The origin of order in random matrices with symmetries

Abstract

From Noether's theorem we know symmetries lead to conservation laws. What is left to nature is the ordering of conserved quantities; for example, the quantum numbers of the ground state. In physical systems the ground state is generally associated with `low' quantum numbers and symmetric, low-dimensional irreps, but there is no \textit{a priori} reason to expect this. By constructing random matrices with nontrivial point-group symmetries, I find the ground state is always dominated by extremal low-dimensional irreps. Going further, I suggest this explains the dominance of J=0 g.s. even for random two-body interactions.

Comments: 5 figures; contribution to "Beauty in Physics" conference in honor of Francesco Iachello, May 2012, Cocoyoc, Mexico