Non self-conjugate strings, singular strings and rigged configurations in the Heisenberg model

Abstract

We observe a different type of complex solutions in the isotropic spin-1/2 Heisenberg chain starting from N = 12, where the central rapidity of some of the odd-length strings becomes complex making not all the strings self-conjugate individually. We show that there are at most (N − 2)/2 singular solutions for M = 4, M = 5 down-spins and at most (N 2 − 6N + 8)/8 singular solutions for M = 6, M = 7 down-spins in an even-length chain with N ≥ 2M . Correspondence of the non self-conjugate string solutions and the singular string solutions to the rigged configurations has also been shown.

Bethe's quantum numbers and rigged configurations

“…The resulting rigged configurations agree with the ones obtained by a geometrical argument [S15] or by the computation of Takahashi's quantum numbers [GD] (see also [DG14]).…”

Singular eigenstates in the even(odd) length Heisenberg spin chain

“…where the string center λ j α for a length j-string is real, α represents the number of j-strings M j and the string deviations are given by ∆ j αa . In the limit that the deviations vanish, ∆ j αa → 0, equations (13) reduce to the equations…”

“…Alternatively, in the language of rigged configurations the singular solutions of an even-length spin chain are flip invariant [37]. Based on these, we in our previous work [13] assumed that the sum of the rapidities for the singular solutions of an even length spin-1/2 chain vanishes. Here, we discuss this assumption in the light of regularization as well as the singular solutions in general.…”

“…{λ α } = {−λ α }. It implies that the sum of rapidities of such a singular solution for even N vanishes [13], i.e,…”

“…Nonetheless, there has been growing interest in the solutions of the spin-1/2 XXX chain in recent years [10][11][12][13]. Although making use of the string hypothesis [14] one can estimate the total number of Bethe eigenstates, its certain assumptions do not always hold for any given finite length spin chain.…”