Processing math: 100%

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 E_n is the corresponding energy. The energy E_n calculated from the Bethe roots is exactly the same as that obtained from the exact diagonalization of the Hamiltonian (2.1).

${\mu }_{1}$	${\mu }_{2}$	${\mu }_{3}$	${\mu }_{4}$	E_n	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