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.

## I. INTRODUCTION

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

*β*= 2) and CII (chiral quaternion Hermitian,

*β*= 4), respectively, in the tenfold way.

^{14,16,17,35}Those Hamiltonians satisfy a chiral symmetry,

*β*= 1), can be dealt in the very same way though the joint probability density of the eigenvalues needed in our computations will be more involved. Thence, we deferred this discussion to a future publication. The index

*β*is the Dyson index indicating the real dimension of the chosen number field.

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 $C$.

*K*has the dimension

*N*×

*N*for AIII and 2

*N*× 2

*N*for CII, and

*K*

^{†}is its (Hermitian) adjoint. Hence, the Hamiltonian is complex Hermitian (

*β*= 2) and quaternion self-dual (

*β*= 4), respectively.

*K*can also be viewed as forming the corresponding Ginibre ensembles.

^{36}We, however, are interested in a parametric dependence

*K*=

*K*(

*p*) and, thus,

*H*=

*H*(

*p*) to investigate topological properties. The real variable

*p*parametrizes the one-dimensional unit circle $S1$, giving the interpretation of

*H*(

*p*) as a Bloch Hamiltonian. Physically, the parameter

*p*is the momentum, which is essentially given by a wavevector in the Brillouin zone. This interpretation has an important consequence for class CII as the time reversal operator $T$ acts on

*K*(

*p*) like

*K*(

*p*) because this matrix will not be real quaternion for a generic $eip\u2208S1$. Only for

*p*= 0, the symmetry directly implies a real quaternion structure for

*K*(0). Hence, for a general $eip\u2208S1$, we can expect that

*K*(

*p*) is a complex 2

*N*× 2

*N*matrix interpolating between real and imaginary quaternions.

The general question addressed in recent studies^{2–5} 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 $eip\u2208S1$ 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*)*K*_{1} + sin(*p*)*K*_{2} with generic complex $K1,K2\u2208C4\xd74$ (AIII) and for the matrix *K*(*p*) = cos(*p*)*K*_{1} + *i* sin(*p*)*K*_{2} with generic real quaternion $K1,K2\u2208H2\xd72\u2282C4\xd74$ (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 2*π* periodic function. In contrast, the eigenvalue spectrum of *K*(*p*) may experience a permutation, meaning when running once from *p* = 0 to *p* = 2*π*, a chosen eigenvalue can become another one. Thence, the eigenvalues of *K*(*p*) might have a different period than 2*π*. 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 2*π* 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.

*w*(

*p*) is more closely related to the winding number since it is the only that part describes the winding around the origin, while the real part always integrates to zero because a closed path described by det

*K*(

*p*), which does not cross the origin, can be always continuously deformed into a path on the unit circle, implying |det

*K*(

*p*)| = 1 and, hence, a vanishing real part of the logarithmic derivative. This quantity is also reasonable for the quaternion (CII) and the real (BDI) case despite that determinants of real and real quaternion matrices are purely real. As mentioned above, the matrix

*K*(

*p*) as a Bloch operator is generally complex. Obviously, the winding number

*W*can directly be related to Cauchy’s argument principle by writing the integral as a contour integral for the complex variable

*s*=

*e*

^{ip}(see Ref. 9). Hence, the winding number is always an integer, $W\u2208Z$.

*K*(

*p*) describes a random field on $S1$, which has its values in $GlC(N)$ for AIII or $GlC(2N)$ for CII. To have an analytically feasible model, we assume a Gaussian random field that is centered. Thus, the model is fully controlled by its variance, which we assume to have the only non-vanishing covariances,

*l*,

*j*, where ⟨·⟩ is the ensemble average. As this choice is independent of the matrix indices

*l*and

*j*,

*S*(

*p*,

*q*) must be a scalar product on a vector space because of

*l*,

*j*. Hitherto, we considered the most general form for the covariance

*S*(

*p*,

*q*). The easiest non-trivial choice is a scalar product of a two-dimensional complex vector space, which can be realized by setting up random matrix fields as the linear combinations,

