{"title":"准动量星上八角形双磁振子态XXX模型能级的伽罗瓦对称","authors":"T. Lulek, M. Łabuz, J. Milewski, R. Stagraczyński","doi":"10.1016/S0034-4877(23)00039-3","DOIUrl":null,"url":null,"abstract":"<div><p>We consider the factor <em>υ</em><span> of the characteristic polynomial </span><em>w<sup>H</sup></em> (<em>x</em><span>) of the Heisenberg Hamiltonian </span><em>Ĥ</em> of the XXX model, corresponding to the generic star [<em>k</em> = ±1, ±3] of quasimomentum <em>k</em> for octagonal (<em>N</em><span><span><span> = 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 </span>Galois group<span><span> as the group of permutations of eigenenergies between the quasimomenta entering the generic star of the </span>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 </span></span>eigenstates are identified by their spectra. Some general remarks are made on Galois symmetries within the XXX integrable model.</span></p></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Galois symmetry of energy levels of the XXX model for the case of octagonal two-magnon states on the generic star of quasimomentum\",\"authors\":\"T. Lulek, M. Łabuz, J. Milewski, R. Stagraczyński\",\"doi\":\"10.1016/S0034-4877(23)00039-3\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We consider the factor <em>υ</em><span> of the characteristic polynomial </span><em>w<sup>H</sup></em> (<em>x</em><span>) of the Heisenberg Hamiltonian </span><em>Ĥ</em> of the XXX model, corresponding to the generic star [<em>k</em> = ±1, ±3] of quasimomentum <em>k</em> for octagonal (<em>N</em><span><span><span> = 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 </span>Galois group<span><span> as the group of permutations of eigenenergies between the quasimomenta entering the generic star of the </span>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 </span></span>eigenstates are identified by their spectra. Some general remarks are made on Galois symmetries within the XXX integrable model.</span></p></div>\",\"PeriodicalId\":49630,\"journal\":{\"name\":\"Reports on Mathematical Physics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2023-06-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Reports on Mathematical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0034487723000393\",\"RegionNum\":4,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Reports on Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0034487723000393","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
Galois symmetry of energy levels of the XXX model for the case of octagonal two-magnon states on the generic star of quasimomentum
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.
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.