Universal finite size scaling around tricriticality between topologically ordered, SPT, 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, $f(w)$, $w\propto mN$, 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 $g\to \infty$. 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 $m=0$.

arXiv comment21 pages, 15 figures

References in corpus (31)

• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Unpaired Majorana fermions in quantum wires
• Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures
• Classification of topological insulators and superconductors in three spatial dimensions
• Classification of topological quantum matter with symmetries
• New directions in the pursuit of Majorana fermions in solid state systems
• Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures
• Search for Majorana fermions in superconductors
• Entanglement entropy and conformal field theory
• Symmetry protected topological orders and the group cohomology of their symmetry group
• Majorana Zero Modes and Topological Quantum Computation
• Topological phases of fermions in one dimension
• Introduction to topological superconductivity and Majorana fermions
• Symmetry Protected Topological phases of Quantum Matter
• The effects of interactions on the topological classification of free fermion systems
• Topological Phases of One-Dimensional Fermions: An Entanglement Point of View
• Detection of Symmetry Protected Topological Phases in 1D
• Exact ground states and topological order in interacting Kitaev/Majorana chains
• Entanglement entropy in quantum impurity systems and systems with boundaries
• Entanglement and boundary critical phenomena
• Topological Luttinger Liquids from Decorated Domain Walls
• Localized Majorana-like modes in a number conserving setting: An exactly solvable model
• The 1D Ising model and topological order in the Kitaev chain
• Topological states in a microscopic model of interacting fermions
• Majorana Fermions and Multiple Topological Phase Transition in Kitaev Ladder Topological Superconductors
• Universal Finite-Size Scaling around Topological Quantum Phase Transitions
• Phase Diagram and Entanglement of two interacting topological Kitaev chains
• Staggered Anisotropy Parameter Modification of the Anisotropic $t-J$ Model
• Entanglement entropy and boundary renormalization group flow - exact results in the Ising universality class
• Mean-field theory for Heisenberg zigzag ladder: Ground state energy and spontaneous symmetry breaking
• Topological degeneracy (Majorana zero-mode) and 1+1D fermionic topological order in a magnetic chain on superconductor via spontaneous Z2 symmetry breaking

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