Journal of the Korean Physical Society, Vol. 67, No. 9, November 2015, pp. 1517∼1523

Ising Antiferromagnet on a Finite Triangular Lattice with Free Boundary Conditions Seung-Yeon Kim∗ School of Liberal Arts and Sciences, Korea National University of Transportation, Chungju 27469, Korea (Received 20 July 2015) The exact integer values for the density of states of the Ising model on an equilateral triangular lattice with free boundary conditions are evaluated up to L = 24 spins on a side for the first time by using the microcanonical transfer matrix. The total number of states is 2Ns = 2300 ≈ 2.037 × 1090 for L = 24, where Ns = L(L + 1)/2 is the number of spins. Classifying all 2300 spin states according to their energy values is an enormous work. From the density of states, the exact partition function zeros in the complex temperature plane of the triangular-lattice Ising model are evaluated. Using the density of states and the partition function zeros, we investigate the properties of the triangularlattice Ising antiferromagnet. The scaling behavior of the ground-state entropy and the form of the correlation length at T = 0 are studied for the triangular-lattice Ising antiferromagnet with free boundary conditions. Also, the scaling behavior of the Fisher edge singularity is investigated. PACS numbers: 05.50.+q, 05.70.−a, 64.60.Cn, 75.50.Ee Keywords: Ising antiferromagnet, Triangular lattice DOI: 10.3938/jkps.67.1517

I. INTRODUCTION The two-dimensional Ising model is the simplest system showing phase transitions and critical phenomena. In particular, the square-lattice and the triangularlattice Ising models have played a central role in our understanding of phase transitions and critical phenomena [1,2]. Onsager and Kaufman [3] have obtained the exact solution of the square-lattice Ising ferromagnet with periodic boundary conditions, instigating the modern theory of phase transitions and critical phenomena. Due to the special symmetry of the ferromagnet and the antiferromagnet for the square-lattice Ising model, the squarelattice Ising antiferromagnet shows the same behavior as the ferromagnet [1,2]. Following the Onsager-Kaufman method, the exact solution of the triangular-lattice Ising ferromagnet with periodic boundary conditions has been obtained [1,2,4]. The results obtained for the triangularlattice Ising ferromagnet exhibit the similar features as the results obtained for the square-lattice ferromagnet do. On the other hand, the triangular-lattice Ising antiferromagnet shows completely different behaviors from those for the triangular-lattice ferromagnet and the square-lattice antiferromagnet. The exact free energy of the triangular-lattice Ising antiferromagnet with periodic boundary conditions has been obtained in the thermodynamic limit [5]. According to the exact solution, ∗ E-mail:

[email protected]

the phase transition of the triangular-lattice Ising antiferromagnet occurs only at zero temperature. Also, the ground state of the triangular-lattice Ising antiferromagnet is highly degenerate, and the ground-state entropy per volume is nonzero in the thermodynamic limit, violating the third law of thermodynamics [5, 6]. The problem of the nonzero ground-state entropy has been one of the most important subjects in statistical physics, condensed-matter physics, and mathematics [7, 8]. The nonzero ground-state entropy of the triangular-lattice Ising antiferromagnet results from (geometrical) frustration. The triangular-lattice Ising antiferromagnet is the simplest of the frustrated systems. Frustration provides insights into various systems, including water ice, amorphous materials, quantum dots, dilute magnets, glasses, spin glasses, neural networks, protein folding, and evolution [7–11]. The free energy of triangular-lattice Ising model is known for periodic boundary conditions [4,5]. However, the exact partition functions of the triangular-lattice Ising model with (more natural) free boundary conditions are not known for systems of arbitrary size. To overcome this, Stosic et al. [12] computed the exact density of states for the Ising model on an equilateral triangular lattice with free boundary conditions up to L = 15 spins on a side (corresponding to 2120 ≈ 1.329 × 1036 states) by using an IBM 3090. In this paper, we evaluate the exact density of states for the Ising model on an equilateral triangular lattice with free boundary conditions up to L = 24 spins on a side for the first time by using the

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Journal of the Korean Physical Society, Vol. 67, No. 9, November 2015

microcanonical transfer matrix [12,13] to count all possible spin states. The total number of states is 2300 ≈ 2.037 × 1090 for L = 24. Classifying all 2300 spin states according to their energy values is an enormous work. From the density of states, the exact partition function is immediately given, and the zeros of the exact partition function in the complex temperature plane of the triangular-lattice Ising model can be evaluated. Using the density of states and the partition function zeros, we investigate the properties of the triangular-lattice Ising antiferromagnet with free boundary conditions.

