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arXiv:1110.4204 · Zeitschrift für Naturforschung A

Spin Hamilton Operators, Symmetry Breaking, Energy Level Crossing, and Entanglement

Willi-Hans Steeb, Yorick Hardy, Jacqueline de Greef

Published Aug 20, 2012• 1 citation
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Abstract

We study finite-dimensional product Hilbert spaces, coupled spin systems, entanglement, and energy level crossing. The Hamilton operators are based on the Pauli group.We show that swapping the interacting term can lead from unentangled eigenstates to entangled eigenstates and from an energy spectrum with energy level crossing to avoided energy level crossing.

DOI
10.5560/zna.2012-0064

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We study finite-dimensional product Hilbert spaces, coupled spin systems, entanglement, and energy level crossing. The Hamilton operators are based on the Pauli group.We show that swapping the interacting term can lead from unentangled eigenstates to entangled eigenstates and from an energy spectrum with energy level crossing to avoided energy level crossing.
Abstract We consider a hierarchy of Hamilton operators Ĥ N in finite dimensional Hilbert spaces C2N \mathbb{C}^{2^N } C2N. We show that the eigenstates of Ĥ N are fully entangled for N even. We also calculate the unitary operator U N(t) = exp(—Ĥ N t/ħ) for the time evolution and show that unentangled states can be transformed into entangled states using this operator. We also investigate energy level crossing for this hierarchy of Hamilton operators.
Entanglement is defined for each vector subspace of the tensor product of two finite-dimensional Hilbert spaces, by applying the notion of operator entanglement to the projection operator onto that subspace. The operator Schmidt decomposition of the projection operator defines a string of Schmidt coefficients for each subspace, and this string is assumed to characterize its entanglement, so that a first subspace is more entangled than a second, if the Schmidt string of the second majorizes the Schmidt string of the first. The idea is applied to the antisymmetric and symmetric tensor products of a finite-dimensional Hilbert space with itself, and also to the tensor product of an angular momentum jjj with a spin 1∕21∕21∕2. When adapted to the subspaces of states of the nonrelativistic hydrogen atom with definite total angular momentum (orbital plus spin), within the space of bound states with a given total energy, this leads to a complete ordering of those subspaces by their Schmidt strings.
Motivated by Heilmann and Lieb's work [Ann. N. V. Acad. Sci. 172, 583 (1971)]. We discuss energy level crossings for the one-dimensional Hubbard model through the Bethe ansatz, constructing explicitly the degenerate eigenstates at the crossing points. After showing the existence of solutions for the Lieb--Wu equations with one-down spin, we solve them numerically and construct Bethe-ansatz eigenstates. We thus verify all the level crossings in the spectral flows observed by the numerical diagonalization method with one down-spin. For each of the solutions we obtain its energy spectral flow along the interaction parameter U. Then, we observe that some of the energy level crossings can not be explained in terms of U-independent symmetries. Dynamical symmetries of the Hubbard model are fundamental for identifying each of the spectral lines at the level crossing points. We show that the Bethe-ansatz eigenstates which degenerate at the points have distinct sets of eigenvalues of the higher conserved operators. We also show a twofold permanent degeneracy in terms of the Bethe-ansatz wave function.