*a*(

*p*) and

*b*(

*p*), that are smooth and 2

*π*-periodic. Arranging the two functions as a vector

*S*(

*p*,

*q*) =

*v*

^{†}(

*p*)

*v*(

*q*). Furthermore, when interpreting our random matrix model as a Bloch Hamiltonian (i.e.,

*p*is a momentum), in the time reversal invariant cases, the functions should satisfy

*K*

_{1}and

*K*

_{2}are either drawn from the complex Ginibre ensemble in the case AIII [see Eq. (22)] or from the real quaternion Ginibre ensemble in the case CII [see Eq. (38)] with probability density

*P*(

*K*

_{1},

*K*

_{2}). As aforementioned, we denote the corresponding ensemble averages of an observable

*F*(

*K*

_{1},

*K*

_{2}) with angular brackets,

*d*[

*K*

_{1},

*K*

_{2}] are simply the products of the differentials of all independent real variables.

*K*(

*p*) to the Hamiltonian

*H*(

*p*), which becomes

*p*

_{1}, …,

*p*

_{l}and

*q*

_{1}, …,

*q*

_{k}in the case

*k*=

*l*. We introduce the more general definition (14) for

*k*and

*l*being different for reasons that will become clear in the sequel. We notice that

*k*and

*l*are the numbers of determinants in the denominator and numerator, respectively.

^{18}since they serve as generators for correlation functions of operator or matrix resolvents. Similarly, we can compute the

*k*-point correlator

*k*-fold derivative

Nevertheless, there is also independent interest in ensemble averages for ratios of characteristic polynomials^{19–30} 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 observation^{40} 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).

## III. RESULTS

*a*(

*p*) and

*b*(

*p*) in terms of the two-dimensional vector

*v*(

*p*). Only then certain inherent symmetries are appropriately reflected in the results. For instance, in the unitary case AIII, labeled

*β*= 2, the partition function $Zk|k(2,N)(q,p)$ [see Eq. (14)], is invariant under the group $SU(2)\xd7GlC(1)$. The part $GlC(1)$ corresponds to the invariance under rescaling

*v*(

*p*) →

*sv*(

*p*) for all $s\u2208GlC(1)=C\{0}$. The scaling factor drops out in the ratio of the characteristic polynomials. The subgroup SU(2) reflects an invariance when rotating

*K*

_{1}and

*K*

_{2}into each other. This carries over to an invariance for the vector

*v*(

*p*); see Sec. IV A for more details. Therefore, the result can only depend on the combinations

*v*

^{†}(

*p*)

*v*(

*q*),

*v*

^{T}(

*p*)

*τ*

_{2}

*v*(

*q*), and their complex conjugates. We emphasize that

*v*

^{T}(

*p*)

*τ*

_{2}

*v*(

*q*) is also an invariant because

*U*=

*τ*

_{2}

*U**

*τ*

_{2}for any

*U*∈ SU(2). Additionally, $Zk|l(2,N)(q,p)$ is a polynomial in

*v*(

*p*

_{j}), while it is quite likely to be not holomorphic in

*v*(

*q*

_{j}). In Sec. IV A, we derive the result

*β*= 4, is slightly simpler in its computation and its results. However, the biggest obstruction is that it respects the smaller invariance group $SO(2)\xd7GlR(1)$. The $GlR(1)$ part is once more the simple rescaling of the two dimensional vector

*v*(

*p*) →

*sv*(

*p*) with $s\u2208GlR(1)=R\{0}$. Yet, the condition that the two matrices

*K*

_{1}and

*K*

_{2}must be real quaternion only allows a rotation of one matrix into the other one via the real special orthogonal group SO(2). Again, more details of this symmetry discussion can be found in Sec. IV B.

*x*,

*y*) = Γ(

*x*)Γ(

*y*)/Γ(

*x*+

*y*) is Euler’s beta function with the gamma function Γ(

*x*). The polynomials are essentially truncated binomial series. The second representation involves Gauss’ hypergeometric function $F12$. The polynomials are actually the skew-orthogonal polynomials of even order corresponding to the quaternion spherical ensemble (see Appendix A for their derivation). In Sec. IV B, we derive the following result:

*k*×

*k*matrix with 2 × 2 matrices of the shown form as matrix entries.