Fig. 1. Equilateral triangular lattice with four spins (L = 4) on a side and free boundary conditions [Ns = L(L+1)/2 = 10 and Nb = 3(Ns − L) = 18].

II. EXACT ENUMERATION The Ising model on a triangular lattice [12,14] with Ns spins and Nb bonds is defined by the Hamiltonian  σ i σj , (1) H = −J i,j

where J is the coupling constant, i, j indicates a sum over all nearest-neighbor pairs of lattice sites, and σi = ±1. We define the density of states, Ω(E), with a given energy  E=− σi σj , (2) i,j

where δ is the Kronecker delta. Next, by introducing (2) spin variables σi (1 ≤ i ≤ 3) for the second row and considering the vertical bonds between the first and the second rows, one can rewrite the array ω (1) as   3  (2) (1) (1) σn(2) (σn(1) +σn+1 ); {σi } . ω ˜ (2) (E; {σi }) = ω (1) E+ n=1

(6) After taking the horizontal bonds of the second-row spins, we obtain

where E is an integer (−Nb ≤ E ≤ Nb /3). Then, the partition function of the Ising model (a sum over 2Ns possible spin configurations) Z=



e−βH ,

(3)

{σn }

where β = 1/kB T (kB is the Boltzmann constant and T is temperature), can be written as Nb /3

Z(T ) =



Ω(E)e−βJE .

The microcanonical transfer matrix [12, 13] is modified to evaluate the exact integer values for the density of states Ω(E) of the Ising model on an equilateral triangular lattice with L spins on a side and free boundary conditions [Ns = L(L + 1)/2 and Nb = 3(Ns − L)]. Here, the microcanonical transfer matrix is briefly described for the Ising model on a triangular lattice with four spins on a side, as shown in Figure 1. First, an array ω (1) , which is indexed by energy E and spin variables (1) σi (1 ≤ i ≤ 4) for the first row, is initialized as ω

(1) (E; {σi })

 =δ E+

3  n=1

(2) (E; {σi })

=ω ˜

(2)

 E+

2  n=1



(2) (2) σn(2) σn+1 ; {σi }

(1) σn(1) σn+1

 ,

(5)

. (7)

(3)

Now, we introduce spin variables σi (1 ≤ i ≤ 2) for the third row, consider the vertical bonds, and obtain

ω ˜

(3)

(3) (E; {σi })

=ω

(2)

 E+

2 

n=1



(2) (2) σn(3) (σn(2) +σn+1 ); {σi }

.

(8)

(4)

E=−Nb

(1)

ω

(2)

Then, the horizontal bond connecting the spins in the third row are taken into account by shifting the energy:   (3) (3) (3) (3) ˜ (3) E + σ1 σ2 ; {σi } . (9) ω (3) (E; {σi }) = ω (4)

Finally, the spin σ1 in the fourth row is introduced with two vertical bonds such as   (4) (4) (3) (3) (3) ω ˜ (4) (E; σ1 ) = ω (3) E +σ1 (σ1 +σ2 ); {σi } . (10) Then, the density of states is given by  (4) ω ˜ (4) (E; σ1 ), Ω(E) = (4) σ1

(11)

Ising Antiferromagnet on a Finite Triangular Lattice · · · – Seung-Yeon Kim

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Table 1. Exact integer values for the density of states Ω(E) for the Ising model on an equilateral triangular lattice with four spins on a side and free boundary conditions. Here, E denotes the given energy. E −18 −6 6

