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We introduce and solve exactly a family of invariant 2x2 random matrices,depending on one parameter \eta, and we show that rotational invariance andreal Dyson index \beta are not incompatible properties. The probability densityfor the entries contains a weight function and a multiple trace-traceinteraction term, which corresponds to the representation of theVandermonde-squared coupling on the basis of power sums. As a result, theeffective Dyson index \beta_{eff} of the ensemble can take any real value in aninterval. Two weight functions (Gaussian and non-Gaussian) are explored indetail and the connections with \beta-ensembles of Dumitriu-Edelman and theso-called Poisson-Wigner crossover for the level spacing are respectivelyhighlighted. A curious spectral twinning between ensembles of differentsymmetry classes is unveiled. The proposed technical tool more genericallyallows for designing actual matrix models which i) are rotationally invariant;ii) have a real Dyson index \beta_{eff}; iii) have a pre-assigned confiningpotential or alternatively level-spacing profile. The analytical results havebeen checked through numerical simulations with an excellent agreement.Eventually, we discuss possible generalizations and further directions ofresearch.

On invariant 2x2 \beta-ensembles of random matrices