Preserving Topology while Breaking Chirality: From Chiral Orthogonal to Anti-symmetric Hermitian Ensemble

Abstract

We consider a parameter dependent ensemble of two real random matrices with Gaussian distribution. It describes the transition between the symmetry class of the chiral Gaussian orthogonal ensemble (Cartan class B$|$DI) and the ensemble of antisymmetric Hermitian random matrices (Cartan class B$|$D). It enjoys the special feature that, depending on the matrix dimension $N$, it has exactly $\nu =0$ $(1)$ zero-mode for $N$ even (odd), throughout the symmetry transition. This "topological protection" is reminiscent of properties of topological insulators. We show that our ensemble represents a Pfaffian point process which is typical for such transition ensembles. On a technical level, our results follow from the applicability of the Harish-Chandra integral over the orthogonal group. The matrix valued kernel determining all eigenvalue correlation functions is explicitly constructed in terms of skew-orthogonal polynomials, depending on the topological index $\nu =0,1$. These polynomials interpolate between Laguerre and even (odd) Hermite polynomials for $\nu =0$ $(1)$, in terms of which the two limiting symmetry classes can be solved. Numerical simulations illustrate our analytical results for the spectral density and an expansion for the distribution of the smallest eigenvalue at finite $N$.

arXiv comment48 pages, 6 figures; v2: 2 references added; v3: Version published in J. Stat. Mech

References in corpus (35)

• Topological Insulators
• Classification of topological quantum matter with symmetries
• Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures
• Periodic table for topological insulators and superconductors
• Non-Abelian statistics and topological quantum information processing in 1D wire networks
• Colloquium: Majorana Fermions in nuclear, particle and solid-state physics
• The space group classification of topological band insulators
• Topological classification of crystalline insulators through band structure combinatorics
• The spectrum of the QCD Dirac operator and chiral random matrix theory: the threefold way
• Random-matrix theory of Majorana fermions and topological superconductors
• Class D spectral peak in Majorana quantum wires
• Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges
• The spectrum of the Dirac operator near zero virtuality for $N_c=2$
• Distribution of the k-th smallest Dirac operator eigenvalue
• Majorana Fermions in Chiral Topological Ferromagnetic Nanowires
• Microscopic Spectrum of the Wilson Dirac Operator
• Spectrum of the Wilson Dirac Operator at Finite Lattice Spacings
• Superbosonization of invariant random matrix ensembles
• Distributions of Dirac Operator Eigenvalues
• Vicious walk with a wall, noncolliding meanders, and chiral and Bogoliubov-deGennes random matrices
• Quasiclassical theory of disordered multi-channel Majorana quantum wires
• Random Matrix Theory for the Hermitian Wilson Dirac Operator and the chGUE-GUE Transition
• Correlation Functions of Harish-Chandra Integrals over the Orthogonal and the Symplectic Groups
• Infinite systems of non-colliding generalized meanders and Riemann-Liouville differintegrals
• A method to calculate correlation functions for $\beta =1$ random matrices of odd size
• Asymptotics of skew orthogonal polynomials
• Completing the picture for the smallest eigenvalue of real Wishart matrices
• Dirac Spectrum of the Wilson Dirac Operator for QCD with Two Colors
• Shift of Symmetries of Naive Fermions in QCD-like Lattice Theories
• Symmetry Transition Preserving Chirality in QCD: A Versatile Random Matrix Model
• WKB-expansion of the HarishChandra-Itzykson-Zuber integral for arbitrary beta
• Individual Eigenvalue Distributions for the Wilson Dirac Operator
• Universality crossover between chiral random matrix ensembles and twisted SU(2) lattice Dirac spectra
• Universal Distribution of Would-be Topological Zero Modes in Coupled Chiral Systems
• Individual eigenvalue distributions of crossover chiral random matrices and low-energy constants of SU(2)$\times$U(1) lattice gauge theory

Cited by in corpus (1)

• Universal Broadening of Zero Modes: A General Framework and Identification