E −14 −2

Ω(E) 2 124 160

Ω(E) 6 282

E −10 2

Ω(E) 36 414

as shown in Table 1. The sum over all densities of states is equal to 2Ns (the number of all possible spin configurations); for example, 210 = 1024 for L = 4. We evaluated the exact integer values for the density of states Ω(E) of the Ising model on a triangular lattice with free boundary conditions up to L = 24 spins on a side. The density of states up to L = 15 has been evaluated by Stosic et al. [12]. In this work, for the first time, we obtain the exact integer values of Ω(E) for the triangular-lattice Ising model for L = 16 ∼ 24. The total number of states is 2Ns = 2120 ≈ 1.329 × 1036 for L = 15 and 2300 ≈ 2.037 × 1090 for L = 24. Classifying all 2300 spin states according to their E values is a very difficult task. The largest densities of states for L = 15 and L = 24 are as follows: Ωmax (L = 15) = 11981932627624701016722664 8878579958 (12) (approximately 1.1982 × 1035 ) and Ωmax (L = 24) = 11289033818834634787439055 54877243038882701491542927356519 16422331440496598605409276708608 (13) (approximately 1.1289 × 1089 ). This kind of a large integer number (I.N.) for L = 24 can be stored in a computer by using a positional numeral system with a radix (or base) of 231 [15,16] such as I.N. = 2

jmax 

Pj (231 )j .

(14)

j=0

For Eq. (13), ten Pj ’s are used: P0 = 1280362240, P1 = 855128988, P2 = 695767265, P3 = 140788095, P4 = 303568244, P5 = 753146653, P6 = 596135797, P7 = 1323626094, P8 = 1940087852, and P9 = 116221.

III. GROUND-STATE ENTROPY Given the density of states Ω(), where  ≡ (Ns − L − E)/2, we obtain the exact entropy s() according to the Boltzmann formula s() =

kB ln Ω(). Ns

(15)

Fig. 2. Exact entropy s() = [ln Ω()]/Ns (in units of kB ) as a function of energy  = (Ns − L − E)/2 (= 0 ∼ 552) of the triangular-lattice Ising model with twenty-four spins on a side and  free boundary conditions (L = 24, Ns = 300, and E = − σi σj ). The ground-state entropy s( = 0) is antiferromagnetic and not zero, and the ground-state entropy s( = 552) is ferromagnetic and becomes zero in the thermodynamic limit.

Figure 2 shows the exact entropy s()/kB = [ln Ω()]/Ns as a function of energy  (= 0 ∼ 552) for the triangularlattice Ising model with 24 spins on a side and free boundary conditions (L = 24 and Ns = 300). As shown in the figure, the entropy is asymmetric due to the complete difference in the behaviors of the antiferromagnet and the ferromagnet in the triangular-lattice Ising model. This difference stems from the geometrical frustration [9, 10] in the ground states of the triangular-lattice Ising antiferromagnet. The ground-state entropy s( = 0)/kB is antiferromagnetic and not zero, and the ground-state entropy s( = 552)/kB = (ln 2)/300 is ferromagnetic and becomes zero in the thermodynamic limit. The second column of Table 2 shows the exact integer values for the number of ground states Ω0 (L) = Ω( = 0, L) for the triangular-lattice Ising antiferromagnet with L spins on a side and free boundary conditions (L = 5 ∼ 24). The number of ground states corresponds approximately to 1.2 × 1048 for L = 24. The third column of Table 2 shows the ground-state entropy s0 (L)/kB = Ns−1 ln Ω0 (L) for the triangular-lattice Ising antiferromagnet [Ns = L(L + 1)/2]. By using the Bulirsch-Stoer (BST) method [17], we have extrapolated the values of the ground-state entropy for finite lattices to the thermodynamic limit (L → ∞), and the extrapolated value is s0 /kB = 0.323066(13), in excellent agreement with the Wannier value s0 = 0.323066 [5] for periodic boundary conditions. The BST error estimates are twice the difference between the (n − 1,1) and (n − 1,2) approximants. The ground-state entropy s0 (L) scales as s0 − s0 (L) = Δs0 (L) ∼ L−ωs ,

(16)

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Journal of the Korean Physical Society, Vol. 67, No. 9, November 2015

Table 2. Exact integer values (the second column) for the number of ground states Ω0 (L) = Ω( = 0, L) and the ground-state entropy (the third column) s0 (L) = Ns−1 ln Ω0 (L) (in units of kB ) of the triangular-lattice Ising antiferromagnet with L spins on a side and free boundary conditions [Ns = L(L + 1)/2]. The fourth column shows the value of the effective scaling exponent ωs (L) defined by Eq. (17). The last row contains the extrapolated values in the thermodynamic limit (L → ∞). L 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 ∞

