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arXiv:1206.5748 · AIP conference proceedings

The origin of order in random matrices with symmetries

Calvin W. Johnson

Published Jan 1, 2012• 0 citations
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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 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.

DOI
10.1063/1.4759388

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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 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.
All nuclei with even numbers of protons and of neutrons have ground states with zero angular momentum. This is ascribed to the pairing force between nucleons, but simulations with random interactions suggest a much broader many-body phenomenon. In this Letter I project out random Hermitian matrices that have good quantum numbers and, computing the width of the Hamiltonian in subspaces, find ground states dominated by low quantum numbers, e.g. J=0. Furthermore I find odd-ZZZ, odd-NNN systems with isospin conservation have relatively fewer J=0 ground states.
Random interactions in nuclei and extension of 0+0^+0+ dominance in ground states to irreps of group symmetries
Random one plus two-body hamiltonians invariant with respect to O(N1)⊕O(N2)O({\cal N}_1) \oplus O({\cal N}_2)O(N1​)⊕O(N2​) symmetry in the group-subgroup chains U(N)⊃U(N1)⊕U(N2)⊃O(N1)⊕O(N2)U({\cal N}) \supset U({\cal N}_1) \oplus U({\cal N}_2) \supset O({\cal N}_1) \oplus O({\cal N}_2)U(N)⊃U(N1​)⊕U(N2​)⊃O(N1​)⊕O(N2​) and U(N)⊃O(N)⊃O(N1)⊕O(N2)U({\cal N}) \supset O({\cal N}) \supset O({\cal N}_1) \oplus O({\cal N}_2)U(N)⊃O(N)⊃O(N1​)⊕O(N2​) chains of a variety of interacting boson models are used to investigate the probability of occurrence of a given (ω1ω2)(ω_1 ω_2)(ω1​ω2​) irreducible representation (irrep) to be the ground state in even-even nuclei; [ω1][ω_1][ω1​] and [ω2][ω_2][ω2​] are symmetric irreps of O(N1)O({\cal N}_1)O(N1​) and O(N2)O({\cal N}_2)O(N2​) respectively. Numerical results obtained for N1≥3,N2=1{\cal N}_1 \geq 3, {\cal N}_2=1N1​≥3,N2​=1 and N1,N2≥3{\cal N}_1, {\cal N}_2 \geq 3N1​,N2​≥3 situations are well explained by an extended Hartree-Bose mean-field analysis.
We investigate the origin of order in the low-lying spectra of many-body systems with random two-body interactions. Contrary to the common belief our study based both on analytical as well as on numerical arguments shows that these are the higher JJJ-sectors whose ground states are more orderly than the ones in the J=0-sector. A predominance of J=0 ground states turns out to be the result of putting on together states with different characteristic energy scales from different JJJ-sectors.
We investigate the origin of order in the low-lying spectra of many-body systems with random two-body interactions. Our study based both on analytical as well as on numerical arguments shows that except for the most J-stretched states, the ground states in the higher J-sectors are more orderly and develop larger energy gaps than the ones in the J=0-sector. Due to different characteristic energy scales in different J-sectors the J=0 ground states may predominate only when all the states are taken together.
Dynamical symmetries in closed systems lead to conservation laws and to selection rules that are based on those conservation laws. However, conservation laws are not applica- ble to open systems that undergo irreversible transformations. More general selection rules are needed to determine whether, given two states, the transition from one state to the other is possible. Characterizing the asymmetry properties of quantum states using quantum information theoretical tools and methods is particularly fruitful for this purpose. This thesis is concerned with the problem of finding new general selection rules that hold for both open and closed systems. The usual approach to this problem relies heavily on group theory and involves a detailed study of the structure of the symmetry group. In this thesis, we approach the problem in a completely new way. Our approach is to use entanglement to investigate the asymmetry properties of quantum states. In order to do that, we embed the system’s Hilbert space in a larger tensor product Hilbert space, whereby all symmetric states are mapped to separable states, asymmetric states are mapped to entangled states, and the symmetric transformations between two states are replaced by local operations on their bipartite images. Our method is not restricted to only a specific group but applies, in general, to all symmetries that are associated with semi-simple compact Lie groups and their associated Lie algebras. The mapping of the original states to bipartite states that act on the larger Hilbert space enables us to use the well studied theory of entanglement to investigate the consequences of dynamic symmetries. For example, the monotonicity condition on measures of entanglement provide us with new selection rules. Under reversible transfor- mations, the entanglement of the bipartite image states becomes a conserved quantity. These entanglement-based conserved quantities are new and different from the conserved quantities based on expectation values of the Hamiltonian symmetry generators.