## IV. DERIVATIONS

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 2*k* 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)

*a*(

*p*),

*b*(

*p*) in terms of the two-dimensional complex vector

*v*(

*p*) [see Eq. (10)]. The reason is that this ensemble actually satisfies an SU(2) symmetry given by

*U*∈ SU(2) acting on the two components of the matrix valued vector $K\u0302$. One can readily verify $P(K\u0302)=P([U\u22971N]K\u0302)$ for any

*U*∈ SU(2). This will become handy when computing the partition function $Zk|k(2,N)(q,p)$ and recognizing that

*v*(

*p*

_{j}) and

*v*(

*q*

_{j}).

*K*(

*p*) as follows:

*b*(

*p*) ≠ 0. This is, however, not very restrictive as the limit

*b*(

*p*) → 0 can be readily carried out in the results. The partition function (14) for

*k*=

*l*has then the form

^{31}The corresponding probability density is

*N*complex eigenvalues $(z1,\u2026,zN)\u2208[C\{0}]N$ is

*x*,

*y*) is Euler’s beta function.

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*(*q*_{j}) + *z*) is integrable in two dimensions, such as the complex plane. However, we need to assume that all *κ*(*q*_{j}) 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.

*k*-point correlation function is a

*k*×

*k*determinant with a single kernel function. This structure actually applies to the partition function (26) as well. In Refs. 20 and 21, it was shown for more general ensembles than the one we study that

*N*asymptotic.

*k*= 1. For this purpose, we finally make use of the SU(2) symmetry we have mentioned previously. The partition function

*v*(

*q*

_{m}) and

*v*(

*p*

_{n}), and the SU(2) symmetry tells us that

*F*(

*v*(

*q*

_{m}),

*v*(

*p*

_{n})) =

*F*(

*U*

^{T}

*v*(

*q*

_{m}),

*U*

^{T}

*v*(

*p*

_{n})) for all

*U*∈ SU(2). Therefore, we can choose the unitary matrix

*Y*→

*e*

^{iφ}

*Y*of the probability density tells us that the average of the characteristic polynomial $det(x1N\u2212Y)$ only yields the monomial

*x*

^{N}. Thus, the final result is

*v*

^{†}(

*q*)

*v*(

*q*),

*v*

^{†}(

*q*)

*v*(

*p*), and

*v*

^{T}(

*q*)

*τ*

_{2}

*v*(

*p*) =

*i*(

*a*(

*p*)

*b*(

*q*) −

*a*(

*q*)

*b*(

*p*)), with

*τ*

_{2}being the second Pauli matrix.

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)

*κ*(

*p*) =

*a*(

*p*)/

*b*(

*p*) defined as before. The matrix

*Y*is now drawn from the quaternion spherical ensemble following the probability density,

^{33}

*z*of

*Y*has a complex conjugate

*z**, which is also an eigenvalue. The corresponding joint probability density of the eigenvalues $z=diag(z1,z1*,z2,z2*,\u2026,zN,zN*)$ is given by

*κ*(

*q*

_{j}),

*κ**(

*q*

_{j})] as we encounter terms of the form $1/[(\kappa (qj)+zj)(\kappa (qj)+zj*)]$. As long as

*κ*(

*q*

_{j}) is not real, the singularities are simple poles. However, when

*κ*(

*q*

_{j}) is real, this term becomes a double pole of the integrand, which is, in general, not integrable even in two dimensions. The fortunate fact that renders also this kind of pole integrable is the factor $|zj\u2212zj*|2$ as it vanishes like a square when

*z*

_{j}becomes real. Therefore, the combination $|zj\u2212zj*|2/[(\kappa (qj)+zj)(\kappa (qj)+zj*)]$ is absolutely integrable even when

*κ*(

*q*

_{j}) becomes real. The condition of pairwise distinct complex pairs [

*κ*(

*q*

_{j}),

*κ**(

*q*

_{j})] can be anew dropped for the final result where the limit

*κ*(

*q*

_{a}) →

*κ*(

*q*

_{b}) as well as

*κ*(

*q*

_{a}) →

*κ**(

*q*

_{b}) is well-defined (see the summary of the results in Sec. III).

