pISSN 0374-4914 eISSN 2289-0041

## Research Paper

New Phys.: Sae Mulli 2023; 73: 150-155

Published online February 28, 2023 https://doi.org/10.3938/NPSM.73.150

## Partition Function Zeros and Phase Transitions of the Ising Model on a Square Lattice with a Ratio of 5:2 between the Nearest-Neighbor and Next-Nearest-Neighbor Interactions

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

Received: November 15, 2022; Revised: December 16, 2022; Accepted: December 16, 2022

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 Ω(E) (as a function of energy E) of the Ising model with a ratio of 5:2 between the nearest-neighbor (J1) and next-nearest-neighbor (J2) interactions on L×2L square lattices with free boundary conditions in the L-direction and periodic boundary conditions in the 2L-direction are evaluated up to L=10 in this study. Given the exact density of states Ω(E), the exact partition function Z(a)=EΩ(E)aE can be constructed, where a=e2βJ1/5 and E=J1E1+J2E2=5E1+2E2. The exact distributions of the partition function zeros in the complex a=e2βJ1/5 plane of the Ising model on a square lattice with a ratio of 5:2 between the nearest-neighbor and next-nearest-neighbor interactions are obtained subsequently. Furthermore, two phase transitions of the ferromagnetic phase transition and antiferromagnetic phase transition are investigated from the exact partition function zeros in the complex a plane.

Keywords: J1-J2 Ising model, Partition function zeros

The Hamiltonian H of the Ising model on a square lattice with the nearest-neighbor (J1) and next-nearest-neighbor (J2) interactions is expressed as

H=2(J1E1+J2E2),

where

E1=12 i,j(1σiσj)

and

E2=12 i,k(1σiσk).

Here, σi(=±1) is the microscopic magnetic spin on a lattice site i. The sum i,j is obtained over all nearest-neighbor i,j pairs. Similarly, the sum i,k is obtained over all next-nearest-neighbor i,k pairs.

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 et al.[13] reported a first-order transition line of the Ising antiferromagnet (J1<0) on a square lattice with the nearest-neighbor and next-nearest-neighbor interactions for 0.5<R=J2/|J1|<0.3340 using effective field theory with nine-spin cluster approximations. The Ising model on a square lattice with the nearest-neighbor and next-nearest-neighbor interactions can be conveniently studied for certain values with the ratio of two interactions J1 and J2, R=J2/|J1|.

In this study, through the exact enumeration method[18-20], we evaluate the exact integer values for the density of states Ω(E) of the Ising model on L×2L square lattices with a ratio of 5:2 between the nearest-neighbor and next-nearest-neighbor interactions up to L=10 for the first time. We use L×2L square lattices with free boundary conditions in the L-direction and periodic boundary conditions in the 2L-direction[14,15]. From the exact density of states, we obtain the exact distributions of the partition function zeros in the complex temperature plane of the Ising model on a square lattice with a ratio of 5:2 between the nearest-neighbor and next-nearest-neighbor interactions for the first time. The theory of partition function zeros has been extensively applied to efficiently understand the properties of various phase transitions and critical phenomena[14-50]. We investigate phase transitions of the Ising ferromagnet (J1>0, R=0.4) and the Ising antiferromagnet (J1<0, R=-0.4) from the exact distributions of the partition function zeros in the complex temperature plane of the Ising model on a square lattice with a ratio of 5:2 between the nearest-neighbor and next-nearest-neighbor interactions.

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].

### II. Partition function and density of states

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

Z(a)=EΩ(E)aE,

where Ω(E) is the density of states, E=5E1+2E2, and a=e2βJ1/5 with the inverse temperature β=1/kBT. The partition function is a polynomial with positive integer coefficients Ω(E).

Table 1 shows the exact integer values for the density of states Ω(E) of the Ising model on 3×6 square lattice with a ratio of 5:2 between the nearest-neighbor and next-nearest-neighbor interactions. Even without the exact enumeration method[18-20], we can obtain the exact integer values, presented in Table 1. However, because the total number of states increases exponentially as the system size increases, we need the exact enumeration method to obtain the density of states for larger systems. The total number of states of the Ising model on 10×20 square lattice with a ratio of 5:2 between the nearest-neighbor and next-nearest-neighbor interactions is

Exact integer values for the density of states Ω(E) (E=J1E1+J2E2=5E1+2E2) of the Ising model on 3×6 square lattice with a ratio of 5:2 between the nearest-neighbor (J1) and next-nearest-neighbor (J2) interactions. For other values of E (not shown in the table), Ω(E)=0.

