On invariant 2×2 β -ensembles of random matrices

We introduce and solve exactly a family of invariant 2×2 random matrices, depending on one parameter η, and we show that rotational invariance and real Dyson index β are not incompatible properties. The probability density for the entries contains a weight function and a multiple trace–trace interaction term, which corresponds to the representation of the Vandermonde-squared coupling on the basis of power sums. As a result, the effective Dyson index βeff of the ensemble can take any real value in an interval. Two weight functions (Gaussian and non-Gaussian) are explored in detail and the connections with β-ensembles of Dumitriu–Edelman and the so-called Poisson–Wigner crossover for the level spacing are respectively highlighted. A curious spectral twinning between ensembles of different symmetry classes is unveiled: as a consequence, the identification between symmetry group (orthogonal, unitary or symplectic) and the exponent of the Vandermonde (β=1,2,4) is shown to be potentially deceptive. The proposed technical tool more generically allows for designing actual matrix models which (i) are rotationally invariant; (ii) have a real Dyson index βeff; (iii) have a pre-assigned confining potential or alternatively level-spacing profile. The analytical results have been checked through numerical simulations with an excellent agreement. Eventually, we discuss possible generalizations and further directions of research.