115 research outputs found
Singular solutions, repeated roots and completeness for higher-spin chains
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.Comment: 17 pages; many tables provided as supplemental material; v2: minor
change
Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity
We consider the sl(2)_q-invariant open spin-1/2 XXZ quantum spin chain of
finite length N. For the case that q is a root of unity, we propose a formula
for the number of admissible solutions of the Bethe ansatz equations in terms
of dimensions of irreducible representations of the Temperley-Lieb algebra; and
a formula for the degeneracies of the transfer matrix eigenvalues in terms of
dimensions of tilting sl(2)_q-modules. These formulas include corrections that
appear if two or more tilting modules are spectrum-degenerate. For the XX case
(q=exp(i pi/2)), we give explicit formulas for the number of admissible
solutions and degeneracies. We also consider the cases of generic q and the
isotropic (q->1) limit. Numerical solutions of the Bethe equations up to N=8
are presented. Our results are consistent with the Bethe ansatz solution being
complete.Comment: 34 pages; v2: reference added; v3: two more references added and
minor correction
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