Let and be even positive integers. Let and be quaternion matrices, which can be represented as ordinary matrices and as¹ (1)

Here the four units of the quaternion field are represented by the unit matrix and Pauli matrices, . Let these matrix elements belong to (2) i.e., is quaternion-real and is not (i.e., is a generic complex matrix). We consider , , and to be independent random variables, distributed according to the Gaussian distributions and , respectively. We define an ensemble of Hermitian matrices of the form (3) where is a real parameter, initially introduced by Dyson as a fictitious time for the Brownian motion of eigenvalues [3]. This ensemble is called a “chiral” random matrix ensemble because it enjoys the chiral symmetry with . This anticommutation relation implies that the eigenvalues of consist of pairs of generically nonzero eigenvalues of equal magnitude and opposite signs (i.e., singular values of ), and zero eigenvalues. The presence of violates the quaternion-reality of and the self-duality of (i.e., ), and lifts the Kramers degeneracy of all nonzero eigenvalues of (i.e., singular values of alone). Accordingly, this random matrix ensemble interpolates between two limiting cases, the chiral Gaussian symplectic ensemble (chGSE) at and the chiral Gaussian unitary ensemble (chGUE) at , depending on a single parameter (at a fixed, finite and ).

In order to make the paper self-contained, below we sketch the derivation of the probability distribution of singular values of , and refer the reader to Refs. [14–16] for rigorous proofs of the relevant formulas. We start from the unnormalized probability measure of the matrix elements of , (4) where , , and . Without loss of generality we assume . We employ the singular value decomposition and with and parametrize the singular values as . Kramers-degenerate pairs of singular values of are ordered such that . The measures and on quaternion-real matrices and complex matrices take the following respective forms (in what follows we suppress all constant factors in the measures): (5)

Here and denote the invariant measures on the respective angular degrees of freedom, and denote the Vandermonde determinants j}^{N/2} (\mu_i^2-\mu_j^2)$]]> and j}^{N} (\lambda_i^2-\lambda_j^2)$]]>. The probability measure of the singular values of follows from Eqs. (4) and (5) by integrating out the unitary matrices . The integrations over symplectic matrices decouple after redefining the unitary matrices by and , leading to the expression (6)

We employ the Berezin–Karpelevich formula [20–22] for the integration over and take the pairwise confluent limit for all : (7)

Here denotes the Bessel function of the pure imaginary argument, . By substituting Eq. (7) into Eq. (6) and performing a change of variables from the singular values of rectangular matrices and to the eigenvalues of Wishart matrices and , and , the probability measure becomes proportional to (8)

Here we have introduced the Laguerre weight , and the symmetric function (9)

With the multiplicative constant chosen as the above, the function admits an alternative interpretation as a one-particle Green’s function for the Brownian motion at time , (10)

with the norm given by and the “one-particle energy” by . Using a lemma by Mehta (A.17 of Ref. [23]), the -fold integral of an determinant in Eq. (8) can be decomposed into an Pfaffian of single integrals: (11) (12)

In this subsection we summarize the procedure presented in Ref. [23], Sect. 14. We introduce a set of arbitrary monic polynomials and arbitrary positive numbers , and define functions by (13) and their -convolutions by (14)

Next we introduce functions , , and , which are bilinear combinations of and : (15) (16) (17) and the corresponding matrices , , and by (18)

Since a antisymmetric matrix is a product of two rectangular matrices of size and : its rank is at most; and also it is at least due to the linear independence of . Accordingly the lower rows are linear combinations of the upper rows . Now consider a Pfaffian of another antisymmetric matrix with . Since adding (or minus its transpose) to the lower rows (the right columns) does not change the determinants due to the aforementioned linear dependence with the upper rows (the left columns), we readily obtain (19)

On the other hand, (20)

Using Eqs. (19) and (20), the probability measure (11) now reads (21)

Since its upper diagonal block is the transpose of the lower diagonal block and both off-diagonal blocks are antisymmetric, can be regarded a complex matrix representative of an quaternion self-dual matrix, which we call . Using Dyson’s lemma for the quaternion determinant (qdet) [24] of a quaternion self-dual matrix , we finally obtain (22)

