The $p$-adic Ising model in an external field on a Cayley tree: periodic Gibbs measures

F. M. Mukhamedov   ; M. M. Rahmatullaev   ; A. M. Tukhtabaev   ; R. Mamadjonov

Teoretičeskaâ i matematičeskaâ fizika, Tome 216 (2023) no. 2, pp. 383-400

Résumé

We consider the generalized Gibbs measures corresponding to the $p$-adic Ising model in an external field on the Cayley tree of order two. It is established that if $p\equiv 1\,(\operatorname{mod}\, 4)$, then there exist three translation-invariant and two $G_2^{(2)}$-periodic non-translation-invariant $p$-adic generalized Gibbs measures. It becomes clear that if $p\equiv 3\,(\operatorname{mod}\, 4)$, $p\neq3$, then one can find only one translation-invariant $p$-adic generalized Gibbs measure. Moreover, the considered model also exhibits chaotic behavior if $|\eta-1|_p<|\theta-1|_p$ and $p\equiv 1\,(\operatorname{mod}\, 4)$. It turns out that even without $|\eta-1|_p<|\theta-1|_p$, one could establish the existence of $2$-periodic renormalization-group solutions when $p\equiv 1\,(\operatorname{mod}\, 4)$. This allows us to show the existence of a phase transition.

Détail

Keywords: $p$-adic numbers, Ising model, $p$-adic generalized Gibbs measure, translation invariance, periodicity
Mots-clés : phase transition.

Bibliographie

[1] Kh.-O. Georgi, Gibbsovskie mery i fazovye perekhody, Mir, M., 1992 | MR

[2] A. H. Kolmogorov, Osnovnye ponyatiya teorii veroyatnostei, Nauka, M., 1974

[3] A. Khrennikov, “$p$-Adic stochastics and Dirac quantization with negative probabilities”, Internat. J. Theor. Phys., 34:12 (1995), 2423–2433 | DOI | MR

[4] A. Yu. Khrennikov, “O rasshirenii chastotnogo podkhoda R. fon Mizesa i aksiomaticheskogo podkhoda A. N. Kolmogorova na $p$-adicheskuyu teoriyu veroyatnostei”, Teoriya veroyatn. i ee primen., 40:2 (1995), 458–464 | DOI | MR | Zbl

[5] I. V. Volovich, “$p$-Adic string”, Class. Quantum Grav., 4:4 (1987), L83–L87 | DOI

[6] S. Albeverio, R. Cianci, A. Yu. Khrennikov, “$p$-Adic valued quantization”, $p$-Adic Numbers Ultrametric Anal. Appl., 1:2 (2009), 91–104 | DOI | MR

[7] V. A. Avetisov, A. H. Bikulov, S. V. Kozyrev, “Application of $p$-adic analysis to models of breaking of replica symmetry”, J. Phys. A: Math. Gen., 32:50 (1999), 8785–8791 | DOI | MR

[8] I. Ya. Aref'eva, B. Dragovich, P. H. Frampton, I. V. Volovich, “The wave function of the Universe and $p$-adic gravity”, Internat. J. Modern Phys. A, 6:24 (1991), 4341–4358 | DOI | MR

[9] E. Arroyo-Ortiz, W. A. Zúñiga-Galindo, “Construction of $p$-adic covariant quantum fields in the framework of white noise analysis”, Rep. Math. Phys., 84:1 (2019), 1–34 | DOI | MR

[10] W. A. Zúñiga-Galindo, “Eigen's paradox and the quasispecies model in a non-Archimedean framework”, Phys. A, 602 (2022), 127648, 18 pp. | DOI | MR

[11] B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, I. V. Volovich, “On $p$-adic mathematical physics”, $p$-Adic Numbers Ultrametric Anal. Appl., 1:1 (2009), 1–17 | DOI | MR

[12] B. Dragovich, A Yu. Khrennikov, S. V. Kozyrev, I. V. Volovich, E. I. Zelenov, “$p$-Adic mathematical physics: the first 30 years”, $p$-Adic Numbers Ultrametric Anal. Appl., 9:2 (2017), 87–121 | DOI | MR

[13] H. García-Compeán, E. Y. López, W. A. Zúñiga-Galindo, “$p$-Adic open string amplitudes with Chan–Paton factors coupled to a constant $B$-field”, Nucl. Phys. B, 951 (2020), 114904, 33 pp. | DOI

