Exact solution of an anisotropic J1–J2 model with the Dzyloshinsky–Moriya interactions at boundaries

Yusong Cao,Jian Wang,Yi Qiao,Junpeng Cao,Wen-Li Yang

Table 1. Numerical solutions of the BAEs (3.17), where $2N=4$, $\eta =0.6$, $a=1.3{\rm{i}}$, ${\xi }_{+}=0.3$, ${\xi }_{-}=1.8$, n indicates the number of energy levels and En is the corresponding energy. The energy En calculated from the Bethe roots is exactly the same as that obtained from the exact diagonalization of the Hamiltonian (2.1).

${\mu }_{1}$ ${\mu }_{2}$ ${\mu }_{3}$ ${\mu }_{4}$ En n $-0.3000+1.1177{\rm{i}}$ $-0.3000+1.3567{\rm{i}}$ ⋯ ⋯ −9.3206 1 $-0.6920-0.5799{\rm{i}}$ $-0.6920+0.5799{\rm{i}}$ $-0.3000+1.3595{\rm{i}}$ ⋯ −5.3779 2 $-0.3000+0.5769{\rm{i}}$ $-0.3000+1.3715{\rm{i}}$ ⋯ ⋯ −4.6013 3 $-0.6958-0.6407{\rm{i}}$ $-0.6958+0.6407{\rm{i}}$ $-0.3000-1.1392{\rm{i}}$ ⋯ −3.2736 4 $-0.3000+0.5994{\rm{i}}$ $-0.3000+1.1743{\rm{i}}$ ⋯ ⋯ −3.1533 5 $-0.3000-1.3867{\rm{i}}$ ⋯ ⋯ ⋯ −2.5986 6 $-0.3000-1.2109{\rm{i}}$ ⋯ ⋯ ⋯ −1.7460 7 $-0.6001-1.1974{\rm{i}}$ $-0.6001+1.1974{\rm{i}}$ ⋯ ⋯ 0.5545 8 $-0.9231-0.7838{\rm{i}}$ $-0.9231+0.7838{\rm{i}}$ $-0.3000-0.8309{\rm{i}}$ ⋯ 2.1627 9 $-0.3000-0.9609{\rm{i}}$ ⋯ ⋯ ⋯ 2.1688 10 $-0.5752-0.8182{\rm{i}}$ $-0.5752+0.8182{\rm{i}}$ ⋯ ⋯ 2.1989 11 −1.4562 −0.9176 $-0.3000+0.0712{\rm{i}}$ 1.8613 3.7768 12 −1.4094 −0.9430 $-0.3000+0.1085{\rm{i}}$ ⋯ 3.8209 13 $-0.6644-0.2051{\rm{i}}$ $-0.6644+0.2051{\rm{i}}$ ⋯ ⋯ 4.1002 14 $-0.3000-0.4181{\rm{i}}$ ⋯ ⋯ ⋯ 4.8439 15 ⋯ ⋯ ⋯ ⋯ 6.4448 16