Determinantal polynomial wave functions induced by random matrices
arXiv:1807.01173 · doi:10.1103/PhysRevA.98.063813
Abstract
Random-matrix eigenvalues have a well-known interpretation as a gas of like-charge particles. We make use of this to introduce a model of vortex dynamics by defining a time-dependent wave function as the characteristic polynomial of a random matrix with a parameterized deformation, the zeros of which form a gas of interacting vortices in the phase. By the introduction of a quaternionic structure, these systems are generalized to include anti-vortices and non-vortical topological defects: phase maxima, phase minima and phase saddles. The commutative group structure for complexes of such defects generates a hierarchy, which undergo topologically-allowed reactions. Several special cases, including defect-line bubbles and knots, are discussed from both an analytical and computational perspective. Finally, we return to the quaternion structures to provide an interpretation of two-vortex fundamental processes as states in a quaternionic space, where annihilation corresponds to scattering out of real space, and identify a time--energy uncertainty principle.
arXiv comment26 pages, 21 figures. The video data referred to in the text will be provided as Supplemental Material to the journal. Changed title and made some changes to the introduction according to referees comments. Also changed colour-scheme for grey-scale printing
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