Geometric Upper Critical Dimensions of the Ising Model
Sheng Fang1,2 , Zongzheng Zhou3* , and Youjin Deng1,2,4*
1 Hefei National Research Center for Physical Sciences at the Microscales, University of Science and Technology of China, Hefei 230026, China2 MinJiang Collaborative Center for Theoretical Physics, College of Physics and Electronic Information Engineering, Minjiang University, Fuzhou 350108, China3 ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), School of Mathematics, Monash University, Clayton, Victoria 3800, Australia4 Shanghai Research Center for Quantum Sciences, Shanghai 201315, China
Abstract ：The upper critical dimension of the Ising model is known to be $d_{\rm c}=4$, above which critical behavior is regarded to be trivial. We hereby argue from extensive simulations that, in the random-cluster representation, the Ising model simultaneously exhibits two upper critical dimensions at $(d_{\rm c}=4,~d_{\rm p}=6)$, and critical clusters for $d \geq d_{\rm p}$, except the largest one, are governed by exponents from percolation universality. We predict a rich variety of geometric properties and then provide strong evidence in dimensions from 4 to 7 and on complete graphs. Our findings significantly advance the understanding of the Ising model, which is a fundamental system in many branches of physics.
收稿日期: 2022-06-12
Express Letter
出版日期: 2022-07-08
[1] Friedli S and Velenik Y 2017 Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction (Cambridge: Cambridge University Press) [2] Ising E 1925 Z. Phys. 31 253 [3] Onsager L 1944 Phys. Rev. 65 117 [4] Onsager first announced the result in a conference in 1942, while the official paper was not published until 1944 [5] Yang C N 1952 Phys. Rev. 85 808 [6] Wilson K G 1971 Phys. Rev. B 4 3174 [7] Wilson K G 1971 Phys. Rev. B 4 3184 [8] Wilson K G and Fisher M E 1972 Phys. Rev. Lett. 28 240 [9] Wilson K G 1975 Rev. Mod. Phys. 47 773 [10] Deng Y and Blöte H W J 2003 Phys. Rev. E 68 036125 [11] Ferrenberg A M, Xu J and Landau D P 2018 Phys. Rev. E 97 043301 [12] Hou P, Fang S, Wang J, Hu H and Deng Y 2019 Phys. Rev. E 99 042150 [13] Kos F, Poland D, Simmons-Duffin D and Vichi A 2016 J. High Energy Phys. 2016(08) 36 [14] Poland D, Rychkov S and Vichi A 2019 Rev. Mod. Phys. 91 015002 [15] Broadbent S R and Hammersley J M 1957 Mathematical Proceedings of the Cambridge Philosophical Society (Cambridge: Cambridge University Press) pp 629–641 [16] Stauffer D and Aharony A 2018 Introduction to Percolation Theory (New York: CRC Press) [17] Chayes J and Chayes L 1987 Commun. Math. Phys. 113 27 [18] Aizenman M and Newman C M 1984 J. Stat. Phys. 36 107 [19] Hara T and Slade G 1990 Commun. Math. Phys. 128 333 [20] Herega A 2015 J. Mater. Sci. Eng. A 5 409 [21] Hopfield J J 1982 Proc. Natl. Acad. Sci. USA 79 2554 [22] Buldyrev S V, Parshani R, Paul G, Eugene S H and Havlin S 2010 Nature 464 1025 [23] Mello I F, Squillante L, Gomes G O, Seridonio A C and de Souza M 2021 Physica A 573 125963 [24] Zeng G, Gao J, Shekhtman L, Guo S, Lv W, Wu J, Liu H, Levy O, Li D, Gao Z, Stanley H E and Havlin S 2020 