Universal finite-size scaling around tricriticality between topologically ordered, symmetry-protected topological, and trivial phases

Abstract

A quantum tricritical point is shown to exists in coupled time-reversal symmetry (TRS) broken Majorana chains. The tricriticality separates topologically ordered, symmetry protected topological (SPT), and trivial phases of the system. Here we demonstrate that the breaking of the TRS manifests itself in an emergence of a new dimensionless scale, $g = \alpha(\xi) B \sqrt{N}$, where $N$ is the system size, $B$ is a generic TRS breaking field, and $\alpha(\xi)$, with $\alpha(0)\equiv 1$, is a model-dependent function of the localization length, $\xi$, of boundary Majorana zero modes at the tricriticality. This scale determines the scaling of the finite size corrections around the tricriticality, which are shown to be {\it universal}, and independent of the nature of the breaking of the TRS. We show that the single variable scaling function, $f(w)$, $w\propto m N$, where $m$ is the excitation gap, that defines finite-size corrections to the ground state energy of the system around topological phase transition at $B=0$, becomes double-scaling, $f=f(w,g)$, at finite $B$. We realize TRS breaking through three different methods with completely different lattice details and find the same universal behavior of $f(w,g)$. In the critical regime, $m=0$, the function $f(0,g)$ is nonmonotonic, and reproduces the Ising conformal field theory scaling only in limits $g=0$ and $g\rightarrow \infty$. The obtained result sets a scale of $N \gg 1/(\alpha B)^2$ for the system to reach the thermodynamic limit in the presence of the TRS breaking. We derive the effective low-energy theory describing the tricriticality and analytically find the asymptotic behavior of the finite-size scaling function. Our results show that the boundary entropy around the tricriticality is also a universal function of $g$ at $m=0$.