EΩ(E)EΩ(E)EΩ(E)EΩ(E)
02192428363748
38108424843484690
47288513365296544
5548561062573205872
6033661960622886496
6517286615006719269384
70283271144072487396
74188475475276173477360
78156794224805184811536
82324831536848132856048
8615968796884140899888
906240911464921320938400
9412276957200961644972856
9810320991209610062761012016
102496210312480104122761056096
10624541074992108103941099912
1105376111284811252021138352
1148132115484811622201172816
1184128119422412030241211440
1221128123158412418841251776
1261110127384128252129408
1303721313601328413348
13496135481369013748
138381392414441502

EΩ(E)=2200,

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 Ωmax(L) of the Ising model on L×2L square lattice for L= 3–10 with a ratio of 5:2 between the nearest-neighbor and next-nearest-neighbor interactions. For example, the maximum value for the densities of states for L=10 is

Maximum integer values for the densities of states Ωmax(L) of the Ising model on L×2L square lattices with a ratio of 5:2 between the nearest-neighbor and next-nearest-neighbor interactions.}

LΩmax(L)
312480
4109057776
519146127630520
662718661425827505000
73531358058070218022029537560
83288528928349183706118352094331199232
949946049806391327094111391445695735994783824892
1012304054332949238819495260659807413163084981913792663009984

Ωmax(L)=12304054332949238819495260659807413163084981913792663009984,

corresponding approximately to 1.2304×1058.

### III. Partition function zeros and phase transitions

Figure 1 shows the exact distributions of the partition function zeros in the complex a=e2βJ1/5 plane of the Ising model on 4×8 square lattice with a ratio of 5:2 between the nearest-neighbor and next-nearest-neighbor interactions. The positive real axis corresponds to the physical interval in the complex a=e2βJ1/5 plane. For the Ising ferromagnet (J1>0, R=J2/|J1|=0.4), the physical interval is a1 with a=1 (corresponding to T=) and a= (T=0). For the Ising antiferromagnet (J1<0, R=-0.4), the physical interval is 0a1 with a=0 (T=0) and a=1 (T=). As shown in the figure, two branches of the partition function zeros approach the positive real axis, indicating two phase transitions of the ferromagnetic phase transition and antiferromagnetic phase transition. In each branch of the partition function zeros, the first partition function zero a1(L) closest to the positive real axis touches the positive real axis at the critical point ac in the thermodynamic limit L.

Figure 1. Exact distributions of the partition function zeros in the complex a=e2βJ1/5 plane of Ising ferromagnet (J1>0) and Ising antiferromagnet (J1<0) on 4×8 square lattice with a ratio of 5:2 between the nearest-neighbor and next-nearest-neighbor interactions.

Figure 2 shows the first partition function zero a1(L), closest to the positive real axis, in the complex a=e2βJ1/5 plane of the Ising ferromagnet (J1>0) on L×2L square lattices for L = 3–10 with a ratio of 5:2 between the nearest-neighbor and next-nearest-neighbor interactions. We extrapolate the first partition function zero a1(L) to the infinite size (L) via the Bulirsch-Stoer (BST) extrapolation method[51,52], and obtain the unknown ferromagnetic critical point ac=1.120840. The convergence (Q) of the BST estimated value can be conveniently measured as twice the difference between the (n-1,1) and (n-1,2) approximants[51,52]. A detailed explanation of the BST extrapolation method is provided in Refs.[49,51,52]. The convergence for the ferromagnetic critical point is Q=0.000001, implying strong stability and high accuracy of the BST estimated value. The BST estimated value for the imaginary part Im[a1] is -0.0000003 with the convergence of Q=0.0000018, clearly indicating the occurrence of a phase transition.

Figure 2. First partition function zero a1(L), closest to the positive real axis, in the complex a=e2βJ1/5 plane of the Ising ferromagnet (J1>0) on L×2L square lattices for L = 3–10 with a ratio of 5:2 between the nearest-neighbor and next-nearest-neighbor interactions.

