Self-Similarity Breaking: Anomalous Nonequilibrium Finite-Size Scaling and Finite-Time Scaling
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Abstract
Symmetry breaking plays a pivotal role in modern physics. Although self-similarity is also a symmetry, and appears ubiquitously in nature, a fundamental question arises as to whether self-similarity breaking makes sense or not. Here, by identifying an important type of critical fluctuation, dubbed ‘phases fluctuations’, and comparing the numerical results for those with self-similarity and those lacking self-similarity with respect to phases fluctuations, we show that self-similarity can indeed be broken, with significant consequences, at least in nonequilibrium situations. We find that the breaking of self-similarity results in new critical exponents, giving rise to a violation of the well-known finite-size scaling, or the less well-known finite-time scaling, and different leading exponents in either the ordered or the disordered phases of the paradigmatic Ising model on two- or three-dimensional finite lattices, when subject to the simplest nonequilibrium driving of linear heating or cooling through its critical point. This is in stark contrast to identical exponents and different amplitudes in usual critical phenomena. Our results demonstrate how surprising driven nonequilibrium critical phenomena can be. The application of this theory to other classical and quantum phase transitions is also anticipated. -
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References
[1] Mandelbrot B B 1983 The Fractal Geometry of Nature New York: Freeman[2] Meakin P 1998 Fractal, Scaling and Growth far from Equilibrium Cambridge: Cambridge University[3] Fisher M E 1982 Scaling, Universality and Renormalization Group Theory, Lecture notes presented at the “Advanced Course on Critical Phenomena” The Merensky Institute of Physics, University of Stellenbosch, South Africa[4] Ma S K 1976 Modern Theory of Critical Phenomena Canada: W. A. Benjamin, Inc.[5] Pelissetto A and Vicari E 2002 Phys. Rep. 368 549 doi: 10.1016/S0370-15730200219-3[6] Kogut J B 1979 Rev. Mod. Phys. 51 659 doi: 10.1103/RevModPhys.51.659[7] Fisher M E and Barber M N 1972 Phys. Rev. Lett. 28 1516 doi: 10.1103/PhysRevLett.28.1516[8] Barber M N 1983 Finite-Size Scaling in Phase Transitions and Critical Phenomena edited by Domb C and Lebowitz J New York: Academic vol 8[9] Cardy J 1988 Finite Size Scaling Amsterdam: North-Holland[10] Privman V 1990 Finite Size Scaling and Numerical Simulations of Statistical Systems Singapore: World Scientific[11] Brézin E 1982 J. Phys. France 43 15 doi: 10.1051/jphys:0198200430101500[12] Brézin E and Zinn-Justin J 1985 Nucl. Phys. B 257 867 doi: 10.1016/0550-32138590379-7[13] Gasparini F M, Kimball M O, Mooney K P and Diaz-Avila M 2008 Rev. Mod. Phys. 80 1009 doi: 10.1103/RevModPhys.80.1009[14] Landau D P and Binder K 2005 A Guide to Monte Carlo Simulations in Statistical Physics 2nd edn Cambridge: Cambridge University[15] Flores-Sola E, Berche B, Kenna R and Weigel M 2016 Phys. Rev. Lett. 116 115701 doi: 10.1103/PhysRevLett.116.115701[16] Grimm J, Elçi E M, Zhou Z, Garoni T M and Deng Y J 2017 Phys. Rev. Lett. 118 115701 doi: 10.1103/PhysRevLett.118.115701[17] Suzuki M 1977 Prog. Theor. Phys. 58 1142 doi: 10.1143/PTP.58.1142[18] Wansleben S and Landau D P 1991 Phys. Rev. B 43 6006 doi: 10.1103/PhysRevB.43.6006[19] Hohenberg P C and Halperin B I 1977 Rev. Mod. Phys. 49 435 doi: 10.1103/RevModPhys.49.435[20] Folk R and Moser G 2006 J. Phys. A 39 R207 doi: 10.1088/0305-4470/39/24/R01[21] Swendsen R H and Wang J S 1987 Phys. Rev. Lett. 58 86 doi: 10.1103/PhysRevLett.58.86[22] Wolff U 1989 Phys. Rev. Lett. 62 361 doi: 10.1103/PhysRevLett.62.361[23] Gong S, Zhong F, Huang X and Fan S 2010 New J. Phys. 12 043036 doi: 10.1088/1367-2630/12/4/043036[24] Zhong F 2011 Applications of Monte Carlo Method in Science and Engineering edited by Mordechai S Intech, Rijeka, Croatia p 469 http://www.dwz.cn/B9Pe2[25] Zhong F and Chen Q Z 2005 Phys. Rev. Lett. 