Weak first-order transition and pseudoscaling behavior in the universality class of the $O(N)$ Ising model

A. O. Sorokin

Teoretičeskaâ i matematičeskaâ fizika, Tome 200 (2019) no. 2, pp. 310-323

Résumé

Using Monte Carlo and renormalization group methods, we investigate systems with critical behavior described by two order parameters: continuous $($vector$)$ and discrete (scalar). We consider two models of classical three-dimensional Heisenberg magnets with different numbers of spin components $N=1,\dots,4$: the model on a cubic lattice with an additional competing antiferromagnetic exchange interaction in a layer and the model on a body-centered lattice with two competing antiferromagnetic interactions. In both models, we observe a first-order transition for all values of $N$. In the case where competing exchanges are approximately equal, the first order of a transition is close to the second order, and pseudoscaling behavior is observed with critical exponents differing from those of the $O(N)$ model. In the case $N=2$, the critical exponents are consistent with the well-known indices of the class of magnets with a noncollinear spin ordering. We also give a possible explanation of the observed pseudoscaling in the framework of the renormalization group analysis.

Détail

Mots-clés : phase transition
Keywords: Monte Carlo method, renormalization group, frustrated magnet, pseudoscaling.

Bibliographie

[1] D. Loison, “Phase transitions in frustrated vector spin systems: numerical studies”, Frustrated Spin Systems, ed. H. T. Diep, World Sci., Singapore, 2004, 177–228 | DOI

[2] B. Delamotte, D. Mouhanna, M. Tissier, “Nonperturbative renormalization-group approach to frustrated magnets”, Phys. Rev. B, 69:13 (2004), 134413, 53 pp., arXiv: cond-mat/0309101 | DOI

[3] O. A. Starykh, “Unusual ordered phases of highly frustrated magnets: a review”, Rep. Prog. Phys., 78:5 (2015), 052502, 24 pp., arXiv: 1412.8482 | DOI

[4] H. Kawamura, “Phase transitions in Heisenberg antiferromagnets on triangular and layered-triangular lattices (invited)”, J. Appl. Phys., 61:8 (1987), 3590–3594 ; “Critical properties of helical magnets and triangular antiferromagnets”, 63:8 (1988), 3086–3088 | DOI | DOI

[5] G. Zumbach, “Almost second order phase transitions”, Phys. Rev. Lett., 71:15 (1993), 2421–2424 | DOI

[6] M. Tiesser, B. Delamotte, D. Mouhanna, “Frustrated heisenberg magnets: a nonperturbative approach”, Phys. Rev. Lett., 84:22 (2000), 5208–5211, arXiv: cond-mat/0001350

[7] M. Plischke, J. Oitmaa, “Ising models with $n>1$: a series-expansion approach”, Phys. Rev. B, 19:1 (1979), 487–493 | DOI

[8] J. R. Banavar, D. Jasnow, D. P. Landau, “Fluctuation-induced first-order transition in a bcc Ising model with competing interactions”, Phys. Rev. B, 20:9 (1979), 3820–3827 | DOI

[9] M. J. Velgakis, M. Ferer, “Fluctuation-induced, first-order transition in a bcc Ising model with competing interactions”, Phys. Rev. B, 27:1 (1983), 401–412 | DOI

[10] P. Butera, M. Comi, “Critical universality and hyperscaling revisited for Ising models of general spin using extended high-temperature series”, Phys. Rev. B, 65:14 (2002), 144431, 25 pp., arXiv: hep-lat/0112049 | DOI

[11] Y. Kamiya, N. Kawashima, C. D. Batista, “Dimensional crossover in the quasi-twodimensional Ising-O(3) model”, Phys. Rev. B, 84:21 (2011), 214429, 12 pp., arXiv: 1108.1599 | DOI

[12] A. O. Sorokin, A. V. Syromyatnikov, Klass universalnosti Izinga–Geizenberga v trekhmernykh frustrirovannykh magnitnykh sistemakh, Soobschenie No 2939, PIYaF, Gatchina, 2013

[13] A. O. Sorokin, “Critical and multicritical behavior in the Ising–Heisenberg universality class”, Phys. Lett. A, 382:48 (2018), 3455–3462 | DOI | MR

[14] A. O. Sorokin, “Perekhod tipa Izing-XY v trekhmernykh frustrirovannykh antiferromagnetikakh s kollinearnym spinovym uporyadocheniem”, Pisma v ZhETF, 109:6 (2019), 423–428 | DOI

