We consider the factor υ of the characteristic polynomial w^H (x) of the Heisenberg Hamiltonian Ĥ of the XXX model, corresponding to the generic star [k = ±1, ±3] of quasimomentum k for octagonal (N = 8) magnetic ring in the two-magnon sector. This factor is recognized as the fourth-degree polynomial with integer coefficients, indecomposable over the prime number field ℚ of rationals. We demonstrate the physical meaning of the corresponding Galois group as the group of permutations of eigenenergies between the quasimomenta entering the generic star of the Brillouin zone of octagon. In particular, we point out the role of intersection of this group with Galois group of the cyclotomic field, responsible for the translational symmetry of octagon. Bound and scattered two-magnon eigenstates are identified by their spectra. Some general remarks are made on Galois symmetries within the XXX integrable model.

我们考虑XXX 模型的海森堡哈密顿量Ĥ的特征多项式w ^H ( x ) 的因子υ ，对应于八角形 ( N = 8)磁环的拟动量k的通用星形 [ k = ±1, ±3]在二磁振子扇区。该因子被认为是具有整数系数的四次多项式，在有理数的素数域 ℚ 上不可分解。我们证明了相应伽罗瓦群的物理意义，即准门塔进入通用恒星之间的本征能排列群。八边形的布里渊区。特别是，我们指出了该群与分圆域的伽罗瓦群的交集的作用，该群负责八边形的平移对称性。束缚和散射的双磁振子本征态通过它们的光谱来识别。对 XXX 可积模型内的伽罗瓦对称性进行了一些一般性评论。