Finite-size correction and bulk hole-excitations for special case of an open XXZ chain with nondiagonal boundary terms at roots of unity

Abstract

Using our solution for the open spin-1/2 XXZ quantum spin chain with N spins and two arbitrary boundary parameters at roots of unity, the central charge and the conformal dimensions for bulk hole excitations are derived from the 1/N correction to the energy (Casimir energy).

BoundarySmatrix of an open XXZ spin chain with nondiagonal boundary terms

“…Reversing the signs of the boundary parameters in the present case leaves the Bethe equation unchanged, hence giving the same one-hole state. Interestingly, a separate one-hole state with 2-string is found [25]. Using a conjectured relation between "extra" roots, hole rapidity and the boundary parameters, which is again confirmed numerically for system up to about 60 sites, we derive the remaining eigenvalue which also agrees with Ghoshal-Zamolodchikov's result.…”

“…Hence, we focus only on the critical and repulsive regime, which corresponds in the sine-Gordon model to β 2 ∈ [4π, 8π) 1 . One-hole excitations for this model occur in even N sector [25] in contrast to the diagonal open spin-1/2 XXZ spin chain where such excitations appear in the odd N sector [11].…”

NLIE and finite size effects of the spin-1/2 XXZ and sine-Gordon models with two boundaries revisited

“…The result for the conformal dimension ∆, turns out to be similar to that of the XXZ spin chain models with diagonal boundary terms rather than the nondiagonal ones, to which the model studied here belongs to. Such a feature however had been encountered before in literature [42]. There it was pointed out that such a behaviour can be possibly attributed to spectral equivalences between diagonal-nondiagonal open XXZ spin chains [39]- [41].…”