*l*−

*k*even and

*M*+ (

*l*−

*k*)/2 <

*N*+ 1:

*M*of integration variables varies.

*g*

^{(4)}(

*z*) with the skew-symmetric two-point weight involving the Dirac delta function for complex numbers,

#### 1. The kernel $K1(4)$

*N*− 2) × (2

*N*− 2) real quaternion Ginibre matrices $K1,K2\u2208GlH(2N\u22122)$. The limits

*a*(

*p*

_{m}),

*b*(

*p*

_{m}),

*a*(

*p*

_{n}), and

*b*(

*p*

_{n}). Hence, we can also consider the average

*b*

_{1}

*a*

_{2}−

*a*

_{1}

*b*

_{2}≠ 0 and then perform an analytic continuation in the result to the complex functions. We need this detour via analytic continuation because we can only rotate real vectors with the SO(2) symmetry similar to what we have done in the complex case AIII. Therefore, the average is equal to

*N*− 2 corresponding to the weight function $g(4)(z)=(z\u2212z*)/(1+|z|2)2N+2$. The skew-orthogonal polynomials have been computed in Appendix A.

*a*(

*p*) and

*b*(

*p*) despite we have derived it for real coefficients due to being a polynomial in these functions. The first kernel function is then

#### 2. The kernel $K2(4)$

*a*(

*p*

_{n}) and

*b*(

*p*

_{n}). With the very same arguments as in Subsection IV B 1, we can exploit the analyticity in these two variables and replace them by two fixed real variables $a1,b1\u2208R$ and analytically continue the result at the end of the day. Unfortunately, we are not allowed to do the same trick for

*a*(

*q*

_{n}) and

*b*(

*q*

_{n}) as the partition function is not holomorphic in these two variables; actually, the result will also depend on their complex conjugates such that we only replace them by two generic but fixed complex variables $a2,b2\u2208C$.

*z*

_{j}as well as their conjugates, each expansion term yields the very same contribution and, hence, an overall factor 2

*N*so that we can also write

*z*

_{1}, …,

*z*

_{N−1}with the Heine-formula (A4) for $q2N\u22122(N)(zN)$.

*a*(

*q*

_{m}) and

*b*(

*q*

_{m}) anywhere in the complex plane. One can apply the standard formula for the derivative in the complex conjugate

*κ** on the integral

*f*(

*z*,

*z**). Considering the integrand in (58), we note that apart from the real line, the integral must be a function of both,

*κ*and

*κ**, which is also what we find. Thus, the analyticity of the integral in

*κ*is violated everywhere.

*L*distinct $\kappa j\u2208C$ and any differentiable measurable function

*f*(

*z*

_{1},

*z*

_{2}), which vanishes at infinity in both arguments and where

*f*(

*z*,

*z**) is singularity free. Collecting everything, we find for the function

*a*

_{1},

*b*

_{1},

*a*

_{2},

*b*

_{2}) → (

*a*(

*p*

_{n}),

*b*(

*p*

_{n}),

*a*(

*q*

_{m}),

*b*(

*q*

_{m})) because only

*a*(

*q*

_{m}) and

*b*(

*q*

_{m}) can be complex conjugated, while

*a*(

*p*

_{n}) and

*b*(

*p*

_{n}) are analytic continuations of

*a*

_{1}and

*b*

_{1}. Therefore, the second kernel is equal to

*a*(

*p*

_{n}) and

*b*(

*p*

_{n}). We only know this from the starting expression in terms of averages over a ratio of two characteristic polynomials of the random matrix

*Y*. Anew, one can check the SO(2) invariance for $Z1|1(4,N)(pn,qm)$, which indeed only depends on the group invariants

*v*

^{T}(

*p*

_{n})

*v*(

*p*

_{n}),

*v*

^{T}(

*q*

_{m})

*v*(

*p*

_{n}),

*v*

^{†}(

*q*

_{m})

*v*(

*p*

_{n}),

*v*

^{T}(

*q*

_{m})

*τ*

_{2}

*v*(

*p*

_{n}), and

*v*

^{†}(

*q*

_{m})

*τ*

_{2}

*v*(

*p*

_{n}).

