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Winding Number Statistics for Chiral Random Matrices: Averaging Ratios of Parametric Determinants in the Orthogonal Case

Nico Hahn, Mario Kieburg, Omri Gat, Thomas Guhr

Abstract

We extend our recent study of winding number density statistics in Gaussian random matrix ensembles of the chiral unitary (AIII) and chiral symplectic (CII) classes. Here, we consider the chiral orthogonal (BDI) case which is the mathematically most demanding one. The key observation is that we can map the topological problem on a spectral one, rendering the toolbox of random matrix theory applicable. In particular, we employ a technique that exploits supersymmetry structures without reformulating the problem in superspace.

Winding Number Statistics for Chiral Random Matrices: Averaging Ratios of Parametric Determinants in the Orthogonal Case

Abstract

We extend our recent study of winding number density statistics in Gaussian random matrix ensembles of the chiral unitary (AIII) and chiral symplectic (CII) classes. Here, we consider the chiral orthogonal (BDI) case which is the mathematically most demanding one. The key observation is that we can map the topological problem on a spectral one, rendering the toolbox of random matrix theory applicable. In particular, we employ a technique that exploits supersymmetry structures without reformulating the problem in superspace.
Paper Structure (11 sections, 105 equations, 1 figure)

This paper contains 11 sections, 105 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Posing the Problem
  3. Result
  4. Derivation
  5. The Kernel
  6. The Kernel
  7. The Kernel
  8. Conclusions
  9. The joint eigenvalue probability density of the real induced spherical ensemble
  10. Characteristic polynomials of the real induced spherical ensemble
  11. Alternative expression of

Figures (1)

  • Figure 1: A realization of a BDI Hamiltonian with $K(p)=\cos(p)K_1+i\sin(p)K_2$ and some fixed $4\times4$ real matrices $K_1$ and $K_2$. The top left plot shows the real eigenvalues of $H(p)$, the top right one shows the generically complex eigenvalues of $K(p)$, and the bottom left plot depicts the determinant $\det K(p)$. The bottom right plot shows the determinant of a different random matrix field $K(p) = (a_1 + a_2 e^{ip} + a_3 e^{2ip}+ a_4 e^{3ip}) K_1 + (b_1 + b_2 e^{ip} + b_3 e^{2ip}+ b_4 e^{3ip}) K_2$ with fixed $4\times4$ real matrices $K_1$ and $K_2$ and real coefficients $a_j$ and $b_j$. All plots show the parametric dependence in $p\in[0,2\pi)$ where we have employed the step size $2\pi/100$ and a B-Spline to obtain the curves. In the parametric plots the starting points $p=0$ are marked by black points and the directions are marked by a color gradient resp. arrows.