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Average Winding Number for Determinantal Curves associated with 2-Matrix Models in the Class AIII

Mathieu Yahiaoui, Mario Kieburg

TL;DR

This work studies the winding number of the determinantal curve det(K(p)) for one-dimensional disordered quantum systems in symmetry class AIII, using non-Hermitian random-matrix models on the unit circle. It generalizes the partition-function framework from the Ginibre ensemble to a broad class of additive 2-matrix Polya ensembles on GL_N(C), obtaining an exact finite-N expression Z^{(N)}_{m} = det(˜Q^{(N)}_{m}[ω])/det(Q_m) and enabling large-N analysis. In the Muttalib–Borodin Laguerre case, the authors derive a precise asymptotic expansion for the mean winding number: E[Wind_N] = (1/(2π i)) ∮ ( u_γ(p)†ν_γ'(p))/||ν_γ(p)||^2 dp · N + (γ−1−2δ)/2 · (1/(2π i)) ∮ (κ'(p)/κ(p)) · (|a(p)|^{2γ}−|b(p)|^{2γ})/(|a(p)|^{2γ}+|b(p)|^{2γ}) dp + o(1). The leading term corresponds to a geometric Aharonov–Anandan phase, while the subleading correction encodes ensemble-dependent tail information and reduces to zero in the Gaussian limit γ=1; the results illuminate universal versus model-specific aspects of topological invariants in disordered quantum systems and point to broader future directions, including CLTs for winding-number fluctuations and extensions to other symmetry classes. The methodology combines Pólya ensemble theory, Mellin transform techniques, and determinant-based integrals to connect random-matrix structure with topological winding on S^1.

Abstract

To classify one-dimensional disordered quantum systems with chiral symmetry, we analyse the winding number of the determinant of a parametrized non-Hermitian random matrix field over the unit circle modelling the off-diagonal block of a disordered chiral Hamiltonian. The associated partition function is computed explicitly for a broad class of additive two-matrix models extending beyond the Ginibre Unitary Ensemble. In the large-dimension limit, we derive an asymptotic expansion of the average winding number whose leading term exhibits universal features, up to the tail behaviour of the underlying random matrix ensemble, and identify a new correction term absent in the previously studied Ginibre case.

Average Winding Number for Determinantal Curves associated with 2-Matrix Models in the Class AIII

TL;DR

This work studies the winding number of the determinantal curve det(K(p)) for one-dimensional disordered quantum systems in symmetry class AIII, using non-Hermitian random-matrix models on the unit circle. It generalizes the partition-function framework from the Ginibre ensemble to a broad class of additive 2-matrix Polya ensembles on GL_N(C), obtaining an exact finite-N expression Z^{(N)}_{m} = det(˜Q^{(N)}_{m}[ω])/det(Q_m) and enabling large-N analysis. In the Muttalib–Borodin Laguerre case, the authors derive a precise asymptotic expansion for the mean winding number: E[Wind_N] = (1/(2π i)) ∮ ( u_γ(p)†ν_γ'(p))/||ν_γ(p)||^2 dp · N + (γ−1−2δ)/2 · (1/(2π i)) ∮ (κ'(p)/κ(p)) · (|a(p)|^{2γ}−|b(p)|^{2γ})/(|a(p)|^{2γ}+|b(p)|^{2γ}) dp + o(1). The leading term corresponds to a geometric Aharonov–Anandan phase, while the subleading correction encodes ensemble-dependent tail information and reduces to zero in the Gaussian limit γ=1; the results illuminate universal versus model-specific aspects of topological invariants in disordered quantum systems and point to broader future directions, including CLTs for winding-number fluctuations and extensions to other symmetry classes. The methodology combines Pólya ensemble theory, Mellin transform techniques, and determinant-based integrals to connect random-matrix structure with topological winding on S^1.

Abstract

To classify one-dimensional disordered quantum systems with chiral symmetry, we analyse the winding number of the determinant of a parametrized non-Hermitian random matrix field over the unit circle modelling the off-diagonal block of a disordered chiral Hamiltonian. The associated partition function is computed explicitly for a broad class of additive two-matrix models extending beyond the Ginibre Unitary Ensemble. In the large-dimension limit, we derive an asymptotic expansion of the average winding number whose leading term exhibits universal features, up to the tail behaviour of the underlying random matrix ensemble, and identify a new correction term absent in the previously studied Ginibre case.

Paper Structure

This paper contains 17 sections, 9 theorems, 206 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Winding number of a determinantal curve
  4. Additive 2-matrix models
  5. Pólya ensembles of multiplicative type on $\mathrm{GL}_{N}(\mathbb{C})$
  6. Main results
  7. The Ginibre random field
  8. Partition function and proof of \ref{['theorem1']}
  9. Asymptotic expansion of the average winding number and proof of \ref{['theorem2']}
  10. Asymptotic behaviour of the tails
  11. Asymptotic expansion of the denominator
  12. Asymptotic expansion of the integral at the edge
  13. Asymptotic expansion of the sum at the edge
  14. Discussion for small parameter amplitudes
  15. Conclusion
  16. ...and 2 more sections

Key Result

Proposition 2.3

We have In particular, for the generalized ratio of $X$ and $Y$ we have $X^{-1}Y\sim\mathrm{P\acute{o}l}_{N}\left [ \check{\omega}_{1}\circledast \omega_{2} \right ]$ and for every complex number $z$ where both $\mathcal{M}\left [ \omega_{1} \right ](z)$ and $\mathcal{M}\left [ \omega_{2} \right ](N+1-z)$ e

Figures (1)

  • Figure 1: On the left: eigenvalue flow of $K(p)=(\frac{1}{2i}(p-p^{-1})+p)K_{1}+(3-\frac{1}{10}(p+p^{-1})+p^{-2})K_{2}$ where the two source random matrices $K_{1}$ and $K_{2}$ are independently drawn from the $\mathrm{GinUE}(5)$. On the right: its associated determinantal curve. The black dots indicate the $5$ eigenvalues of $K(0)$ and $\det(K(0))$ while the red arrows mark the points where $\mathrm{Arg}(p)=\pi$. The winding number around the origin of the determinantal curve equals $+1$. The eigenvalue flow is not $2\pi$-periodic: as $p$ winds once around $\mathrm{S}^{1}$, one eigenvalue traces a closed orbit whereas the remaining four form a $4$-cycle. As a result, the flow is $8\pi$-periodic. Plots generated with Mathematica

Theorems & Definitions (23)

  • Remark 2.2
  • Proposition 2.3: see Eq. (3.10), Rem. 3.5 in kieburgProductsRandomMatrices2019
  • Example 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Remark 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Remark 2.10
  • ...and 13 more