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

This paper classifies one-dimensional disordered quantum systems with chiral symmetry by computing the partition function for a broad class of additive two-matrix models and deriving an asymptotic expansion for the average winding number that reveals universal leading features and a novel correction term distinct from the Ginibre case.

Original authors: Mathieu Yahiaoui, Mario Kieburg

Published 2026-10-23

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CC BY 4.0

Original authors: Mathieu Yahiaoui, Mario Kieburg

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the hidden world of quantum materials, some substances act as perfect insulators in their interior while conducting electricity effortlessly along their edges. This counterintuitive behavior, known as a topological phase, was first discovered in the 1980s when scientists observed that the electrical conductance of a two-dimensional electron gas forms precise, unchangeable steps. These steps remain fixed even when the material is riddled with impurities or defects, suggesting that the stability of these states comes from a deep, global property of the system rather than the specific arrangement of its atoms. To understand why this happens, physicists often look at the mathematical "shape" of the energy levels within the material. In one-dimensional systems that possess a specific type of symmetry called chiral symmetry, this shape can be described by a curve that winds around a central point. The number of times this curve loops around the center is a whole number called a winding number, which acts as a fingerprint for the material's topological state.

However, real-world materials are never perfectly ordered; they are always subject to disorder, such as random variations in atomic spacing or the presence of impurities. A major question in physics has been whether these topological fingerprints survive when the underlying system is randomized. If the disorder is strong enough, it might smear out the energy gaps that protect these states, potentially destroying the topological identity. Recent work has shown that even in disordered systems, a specific mathematical object—the determinant of a matrix representing the system's energy—still traces out a curve with a well-defined winding number. This suggests that the topological protection is robust, but calculating exactly what that number is in a disordered environment has remained a difficult challenge, especially when the disorder does not follow simple, standard statistical patterns.

Mathieu Yahiaoui and Mario Kieburg, researchers at the University of Melbourne, have now taken a significant step forward by solving this problem for a broad new class of disordered systems. Instead of limiting their study to the simplest, most symmetric type of random matrix, which has been the standard model for decades, they extended their analysis to a much wider family of mathematical ensembles known as Pólya ensembles. These ensembles allow for different types of "tails" in their probability distributions, meaning the random fluctuations in the system can be more extreme or behave differently than in the standard models. By developing a new mathematical framework to handle these complex, non-Gaussian distributions, the authors were able to calculate the average winding number for these systems as the size of the system grows very large.

Their findings reveal that the average winding number is composed of two distinct parts. The first part, which dominates the result, is a universal feature that depends only on the geometry of the system's parameters. This leading term behaves like a geometric phase, a concept familiar in quantum mechanics where a system acquires a specific shift in its state after traveling a closed loop. This part of the result is robust and appears regardless of the specific details of the disorder, confirming that the topological nature of the system is preserved even in the presence of randomness.

The second part of their discovery is a correction term that appears only when the disorder follows these more complex, non-standard statistical patterns. In the simpler, standard models used in previous research, this correction term vanishes completely. However, the authors found that in their broader class of models, this term persists and contributes a specific, calculable amount to the total winding number. This correction depends on how the random fluctuations decay at their extremes and how strongly the system repels values near zero. It acts as a subtle adjustment to the topological fingerprint, revealing that while the main topological feature is universal, the fine details of the disorder do leave a measurable imprint on the system's behavior.

The researchers demonstrated this by applying their new formulas to a specific, well-known family of random matrices called the Muttalib-Borodin ensemble. They showed that as the system size increases, the average winding number approaches a value determined by the leading geometric term plus this new, non-zero correction. This result is significant because it proves that the topological invariant is not just a binary switch that is either on or off, but a quantity that can be finely tuned by the specific statistical properties of the disorder. The work confirms that while the broad strokes of topological protection are universal, the precise value of the winding number in a disordered system carries a signature of the specific type of randomness present.

By moving beyond the standard models, this study provides a more complete picture of how topology survives in the messy reality of disordered materials. It shows that the mathematical tools used to describe these systems can be generalized to handle a much wider range of physical scenarios. The authors' ability to derive exact formulas for these complex cases suggests that the interplay between disorder and topology is richer and more nuanced than previously thought. Their work lays the groundwork for understanding other symmetry classes of quantum systems, where similar but more complex mathematical structures might govern the behavior of edge states. Ultimately, the paper establishes that even in a chaotic environment, the fundamental topological character of a quantum system remains intact, though its precise measurement is influenced by the unique statistical landscape of the disorder itself.

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

Problem Statement
The paper addresses the classification of topological phases in one-dimensional disordered quantum systems belonging to the chiral symmetry class AIII of the Altland-Zirnbauer classification. In such systems, the topological invariant is the winding number of the determinant of the off-diagonal block of a chiral Hamiltonian, $H(p)$ , as the quasi-momentum $p$ traverses the unit circle $S^1$ . While this invariant is well-defined for periodic Bloch Hamiltonians, disorder typically breaks periodicity. To recover a well-defined invariant, the authors model the system as a random matrix field $H(p)$ over $S^1$ , where the randomness encodes disorder. The primary objective is to compute the average winding number and its moments for a broad class of non-Hermitian random matrix ensembles, extending beyond the previously studied Ginibre Unitary Ensemble (GinUE).

