^{1}

^{2}

^{3}.

In non-Hermitian random matrix theory there are three universality classes for local spectral correlations: the Ginibre class and the nonstandard classes

The profound relationship between random matrix theory (RMT) and natural sciences has been the subject of research over decades

RMT ensembles that interpolate between WD classes have also been discussed

In the classification of Hermitian RMT

In mathematics, the universality of RMT refers to the invariance of the spectral statistics for general matrix potentials

Recent years have witnessed a surge of interest in non-Hermitian quantum systems

Earlier classifications

Despite the diversity of non-Hermitian RMT, only one universality class has been known until recently: the Ginibre universality class

It has recently been pointed out

Here, the naming scheme of

In view of the aforementioned developments it is natural to ask the following questions. First, is there any application of the 19 nonstandard non-Hermitian chiral RMT classes to high-energy physics? Second, can the non-Ginibre universal statistics be observed in the Dirac spectrum? We answer both questions in the affirmative. Our contributions in this paper are as follows. (1) We show that Dirac operators in two-color QCD and adjoint QCD coupled to a

Our findings are in stark contrast to the Dirac spectrum at finite baryon chemical potential, which exhibits correlations in the Ginibre class. This work provides a new point of view on quantum chaos in gauge theories.

This paper is organized as follows. In Sec.

In the following, the Euclidean Dirac matrices are denoted by

Let us first consider quarks in the fundamental representation of the gauge group SU(2). The Euclidean Dirac operator coupled to a chiral U(1) gauge field is given by

Next, we turn to quarks in a real (e.g., adjoint) representation of a non-Abelian compact gauge group. In the absence of

Note that, if there is nonzero baryon chemical potential, the Dirac operator

If we drop the spatial components of

On the basis of the transposition symmetries

In Sec.

The fermionic partition function, i.e., the average of the product of characteristic polynomials, is of special interest in RMT. The arguments for pseudoreal and real quarks can be run in parallel, so we shall concentrate on pseudoreal quarks in the following. At first sight it may seem natural to consider the partition function

Let us consider the special case

In this section we define the lattice Dirac operators that will be used in the numerical simulations. To keep the notation simple we will use the same symbol

The massless staggered Dirac operator coupled to a lattice gauge field

The continuum chiral symmetry is replaced by the remnant chiral symmetry

We now introduce a chiral U(1) gauge field

For

We now consider the SU(2) gauge field in the fundamental and adjoint representation, denoted by

All statements made in this section so far hold for both representations. The difference between the representations lies in the transposition symmetries. Using the pseudoreality

The minus signs in

The eigenvalues of

To introduce the chiral potential

The eigenvalues of

The aim of this section is to test our proposal by numerical simulations of the corresponding lattice quantum field theories. As mentioned above, we use staggered fermions. We work in the quenched approximation since it is numerically cheap and sufficient for the point we are making. The inclusion of dynamical fermions does not change the universality class and therefore makes no difference for the quantities we compute.

Our simulations were done in the

Numerically, the most expensive operation is the calculation of all eigenvalues of the Dirac operator, which scales like

Number of configurations for the cases simulated. U(1) refers to the operator in

Scatter plots of the Dirac eigenvalues for a single configuration are shown in Fig.

Scatter plots of the staggered Dirac eigenvalues for a single configuration in SU(2) lattice gauge theory on an

Once the eigenvalues

Writing

We use the same symbol

Our numerical results for gauge group SU(2) and fermions in the fundamental representation are summarized in Figs.

Heatmaps of

Results for

Moments of

We also computed the Jensen-Shannon distance between pairs of the two-dimensional distributions shown in Fig.

Jensen-Shannon distance between the distributions

Our numerical results for gauge group SU(2) and fermions in the adjoint representation are summarized in Figs.

Same as Fig.

Same as Fig.

Same as Fig.

Same as Fig.

We have shown that the two nonstandard universality classes

There are several avenues for future work. Analytically, it would be interesting to study the integral representation

This work was supported in part by DFG Grant No. SFB TRR 55.

Spatially inhomogeneous bilinear condensates arise in QCD and QCD-like theories under various conditions

It is well known

For greater generality we formally include both

Now let

Adding a mass term

In this section we give a brief derivation of the sum rules for the squared staggered Dirac operator. Again, for greater generality we include both the chiral U(1) field

We split the massive operator

Here and below we assume that the lattice has more than two sites in every direction because otherwise some Kronecker deltas could be nonzero due to the toric nature of the lattice.

Hence