[15] M. K. Ramazanov, A. K. Murtazaev, “Fazovye perekhody i kriticheskie svoistva v antiferromagnitnoi sloistoi modeli Izinga s uchetom vnutrisloinykh vzaimodeistvii vtorykh blizhaishikh sosedei”, Pisma ZhETF, 101:10 (2015), 793–798 | DOI

[16] A. K. Muratazaev, M. K. Ramazanov, F. A. Kasan-Ogly, D. R. Kurbanova, “Fazovye perekhody v antiferromagnitnoi modeli Izinga na ob'emno-tsentrirovannoi kubicheskoi reshetke s vzaimodeistviyami vtorykh blizhaishikh sosedei”, ZhETF, 147:1 (2015), 127–131 | DOI | DOI

[17] A. K. Muratazaev, M. K. Ramazanov, D. R. Kurbanova, M. K. Badiev, Ya. K. Abuev, “Issledovanie kriticheskikh svoistv modeli Izinga na ob'emno tsentrirovannoi kubicheskoi reshetke s uchetom vzaimodeistviya sleduyuschikh za blizhaishimi sosedei”, FTT, 59:6 (2017), 1082–1088 | DOI | DOI

[18] M. K. Ramazanov, A. K. Muratazaev, “Fazovye perekhody i kriticheskie svoistva v antiferromagnitnoi modeli Geizenberga na sloistoi kubicheskoi reshetke”, Pisma ZhETF, 106:2 (2017), 72–77 | DOI

[19] A. O. Sorokin, A. V. Syromyatnikov, “Transitions in there-dimensional magnets with extra broken symmetry”, Solid State Phenomena, 190 (2012), 63–66 | DOI

[20] C. L. Henley, “Ordering due to disorder in a frustrated vector antiferromagnet”, Phys. Rev. Lett., 62:17 (1989), 2056–2059

[21] E. F. Shender, “Antiferromagnitnye granaty s fluktuatsionno vzaimodeistvuyuschimi podreshetkami”, ZhETF, 83:1 (1982), 326–337

[22] E. Brezin, J. C. Le Guillou, J. Zinn-Justin, “Discussion of critical phenomena for general $n$-vector models”, Phys. Rev. B, 10:3 (1974), 892–900 | DOI

[23] K. De'Bell, D. J. W. Geldart, “Coefficients to $\mathrm{O}(\varepsilon^3)$ for the mixed fixed point of the $nm$-component field model”, Phys. Rev. B, 32:7 (1985), 4763–4765 | DOI

[24] Y. M. Pis'mak, A. Weber, F. J. Wegner, “Critical behavior of a general $O(n)$-symmetric model of two $n$-vector fields in $D = 4 - 2$”, J. Phys. A: Math. Theor., 42:9 (2009), 095003, 24 pp., arXiv: 0809.1568 | DOI

[25] P. Bak, D. Mukamel, “Physical realizations of $n\ge4$-component vector models. III. Phase transitions in Cr, Eu, MnS$_{2}$, Ho, Dy, and Tb”, Phys. Rev. B, 13:11 (1976), 5086–5094 | DOI

[26] S. A. Brazovskii, I. E. Dzyaloshinskii, B. G. Kukharenko, “Magnitnye fazovye perekhody pervogo roda i fluktuatsii”, ZhETF, 70:6 (1976), 2257–2267

[27] I. E. Dzyaloshinskii, “O kharaktere fazovykh perekhodov v gelikoidalnoe ili sinusoidalnoe sostoyanie magnetikov”, ZhETF, 72:5 (1977), 1930–1945

[28] D. R. Nelson, J. M. Kosterlitz, M. E. Fisher, “Renormalization-group analysis of bicritical and tetracritical points”, Phys. Rev. Lett., 33:14 (1974), 813–817 | DOI

[29] I. F. Lyuksyutov, V. L. Pokrovskii, D. E. Khmelnitskii, “Peresechenie linii perekhodov vtorogo roda”, ZhETF, 69:5 (1975), 1817–1824

[30] J. M. Kosterlitz, D. R. Nelson, M. E. Fisher, “Bicritical and tetracritical points in anisotropic antiferromagnetic systems”, Phys. Rev. B, 13:1 (1976), 412–432 | DOI

