2,793 research outputs found
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
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