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Estimation of Magnetization, Susceptibility and Specific heat for the two-dimensional Ising Model in a Non-zero Magnetic field

The partition functions for two-dimensional nearest neighbour Ising model in a non-zero magnetic field have been derived for finite square lattices with the help of graph theoretical procedures, show-bit algorithm, enumeration of configurations and transfer matrix methods. A new recurrence relation valid for all lattice sizes is proposed. A novel method of computing the critical temperature has been demonstrated. The magnetization and susceptibility for infinite lattices in the presence of magnetic field is estimated. The critical exponent {\delta} pertaining to the magnetization has also been computed.


INTRODUCTION
The analysis of Ising models constitutes a central theme in statistical mechanics and has diverse applications [1][2] .Although the one-dimensional nearest neighbour Ising model did not exhibit phase transitions, the exact solution of the two-dimensional Ising model for zero magnetic field given by Onsager [3] yielded an explicit expression for critical temperatures. Lee and Yang [4] provided a complete solution of the twodimensional Ising model for vanishing magnetic fields. However, subsequent attempts of solving the two-dimensional Ising models when the magnetic field is non-zero has remained elusive till now; nevertheless methodologies employing Bragg-Williams approximation [5], Bethe ansatz [6], series expansions [7], renormalization group [8] scaling hypothesis [9], etc. have been investigated extensively. Due to the equivalence with binary alloys and lattice gas description of fluids [10], the results derived in the context of Ising models become applicable mutatis mutandis to various topics in solid state and condensed matter physics. Among several applications of the two-dimensional Ising models [11], mention may be made of the following: order-disorder transitions [12], electrochemical interfaces [13], phase separation in self-assembled monolayer films [14], protein folding [15], free energies of surface steps [16] etc. In contrast to the one-dimensional case, the analysis of two-dimensional Ising models employing the transfer matrix method is considered almost impossible since the total number of spin configurations for a square lattice of N sites is 2 N and even for N=16, this number is 65,536. As has been pointed out elsewhere [17] "In terms of the number of papers published, Ising model ranks as probably the most celebrated model in mathematical physics." In this Communication, we derive the canonical partition function of the two-dimensional nearest neighbour Ising models for a non-zero magnetic field pertaining to a square lattice of 16 sites using (i) enumeration of all the 2 16 configurations and the corresponding energies; (ii) 'show-bit' algorithm (decimal to binary conversion); (iii) graph theoretical approach of counting of 'black-white' edges and (iv) transfer matrix formulation. The partition functions are also deduced for N =25; N=36 and N =64 sites. A general recurrence relation among the partition functions valid for all lattice sizes is also proposed. We also demonstrate a new method of obtaining the critical temperature for finite lattice sizes directly from the energetics of the system, without recourse to the expression for the magnetization. The exact values for the specific heat, magnetization and susceptibilities are reported for N=16, N=36 and N=64 sites for non-zero magnetic field and the extrapolated values for infinite lattices are also provided.

Methodology
For the two-dimensional nearest neighbour Ising model on a square lattice [18], the Hamiltonian is given by , , where J is the nearest neighbour interaction energy, H being the external magnetic field. i and j denote the row and column index respectively.
The periodic boundary conditions are assumed. The corresponding canonical partition function is defined as [19] ( ) where k denotes the Boltzmann constant , T being the absolute temperature.
Partition function for a square lattice of 16 sites
where x=e J/kT and y=e H/kT .

(B) 'Show-bit' algorithm
The earlier method of summation of individual energy states becomes tedious for larger lattice sizes and an alternate method commonly referred to as 'Show-bit' algorithm is employed herein for the square lattices (N=25, 36). In this algorithm, a decimal number is converted to its corresponding binary number where the 0s are replaced by -1s. The energy of each configuration is estimated and progressively summed in order to compute the corresponding partition function for each N. The partition functions for N=16 are entirely in agreement both numerically and analytically with that obtained from (A). In the case of N =25 and N =36, the energy pertaining to each of the 2 25 and 2 36 configurations becomes computationally complex [20]. Hence MPI (Message Passing Interface) coding was employed using C++ programming in a 64-bit Linux cluster. The partition functions for N=25 and N=36 are given below: Partition function for N = 25   50  25  25  42  23  23  38  34  21  21  34  30  26  25, 1   19  19  34  30  26  22  18  17  17  30  26  22   18  14  10 In view of the computational complexity, the partition function for N=64 was not attempted using the show-bit algorithm.

