Topological invariance is a powerful concept in different branches of physics as they are particularly robust under perturbations. We generalize the ideas of computing the statistics of winding numbers for a specific parametric model of the chiral Gaussian unitary ensemble to other chiral random matrix ensembles. In particular, we address the two chiral symmetry classes, unitary (AIII) and symplectic (CII), and we analytically compute ensemble averages for ratios of determinants with parametric dependence. To this end, we employ a technique that exhibits reminiscent supersymmetric structures, while we never carry out any map to superspace.
The idea of classifying Hamilton operators that reveal spectral gaps through topological lenses has been very successful in physical systems as those classes are very robust with respect to perturbations. This robustness has been theoretically and experimentally verified in various systems (see, e.g., Refs. 1–5). One specific topological index is the winding number for chiral operators. It is indeed a winding number in the classical sense when considering the spectral flow of the complex eigenvalues of the off-diagonal block of the chiral Hamiltonian in the chiral representation with respect to the momentum/wavevector in the Brillouin zone. Due to periodicity and continuity of the eigenvalues as functions of the momentum and the condition of a spectral gap, the eigenvalues will move around the origin in closed contours.
Physically, a nonzero winding number yields the number of localized modes at the boundaries and, thus, indicates topologically nontrivial systems.6–8 If disorder comes into play, the winding number can become random and a statistical analysis is called for. We refer to the reader to Ref. 9 for further discussion of the physics aspects. Here, we consider simple schematic models of chiral systems with a parametric dependence. We are guided by the long-standing experience that random matrix theory is often capable of modeling universal statistical properties.10,11 It is worthwhile mentioning that the winding number statistics is not related to the parametric spectral correlations introduced and investigated in Refs. 12 and 13. Although the random matrix models are apart from chirality, very similar, the statistical observables are different.
In a previous study,9 three of the authors studied chiral unitary symmetry and evaluated the winding number distribution as well as the correlators of the winding number density. Here, we investigate two of the five chiral symmetry classes, which are among the ten symmetry classes known as tenfold way.14–17 More precisely, we work with the chiral unitary (AIII) and symplectic (CII) symmetry. Our objectives are ensemble averages for ratios of determinants with parametric dependence. This is related to averages for ratios of characteristic polynomials in the context of classical random matrix theory. Apart from the crucial importance of the latter in the supersymmetry method,18 they are also interesting quantities in their own right for mathematical physics (see the by far not exhaustive list of Refs. 19–30).
To carry out our study, we employ and extend a method put forward some years ago by two of the present authors.21,22 Jokingly, but not deceptively, it has been coined “supersymmetry without supersymmetry” because it uncovers supersymmetric structures deeply rooted in the ensemble averages without actually mapping the integrals to be considered to superspace. This method proceeds as follows: First, we map the average for ratios of determinants with parametric dependence to averages for ratios of characteristic polynomials over another random matrix ensemble, referred to as spherical.31–33 Second, we reformulate the integrals by introducing superspace Jacobians, also known as Berezinians, which are in the present case mixtures of Vandermonde and Cauchy determinants.34 This facilitates a decomposition and direct formal computation of all integrals, leading to determinants or Pfaffians. Third, we exploit the results of Refs. 21 and 22 where the kernels of these determinants and Pfaffians have been identified as averages for ratios or products of only two determinants with parametric dependence. Finally, we evaluate these simplified averages over the spherical ensemble with the help of orthogonal and skew-orthogonal polynomials. Here, we show only the first and the last step and refer to Refs. 21 and 22 for the intermediate steps with general validity.
II. POSING THE PROBLEM
We employ and extend the conventions and notations of Ref. 9. Importantly, all matrix elements in the symplectic case CII are 2 × 2 quaternions, effectively doubling the dimension of H and .
The general question addressed in recent studies2–5 is about the stability of the spectral properties of Hamiltonians under perturbations. In the present case, this is a question about the topology of subsets of chiral operators, which can be quantified by the eigenvalues of the block matrix K(p), which are also parametrically depending on p. In Ref. 8, it has been proposed that assuming a gaped Hamiltonian, also the eigenvalues of K(p) exhibit a spectral gap to the origin. However, they are generically complex such that trajectories of the eigenvalues with respect to describe paths around the origin without crossing it due to the spectral gap. This is not only true for class AIII but also for the other chiral symmetry classes.
In Figs. 1 and 2, we illustrate the spectral flow for the matrix K(p) = cos(p)K1 + sin(p)K2 with generic complex (AIII) and for the matrix K(p) = cos(p)K1 + i sin(p)K2 with generic real quaternion (CII). In these figures, we illustrate the spectral flow of the eight eigenvalues of H(p), the four complex eigenvalues of K(p), and the determinant det K(p), which will play a crucial role when defining the winding number. Two important observations can be made, which hold generically true. The order of the eigenvalues of H(p) remains the same for all p when level repulsion governs the spectral statistics, which implies that each eigenvalue of H(p) is a 2π periodic function. In contrast, the eigenvalue spectrum of K(p) may experience a permutation, meaning when running once from p = 0 to p = 2π, a chosen eigenvalue can become another one. Thence, the eigenvalues of K(p) might have a different period than 2π. Therefore, the eigenvalues of K(p) are not suitable for classifying Hamiltonians. The determinant det K(p) is more suitable as this quantity must be 2π periodic. For the specific choice of the parametric dependence in Figs. 1 and 2, we have K(p + π) = −K(p), which restricts the amount of times det K(p) winds around the origin to be an even resp. odd number for even resp. odd matrix dimensions. These symmetries seen in the spectral flows are spurious and may not exist, in general.