95 175701 doi: 10.1103/PhysRevLett.95.175701[26] Yin S, Qin X, Lee C and Zhong F 2012 arXiv:1207.1602 [cond-mat.stat-mech][27] Yin S, Mai P and Zhong F 2014 Phys. Rev. B 89 094108 doi: 10.1103/PhysRevB.89.094108[28] Huang Y, Yin S, Feng B and Zhong F 2014 Phys. Rev. B 90 134108 doi: 10.1103/PhysRevB.90.134108[29] Liu C W, Polkovnikov A and Sandvik A W 2014 Phys. Rev. B 89 054307 doi: 10.1103/PhysRevB.89.054307[30] Liu C W, Polkovnikov A, Sandvik A W and Young A P 2015 Phys. Rev. E 92 022128 doi: 10.1103/PhysRevE.92.022128[31] Liu C W, Polkovnikov A and Sandvik A W 2015 Phys. Rev. Lett. 114 147203 doi: 10.1103/PhysRevLett.114.147203[32] Feng B, Yin S and Zhong F 2016 Phys. Rev. B 94 144103 doi: 10.1103/PhysRevB.94.144103[33] Pelissetto A and Vicari E 2016 Phys. Rev. E 93 032141 doi: 10.1103/PhysRevE.93.032141[34] Xu N, Castelnovo C, Melko R G, Chamon C and Sandvik A W 2018 Phys. Rev. B 97 024432 doi: 10.1103/PhysRevB.97.024432[35] Xue M, Yin S and You L 2018 Phys. Rev. A 98 013619 doi: 10.1103/PhysRevA.98.013619[36] Cao X, Hu Q and Zhong F 2018 Phys. Rev. B 98 245124 doi: 10.1103/PhysRevB.98.245124[37] Gerster M, Haggenmiller B, Tschirsich F, Silvi P and Montangero S 2019 Phys. Rev. B 100 024311 doi: 10.1103/PhysRevB.100.024311[38] Li Y, Zeng Z and Zhong F 2019 Phys. Rev. E 100 020105R doi: 10.1103/PhysRevE.100.020105[39] Mathey S and Diehl S 2020 Phys. Rev. Res. 2 013150 doi: 10.1103/PhysRevResearch.2.013150[40] Clark L W, Feng L and Chin C 2016 Science 354 606 doi: 10.1126/science.aaf9657[41] Keesling A, Omran A, Levine H, Bernien H, Pichler H, Choi S, Samajdar R, Schwartz S, Silvi P, Sachdev S, Zoller P, Endres M, Greiner M, Vuletić V and Lukin M D 2019 Nature 568 207 doi: 10.1038/s41586-019-1070-1[42] Zhong F 2006 Phys. Rev. E 73 047102 doi: 10.1103/PhysRevE.73.047102[43] Huang Y, Yin S, Hu Q and Zhong F 2016 Phys. Rev. B 93 024103 doi: 10.1103/PhysRevB.93.024103[44] Metropolis N, Rosenbluth A W, Rosenbluth M N, Teller A M and Teller E 1953 J. Chem. Phys. 21 1087 doi: 10.1063/1.1699114[45] Glauber R J 1963 J. Math. Phys. 4 294 doi: 10.1063/1.1703954[46] Nightingale M P and Blote H W J 2000 Phys. Rev. B 62 1089 doi: 10.1103/PhysRevB.62.1089[47] Ferrenberg A M and Landau D P 1991 Phys. Rev. B 44 5081 doi: 10.1103/PhysRevB.44.5081[48] Kleinert H 1999 Phys. Rev. D 60 085001 doi: 10.1103/PhysRevD.60.085001[49] Kikuchi M and Ito N 1993 J. Phys. Soc. Jpn. 62 3052 doi: 10.1143/JPSJ.62.3052[50] Grassberger P 1995 Physica A 214 547 doi: 10.1016/0378-43719400285-2[51] Landau D P 1976 Phys. Rev. B 7 2997 doi: 10.1103/PhysRevB.13.2997[52] Yuan W and Zhong F 2020 in preparation[53] Wegner F W 1972 Phys. Rev. B 5 4529 doi: 10.1103/PhysRevB.5.4529 -
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1. Sieke, L.J., Harhoff, M., Schlichting, S. et al. Universal non-equilibrium scaling of cumulants across a critical point. Nuclear Physics B, 2025. DOI:10.1016/j.nuclphysb.2025.116808 2. Zhong, F.. Complete Universal Scaling in First-Order Phase Transitions. Chinese Physics Letters, 2024, 41(10): 100502. DOI:10.1088/0256-307X/41/10/100502 3. Zeng, S., Zhong, F. Theory of critical phenomena with long-range temporal interaction. Physica Scripta, 2023, 98(7): 075017. DOI:10.1088/1402-4896/acdcc0 4. Zeng, S., Szeto, S.P., Zhong, F. Theory of Critical Phenomena with Memory. Chinese Physics Letters, 2022, 39(12): 120501. DOI:10.1088/0256-307X/39/12/120501 5. Zuo, Z., Yin, S., Cao, X. et al. Scaling theory of the Kosterlitz-Thouless phase transition. Physical Review B, 2021, 104(21): 214108. DOI:10.1103/PhysRevB.104.214108 6. Yuan, W., Zhong, F. Phases fluctuations, self-similarity breaking and anomalous scalings in driven nonequilibrium critical phenomena. Journal of Physics Condensed Matter, 2021, 33(38): 385401. DOI:10.1088/1361-648X/ac0f9d 7. Yuan, W., Zhong, F. Phases fluctuations and anomalous finite-time scaling in an externally applied field on finite-sized lattices. Journal of Physics Condensed Matter, 2021, 33(37): 375401. DOI:10.1088/1361-648X/ac0ea8 Other cited types(0)