(C) Combinatorial approach
The connection of the Ising model problem to the combinatorial graph theory is well known [21]. The essential premise in this context is that the two-dimensional Ising model can be visualized as a bipartite graph G consisting of the set of vertices V and edges E viz. G = {V,E}. The vertices represent the arrangement of + 1 ('black') and -1 ('white') spins and the edges correspond to the connectivity between the vertices. The black-white edges correspond to the perfect matching of the graph. The number of ways in which a blackwhite edge can arise, correspond to its weight. This strategy is analogous to the weighted graph colorings problem wherein adjacent vertices have different colours [22].
For brevity, let p denote the number of +1 spins (black vertex) and q represent the blackwhite edge. Thus, the estimation of the partition function in non-zero magnetic field is now transformed as counting of the black-white edges of a square lattice of N sites wherein adjacent and non-adjacent black-white edges along with their identities need to be included.
along with the periodic boundary conditions.
As an illustration for a square lattice of N sites, the number of black sites may be denoted as p and hence that of white sites is N-p. Let q be the number of black-white edges formed.
Then, A(p,q) denotes the number of times a configuration occurs with p number of black sites and q number of black-white edges. For example, when there are 2 adjacent black sites, the number of black-white edges will be 6, taking into account, the periodic boundary conditions, although the 2 black sites can occur at any place among the 4 rows and 4 columns(for N =16). The value of p may vary from 0 to N since there can be no black sites, 1 black site, 2 black sites etc. Accordingly, the value of q also changes. The number of black-white edges (q) changes from 0 to 2N and is also dictated by the value of p, since for a given value of p, the black sites can be adjacent or non-adjacent. For example, if p=3 (3 black sites), the following cases arise: (i) all the three can be adjacent, (ii) two of them can be adjacent and the remaining site is present elsewhere within the square and (iii) all the three may be scattered. Table 1 denotes A(p,q) for a square lattice of 16 sites deduced using a program written in C++. The number of black-white edges A(p,q) is related to the canonical partition function in two dimensions as [23] ( ) where x=e J/kT and y=e H/kT as before. Incorporating A(p,q) and the corresponding p and q values for N =16, the partition function is derived for N =16. This equation is identical with that arising from the other two approaches (A) and (B) mentioned above. The methodology has also been extended to N =25 (Table 2)and N =36 (Appendix). This enumeration method for counting of vertices and edges has tremendous potentialities and among them, mention may be made of the following: protein folding [24], stability of polycyclic benzenoid hydrocarbons [25], organization of biochemical networks [26], percolation on two dimensional lattices [27] etc.

(D) Transfer matrix formulation
For one-dimensional Ising models, the basic transfer matrix turns out to be 2×2, whose eigenvalues can then be employed for all lattice sizes. This simplicity is no longer present for the two-dimensional case. Furthermore, for a square lattice of n 2 sites, the transfer matrix [28] is of dimension 2 n ×2 n where n is the number of rows or columns, thus implying that the matrix needs to be constructed separately for each lattice size in general.
It is of interest to point out that the procedure of Kramers and Wanniers [29] for twodimensional Ising models results in a double-shifted circulant matrix and is valid irrespective of the lattice size albeit at H =0. Inspired by this rigorous methodology, it is tempting to formulate a matrix version for H ≠ 0 too. In the case of one-dimensional Ising models, a Toeplitz matrix does exist [30] for H ≠ 0 and J ≠ 0 which can be obtained from the Discrete Fourier Transformation [31]. However, for two-dimensional Ising models, this computation is tedious even for N =16.Although the methodology for constructing such Toeplitz matrices for N ≥16 is quite tedious, this task when accomplished renders it possible to exploit well-known theorems pertaining to asymptotic limits for eigenvalues of circulant [32] matrices. Thus, the thermodynamic limit of N → ∞ for non-zero magnetic field may possibly be reached from the eigenvalues of the circulant matrices. If the correct matrix for a square lattice of N sites is constructed, the partition function follows from where n denotes the number of rows or columns.
While there are 65,536 configurations for a square lattice of 4×4 sites, the transfer matrix is indeed a 16×16 matrix consisting of 256 elements. The ferreting out of 256 elements out of the entire set of 65,536 elements in order to construct the transfer matrix appears at first sight to be daunting. However, as has been demonstrated by Kramers and Wanniers [29] there exists an inherent symmetry of the two-dimensional Ising models which can be exploited. Out of 2 N configurations, only those configurations which have reflection symmetry are constituents of the matrix. If the condition, site (i) = site (N/2+i) is satisfied in a configuration, the configuration is said to obey the reflection symmetry. For example, in a square lattice of 16 sites, among all the 65536 elements, only 256 elements satisfy this reflection symmetry property and become the constituents of the matrix, i.e., only those configurations, wherein, site(1)=site (9), site (2)= site (10) and site (8)= site (16) etc satisfy the reflection symmetry and constitute the elements of the matrix. As an illustration, the following arrangement denotes the 2314 th configuration out of 65536 for N=16. (10) (11) (12) (13) (14) (15) (16) The energy pertaining to the above configuration is E 2314 = -8J -8H.
Hence e -E 2314 / kT = e 8J/kT+8H/kT = x 8 y 8 . Subsequently, equation (3)  which arise from the reflection symmetry method. This is obvious since the partition function is the trace of the matrix corresponding to the Hamiltonian. The matrix pertaining to this lattice is provided in Figure 1. When H =0, i.e y =1, this matrix will correspond to the zero magnetic field case. Thus, it follows that the matrix for a square lattice of 16 sites pertaining to the two-dimensional nearest neighbour Ising model for a non-zero magnetic field along with the algebraic expression for the corresponding partition function is not unattainable! The partition function equation deduced from the matrix is identical with that arising from the approaches (A),(B) and (C).When N is an odd number, this reflection property method becomes inapplicable and hence Q could not be obtained for N=25. In the case of N= 64, the 2 64 elements are first arranged into a 2 32 ×2 32 matrix, which is further reduced to 2 16 ×2 16 using the diagonal elements of the former and then expressed as the 2 8 ×2 8 using C++ programming. The expression for the partition function could not be obtained for the large symbolic matrix. However, for various values of H and J, the numerical value of the partition function was obtained easily (Figure 2).But the same procedure cannot be applied to arrive at a matrix for 36 sites. It is well known that the matrix dimension that corresponds to 36 sites is 2 6 ×2 6 , i.e., the number of matrix elements should be 2 12 , i.e., 4096, out of 2 36 elements. Hence, the various combinations obtained from 12 (i.e., 2n) +1s and -1s are reflected 3 times to get 36 +1s and -1s combinations such that site (1) =site (13) =site (25), site (2) = site (14) = site (26) and so on, till, site (12)

