Ex) Article Title, Author, Keywords
Ex) Article Title, Author, Keywords
New Phys.: Sae Mulli 2023; 73: 150-155
Published online February 28, 2023 https://doi.org/10.3938/NPSM.73.150
Copyright © New Physics: Sae Mulli.
Seung-Yeon Kim*
School of Liberal Arts and Sciences, Korea National University of Transportation, Chungju 27469, Korea
Correspondence to:*E-mail: sykimm@ut.ac.kr
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License(http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
The exact integer values for the density of states
Keywords: J1-J2 Ising model, Partition function zeros
The Hamiltonian H of the Ising model on a square lattice with the nearest-neighbor (
where
and
Here,
The Ising model on a square lattice with the nearest-neighbor and next-nearest-neighbor interactions has not been solved yet[1,2]. Even the exact critical line of the Ising model on a square lattice with the nearest-neighbor and next-nearest-neighbor interactions still remains unknown[3,4]. Until now, numerous mean-field theories and computational methods have been applied to understand the properties of the Ising model on a square lattice with the nearest-neighbor and next-nearest-neighbor interactions[3-17]. Recently, Bobak
In this study, through the exact enumeration method[18-20], we evaluate the exact integer values for the density of states
Yang and Lee[21,22] introduced the concept of the partition function zeros in the complex fugacity plane to explain naturally the singularities of the various thermodynamic functions appearing in phase transitions and critical phenomena. Using partition function zeros in the complex fugacity plane, Yang and Lee provided an important insight into the Ising ferromagnet in an external magnetic field[21,22]. Similarly, Fisher[23] introduced the concept of the partition function zeros in the complex temperature plane, based on the Onsager solution[1] of the square-lattice Ising model in the absence of an external magnetic field. Recently, the partition function zeros in the complex temperature plane have been studied for the Ising model on a square lattice with the nearest-neighbor and next-nearest-neighbor interactions[14-17].
The partition function of the Ising model on a square lattice with a ratio of 5:2 between the nearest-neighbor and next-nearest-neighbor interactions can be simply written as
where
Table 1 shows the exact integer values for the density of states
Table 1 Exact integer values for the density of states
0 | 2 | 19 | 24 | 28 | 36 | 37 | 48 |
38 | 108 | 42 | 48 | 43 | 48 | 46 | 90 |
47 | 288 | 51 | 336 | 52 | 96 | 54 | 4 |
55 | 48 | 56 | 1062 | 57 | 320 | 58 | 72 |
60 | 336 | 61 | 960 | 62 | 288 | 64 | 96 |
65 | 1728 | 66 | 1500 | 67 | 192 | 69 | 384 |
70 | 2832 | 71 | 1440 | 72 | 48 | 73 | 96 |
74 | 1884 | 75 | 4752 | 76 | 1734 | 77 | 360 |
78 | 156 | 79 | 4224 | 80 | 5184 | 81 | 1536 |
82 | 324 | 83 | 1536 | 84 | 8132 | 85 | 6048 |
86 | 1596 | 87 | 96 | 88 | 4140 | 89 | 9888 |
90 | 6240 | 91 | 1464 | 92 | 1320 | 93 | 8400 |
94 | 12276 | 95 | 7200 | 96 | 1644 | 97 | 2856 |
98 | 10320 | 99 | 12096 | 100 | 6276 | 101 | 2016 |
102 | 4962 | 103 | 12480 | 104 | 12276 | 105 | 6096 |
106 | 2454 | 107 | 4992 | 108 | 10394 | 109 | 9912 |
110 | 5376 | 111 | 2848 | 112 | 5202 | 113 | 8352 |
114 | 8132 | 115 | 4848 | 116 | 2220 | 117 | 2816 |
118 | 4128 | 119 | 4224 | 120 | 3024 | 121 | 1440 |
122 | 1128 | 123 | 1584 | 124 | 1884 | 125 | 1776 |
126 | 1110 | 127 | 384 | 128 | 252 | 129 | 408 |
130 | 372 | 131 | 360 | 132 | 84 | 133 | 48 |
134 | 96 | 135 | 48 | 136 | 90 | 137 | 48 |
138 | 38 | 139 | 24 | 144 | 4 | 150 | 2 |
corresponding approximately to 1.6069×1060. Counting all these 2200 states is extremely difficult. Table 2 shows only the maximum values for the densities of states
Table 2 Maximum integer values for the densities of states
3 | 12480 |
4 | 109057776 |
5 | 19146127630520 |
6 | 62718661425827505000 |
7 | 3531358058070218022029537560 |
8 | 3288528928349183706118352094331199232 |
9 | 49946049806391327094111391445695735994783824892 |
10 | 12304054332949238819495260659807413163084981913792663009984 |
corresponding approximately to 1.2304×1058.
Figure 1 shows the exact distributions of the partition function zeros in the complex
Figure 2 shows the first partition function zero
Following the finite-size scaling law[28,45], as
Table 3 shows the thermal scaling exponent
Table 3 Thermal scaling exponent
3 | 1.10211205 |
4 | 1.07536327 |
5 | 1.05967472 |
6 | 1.04940245 |
7 | 1.04216417 |
8 | 1.03679064 |
9 | 1.03264339 |
Figure 3 shows the first partition function zero
Table 4 Thermal scaling exponent
3 | 0.95176092 |
4 | 0.94559473 |
5 | 0.93829827 |
6 | 0.93518510 |
7 | 0.93481865 |
8 | 0.93578778 |
9 | 0.93729744 |
Through the exact enumeration method, we obtained the exact integer values for the density of states
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (grant number: NRF-2017R1D1A3B06035840).