^{1}

^{2}

^{,*}

^{3}.

We consider a random matrix model which interpolates between the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble while preserving chiral symmetry. This ensemble describes flavor symmetry breaking for staggered fermions in 3D QCD as well as in 4D QCD at high temperature or in 3D QCD at a finite isospin chemical potential. Our model is an Osborn-type two-matrix model which is equivalent to the elliptic ensemble but we consider the singular value statistics rather than the complex eigenvalue statistics. We report on exact results for the partition function and the microscopic level density of the Dirac operator in the

Random matrix theory (RMT) has been applied in several areas of physics. To name only a few: quantum chaos, condensed matter theory of disordered systems, and quantum information. Introductions to those and other applications can be found in Ref.

RMT has been successfully applied to quantum chromodynamics (QCD) in the

Two main ideas have driven the RMT approach in QCD. First, the random matrix models are simple enough to provide analytical formulas relating observables like the microscopic level density of the Dirac operator to low energy constants in QCD, e.g., see Refs.

The applicability domain of RMT with chiral symmetry is not limited to QCD but includes all areas where fermions with chirality emerge. For example, Dirac and Weyl fermions appear in the low-energy effective field theories of graphene, topological insulators, semimetals, and

In the present work we wish to consider the chiral random matrix

There are at least three major connections between the model

Another application of Eq.

The third application of the model Eq.

Before closing the introduction, let us briefly review preceding works that are closely related to ours. The random matrix

Comparison of the analytical result

To derive the chiral Lagrangian of Eq.

In the next step we apply the superbosonization formula

First we want to consider the partition function with

Next we want to derive the microscopic level density in the quenched limit. Its relation to the partially quenched partition function for

This is a new result. We normalized it to the asymptotics

We sketched the derivation of the chiral Lagrangian, see Eq.

The results we presented here can be readily generalized to the situation of

Given the connection of RMT to various fields of physics we anticipate that the solution presented in the present work would be of value outside of QCD as well, especially in the field of condensed matter systems with emergent Dirac fermions.

We acknowledge support by the RIKEN iTHES project (T. K.) and the German research council (DFG) via the CRC 1283 (M. K.).

We count the number of four-component Dirac fermions by