Statistical Topology -- Distribution and Density Correlations of Winding Numbers in Chiral Systems
arXiv:2301.06550 · doi:10.3390/e25020383
Abstract
Statistical Topology emerged since topological aspects continue to gain importance in many areas of physics. It is most desirable to study topological invariants and their statistics in schematic models that facilitate the identification of universalities. Here, the statistics of winding numbers and of winding number densities are addressed. An introduction is given for readers with little background knowledge. Results that my collaborators and I obtained in two recent works on proper random matrix models are reviewed, avoiding a technically detailed discussion. A special focus is on the mapping of topological problems to spectral ones as well as on the first glimpse on universality.
arXiv commentReview article, dedicated to Giulio Casati on the occasion of his 80th birthday
References in corpus (31)
- Anyons in an exactly solved model and beyond
- Unpaired Majorana fermions in quantum wires
- Topological Photonics
- 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
- Random Matrix Theories in Quantum Physics: Common Concepts
- Periodic table for topological insulators and superconductors
- Random matrix theory and spectral sum rules for the Dirac operator in QCD
- The spectrum of the QCD Dirac operator and chiral random matrix theory: the threefold way
- Spectral Density of the QCD Dirac Operator near Zero Virtuality
- Random Matrix Theory and Chiral Symmetry in QCD
- Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics
- Topological characterization of chiral models through their long time dynamics
- Symmetry classes of disordered fermions
- A random matrix model for chiral symmetry breaking
- The chiral phase transition in a random matrix model with molecular correlations
- An elementary rigorous proof of bulk-boundary correspondence in the generalized Su-Schrieffer-Heeger model
- Random-Matrix Theory of Parametric Correlations in the Spectra of Disordered Metals and Chaotic Billiards
- Particle-Hole Symmetries in Condensed Matter
- Derivation of determinantal structures for random matrix ensembles in a new way
- Derivation of an eigenvalue probability density function relating to the Poincare disk
- Role of Majorana Fermions in high-harmonic generation from Kitaev chain
- A new approach to derive Pfaffian structures for random matrix ensembles
- The Bulk-Edge Correspondence in Three Simple Cases
- A real quaternion spherical ensemble of random matrices
- Stochastic field theory for a Dirac particle propagating in gauge field disorder
- Correlations of quantum curvature and variance of Chern numbers
- Winding Number Statistics of a Parametric Chiral Unitary Random Matrix Ensemble
- Winding Number Statistics for Chiral Random Matrices: Averaging Ratios of Determinants with Parametric Dependence