Nevertheless, there is also independent interest in ensemble averages for ratios of characteristic polynomials19–30 in classical random matrix theory. For the Gaussian orthogonal, unitary, and symplectic ensemble, a direct connection between averages corresponding to k = l = 1 and the kernels of the k-point correlation functions was found in Ref. 39, generalizing some implicit observation40 for the unitary case in a supersymmetry context. For classical random matrix theory, the decomposition of ensemble averages for ratios of characteristic polynomials in the case of arbitrary k and l into ensemble averages for small k and l with k + l = 2 was derived in Ref. 20, employing a discrete approximation method related to representation theory. In Refs. 21 and 22, two of the present authors presented a very direct solution of this type of problem. They extended a method put forward in Ref. 34 by establishing a connection with supersymmetry without mapping on superspace. More precisely, Jacobians or Berezinians for the radial coordinates on certain symmetric superspaces were identified in the integrals, considerably facilitating the calculations. Here, we exploit the results of Refs. 21 and 22 to explicitly compute the functions (14).
In Secs. IV A and IV B, we first analyze the symmetries of the partition function (14) for the symmetry classes AIII and CII. Those symmetries become handy when simplifying the computations. Furthermore, we trace the ensemble average over the two independent Ginibre matrices back to the spherical ensembles that have been studied in Refs. 31 and 33. Using results from Refs. 21 and 22, we make use of determinantal and Pfaffian structures that reduce the problem of averaging over a ratio of 2k characteristic polynomials to averages of only two characteristic polynomials. In combination with the techniques of orthogonal and skew-orthogonal polynomials as well as some complex analysis tools, we find the results summarized in Sec. III.
A. Unitary case (AIII)
An important remark about the integrability of the partition function is in order. We certainly make use of the fact that a simple pole like 1/(κ(qj) + z) is integrable in two dimensions, such as the complex plane. However, we need to assume that all κ(qj) are pairwise distinct. Despite this, it is rather remarkable that the final result can be nonetheless analytically continued to these singular points without any problems.
The SU(2) invariance is actually also reflected in the symmetry of the eigenvalue spectrum of the complex spherical ensemble. In Ref. 31, it was pointed out that the complex spectrum is uniformly distributed on a two-dimensional sphere after a stereographic projection. It is the adjoint representation of SU(2), which is the special orthogonal group SO(3) that highlights the uniform distribution as it is the invariance group of a two-dimensional sphere.
B. Symplectic case (CII)
1. The kernel
2. The kernel
3. The kernel
We studied statistical aspects of the winding number, which is a fundamental topological invariant for chiral Hamilton operators. To do so, we set up schematic models involving two matrices with chiral unitary (AIII) and symplectic (CII) symmetry and one-dimensional parametric dependence. In particular, ensemble averages for ratios of determinants with parametric dependence were computed and related to the k-point correlators of the winding number densities. We mapped this problem to averages for ratios of characteristic polynomials for the respective spherical ensembles and employed techniques from orthogonal and skew-orthogonal polynomial theory. We verified our analytical results carefully with numerical calculations. We are certain that similar techniques may help to unravel the technically more involved chiral orthogonal symmetry class (BDI). One problem that needs to be addressed in this class is the splitting of the eigenvalues into real and complex conjugate pairs. The k-point correlation functions of the corresponding spherical ensemble have been already computed in Refs. 31 and 33.
In a previous study,9 we also addressed the important issue of universality, suggesting for the chiral unitary case that the two-point correlator of the winding number density and the winding number distribution are universal on proper scales when taking the limit of infinite matrix dimension. Universality is the crucial feature making random matrix theory so powerful (see Refs. 10 and 11). Consequently, universality is also a crucial issue in the new context of statistics for winding numbers and other topological quantities. At least two questions become relevant: First, which probability densities of the random matrices are compatible with universal results and, second, which realizations of the parametric dependence are admissible? Thorough investigation is beyond the scope of the present contribution. This includes the rather involved evaluation of all k-point correlators for the winding number density by calculating the proper derivatives of the formulas we obtained here. In addition, the large N-limit in all possible double scaling limits has to be performed. In a future work, we want to address this in combination with universality studies.
Related to the analyzing universality is the following observation. Our method to explicitly calculate the ensemble averages would also work for other joint probability density functions of the eigenvalues, provided the underlying symmetries are the same. This is a considerable advantage when tackling the problem of universality. In the “true” supersymmetry method that actually employs superspace, non-Gaussian probability densities for the random matrices can be treated, too.44–47 Nevertheless, the resulting formulas are less explicit. It is tempting to speculate that studies along the lines just sketched might help to improve these results for the “true” supersymmetry method.
We thank Boris Gutkin for fruitful discussions. This work was funded by the German-Israeli Foundation within the project Statistical Topology of Complex Quantum Systems (Grant No. GIF I-1499-303.7/2019) (N.H., O.G., and T.G.). Furthermore, M.K. acknowledges support by the Australian Research Council via Discovery Project Grant No. DP210102887.
Conflict of Interest
The authors have no conflicts to disclose.
Nico Hahn: Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Mario Kieburg: Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Omri Gat: Conceptualization (equal); Funding acquisition (equal). Thomas Guhr: Conceptualization (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).
Data sharing is not applicable to this article as no new data were created or analyzed in this study.