Following the finite-size scaling law[28,45], as L increases, the imaginary part Im[a1(L)] decreases as Im[a1(L)] L1/ν=Lyt, where ν is the correlation-length critical exponent, and yt is the thermal scaling exponent. The thermal scaling exponent is 1yt<2 for a second-order phase transition whereas it is yt=2 for a first-order phase transition[53]. The thermal scaling exponent yt(L) on a finite lattice can be calculated as

yt(L)=ln{Im[a1(L+1)]/Im[a1(L)]}ln[(L+1)/L].

Table 3 shows the thermal scaling exponent yt(L) of the Ising ferromagnet (J1>0) on L×2L square lattices with a ratio of 5:2 between the nearest-neighbor and next-nearest-neighbor interactions. The BST estimated value is yt=1.00003 (with the convergence of Q=0.00008), clearly indicating that the ferromagnetic phase transition is second order.

Thermal scaling exponent yt(L) of the Ising ferromagnet (J1>0) on L×2L square lattices with a ratio of 5:2 between the nearest-neighbor and next-nearest-neighbor interactions.}

Lyt(L)
31.10211205
41.07536327
51.05967472
61.04940245
71.04216417
81.03679064
91.03264339

Figure 3 shows the first partition function zero a1(L), closest to the positive real axis, in the complex a=e2βJ1/5 plane of the Ising antiferromagnet (J1<0) on L×2L square lattices for L = 3–10 with a ratio of 5:2 between the nearest-neighbor and next-nearest-neighbor interactions. The BST estimated value for the unknown antiferromagnetic critical point is ac=0.630 (with the convergence of Q=0.012). Table 4 shows the thermal scaling exponent yt(L) of the Ising antiferromagnet (J1<0, R=-0.4) on L×2L square lattices with a ratio of 5:2 between the nearest-neighbor and next-nearest-neighbor interactions. The signal of the phase transition for the Ising antiferromagnet is inherently considerably weaker[20] than that for its counterpart, the Ising ferromagnet. Similarly, the finite-size scaling behavior of the thermal scaling exponent yt(L) in Table 4 is less clear for the Ising antiferromagnet (J1<0, R=-0.4), compared with that of yt(L) for the Ising ferromagnet (J1>0, R=0.4) in Table 3. The BST estimated value for the Ising antiferromagnet (J1<0, R=-0.4) is yt=0.965 (with the convergence of Q=0.010) that is distant from the value yt=2 of a first-order phase transition. Bobak et al.[13] assumed a first-order transition line of the Ising antiferromagnet for 0.5<R=J2/|J1|<0.3340. However, the BST estimated value rules out the possibility of a first-order transition line for 0.4R<0.3340.

Thermal scaling exponent yt(L) of the Ising antiferromagnet (J1<0) on L×2L square lattices with a ratio of 5:2 between the nearest-neighbor and next-nearest-neighbor interactions.

Lyt(L)
30.95176092
40.94559473
50.93829827
60.93518510
70.93481865
80.93578778
90.93729744

Figure 3. First partition function zero a1(L), closest to the positive real axis, in the complex a=e2βJ1/5 plane of the Ising antiferromagnet (J1<0) on L×2L square lattices for L = 3–10 with a ratio of 5:2 between the nearest-neighbor and next-nearest-neighbor interactions.

Through the exact enumeration method, we obtained the exact integer values for the density of states Ω(E) (as a function of energy E) of the Ising model with a ratio of 5:2 between the nearest-neighbor and next-nearest-neighbor interactions on L×2L square lattices with free boundary conditions in the L-direction and periodic boundary conditions in the 2L-direction up to L=10. From the exact density of states Ω(E), we constructed the exact partition function as Z(a)=EΩ(E)aE, where a=e2βJ1/5. Using the exact partition function, we evaluated the exact distributions of the partition function zeros in the complex a=e2βJ1/5 plane of the Ising model on a square lattice with a ratio of 5:2 between the nearest-neighbor and next-nearest-neighbor interactions. Subsequently, we investigated two phase transitions of the ferromagnetic phase transition and antiferromagnetic phase transition, based on the exact partition function zeros in the complex a plane. For the Ising ferromagnet (J1>0, R=0.4), we estimated the critical point ac=1.120840(1) and the thermal scaling exponent yt=1.00003(8). For the Ising antiferromagnet (J1<0, R=-0.4), we obtained the critical point ac=0.630(12) and the thermal scaling exponent yt=0.965(10) that was clearly different from the value yt=2 of a first-order phase transition.

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).

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