their behavior in the ordered region, distinguishing between the tower, the Goldstone modes, and the singlet states. Note that the three sectors (A,P), (P,A), and (A,A) are degenerate in this geometry. Our choice of states is not exhaustive, but highlights the main features of the proximate phases.