Emergent geometric description for a topological phase transition in the Kitaev superconductor model

Resorting to Wilsonian renormalization group (RG) transformations, we propose an emergent geometric description for a topological phase transition in the Kitaev superconductor model. An effective field theory consists of an emergent bulk action with an extra dimension, an ultraviolet (UV) boundary condition for an initial value of a coupling function, and an infrared (IR) effective action with a fully renormalized coupling function. The bulk action describes the evolution of the coupling function along the direction of the extra dimension, where the extra dimension is identified with an RG scale and the resulting equation of motion is nothing but a $\beta$ function. In particular, the IR effective field theory turns out to be consistent with a Callan-Symanzik equation which takes into account both the bulk and IR boundary contributions. This derived Callan-Symanzik equation gives rise to a metric structure. Based on this emergent metric tensor, we uncover the equivalence of the entanglement entropy between the emergent geometric description and the quantum field theory in the vicinity of the quantum critical point.

Figure 1

Emergent Ricci curvature given by Eqs. (31) and (32) for the quantum critical point ($\lambda =2.0$), the topologically trivial ($\lambda =2.05$), and nontrivial ($\lambda =1.95$) superconducting phases. We emphasize that the Ricci curvature diverges at $z=z_c$ in the topological superconducting phase, which may be identified with a horizon. The emergence of such a horizon in a dense phase is consistent with a recent study [23], where the existence of the horizon is a fingerprint of a quantum phase transition.

Figure 2

Comparison between the Ryu-Takayanagi formula based on the emergent metric [Eqs. (31) and (32)] and the entanglement entropy based on the lattice model [Eq. (2)]. The Ryu-Takayanagi formula describes the thick red line and the field-theory entanglement entropy is represented by the blue dots. Here, we have $\mu =\frac{\lambda +1}{2}$. The total number of lattice sites is $L=400$. See Refs. [38, 39] for how to calculate the entanglement entropy from the lattice model directly. It is surprising for both quantities to match almost perfectly not only near the quantum critical point but also far away from it in the topological superconducting phase. We note that a constant value of ${\log }_{2}2$ is added to the Ryu-Takayanagi formula, where our renormalization group procedure does not take into account effects of boundary Majorana-fermion zero modes. This comparison leads us to suspect that the Ryu-Takayanagi formula based on the emergent metric [Eqs. (31) and (32)] may be identified with an entanglement entropy although we did not prove that this emergent metric is governed by the Einstein equation.