Dynamical stability of the quantum Lifshitz theory in 2+1 dimensions

The roles of magnetic and electric perturbations in the quantum Lifshitz model in $2+1$ dimensions are examined in this paper. The quantum Lifshitz model is an effective field theory for quantum multicritical systems, which include generalized two-dimensional (2D) quantum dimer models in bipartite lattices and their generalizations. It describes a class of quantum phase transitions between ordered and topological phases in $2+1$ dimensions. Magnetic perturbations break the dimer conservation law. Electric excitations, the condensation of which leads to ordered phases, have been studied extensively both in the classical three-dimensional model and in the quantum 2D model. The role of magnetic vortex excitations, the condensation of which drives these systems into a ${\mathbb{Z}}_2$ topological phase, has been largely ignored. Recent numerical studies claim that the quantum theory has a peculiar feature: the dynamical exponent $z$ flows continuously and the quantum theory is hence unstable to magnetic vortices. To study the interplay of both excitations, we perform a perturbative renormalization group study to one-loop order and study the stability of the theory away from quantum multicriticality. This is done by generalizing the operator-product expansion to anisotropic models. It is found that the dynamical exponent does not appear to flow, in contrast to the classical Monte Carlo study. Possible reasons for this difference are discussed at length.

Figure 1

The amplitude of the wave function depends on the type of vertices that enter in each configuration. Vertices in the (a) six-vertex model with amplitudes (“fugacities”) $a$, $b$, and $c$, respectively, and (b) the additional vertex in the eight-vertex model with fugacity $d$. These vertices and their inverses with all arrows flipped form the six vertex and eight vertex models, respectively. The heights are displayed in quadrant. To do so, one arbitrarily picks a starting plaquette, and moving in a counterclockwise direction raises the height variable by 1 if an arrow going into the vertex is crossed or decreases the height variable by one if an arrow going out of the vertex is crossed. The configurations in (b) break the ice rule of the six-vertex model, and are vortices of the height field.

Figure 2

Phase diagram of the quantum eight-vertex model (from Ref. 21). Bold lines represent quantum critical lines. The effective field theory along the lines labeled “six-vertex model” is the quantum Lifshitz model. Here, $c$ and $d$ represent amplitudes of the wave function (see text). The quantum disordered phase is a ${\mathbb{Z}}_2$ topological phase and the point labeled “Kitaev” is the toric code. At the level of the wave function, the two critical lines are related by Kramers-Wannier duality. We will be studying the theory near the multicritical point $X$ into the full eight-vertex model along the critical line with $d^2,c^2\ne 0$.

Figure 3

The renormalization group flows in the full quantum eight-vertex model. The renormalization group flows are reminiscent of Kosterlitz-Thouless physics. Dashed lines are flow lines in the plane where $\tilde{\alpha }=0$, while the solid lines are in the plane with $\alpha =0$. There is a single fixed point at $(\delta ,\alpha ,\tilde{\alpha })=(0,0,0)$.