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

A quantum tricritical point is shown to exist 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 the emergence of a dimensionless scale, g = proportional to(xi)B root N, where N is the system size, B is a generic TRS-breaking field, and proportional to(xi), with proportional to (0) 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 universal, and independent of the nature of the breaking of the TRS. We show that the single-variable scaling function, f (w), w proportional to mN, 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 -> infinity. The obtained result sets a scale N >> 1(proportional to 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.