We would like to choose and (which are so far arbitrary) in such a way that the matrix representing the quaternion kernel enjoys the quasi-projectivity (23) and is correctly normalized: (24)

These two relationships would yield a crucial property that the restricted quaternion matrix satisfies the recursion relation (25) from which a -level correlation function is expressed as . It is well established that the properties (23), (24) are fulfilled by requiring to be skew-orthogonal with respect to the skew inner product and to be their skew-norms, (26) (27)

Under this choice, itself is expressed as (28)

Thus the matrix element of takes the form (29) due to the definition (17).

One can verify that (30) satisfy the skew-orthogonality (27) at (i.e., chGSE, with the skew inner product denoted by ) with [25]. Due to a lemma [26], (31)=e^{-(\gamma_m+\gamma_n)\tau} \bigl< R_m^{(0)}, R_n^{(0)} \bigr>_{(0)}, \end{equation}]]> which directly follows from the definition (26), (32) satisfy the skew-orthogonality (27) at 0$]]> with [14]. Together with the definitions (13)–(16) and (29), the quaternion kernel elements at finite are completely determined.

Now we concentrate on the case in which the Kramers degeneracy is weakly broken by the small parameter . Then the spectral density of in the large- limit is identical to that of chGSE (), i.e., Wigner’s semicircle . We magnify the vicinity of the origin by introducing the rescaled variables² , which measure the eigenvalues in units of the mean level spacing at the origin, . Moreover, in order to realize a nontrivial crossover behavior we take the triple-scaling limit while keeping the combination and fixed finite. In this limit, sums over turn to integrals over and Laguerre polynomials reduce to Bessel functions, as . Accordingly, the quaternion kernel elements (15), (16), (29) reduce to (see footnote 2) [14] (33) (34) (35)

Thus the correlation function of positive rescaled eigenvalues of in the vicinity of the origin is finally expressed as (36)

It is well known that, for a determinant process in which -point correlation functions are expressed in terms of a scalar kernel as the probability for an interval to contain exactly points is expressed as a Fredholm determinant (Det) of over [23]: (37)

Here acts on -functions over as . This argument directly carries over to our case of the quaternion determinant process in which -point correlation functions are expressed in terms of a quaternion kernel as , leading to (38)

This time, acts on 2-component -functions over as . By differentiating in , the first few are expressed in terms of the Fredholm determinant and the resolvents of the operator , as [19] (39)

After specializing to and abbreviating , the probability distribution of the th smallest positive eigenvalue is given in terms of as (40)

This relationship follows from a simple observation that, for a joint of two intervals , the probability that the narrower interval contains more than one eigenvalue is of order , so to the order one has (41)

Subtracting Eq. (41) from the definition of gives (42) which is equivalent, in the limit , to (43) with understood. Summing over gives Eq. (40).

An efficient way of numerically evaluating the Fredholm determinant of a trace-class operator is the Nyström-type discretization [17,18] (44) is an matrix evaluated with a quadrature rule consisting of a set of points and associated weights such that . As the order of the quadrature increases, the RHS of Eq. (44) is proven to converge to its LHS uniformly and exponentially fast in [17,18]. We also need to evaluate the resolvents in Eq. (39), which are likewise approximated as (45)

Obviously these formulae hold for a -matrix-valued kernel as well. For our purpose we employ the Gauss–Legendre quadrature rule in which are the nodes of the Legendre polynomial on a shifted and rescaled domain . We have applied the Nyström-type method to the kernel (33)–(36) for the chGSE–chGUE crossover at , bearing in mind that the topological charge is washed away in the staggered Dirac operator that we shall employ in Sect. 3. We have numerically evaluated with at least 20 for far-more-than-sufficient precision, and confirmed the stability of the results for increasing . Plots of for are exhibited in Fig. 1 (left).

