Analytic Approach to Calculating Vison Gaps in Kitaev spin liquids and odd-frequency superconductors

We present an analytic approach to calculating the vison gap in Kitaev and Yao Lee spin liquids, in which the energy for flipping a $Z_2$ bond can be computed in terms of a scattering phase shift. Our result reproduces the known numerical result E = 0.2672 J for the Kitaev spin liquid[1,2], and is readily generalized to infinitely separated visons, allowing us to demonstrate the development of a topological state in the toric code. Using this method, we examine the vison energy in the CPT model[2], where a Kondo coupling to a 3D Yao-Lee spin liquid drives a spontaneous pairing instability into an odd-frequency triplet superconductor S=1/2 spinor order. The vison energy remains finite even at strong coupling, where fractionalized superconductivity is replaced by a conventional Kondo insulator embedded in a gapless orbital spin liquid.