Ω0 (L) 1386 16814 284724 6715224 220240306 10032960146 634271091558 55607968072800 6757401238296442 1137661035904122264 265265658215457903864 85635780217381861437248 38267278120418832223426206 23665003296449525435806996826 20249495559177400885698315743146 23970844200912116051104028706840960 39251450758817189028672632591398756000 88895661772889964647310452601287567856526 278428258514959344899126829670051187539796250 1205909184032277557981022320993508668213426605000

where ωs is the scaling exponent controlling the finitesize scaling behavior of the ground-state entropy [18], and its value has never been determined for the triangular-lattice Ising antiferromagnet. Then, for finite lattices, we can define the effective scaling exponent as ωs (L) = −

ln[Δs0 (L + 1)/Δs0 (L)] . ln[(L + 1)/L]

(17)

The fourth column of Table 2 shows the values of the effective scaling exponent ωs (L). By using the BST extrapolation, in the thermodynamic limit, we have ωs = 0.997001(9), indicating the possibility of ωs = 1.

s0 (L) 0.48227848 0.46333177 0.44854556 0.43666355 0.42689400 0.41871166 0.41175367 0.40576087 0.40054302 0.39595720 0.39189386 0.38826757 0.38501070 0.38206898 0.37939833 0.37696260 0.37473183 0.37268096 0.37078889 0.36903773 0.323066(13)

0.997001(9)

more clearly understand the phase transitions and critical phenomena. Because the partition function zeros of a given system provide valuable information on its exact solution, earlier studies on partition function zeros were mainly performed in the fields of mathematical physics and rigorous statistical mechanics. Nowadays, the concept of partition function zeros is applied to all fields of physics from particle physics to biophysics, and they are popularly used as one of the most effective methods to determine the critical points and exponents [21,22]. Given the density of states Ω(), the partition function of the triangular-lattice Ising model is written as Z(a) = eβJ(L−Ns )

IV. PARTITION FUNCTION ZEROS Phase transitions and critical phenomena can be understood based on the concept of partition function zeros [19]. Fisher introduced the partition function zeros in the complex temperature plane and showed that the partition function zeros of the square-lattice Ising model determined its phase transitions [20]. By calculating the partition function zeros and examining the behavior of the first partition function zero (partition function zero closest to the positive real axis), we can much

ωs (L) 0.69493148 0.72264641 0.74500252 0.76349353 0.77909243 0.79246288 0.80407475 0.81427103 0.82330865 0.83138429 0.83865131 0.84523132 0.85122206 0.85670309 0.86173993 0.86638704 0.87069009 0.87468772 0.87841286

2(Ns −L)



Ω a ,

(18)

=0

where a = e2βJ . For antiferromagnetic interactions (J < 0), the physical interval is 0 ≤ a ≤ 1 (0 ≤ T ≤ ∞) while for ferromagnetic interactions (J > 0), it is 1 ≤ a ≤ ∞ (∞ ≥ T ≥ 0). The partition function can be expressed as its zeros {ai } in the complex temperature (a = e2βJ ) plane: 2(Ns −L)

Z(a) = C



i=1

(a − ai ),

(19)

Ising Antiferromagnet on a Finite Triangular Lattice · · · – Seung-Yeon Kim

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Table 3. First zero a1 (L) in the complex temperature (a = e2βJ ) plane of the triangular-lattice Ising antiferromagnet with L spins on a side and free boundary conditions. The third column shows the value of the effective thermal scaling exponent yt (L) defined by Eq. (22). The last row contains the extrapolated values in the L → ∞ limit.