#### 3. The kernel $K3(4)$

*κ*

_{j}times the difference

*κ*

_{2}−

*κ*

_{1}can be written in terms of a Berezinian (see Ref. 21)As before, the vertical lines should highlight the two last columns, while the odd rows only comprise

*z*

_{a}and the even rows $za*$. We may choose the skew-orthogonal polynomials

*q*

_{j}(

*x*) in the entries of this determinant instead of the monomials,This allows us to apply the generalized de Bruijn theorem to carry out the integral (see Ref. 21), yieldingwhere we have employed the skew-symmetric product,

*a*is the row index for the first 2

*N*rows and

*b*the column index for the first 2

*N*columns. The skew-orthogonality of the polynomials simplifies the upper left 2

*N*× 2

*N*block drastically, which becomes a 2 × 2 block-diagonal matrix. This can be exploited in combination with the standard identity

*h*

_{j}= 1/[

*π*B(2

*j*+ 2, 2

*N*− 2

*j*)] being the normalization of the skew-orthogonal polynomials. Plugging in the explicit expressions of the skew-symmetric product and the skew-orthogonal polynomials, we have

*v*(

*p*) in Sec. III to underline the invariance under SO(2) transformations.

## V. CONCLUSIONS

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.

## ACKNOWLEDGMENTS

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.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**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 AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

### APPENDIX A: SKEW-ORTHOGONAL POLYNOMIALS OF THE CASE CII

*n*and the relations

*c*

^{(4)}of the joint probability density (41) in the standard way (see Refs. 11 and 43), namely, by

*a*is the running index for the last 2

*n*+ 1 rows and

*b*those of the columns. The Pfaffian involves an antisymmetric (2

*n*+ 1) × (2

*n*+ 1)-kernel with the elements

*n*odd and even rows comprise

*z*

_{a}and $za*$, respectively. The polynomials of odd degree are thenwhere we anew applied the generalized de Bruijn theorem (see Refs. 21 and 48). This time the two vertical and horizontal lines underline the particular role of the first and last columns and rows. The antisymmetric kernel is the same as in the even case (A6) for 1 ≤

*a*,

*b*≤ 2

*n*. The integrals in the last row and column are the skew-symmetric product ⟨

*z*

^{a−1}|

*z*

^{2n+1}⟩ with

*a*= 1, …, 2

*n*and, thus, vanish. Expanding the Pfaffian in the last row and column yields the monomial

### APPENDIX B: EVALUATING Ξ_{3}

*z*

_{1}and

*z*

_{2}for the left hand side of the equation above, we find

*N*× 2

*N*highlights that we average over 2

*N*× 2

*N*real quaternion Ginibre matrices $K1,K2\u2208GlH(2N)$. We emphasize that we can exploit the results of the first kernel function $K1(4)(pm,pn)$ [see Eq. (53)], with the difference that the matrix dimension is larger. Thus, it is

*z** +

*κ*

_{1}=

*z** +

*κ*

_{2}+

*κ*

_{1}−

*κ*

_{2}in the numerator, it is straightforward to show that the integral

*z*= 0 and

*z*= −

*κ*

_{1}cancel each other.

*φ*∈ [0, 2

*π*], exploiting the partial fraction decomposition

*J*. Therefore, we arrive at

*κ*

_{1}↔

*κ*

_{2}, this final result reflects this symmetry.

## REFERENCES

_{2}Se

_{3}upon deposition of gold

*Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics*

*L*

^{2}-phase

*A Short Course on Topological Insulators*

*k*-point random matrix kernels obtained from one-point supermatrix models

*The Oxford Handbook of Random Matrix Theory*