Preserving topology while breaking chirality: from chiral orthogonal to anti-symmetric Hermitian ensemble
Akemann, Gernot
Akemann
Gernot
Kieburg, Mario
Kieburg
Mario
Mielke, Adam
Mielke
Adam
Vidal, Pedro
Vidal
Pedro
We consider a parameter dependent ensemble of two real random matrices with
Gaussian distribution. It describes the transition between the symmetry class
of the chiral Gaussian orthogonal ensemble (Cartan class B$|$DI) and the
ensemble of antisymmetric Hermitian random matrices (Cartan class B$|$D). It
enjoys the special feature that, depending on the matrix dimension $N$, it has
exactly $\nu=0$ $(1)$ zero-mode for $N$ even (odd), throughout the symmetry
transition. This "topological protection" is reminiscent of properties of
topological insulators. We show that our ensemble represents a Pfaffian point
process which is typical for such transition ensembles. On a technical level,
our results follow from the applicability of the Harish-Chandra integral over
the orthogonal group. The matrix valued kernel determining all eigenvalue
correlation functions is explicitly constructed in terms of skew-orthogonal
polynomials, depending on the topological index $\nu=0,1$. These polynomials
interpolate between Laguerre and even (odd) Hermite polynomials for $\nu=0$
$(1)$, in terms of which the two limiting symmetry classes can be solved.
Numerical simulations illustrate our analytical results for the spectral
density and an expansion for the distribution of the smallest eigenvalue at
finite $N$.
2019
2
IOPScience
2019