For comparison, the spectral densities that comprise the former, in Eq. (33), are plotted in Fig. 1 (right). The practical advantage of adopting individual eigenvalue distributions over -level correlation functions (including ) for fitting is clear from the figures: As the oscillation of the latter consists of overlapping multiple peaks, the characteristic shape of each peak is inevitably smeared, resulting in a rather structureless curve for which an accurate fit is difficult. On the other hand, the shape of the former is clearly distinguishable and is extremely sensitive to the parameter, because the ratio of the two of the chGUE–chGSE crossover at different grows as for large . Therefore, the are expected to admit very precise one-parameter fitting of data by the least-squares method (as will be shown in Tables 3–6 of Sect. 3).

Now we shall relate the crossover random matrix ensemble to the nonlinear models originating from gauge theory. In continuum, QCD-like theories with flavors of quarks in a real representation have pseudoreal Dirac operators, as does our case of SU(2) lattice gauge theory with fundamental staggered fermions (Eq. (52) in Sect. 3). The low-energy effective Lagrangian of these theories is universally determined by the spontaneous breaking of the Pauli–Gürsey extended flavor group down to its vector subgroup and takes the form (46) in the leading order of the -expansion. Here is a symmetric SU() matrix-valued Nambu–Goldstone field (called a nonlinear model of class AI), the degenerate quark mass, and . It contains two phenomenological constants: the pseudo-scalar decay constant and the chiral condensate (both measured in the chiral and zero-chemical potential limit ). The effect of introducing the quark number chemical potential to the fundamental theory is unambiguously incorporated in through flavor covariantization in the symmetric rank-2 tensor representation [27], (47) with . If the theory is in a finite volume and the Thouless energy defined as is much larger than (called the -regime), the path integral is dominated by the zero mode only and takes the form (48)

The above form (not the concrete symmetric space on which takes values) of action is in fact common for all classes of nonlinear models (in even or odd dimensions, with Dyson indices ) in which the global symmetry is broken by the chemical potential [28].

On the other hand, the characteristic polynomial of a random matrix (3) in the microscopic crossover scaling limit explained in Sect. 2.5 can be evaluated by exponentiating the determinant with flavors of Grassmannian vectors and using a standard technique of Hubbard–Stratonovich transformations (see, e.g., Ref. [29]) for matrices and . One can easily show that the outcome is the same nonlinear model as Eq. (48) with parameters replaced by (49) respectively. Note that the above identification can also be read off from the exponents in the quaternion kernel elements (33)–(35).

(a) gauge theory with the imaginary chemical potential (ICP): The variables on temporal links of a hypercubic lattice of size are multiplied by a constant phase: (50)

The phase can be regarded as the Aharonov–Bohm (AB) flux [4] penetrating the temporal circle and is gauge-equivalent to the imaginary chemical potential , i.e., a fixed U(1) background . We chose antiperiodic/periodic boundary conditions in the temporal/spatial directions, respectively, and consider a small twisting along the temporal direction ().

(b) gauge theory: Following Ref. [31], non-compact link variables are generated under the Coulomb gauge-fixing condition (with an additional constraint for the ) and at the unit coupling constant, and are multiplied to link variables , (51) with denoting the bare coupling constant. Fermions are quenched both for the SU(2) and U(1) gauge fields. For the pure SU(2) case (or ), the staggered Dirac operator in the fundamental representation (52) is pseudoreal, i.e., satisfies (53) ( denotes complex conjugation) for either choice of periodic or antiperiodic boundary condition in each direction, and we impose periodic boundary conditions in all four directions and consider the part as a small perturbation (i.e., ).

As the presence of the AB flux or the coupling breaks the pseudoreality (53), they parametrize antiunitary symmetry breaking in Dirac operators and are anticipated to be the lattice gauge theory counterpart of the crossover parameter in random matrix ensemble interpolating chGSE and chGUE. We note that while the effect of the ICP on the low-energy effective Lagrangian is completely dictated on the symmetry ground [32,33] and is related to the pseudo-scalar decay constant as for the real chemical potential [27], the effect of cannot be directly related to , because integrating over the dynamical U(1) gauge field in the effective Lagrangian would lead to nonlocal self-couplings of pseudo-scalar mesons.