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Singular solutions, repeated roots and completeness for higher-spin chains

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arxiv 1312.2982 v2 pith:ZCIUDMR3 submitted 2013-12-10 math-ph hep-thmath.MP

Singular solutions, repeated roots and completeness for higher-spin chains

classification math-ph hep-thmath.MP
keywords solutionsrootssingularrepeatedstrangebethecompletenessconditions
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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We investigate the completeness of the solutions of the Bethe equations for the integrable spin-s isotropic (XXX) spin chain with periodic boundary conditions. Solutions containing the exact string i s, i (s-1), ..., -i(s-1), -is are singular. For s>1/2, there exist also "strange" solutions with repeated roots, which nevertheless are physical (i.e., correspond to eigenstates of the Hamiltonian). We derive conditions for the singular solutions and the solutions with repeated roots to be physical. We formulate a conjecture for the number of solutions with pairwise distinct roots in terms of the numbers of singular and strange solutions. Using homotopy continuation, we solve the Bethe equations numerically for s=1 and s=3/2 up to 8 sites, and find some support for the conjecture. We also exhibit several examples of strange solutions.

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  1. Spin-$s$ $U(1)$-eigenstate preparation

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    A Gray-code-based quantum circuit prepares arbitrary fixed-digit-sum (U(1)) eigenstates of spin-s chains, yielding the first preparation of spin-s XXX Bethe states.