[14] A. Yu. Khrennikov, $p$-Adic Valued Distributions in Mathematical Physics, Mathematics and Its Applications, 309, Kluwer, Dordrecht, 1994 | DOI

[15] A. Yu. Khrennikov, S. V. Kozyrev, W. A. Zúñiga-Galindo, Ultrametric Pseudodifferential Equations and Applications, Encyclopedia of Mathematics and its Applications, 168, Cambridge Univ. Press, Cambridge, 2018 | DOI

[16] B. C. Vladimirov, I. V. Volovich, E. I. Zelenov, $p$-Adicheskii analiz i matematicheskaya fizika, Nauka, M., 1994 | MR | MR | Zbl

[17] W. A. Zúñiga-Galindo, “Non-Archimedean statistical field theory”, Rev. Math. Phys., 34:8 (2022), 2250022, 41 pp., arXiv: 2006.05559 | DOI | Zbl

[18] W. A. Zúñiga-Galindo, S. M. Torba, “Non-Archimedean Coulomb gases”, J. Math. Phys., 61:1 (2020), 013504, 16 pp. | DOI | MR

[19] A. Yu. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Mathematics and Its Applications, 427, Springer, Dordrecht, 1997 | DOI

[20] A. C. M. van Rooij, Non-Archimedean Functional Analysis, Monographs and Textbooks in Pure and Applied Mathematics, 51, Marcel Dekker, New York, 1978 | Zbl

[21] A. Yu. Khrennikov, Nearkhimedov analiz i ego prilozheniya, Fizmatlit, M., 2003 | MR | Zbl

[22] A. Yu. Khrennikov, “Generalized probabilities taking values in non-Archimedean fields and topological groups”, Russ. J. Math. Phys., 14:2 (2007), 142–159 | DOI | MR

[23] A. Khrennikov, S. Ludkovsky, “Stochastic processes on non-Archimedean spaces with values in non-Archimedean fields”, Markov Process. Related Fields, 9:1 (2003), 131–162, arXiv: math/0110305 | MR

[24] F. Mukhamedov, O. Khakimov, “Chaos in $p$-adic statistical lattice models: Potts model”, Advances in Non-Archimedean Analysis and Applications. The $p$-adic Methodology in STEAM-H, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics Health, eds. W. A. Zúñiga-Galindo, B. Toni, Springer Nature, Cham, 2022, 115–165 | DOI

[25] R. Bekster, Tochno reshaemye modeli v statisticheskoi mekhanike, Mir, M., 1985 | MR | MR | Zbl

[26] T. P. Eggarter, “Cayley trees, the Ising problem, and the thermodynamic limit”, Phys. Rev. B, 9:7 (1974), 2989–2992 | DOI

[27] O. N. Khakimov, “On $p$-adic Gibbs measures for Ising model with four competing interactions”, $p$-Adic Numbers Ultrametric Anal. Appl., 5:3 (2013), 194–203 | DOI | MR

[28] O. N. Khakimov, “On a generalized $p$-adic Gibbs measure for Ising model on trees”, $p$-Adic Numbers Ultrametric Anal. Appl., 6:3 (2014), 207–217 | DOI | MR

[29] M. Khamraev, F. M. Mukhamedov, “On $p$-adic $\lambda$-model on the Cayley tree”, J. Math. Phys., 45:11 (2004), 4025–4034 | DOI | MR

[30] F. Mukhamedov, O. Khakimov, “Translation-invariant generalized $p$-adic Gibbs measures for the Ising model on Cayley trees”, Math. Methods Appl. Sci., 44:16 (2021), 12302–12316 | DOI | MR

[31] F. Mukhamedov, O. Khakimov, “On Julia set and chaos in $p$-adic Ising model on the Cayley tree”, Math. Phys. Anal. Geom., 20:4 (2017), 23, 14 pp. | DOI

[32] M. M. Rakhmatullaev, O. N. Khakimov, A. M. Tukhtaboev, “O $p$-adicheskoi obobschennoi mere Gibbsa dlya modeli Izinga na dereve Keli”, TMF, 201:1 (2019), 126–136 | DOI | DOI | MR

