Correlation Length, Universality Classes, and Scaling Laws Associated with Topological Phase Transitions

Abstract

The correlation functions related to topological phase transitions in inversion-symmetric lattice models described by $2\times 2$ Dirac Hamiltonians are discussed. In one dimension, the correlation function measures the charge-polarization correlation between Wannier states at different positions, while in two dimensions it measures the itinerant-circulation correlation between Wannier states. The correlation function is nonzero in both the topologically trivial and nontrivial states, and allows to extract a correlation length that diverges at topological phase transitions. The correlation length and the curvature function that defines the topological invariants are shown to have universal critical exponents, allowing the notion of universality classes to be introduced. Particularly in two dimensions, the universality class is determined by the orbital symmetry of the Dirac model. The scaling laws that constrain the critical exponents are revealed, and are predicted to be satisfied even in interacting systems, as demonstrated in an interacting topological Kondo Insulator.

13 pages, 7 figures

References in corpus (55)

• Topological insulators and superconductors
• Quantum Spin Hall Effect in Graphene
• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Berry Phase Effects on Electronic Properties
• Maximally-localized generalized Wannier functions for composite energy bands
• Unpaired Majorana fermions in quantum wires
• Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices
• Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures
• Classification of topological insulators and superconductors in three spatial dimensions
• Maximally localized Wannier functions: Theory and applications
• Helical liquids and Majorana bound states in quantum wires
• Experimental realisation of the topological Haldane model
• Periodic table for topological insulators and superconductors
• Chern semi-metal and Quantized Anomalous Hall Effect in HgCr2Se4
• Classification of stable three-dimensional Dirac semimetals with nontrivial topology
• Topological Kondo Insulators
• The Quantum Spin Hall Effect: Theory and Experiment
• Surface States of the Topological Insulator Bi_{1-x}Sb_x
• Majorana fermions in chains of magnetic atoms on a superconductor
• Multi-Weyl Topological Semimetals Stabilized by Point Group Symmetry
• Massive Dirac fermions and spin physics in an ultrathin film of topological insulator
• Berry phase correction to electron density of states in solids
• Ab initio calculation of the anomalous Hall conductivity by Wannier interpolation
• Topological Superconductivity and Majorana Fermions in RKKY Systems
• Topological superconducting phase in helical Shiba chains
• Finite size effects of helical edge states in HgTe/CdTe quantum wells
• Orbital magnetization in periodic insulators
• Majorana fermions emerging from magnetic nanoparticles on a superconductor without spin-orbit coupling
• A new type of Weyl semimetal with quadratic double Weyl fermions in SrSi2
• Interplay between classical magnetic moments and superconductivity in quantum one-dimensional conductors: toward a self-sustained topological Majorana phase
• Anomalous Finite Size Effects on Surface States in the Topological Insulator Bi${}_2$Se${}_3$
• Self-Organized Topological State with Majorana Fermions
• Orbital magnetization in crystalline solids: Multi-band insulators, Chern insulators, and metals
• Quantum Theory of Orbital Magnetization and its Generalization to Interacting Systems
• An Aharonov-Bohm interferometer for determining Bloch band topology
• Dichroic f-sum rule and the orbital magnetization of crystals
• Interferometric approach to measuring band topology in 2D optical lattices
• Helical order in one-dimensional magnetic atom chains and possible emergence of Majorana bound states
• Emergence of a Chern-insulating state from a semi-Dirac dispersion
• Thermoelectric transport in double-Weyl semimetals
• Correlated double-Weyl semimetals with Coulomb interactions: possible applications to HgCr2Se4 and SrSi2
• Phenomenology of a semi-Dirac semi-Weyl semi-metal
• Correlation effects in double-Weyl semimetals
• Majorana Edge States in Superconductor/Noncollinear Magnet Interfaces
• Topological invariants, surface states, and interaction-driven phase transitions in correlated Kondo insulators with cubic symmetry
• Flat Majorana bands in 2-d lattices with inhomogeneous magnetic fields: topology and stability
• Scaling Theory of Topological Phase Transitions
• Tuning topological superconductivity in helical Shiba chains by supercurrent
• Dynamical polarization and plasmons in a two-dimensional system with merging Dirac points
• Collective modes in multi-Weyl semimetals
• Scaling Theory of ${\mathbb{Z}}_2$ Topological Invariants
• Gap closing and universal phase diagrams in topological insulators
• Charge transport in gapless chiral electron systems with arbitrary band dispersion

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• A supervised learning algorithm for interacting topological insulators based on local curvature
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• Universal Hall conductance scaling in non-Hermitian Chern insulators
• Detection of topological phases by quasi-local operators
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• Band geometry from position-momentum duality at topological band crossings
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• Opacity of graphene independent of light frequency and polarization due to the topological charge of the Dirac points
• Unconventional quantum criticality in a non-Hermitian extended Kitaev chain
• Reentrant topological phases and entanglement scalings in moiré-modulated extended Su-Schrieffer-Heeger Model
• Kibble-Zurek behavior in a topological phase transition with a quadratic band crossing
• Precise determination of critical points of topological phase transitions via shift current in two-dimensional inversion asymmetric insulators
• Kibble-Zurek mechanism and errors of gapped quantum phases
• Topological invariant of velocity field in quantum systems
• Electron density and compressibility in the Kitaev model with a spatially modulated Phase in the superconducting pairing
• Scaling of Quantum Geometry Near the Non-Hermitian Topological Phase Transitions