Methodology
The authors employ Random Matrix Theory (RMT) to analyze the statistical properties of the winding number. The core methodology involves the following steps:

Model Construction: The system is modeled by an additive 2-matrix model $K(p) = a(p)K_1 + b(p)K_2$ , where $a(p)$ and $b(p)$ are parameter functions on $S^1$ , and $K_1, K_2$ are independent random matrices.
Ensemble Generalization: Instead of restricting $K_1$ and $K_2$ to the GinUE, the authors generalize the model to Polya ensembles on the complex general linear group $GL_N(\mathbb{C})$ . These ensembles are characterized by bi-unitary invariance and specific joint probability density functions (jpdfs) for eigenvalues and singular values, determined by a Polya weight function $\omega$ .
Reduction to 1-Matrix Model: A crucial step involves utilizing the closure properties of Polya ensembles under matrix multiplication and inversion. The ratio $K_1^{-1}K_2$ (or generalized ratio) is shown to be distributed as a Polya ensemble with a "self-inverse-convoluted" weight $\hat{\omega}$ . This reduction transforms the 2-matrix problem into an effective 1-matrix problem involving the product of the random matrix with a scalar parameter.
Partition Function Computation: The authors compute the generalized partition function $Z_m^{(N)}$ , which generates the $m$ -point correlation functions of the winding number density. By leveraging determinantal identities (specifically the Andreief identity) and complex analysis techniques (contour integration, residue calculus), they derive an exact closed-form expression for the partition function in terms of determinants of specific kernels.
Asymptotic Analysis: For the specific case of the Muttalib-Borodin ensemble (a subclass of Polya ensembles of Laguerre type), the authors perform a large- $N$ $N$ asymptotic expansion. This involves:
- Analyzing the auxiliary function $\Upsilon_N$ via Laplace-type integrals.
- Decomposing the summation over eigenvalues into bulk and boundary (tail) regimes.
- Applying the Euler-Maclaurin formula and saddle-point approximations to derive the asymptotic behavior of the average winding number.
- Carefully handling singularities where parameter functions $a(p)$ or $b(p)$ vanish, utilizing symmetrization and Cauchy principal value integrals.

Key Contributions and Results

Exact Partition Function (Theorem 2.8): The paper derives an exact expression for the partition function of the additive 2-matrix model where source matrices are drawn from any Polya ensemble of size $N$ . The result is expressed as a ratio of determinants involving a deformed kernel $\tilde{Q}_m^{(N)}[\omega]$ and a Cauchy-like kernel $Q_m$ . This generalizes previous results known only for the GinUE.
Asymptotic Expansion of Average Winding Number (Theorem 2.9): For the Muttalib-Borodin ensemble, the authors derive the large- $N$ $N$ asymptotic expansion of the average winding number $\mathbb{E}[\text{Wind}_N]$ $E [Wind_{N}]$ . The expansion takes the form:
$\mathbb{E}[\text{Wind}_N] = N \cdot \Phi + C + o(1)$
- Leading Term ( $N \cdot \Phi$ ): The leading term is proportional to $N$ and corresponds to an Aharonov-Anandan phase angle associated with a deformed vector field $\nu_\gamma$ . This term exhibits universal features, depending on the geometry of the parameter functions $a(p)$ and $b(p)$ .
- Subleading Term ( $C$ ): A novel contribution is identified in the $O(1)$ correction term. Unlike the Gaussian (GinUE) case where this term vanishes, for non-Gaussian Polya ensembles, this term is non-zero. It is interpreted as a weighted winding number of the curve $\kappa(p) = a(p)/b(p)$ around the origin. The weight depends on the local asymmetry between $|a(p)|$ and $|b(p)|$ and is controlled by the tail parameter $\gamma$ and the repulsion parameter $\delta$ of the ensemble.
Gaussian Random Fields (Proposition 2.14): The authors extend the analysis to general complex Ginibre random fields (not limited to the additive 2-matrix structure) and show that the partition function and average winding number can be expressed in terms of the covariance kernel of the field, recovering the additive 2-matrix results as a special case.
Handling Singularities: The paper provides a rigorous treatment of the logarithmic derivatives of the determinant when $a(p)$ or $b(p)$ vanish, demonstrating that the singularities are integrable via the Cauchy principal value and that the symmetrization of the correlation function ensures convergence.

Significance and Claims
The paper claims to establish a unified framework for calculating topological invariants in disordered chiral systems beyond the Gaussian limit. Its primary significance lies in:

Universality vs. Model Dependence: It demonstrates that while the leading-order behavior of the average winding number is universal (geometric), the subleading corrections are sensitive to the specific statistical properties of the disorder (specifically the tail behavior and origin repulsion of the eigenvalue distribution).
New Correction Term: The identification of a non-vanishing $O(1)$ correction term for non-Gaussian ensembles is a key finding, distinguishing these systems from the previously studied GinUE case.
Analytical Framework: By extending the exact solvability of 2-matrix models to the broader class of Polya ensembles, the work provides a tractable tool for studying topological phases in systems with non-Gaussian disorder.

The authors note that while the mean winding number shows this layered asymptotic structure, the universality of the full winding number distribution (e.g., a Central Limit Theorem) remains an open question, though they expect the variance to depend on ensemble-specific details similar to the mean. They also acknowledge that extending these results to other symmetry classes (like CII and BDI) is challenging due to the lack of analogous Polya ensembles and the more complex Pfaffian structure of correlations in those classes.

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