The origin of order in random matrices with symmetries

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.Comment: 5 figures; contribution to "Beauty in Physics" conference in honor of Francesco Iachello, May 2012, Cocoyoc, Mexic

Tracing the evolution of nuclear forces under the similarity renormalization group

I examine the evolution of nuclear forces under the similarity renormalization group (SRG) using traces of the many-body configuration-space Hamiltonian. While SRG is often said to "soften" the nuclear interaction, I provide numerical examples which paint a complementary point of view: the primary effect of SRG, using the kinetic energy as the generator of the evolution, is to shift downward the diagonal matrix elements in the model space, while the off-diagonal elements undergo significantly smaller changes. By employing traces, I argue that this is a very natural outcome as one diagonalizes a matrix, and helps one to understand the success of SRG.Comment: 6 pages, 3 figures, 1 tabl

Systematics of strength function sum rules

Sum rules provide useful insights into transition strength functions and are often expressed as expectation values of an operator. In this letter I demonstrate that non-energy-weighted transition sum rules have strong secular dependences on the energy of the initial state. Such non-trivial systematics have consequences: the simplification suggested by the generalized Brink-Axel hypothesis, for example, does not hold for most cases, though it weakly holds in at least some cases for electric dipole transitions. Furthermore, I show the systematics can be understood through spectral distribution theory, calculated via traces of operators and of products of operators. Seen through this lens, violation of the generalized Brink-Axel hypothesis is unsurprising: one \textit{expects} sum rules to evolve with excitation energy. Furthermore, to lowest order the slope of the secular evolution can be traced to a component of the Hamiltonian being positive (repulsive) or negative (attractive).Comment: 5 pages, 4 figures; minor revisions; references updated; title revised; matches accepted versio

Collectivity, chaos, and computers

Two important pieces of nuclear structure are many-body collective deformations and single-particle spin-orbit splitting. The former can be well-described microscopically by simple SU(3) irreps, but the latter mixes SU(3) irreps, which presents a challenge for large-scale, ab initio calculations on fast modern computers. Nonetheless, SU(3)-like phenomenology remains even in the face of strong mixing. The robustness of band structure is reminiscent of robust, pairing collectivity that arises from random two-body interactions.Comment: 9 pages, invited talk at Computational and Group Theoretical Methods in Nuclear Physics, Playa del Carmen, Mexico, February 18-21, 200