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Winding Number Statistics for Chiral Random Matrices: Universal Correlations and Statistical Moments in the Unitary Case

Nico Hahn, Mario Kieburg, Omri Gat, Thomas Guhr

TL;DR

This paper addresses the statistics of the winding number for one-dimensional parametric Hamiltonians in the chiral unitary class AIII using a two-matrix Gaussian random matrix model. The authors center the winding number via a local gauge transformation and unfold correlations on the scale $p_j=p_0+\psi_j/\sqrt{N}$, showing universal dependence on $|\Delta(q)|=\sqrt{\partial_1\partial_2 S(q,q)}$. In the large-$N$ limit, the $k$-point winding-density correlators factorize into parallel blocks, yielding a Gaussian distribution for $W$ with variance $\propto \sqrt{N}$ and vanishing odd cumulants. They provide explicit expressions for the unfolded two-point function, show universality with a finite set of parameters (I$_2$ controls the variance), and give an example where $I_2=2/\sqrt{\pi}$. Numerical simulations corroborate the analytic results, and the work highlights universality that should extend to other chiral classes, while noting limitations and avenues for future extension to Pfaffian cases.

Abstract

The winding number is the topological invariant that classifies chiral symmetric Hamiltonians with one-dimensional parametric dependence. In this work we complete our study of the winding number statistics in a random matrix model belonging to the chiral unitary class AIII. We show that in the limit of large matrix dimensions the winding number distribution becomes Gaussian. Our results include expressions for the statistical moments of the winding number and for the k-point correlation function of the winding number density.

Winding Number Statistics for Chiral Random Matrices: Universal Correlations and Statistical Moments in the Unitary Case

TL;DR

This paper addresses the statistics of the winding number for one-dimensional parametric Hamiltonians in the chiral unitary class AIII using a two-matrix Gaussian random matrix model. The authors center the winding number via a local gauge transformation and unfold correlations on the scale , showing universal dependence on . In the large- limit, the -point winding-density correlators factorize into parallel blocks, yielding a Gaussian distribution for with variance and vanishing odd cumulants. They provide explicit expressions for the unfolded two-point function, show universality with a finite set of parameters (I controls the variance), and give an example where . Numerical simulations corroborate the analytic results, and the work highlights universality that should extend to other chiral classes, while noting limitations and avenues for future extension to Pfaffian cases.

Abstract

The winding number is the topological invariant that classifies chiral symmetric Hamiltonians with one-dimensional parametric dependence. In this work we complete our study of the winding number statistics in a random matrix model belonging to the chiral unitary class AIII. We show that in the limit of large matrix dimensions the winding number distribution becomes Gaussian. Our results include expressions for the statistical moments of the winding number and for the k-point correlation function of the winding number density.

Paper Structure

This paper contains 11 sections, 112 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Concepts
  3. Posing the Problem and Previous Results
  4. Centering the Winding Number
  5. Scales
  6. The "Lagrangian" and its Expansion
  7. Results
  8. Conclusions
  9. Derivations
  10. Derivation of the Unfolded Point Correlation Function
  11. Decomposition of the Correlation Function and Statistical Moments

Figures (1)

  • Figure 1: Winding number distribution for $10^4$ realizations of a random matrix model \ref{['model']} with $N = 1500$ and parameters $a_1 = 0.92 + 0.82 i,\, a_2 = 0.91 - 0.77 i,\, b_1 = 0.41 - 0.95 i,\, b_2 = -0.84 - 0.70 i$. The left ordinate shows the frequency of each value and the right ordinate the value of the associated probability density function, plotted in red. The inset shows the mean $\mu_1$, the variance $\mu_2$, the skewness $\mu_3$ and the kurtosis $\mu_4$, rounded to two decimals.