Results and Discussion
The foregoing analysis demonstrates that the partition function for a square lattice of 16 sites can be obtained by different methods even in the presence of a magnetic field.
However, difficulties emerge for the next square lattice viz N = 25. In this case, the matrix method by reflection symmetry becomes invalid since the number of sites is odd; further, the enumeration procedure described in (A) is tedious. The combinatorial approach for N=25 using A(p,q) values is feasible. The show bit algorithm is convenient for N =16, N =25 and N =36; however, the matrix formulation enables the estimation of the partition function for N =64 but a closed form expression could not be obtained.

Recurrence Relation
Thus, the computation of the partition function becomes more complex as the lattice size increases and hence it is preferable to have a valid generalization for large lattices so that the conversion into an integral may become feasible. A careful scrutiny of the partition function estimates pertaining to N =16 and N = 25 reveals a recurrence relation as follows: ( ) where n is the number of rows or columns. The partition functions predicted from the above equation for N =36 are almost identical with the values arising directly from the partition function equation for N =36 for all values of H and J. (In the numerical calculations for the partition function, magnetization, specific heat as well as the susceptibility, the magnetic field has been non-dimensionalised with respect to kT). Thus, the partition function for a chosen (n+2) 2 value is obtainable from those pertaining to (n+1) 2 and n 2 sizes. Furthermore, the partition functions for N =16 and N =25 alone are adequate to deduce the partition functions for any other value of N. The free energy for 320*320lattice for H=0, at the critical temperature Tc (J/ kTc = 0.4407) predicted from the above recurrence relation is 0.929429 while the corresponding value reported by Roland et al [33] is 0.929701. The computed values of lnQ for N = 320*320 from the recurrence relation employing the values for N = 16 and N =25 are provided in Table 3 for H = 0 and H = 0.1. Table 3 The partition function for a square lattice of 102400 sites arising from the recurrence relation (7)