Fig. 3. Partition function zeros in the complex temperature (a = e2βJ ) plane of the triangular-lattice Ising model with twenty-four spins on a side and free boundary conditions.

where C is a term independent of {ai }. Figure 3 shows the exact partition function zeros in the complex a plane of the triangular-lattice Ising model with twenty-four spins on a side and free boundary conditions. Some partition function zeros lie close to the antiferromagnetic 0 (J < 0) and the ferromagnetic critical point aaf c =√ critical point afm 3 (J > 0), as shown in the figure. c = The properties of the triangular-lattice Ising ferromagnet are known well [2,4], so we √ do not pursue the partition fm function zeros near ac = 3. In particular, the first zero a1 (L) on the imaginary axis of the triangular-lattice Ising antiferromagnet with free boundary conditions approaches the antiferromagFor netic critical point aaf c , as shown in Table 3. the triangular-lattice Ising antiferromagnet with periodic boundary conditions, the finite-size series L = 3, 6, ..., the finite-size series L = 4, 7, ..., and the finite-size series L = 5, 8, ... exhibit different finite-size behaviors. Therefore, we choose generally the finite-size series L = 3, 6, ... for the extrapolation to the L → ∞ limit, reducing the efficiency of extrapolation [14]. Remarkably, no different behaviors exist for free boundary conditions, as shown in Table 3, enabling us to use all the data. The BST extrapolated value for the first zero of the triangular-lattice Ising antiferromagnet with free boundary conditions is a1 = (−0.6 ± 1.2) × 10−9 i, in excellent agreement with the antiferromagnetic critical point. The antiferromagnetic first zero a1 (L) scales as [23,24] |a1 (L)| ∼ L−yt ,

(20)

where yt is the thermal scaling exponent with the corre-

L 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 ∞

a1 (L) 0.59862250i 0.50521503i 0.43856854i 0.38825661i 0.34875311i 0.31681987i 0.29041843i 0.26819450i 0.24920959i 0.23279094i 0.21844262i 0.20579038i 0.19454614i 0.18448428i 0.17542544i 0.16722514i 0.15976563i 0.15294994i 0.14669744i 0.14094052i −0.6(12) × 10−9 i

yt (L) 0.93048260 0.91772574 0.91251670 0.91101595 0.91145033 0.91292067 0.91494114 0.91723760 0.91965091 0.92208720 0.92449105 0.92683017 0.92908654 0.93125095 0.93331978 0.93529284 0.93717208 0.93896071 0.94066266 1.000000(1)

lation length ξ ∼ a−ν = e2νβ|J| ,

(21)

where ν(= 1/yt ) is the correlation-length critical exponent. From Eq. (20), for finite lattices, we can define the effective thermal scaling exponent as yt (L) = −

ln[|a1 (L + 1)|/|a1 (L)|] . ln[(L + 1)/L]

(22)

The third column of Table 3 shows the value of the effective thermal scaling exponent yt (L). By using the BST extrapolation, in the L → ∞ limit, we obtain yt = 1.000000 ± 0.000001, clearly indicating that the correlation length of the triangular-lattice Ising antiferromagnet at T = 0 diverges exponentially as ξ ∼ e2β|J| .

(23)

Thus, the form, Eq. (23), of the correlation length for the triangular-lattice Ising antiferromagnet with free boundary conditions at T = 0 is obtained for the first time. Stephenson [25] showed that the zero-temperature behavior of the pair correlation for the triangular-lattice

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Journal of the Korean Physical Society, Vol. 67, No. 9, November 2015

Table 4. Fisher edge singularity ae (L) in the complex temperature (a = e2βJ ) plane of the triangular-lattice Ising model with L spins on a side and free boundary conditions. The third column shows the value of the effective edge scaling exponent ye (L) defined by Eq. (26). The last row contains the extrapolated values in the L → ∞ limit. L 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 ∞

ae (L) 1.43733076i 1.51599795i 1.56385494i 1.59639448i 1.61990375i 1.63758682i 1.65128629i 1.66214656i 1.67091804i 1.67811338i 1.68409429i 1.68912283i 1.69339310i 1.69705164i 1.70021087i 1.70295833i 1.70536303i 1.7320508(3)i

ye (L) 2.63622723 2.37654419 2.25583961 2.18722736 2.14375975 2.11422503 2.09313358 2.07749553 2.06555354 2.05621336 2.04876188 2.04271688 2.03774207 2.03359679 2.03010488 2.02713490 1.999999(2)

Ising antiferromagnet with periodic boundary conditions was of the form 1 2 σ0 σr  ∼ r− 2 cos πr, 3