[33] U. A. Rozikov, O. N. Khakimov, “$p$-Adicheskie mery Gibbsa i markovskie sluchainye polya na schetnykh grafakh”, TMF, 175:1 (2013), 84–92 | DOI | DOI | MR | Zbl

[34] H. Diao, C. E. Silva, “Digraph representations of rational functions over the $p$-adic numbers”, $p$-Adic Numbers, Ultametric Anal. Appl., 3:1 (2011), 23–38 | DOI

[35] M. L. Lapidus, L. Hùng, M. van Frankenhuijsen, “$p$-Adic fractal strings of arbitrary rational dimensions and Cantor strings”, $p$-Adic Numbers, Ultametric Anal. Appl., 13:3 (2021), 215–230 | DOI

[36] N. Memić, “Sets of minmality of $(1-1)$-rational functions”, $p$-Adic Numbers, Ultametric Anal. Appl., 10:3 (2018), 209–221 | DOI | MR

[37] F. Mukhamedov, “Renormalization method in $p$-adic $\lambda$-model on the Cayley tree”, Internat. J. Theor. Phys., 54:10 (2015), 3577–3595 | DOI

[38] F. Mukhamedov, O. Khakimov, “Phase transition and chaos: $p$-adic Potts model on a Cayley tree”, Chaos Solitons Fractals, 87 (2016), 190–196 | DOI | MR

[39] F. Mukhamedov, O. Khakimov, “Chaotic behavior of the $p$-adic Potts–Bethe mapping”, Discrete Contin. Dyn. Syst., 38:1 (2018), 231–245 | DOI | MR

[40] O. Khakimov, F. Mukhamedov, “Chaotic behavior of the $p$-adic Potts-Bethe mapping II”, Ergod. Theory Dyn. Syst., 42:11 (2022), 3433–3457 | DOI

[41] F. Mukhamedov, H. Akin, “On non-Archimedean recurrence equations and their applications”, J. Math. Anal. Appl., 423:2 (2015), 1203–1218 | DOI | MR

[42] A. Le Ny, L. Liao, U. A. Rozikov, “$p$-Adic boundary laws and Markov chains on trees”, Lett. Math. Phys., 110:10 (2020), 2725–2741 | DOI | MR

[43] F. M. Mukhamedov, M. Saburov, O. N. Khakimov, “On $p$-adic Ising–Vannimenus model on an arbitrary order Cayley tree”, J. Stat. Mech., 2015 (2015), P05032, 26 pp. | DOI

[44] M. Rahmatullaev, A. Tukhtabaev, “Non periodic $p$-adic generilazed Gibbs measure for the Ising model”, $p$-Adic Numbers Ultrametric Anal. Appl., 11:4 (2019), 319–327 | DOI | MR

[45] M. Rakhmatullaev, A. Tukhtabaev, “On periodic $p$-adic generalized Gibbs measures for Ising model on a Cayley tree”, Lett. Math. Phys., 112:6 (2022), 112, 18 pp. | DOI | Zbl

[46] F. Mukhamedov, H. Akin, M. Dogan, “On chaotic behaviour of the $p$-adic generalized Ising mapping and its application”, J. Difference Equ. Appl., 23:9 (2017), 1542–1561 | DOI | MR

[47] N. Koblits, $p$-Adicheskie chisla, $p$-adicheskii analiz i dzeta-funktsii, Sovremennaya matematika. Vvodnye kursy, Mir, M., 1982 | MR | MR | Zbl | Zbl

[48] F. M. Mukhamedov, O. N. Khakimov, “$p$-Adicheskie monomialnye uravneniya i ikh vozmuscheniya”, Izv. RAN. Ser. matem., 84:2 (2020), 152–165 | DOI | DOI | MR

[49] U. A. Rozikov, Gibbs Measures on Cayley Trees, World Sci., Singapore, 2013 | MR

[50] F. Mukhamedov, B. Omirov, M. Saburov, “On cubic equations over $p$-adic fields”, Int. J. Number Theory, 10:5 (2014), 1171–1190 | DOI

[51] K. H. Rosen, Elementary Number Theory and Its Applications, Addison Wesley, Pearson, 2011 | MR

[52] F. Mukhamedov, M. Dogan, “On $p$-adic $\lambda$-model on the Cayley tree II: Phase transitions”, Rep. Math. Phys., 75:1 (2015), 25–46 | DOI | MR