5,569 research outputs found
Slavnov and Gaudin-Korepin Formulas for Models without Symmetry: the Twisted XXX Chain
We consider the XXX spin- Heisenberg chain on the circle with an
arbitrary twist. We characterize its spectral problem using the modified
algebraic Bethe anstaz and study the scalar product between the Bethe vector
and its dual. We obtain modified Slavnov and Gaudin-Korepin formulas for the
model. Thus we provide a first example of such formulas for quantum integrable
models without symmetry characterized by an inhomogenous Baxter
T-Q equation
Universal Bethe ansatz solution for the Temperley-Lieb spin chain
We consider the Temperley-Lieb (TL) open quantum spin chain with "free"
boundary conditions associated with the spin- representation of
quantum-deformed . We construct the transfer matrix, and determine its
eigenvalues and the corresponding Bethe equations using analytical Bethe
ansatz. We show that the transfer matrix has quantum group symmetry, and we
propose explicit formulas for the number of solutions of the Bethe equations
and the degeneracies of the transfer-matrix eigenvalues. We propose an
algebraic Bethe ansatz construction of the off-shell Bethe states, and we
conjecture that the on-shell Bethe states are highest-weight states of the
quantum group. We also propose a determinant formula for the scalar product
between an off-shell Bethe state and its on-shell dual, as well as for the
square of the norm. We find that all of these results, except for the
degeneracies and a constant factor in the scalar product, are universal in the
sense that they do not depend on the value of the spin. In an appendix, we
briefly consider the closed TL spin chain with periodic boundary conditions,
and show how a previously-proposed solution can be improved so as to obtain the
complete (albeit non-universal) spectrum.Comment: v2: 21 pages; minor revisions, references added, publishe
Algebraic Bethe ansatz for the Temperley-Lieb spin-1 chain
We use the algebraic Bethe ansatz to obtain the eigenvalues and eigenvectors
of the spin-1 Temperley-Lieb open quantum chain with "free" boundary
conditions. We exploit the associated reflection algebra in order to prove the
off-shell equation satisfied by the Bethe vectors.Comment: v2: 28 pages; minor revisions, publishe
The integrable quantum group invariant A_{2n-1}^(2) and D_{n+1}^(2) open spin chains
A family of A_{2n}^(2) integrable open spin chains with U_q(C_n) symmetry was
recently identified in arXiv:1702.01482. We identify here in a similar way a
family of A_{2n-1}^(2) integrable open spin chains with U_q(D_n) symmetry, and
two families of D_{n+1}^(2) integrable open spin chains with U_q(B_n) symmetry.
We discuss the consequences of these symmetries for the degeneracies and
multiplicities of the spectrum. We propose Bethe ansatz solutions for two of
these models, whose completeness we check numerically for small values of n and
chain length N. We find formulas for the Dynkin labels in terms of the numbers
of Bethe roots of each type, which are useful for determining the corresponding
degeneracies. In an appendix, we briefly consider D_{n+1}^(2) chains with other
integrable boundary conditions, which do not have quantum group symmetry.Comment: 47 pages; v2: two references added and minor change
A tale of two Bethe ans\"atze
We revisit the construction of the eigenvectors of the single and double-row
transfer matrices associated with the Zamolodchikov-Fateev model, within the
algebraic Bethe ansatz method. The left and right eigenvectors are constructed
using two different methods: the fusion technique and Tarasov's construction. A
simple explicit relation between the eigenvectors from the two Bethe ans\"atze
is obtained. As a consequence, we obtain the Slavnov formula for the scalar
product between on-shell and off-shell Tarasov-Bethe vectors.Comment: 28 pages; v2: 30 pages, added proof of (4.40) and (5.39), minor
changes to match the published versio
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