[31] T. Garel, P. Pfeuty, “Commensurability effects on the critical behaviour of systems with helical ordering”, J. Phys. C, 9:10 (1976), L245–L249 | DOI

[32] Z. Barak, M. B. Walker, “First-order phase transitions in Tb, Dy, and Ho”, Phys. Rev. B, 25:3 (1982), 1969–1972 | DOI

[33] H. Kawamura, “Renormalization-group analysis of chiral transitions”, Phys. Rev. B, 38:7 (1988), 4916–4928 | DOI

[34] P. Calabrese, A. Pelissetto, E. Vicari, “Multicritical phenomena in $\mathrm{O}{(n}_{1})\oplus\mathrm{O}{(n}_{2})$-symmetric theories”, Phys. Rev. B, 67:5 (2003), 054505, 12 pp., arXiv: cond-mat/0209580 | DOI

[35] S. A. Antonenko, A. I. Sokolov, K. B. Varnashev, “Chiral transitions in three-dimensional magnets and higher order $\varepsilon$ expansion”, Phys. Lett. A, 208:1 (1995), 161–164, arXiv: cond-mat/9803377 | DOI

[36] A. Pelissetto, P. Rossi, E. Vicari, “Large-$n$ critical behavior of ${\rm O}(n)\times{\rm O}(m)$ spin models”, Nucl. Phys. B, 607:3 (2001), 605–634, arXiv: hep-th/0104024 | DOI | MR

[37] P. Calabrese, P. Parruccini, “Five-loop $\varepsilon$ expansion for ${\rm O}(n)\times{\rm O}(m)$ spin models”, Nucl. Phys. B, 679:3 (2004), 568–596, arXiv: cond-mat/0308037 | DOI

[38] F. R. Brown, T. J. Woch, “Overrelaxed heat-bath and Metropolis algorithms for accelerating pure gauge Monte Carlo calculations”, Phys. Rev. Lett., 58:23 (1987), 2394–2396 | DOI

[39] M. Creutz, “Overrelaxation and Monte Carlo simulation”, Phys. Rev. D, 36:2 (1987), 515–519 | DOI

[40] K. Binder, “Finite size scaling analysis of Ising model block distribution functions”, Z. Phys. B, 43:2 (1981), 119–140 | DOI

[41] A. M. Ferrenberg, D. P. Landau, “Critical behavior of the three-dimensional Ising model: a high-resolution Monte Carlo study”, Phys. Rev. B, 44:10 (1991), 5081–5091 | DOI

[42] H. Kawamura, “Monte Carlo study of chiral criticality – XY and Heisenberg stacked-triangular antiferromagnets”, J. Phys. Soc. Japan, 61:4 (1992), 1299–1325 | DOI

[43] A. O. Sorokin, “Kriticheskoe povedenie trekhmernykh frustrirovannykh spiralnykh magnetikov”, ZhETF, 145:3 (2014), 481–490 | DOI | DOI

[44] A. O. Sorokin, A. V. Syromyatnikov, “Perekhod pervogo roda v trekhmernykh sistemakh s polnostyu narushennoi $O(3)$-simmetriei”, ZhETF, 139:6 (2011), 1148–1157 | DOI

[45] A. O. Sorokin, A. V. Syromyatnikov, “Perekhody v trekhmernykh XY-magnetikakh s dvumya kiralnymi parametrami poryadka”, ZhETF, 140:4 (2011), 771–776 | DOI

[46] M. Tiesser, B. Delamotte, D. Mouhanna, “XY frustrated systems: continuous exponents in discontinuous phase transitions”, Phys. Rev. B, 67:13 (2003), 134422, 11 pp., arXiv: cond-mat/0107183 | DOI

[47] M. L. Plumer, A. Mailhot, “Tricritical behavior of the frustrated XY antiferromagnet”, Phys. Rev. B, 50:21 (1994), 16113–16116, arXiv: cond-mat/9405009 | DOI

[48] S. Fujimoto, “Low-energy properties of two-dimensional quantum triangular antiferromagnets: nonperturbative renormalization group approach”, Phys. Rev. B, 73:18 (2006), 184401, 11 pp., arXiv: cond-mat/0511215 | DOI

[49] N. Tetradis, C. Wetterich, “Critical exponents from the effective average action”, Nucl. Phys. B, 422:3 (1994), 541–592, arXiv: hep-ph/9308214 | DOI