S.M. Tashpulatov. The structure of the essential spectrum and the discrete spectrum of the energy operator for six-electron systems in the Hubbard model. The second singlet state ... P. 210-230

We consider the energy operator of six-electron systems in the Hubbard model and study the structure of the essential spectrum and the discrete spectrum of the system for the second singlet state of the system. In the one- and two-dimensional cases, it is shown that the essential spectrum of the six-electron second singlet state operator consists of unions of seven segments, and the discrete spectrum of the system consists of a single eigenvalue lying below (above) the domain of the lower (upper, respectively) edge of the essential spectrum of this operator. In the three-dimensional case, there are the following situations: (a) the essential spectrum of the six-electron second singlet state operator consists of unions of seven segments, and the discrete spectrum of this operator consists of a single eigenvalue; (b) the essential spectrum of the six-electron second singlet state operator consists of unions of four segments, and the discrete spectrum of this operator is empty; (c) the essential spectrum of the six-electron second singlet state operator consists of unions of two segments, and the discrete spectrum of this operator is empty; (d) the essential spectrum of the six-electron second singlet state operator consists of a single segment, and the discrete spectrum of this operator is empty. Conditions are found under which each of the situations takes place.

Keywords: Hubbard model of six-electron systems, spectrum, essential spectrum, discrete spectrum, octet state, quintet state, triplet state, singlet state

Received March 30, 2023

Revised May 29, 2023

Accepted July 19, 2023

Sa’dulla M. Tashpulatov, Dr. Phys.-Math. Sci., leading researcher of the laboratory “Physics of many-particle system”, Institute of Nuclear Physics of the Academy of Sciences of the Republic of Uzbekistan, Tashkent, Republic of Uzbekistan, e-mail: sadullatashpulatov@yandex.com, toshpul@mail.ru

REFERENCES

1.   Hubbard J. Electron correlations in narrow energy bands. Proc. Roy. Soc. A., 1963, vol. 276, pp. 238–257. doi: 10.1098/rspa.1963.0204

2.   Gutzwiller M.C. Effect of correlation on the ferromagnetism of transition metals. Phys. Rev. Lett., 1963, vol. 10, no. 159, pp. 159–162. doi: 10.1103/PhysRevLett.10.159

3.   Kanamori J. Electron correlation and ferromagnetism of transition metals. Prog. Theor. Phys., 1963, vol. 30, no. 3, pp. 275–289. doi: 10.1143/PTP.30.275

4.   Anderson P.W. Localized magnetic states in metals. Phys. Rev., 1961, vol. 124, pp. 41–53. doi: 10.1103/PhysRev.124.41

5.   Shubin S.P., Wonsowsky S.V. On the electron theory of metals. Proc. Roy. Soc. A., 1934, vol. 145, pp. 159–172. doi: 10.1098/rspa.1934.0089

6.   Izyumov Yu.A. Hubbard model of strong correlations. Physics-Uspekhi, 1995, vol. 38, no. 4, pp. 385–408. doi: 10.1070/PU1995v038n04ABEH000081

7.   Arovas D.P., Berg E., Kivelson S.A., Raghy S. The Hubbard model. Annu. Rev. Condens. Matter Physics, 2022, vol. 13, pp. 239–274. doi: 10.1146/annurev-conmatphys-031620-102024

8.   Ovchinnikov S.G., Shneider E.I. Spectral functions in the Hubbard model with half-filling. Physics of the Solid State, 2004, vol. 46, no. 8, pp. 1469–1473. doi: 10.1134/1.1788780

9.   Izyumov Yu.A., Chashchin N.I., Alekseev D.S. Teoriya sil’no korrelirovannykh sistem. Metod proizvodyashchego funktsionala [Theory of strongly correlated systems. Generating functional method], Moscow: Inst. Komp. Issled., Izhevsk: Regul. Khaot. Dinamika Publ., 2006, 381 p. ISBN: 5-93972-502-3.

