Galois symmetry of energy levels of the XXX model for the case of octagonal two-magnon states on the generic star of quasimomentum

IF 1 4区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL Reports on Mathematical Physics Pub Date : 2023-06-01 DOI:10.1016/S0034-4877(23)00039-3
T. Lulek, M. Łabuz, J. Milewski, R. Stagraczyński
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Abstract

We consider the factor υ of the characteristic polynomial wH (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.

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准动量星上八角形双磁振子态XXX模型能级的伽罗瓦对称
我们考虑了XXX模型的海森堡哈密顿量Ĥ的特征多项式wH (x)的因子υ,对应于双磁振子扇区中八角形(N = 8)磁环准动量k的一般星[k =±1,±3]。该因子被认为是具有整数系数的四次多项式,在有理数的素数域上不可分解。我们证明了相应伽罗瓦群的物理意义,即进入八边形布里渊带的一般星的准动量之间的本征能置换群。特别地,我们指出了这个群与环切场的伽罗瓦群的交点对八边形平移对称的作用。结合和散射双磁振子本征态由它们的光谱来识别。对XXX可积模型中的伽罗瓦对称性作了一般性的评述。
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来源期刊
Reports on Mathematical Physics
Reports on Mathematical Physics 物理-物理:数学物理
CiteScore
1.80
自引率
0.00%
发文量
40
审稿时长
6 months
期刊介绍: Reports on Mathematical Physics publish papers in theoretical physics which present a rigorous mathematical approach to problems of quantum and classical mechanics and field theories, relativity and gravitation, statistical physics, thermodynamics, mathematical foundations of physical theories, etc. Preferred are papers using modern methods of functional analysis, probability theory, differential geometry, algebra and mathematical logic. Papers without direct connection with physics will not be accepted. Manuscripts should be concise, but possibly complete in presentation and discussion, to be comprehensible not only for mathematicians, but also for mathematically oriented theoretical physicists. All papers should describe original work and be written in English.
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