Proc. Natl. Acad. Sci. USA 117 17528 [25] Ma Y P, Sudakov I, Strong C and Golden K M 2019 New J. Phys. 21 063029 [26] Brunk N E and Twarock R 2021 ACS Nano 15 12988 [27] Zhang L, Zeng G, Li D, Huang H J, Stanley H E and Havlin S 2019 Proc. Natl. Acad. Sci. USA 116 8673 [28] Grimmett G R 2006 The Random-Cluster Model (Berlin: Springer) [29] Wu F Y 1982 Rev. Mod. Phys. 54 235 [30] Swendsen R H and Wang J S 1987 Phys. Rev. Lett. 58 86 [31] Wolff U 1989 Phys. Rev. Lett. 62 361 [32] Chayes L and Machta J 1998 Physica A 254 477 [33] Zhang L, Michel M, Elçi E M and Deng Y 2020 Phys. Rev. Lett. 125 200603 [34] A complete graph with $V$ vertices is a graph in which each vertex is connected to all others. [35] Chayes L, Coniglio A, Machta J and Shtengel K 1999 J. Stat. Phys. 94 53 [36] Luczak M and Łuczak T 2006 Random Struct. & Algorithms 28 215 [37] Fang S, Zhou Z and Deng Y 2021 Phys. Rev. E 103 012102 [38] Fang S, Grimm J, Zhou Z and Deng Y 2020 Phys. Rev. E 102 022125 [39] Lv J P, Xu W, Sun Y, Chen K and Deng Y 2020 Natl. Sci. Rev. 8 nwaa212 [40] Lundow P H and Markström K 2015 Nucl. Phys. B 895 305 [41] Mertens S and Moore C 2018 Phys. Rev. E 98 022120 [42] Wittmann M and Young A P 2014 Phys. Rev. E 90 062137 [43] Flores-Sola E, Berche B, Kenna R and Weigel M 2016 Phys. Rev. Lett. 116 115701 [44] Grimm J, Elçi E M, Zhou Z, Garoni T M and Deng Y 2017 Phys. Rev. Lett. 118 115701 [45] Zhou Z, Grimm J, Fang S, Deng Y and Garoni T M 2018 Phys. Rev. Lett. 121 185701 [46] Fang S, Deng Y and Zhou Z 2021 Phys. Rev. E 104 064108 [47] Papathanakos V 2006 Ph.D. Dissertation (Princeton University) [48] Grimm J, Elçi E M, Zhou Z, Garoni T M and Deng Y 2017 Phys. Rev. Lett. 118 115701 [49] Zhou Z, Grimm J, Fang S, Deng Y and Garoni T M 2018 Phys. Rev. Lett. 121 185701 [50] Bollobás B, Grimmett G and Janson S 1996 Probab. Theory Relat. Fields 104 283 [51] Kenna R 2013 Order, Disorder and Criticality: Advanced Problems of Phase Transition Theory Volume 3 (Singapore: World Scientific) p 1 [52] Kenna R 2004 Nucl. Phys. B 691 292 [53] Aharony A, Gefen Y and Kapitulnik A 1984 J. Phys. A 17 L197 [54] Nahum A, Chalker J, Serna P, Ortuno M and Somoza A 2013 Phys. Rev. B 88 134411
[1]
.
[2]
.
[3]
.
[4]
.
[5]
.
[6]
.
[7]
.
[8]
.
[9]
LU Yun-Bin;LIAO Shu-Zhi**;PENG Hao-Jun;ZHANG Chun;ZHOU Hui-Ying;XIE Hao-Wen;OUYANG Yi-Fang;ZHANG Bang-Wei;
. Size Model of Critical Temperature for Grain Growth in Nano V and Au
[10]
WANG Ping;ZHENG Qiang;WANG Wen-Ge. Decay of Loschmidt Echo at a Critical Point in the Lipkin-Meshkov-Glick model
[11]
HU Bin;LI Fang;ZHOU Hou-Shun. Robustness of Complex Networks under Attack and Repair
[12]
GU Shi-Jian;TIAN Guang-Shan;LIN Hai-Qing. Pairwise Entanglement and Quantum Phase Transitions in Spin Systems
[13]
WU Jun;TAN Yue-Jin;DENG Hong-Zhong;LI Yong. A Robustness Model of Complex Networks with Tunable Attack Information Parameter
[14]
WANG Lin-Cheng;CUI Hai-Tao;YI Xue-Xi. Decoherence Induced by Spin Chain Environment
[15]
PAN Gui-Jun;ZHANG Duan-Ming;YIN Yan-Ping;HE Min-Hua. Avalanche Dynamics in Quenched Random Directed Sandpile Models
2022, Vol. 39 
