The origin of order in random matrices with symmetries

TL;DR

This paper investigates how symmetries in random matrices influence the ordering of conserved quantities, revealing that ground states tend to be dominated by low-dimensional irreps, which explains observed physical phenomena.

Contribution

It demonstrates that symmetries in random matrices lead to ground states dominated by extremal low-dimensional irreps, providing a new understanding of order in symmetric systems.

Findings

Ground states are dominated by extremal low-dimensional irreps.

Symmetries induce order in the ground state structure.

Explains the dominance of J=0 ground states in random interactions.

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.