10.   Karpenko B.V., Dyakin V.V., Budrina G.L. Two electrons in Hubbard model. Fizika Metallov i Metallovedenie, 1986, vol. 61, no. 4, pp. 702–706 (in Russian).

11.   Tashpulatov S.M. Spectral properties of three-electron systems in the Hubbard model. Theor. Math. Phys., 2014, vol. 179, no. 3, pp. 712–728. doi: 10.1007/s11232-014-0173-y

12.   Tashpulatov S.M. Spectra of the energy operator of four-electron systems in the triplete state in the Hubbard model. J. Phys. Conf. Ser., 2016, vol. 697, article no. 012025, pp. 1–25. doi: 10.1088/1742-6596/697/1/012025

13.   Tashpulatov S.M. The structure of essential spectra and discrete spectrum of four-electron systems in the Hubbard model in a singlet state. Lobachevskii J. Math., 2017, vol. 38, no. 3, pp. 530–541. doi: 10.1134/S1995080217030246

14.   Tashpulatov  S.M. Structure of essential spectrum and discrete spectra of the energy operator of five-electron systems in the Hubbard model-doublet state. In: Operator theory and differential equations, Kusraev A.G., Totieva Z.D. (eds.), Cham, BirkhЈauser, 2021, pp. 275–301. doi: 10.1007/978-3-030-49763-7_19

15.   Tashpulatov S.M. Structure of essential spectra and discrete spectra of the energy operator of six-electron systems in the Hubbard model. First quintet and first singlet states. J. Appl. Math. Phys., 2022, vol. 10, no. 11, pp. 3424–3461. doi: 10.4236/jamp.2022.1011227

16.   Reed M., Simon B. Methods of modern mathematical physics. Vol. 1. Functional Analysis, NY, Acad. Press, 1972. doi: 10.1016/B978-0-12-585001-8.X5001-6 Translated to Russian under the title Metody sovremennoi matematicheskoi fiziki. Tom 1. Functsional’nyi analiz, Moscow, Mir Publ., 1977, 360 p.

17.   Reed M., Simon B. Methods of modern mathematical physics. Vol. 4. Operator analysis, NY, Acad. Press, 1978. ISBN: 9780125850049. Translated to Russian under the title Metody sovremennoi matematicheskoi fiziki. Tom 4. Analiz operatorov, Moscow, Mir Publ., 1982, 432 p.

18.   Ichinose T. Spectral properties of tensor products of linear operators. I. Trans. Amer. Math. Soc., 1978, vol. 235, pp. 75–113. doi: 10.1090/S0002-9947-1978-0472915-2

19.   Ichinose T. Spectral properties of tensor products of linear operators. II: the approximate point spectrum and Kato essential spectrum. Trans. American Math. Soc., 1978, vol. 237, pp. 223–254. doi: 10.1090/S0002-9947-1978-0479372-0

20.   Ichinose T. On the spectral properties of tensor products of linear operators in Banach spaces. Banach Center Publications, 1982, vol. 8, pp. 295–300. doi: 10.4064/-8-1-295-300

21.   Naimark M. A. Normed Rings, Wolters-Noordhoff, 1970, 572 p. Original Russian text was published in Naimark M. A., Normirovannye kol’tsa, Moscow, Nauka publ., 1968.

22.   Val’kov V.V., Ovchinnikov S. G., Petrakovskii O. G. Excitation spectrum of two-magnon systems in the easy-axis quasi-dimensional ferromagnet. Fizika tverdogo tela, 1988, vol. 30, no. 10, pp. 3044–3047 (in Russian).

Cite this article as: S.M. Tashpulatov. The structure of the essential spectrum and the discrete spectrum of the energy operator for six-electron systems in the Hubbard model. The second singlet state. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 3, pp. 210–230; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2023, Vol. 323, Suppl. 1, pp. S279–S299.