Effects of time reversal symmetry breaking on localization: Band random matrix description

We investigate two kinds of band random matrix ensemble (BRM) with differen t time reversal symmetry, namely, BRM of complex and quaternion real (C-BRM and QR-BRM) Hermitians. Their scaling behaviors of localization are numeri cally studied. Results show that the removal of time reversal symmetry yiel ds a multiplication of the localization length by a factor 2 (or, 1/2) in t he absence (or, presence) of Kramers' degeneracy This leads us to expect a simple and universal description of the effects of time-reversal symmetry b reaking on localization that the localization length xi is given by xi = be ta b(2)/4 where b is the band-width parameter and beta the universality par ameter.