(24)

indicating oscillatory correlation. Because of this oscillatory behavior, the finite-size series for the triangularlattice Ising antiferromagnet with periodic boundary conditions may exhibit different finite-size behaviors, reducing the efficiency of extrapolation. In addition, as shown in Fig. 3, the partition function zeros on the imaginary √axis approach the Fisher edge singularity [26,27] ae = 3i ≈ 1.7320508i. The edge singularities provide exact information on the analytic structure of a given system. Table 4 shows the Fisher edge singularity ae (L) of the triangular-lattice Ising model with L spins on a side and free boundary conditions. The BST extrapolated value for ae (L) in the L → ∞ limit in excellent agreement is ae = (1.7320508 ± 0.0000003)i, √ with the exact value 3i. Following the behavior of the Yang-Lee edge singularity [28, 29], we assume that the Fisher edge singularity ae (L) scales as |ae (L)| ∼ L−ye ,

(25)

where ye is the edge scaling exponent. From Eq. (25), for finite lattices, we can define the effective edge scaling

exponent as ye (L) = −

ln[|ae (L + 1)|/|ae (L)|] . ln[(L + 1)/L]

(26)

The third column of Table 4 shows the value of the effective edge scaling exponent ye (L) defined by Eq. (26). By using the BST extrapolation, in the L → ∞ limit, we obtain ye = 1.999999 ± 0.000002, strongly implying ye = 2. Thus, the value of the edge scaling exponent for the triangular-lattice Ising model is obtained for the first time. If we assume that the usual scaling relation α = 2 − d/yt , where α is the specific-heat critical exponent and d is the dimension, applies to the Fisher edge singularity, we can obtain αe = 2 − d/ye = 1 from ye = 2 for the triangular-lattice Ising model, explaining the result (αe = 1) of Ref. [27], in the unified viewpoint of the edge scaling exponent.

V. CONCLUSION We have evaluated the exact density of states for the Ising model on an equilateral triangular lattice with free boundary conditions up to L = 24 spins on a side for the first time by counting all possible spin states. For L = 24, the task to classify all 2300 spin states according to their energy values is very difficult. From the density of states, we have evaluated the exact partition function zeros in the complex temperature (a = e2βJ ) plane of the triangular-lattice Ising model. Using the density of states and the partition function zeros, we have investigated the properties of the triangular-lattice Ising antiferromagnet whose ground-state entropy per volume is nonzero in the L → ∞ limit, resulting from frustration. The finite-size scaling behavior of the ground-state entropy for the triangular-lattice Ising antiferromagnet has been studied from the density of states. Also, using the partition function zeros, we obtained the form of the correlation length for the triangular-lattice Ising antiferromagnet with free boundary conditions at T = 0, and we investigated finite-size scaling behavior of the Fisher edge singularity for the triangular-lattice Ising model.

ACKNOWLEDGMENTS This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (grant number NRF-2014R1A1A2056127).

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Ising Antiferromagnet on a Finite Triangular Lattice · · · – Seung-Yeon Kim [2] H. N. V. Temperley, in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green (Academic Press, London, 1972), Vol. 1, p. 227, and references therein. [3] L. Onsager, Phys. Rev. 65, 117 (1944); B. Kaufman, Phys. Rev. 76, 1232 (1949). [4] M.-C. Wu and C.-K. Hu, J. Phys. A 35, 5189 (2002), and references therein. [5] G. H. Wannier, Phys. Rev. 79, 357 (1950); Phys. Rev. B 7, 5017 (1973). [6] R. P. Millane and R. M. Clare, Phys. Rev. E 74, 051101 (2006). [7] A. P. Ramirez, A. Hayashi, R. J. Cava and R. Siddharthan, Nature 399, 333 (1999), and references therein. [8] S.-Y. Kim, J. Korean Phys. Soc. 52, 551 (2008), and references therein. [9] J.-F. Sadoc and R. Mosseri, Geometrical Frustration (Cambridge Universiy Press, New York, 1999). [10] S.-Y. Kim, S. J. Lee and J. Lee, Phys. Rev. B 76, 184412 (2007). [11] S.-Y. Kim, J. Chem. Phys. 133, 135102 (2010). [12] B. Stosic, S. Milosevic and M. E. Stanley, Phys. Rev. B 41, 11466 (1990).

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