Spontaneous Magnetization and critical temperature
Critical Temperature of finite lattices from energy considerations It is well-known that the spontaneous magnetization vanishes for any finite sized lattice [34]. Hence, despite the availability of the partition functions for H ≠ 0 and J ≠ 0 pertaining to N =16, 25, 36, 64 etc, the same is not adequate to deduce the critical temperature from magnetization. In view of this limitation, an alternate procedure for estimating the critical temperature without resorting to the magnetization route is suggested here. Since the critical temperature refers to the temperature at which the spontaneous magnetization changes from +1 (or -1) to 0, it is legitimate to equate the differences in their Helmholtz free energies (A N ) for M =1 (all spins +1) and M =0( equal +1 and -1 spins) to zero viz From the final expression for the partition functions which is a sum over energy states, it is not possible to ferret out the energies with the chosen spin arrangements, even for N =16, implying that the complete algebraic expressions for Q, even if available for large lattice size will be inadequate unless the individual terms are identified with the configurational states. In the present analysis, the graph theoretical procedure, show-bit algorithm and enumeration of individual energies using EXCEL have the feature wherein the energies corresponding to each arrangement can be deciphered. We illustrate the procedure for deducing Tc from the counting of black-white edges using A(p,q) values of Table 1, pertaining to a square lattice of 16 sites. The spontaneous magnetization M =1 corresponds to all spins equal to +1 ('black' sites) and there is only one way of obtaining this configuration, whose energy is exp(-32J/kT). In the terminology of black-white edges, this corresponds to A(8,0) = 1, yielding A 16 (M =1) = (-32J).
However, several the configuration states can yield M =0 i.e q can vary from 8 to 32 for p=8 (number of +1 spins = number of -1 spins) as shown in Table 1 Equating the two free energies, the transcendental equation given below results viz.
The solution of the above equation (8)     predicted by Huang [28] and Baxter [37]. In view of the availability for magnetization for various magnetic fields, it is imperative to deduce the critical exponent δ. For this purpose, the variation of ln M with ln H is analyzed when T/T c is ~ 1. As shown in Figure   6, the value of 1/δ is found to be 0.06282 ± 0.0044 in gross agreement 38 with the critical exponent of two-dimensional Ising models viz. i.e. 1/δ =0.0667 .

Magnetic Susceptibility
The analysis of magnetic susceptibility of Ising models is very intricate in view of the sensitive nature of the series coefficients as was demonstrated by Fisher [39] and Sykes [40]. Consequently, our present analysis focuses only on the susceptibility values for nonzero magnetic field for N =16, N =36 and N =64. Since the magnetization pertaining to the infinite lattice itself has been deduced from the extrapolation procedure, differentiation of the same leads to the variation of the susceptibility predicted in Figure 8. In view of the sensitivity of the numerical differentiation of the extrapolated data, these estimates may involve an error of ca. 5%. It is of interest to note from Figure 8 that the maximum in susceptibility arises only when H → 0. As mentioned earlier, the magnetization equation itself is not employed to deduce

Specific Heat
The specific heat of two-dimensional Ising models constitutes a sensitive test of validating any computational procedure [11]. The specific heat is related to the internal energy as Employing the ln Q values for N = 16, N= 36 and N =64 at H =0, the predicted variation of C v with kT/J is depicted in Figure 9(a) for H =0.While the C v variation for 16 and 36 sites were obtained by analytical differentiation of the partition function expression, the Cv variation pertaining to 8×8 lattice was obtained by numerical differentiation of the partition function for 64 sites. An empirical equation pertaining to C v for H ≠ 0 is given below: Figure 9 The dependence of the specific heat on (kT/J) for N=16, 36 and 64 at different magnetic fields :(a) H=0.1 and (b) H=0.5.The points denote the values arising from the partition function while the line denotes the prediction of equation (10).The inset depicts the specific heat at H =0.
Equation (10) when H =0 (i) leads to the singularity at the critical temperature and (ii) is similar to the expression reported by Ferdinand and Fisher [41]. Figure 9(b) depicts the  It is intriguing to note that the vanishing of the spontaneous magnetization, maximum in susceptibility as well as its temperature derivative all occur at the same point viz Tc. This arises on account of the symmetry underlying the magnetization (order parameter) equation. Since the order parameter is an odd function of the magnetic field, the susceptibility viz (dM/dH) is an even function of H. The partial derivative of magnetization with respect to temperature is also an odd function while the derivative with respect to H is an even function. Thus, it follows that the susceptibility and its temperature derivative will have the extrema at the same temperature viz Tc, where the spontaneous magnetization becomes zero. This insight has an interesting connotation in the statistical mechanical theory of the electrical double layer wherein the vanishing of the dipole potential and the extrema of the differential capacitance and its temperature coefficient occur at the same surface charge density [42].While the sizes of the lattice chosen here are quite small in comparison with the Monte Carlo simulations [36] and other numerical procedures [43], the present strategy is entirely new since all the thermodynamic quantities are obtained from the corresponding partition functions-a central quantity in statistical mechanics.
Furthermore, the partition functions reported herein for N =16, N =25, N =36 and N =64 are exact, these values being deduced for zero and non-zero magnetic fields. While the recurrence relation needs further refinement in order to yield accurate thermodynamic parameters, the algebraic expressions for the canonical partition functions are reported for the first time. It appears therefore that the Holy Grail is not entirely un-surmountable.
In Summary, the partition functions for two-dimensional Ising models for small square lattices in the presence of the magnetic field are computed by different methods and the magnetization, specific heat and susceptibilities are deduced therefrom. The extrapolation to infinite lattice sizes for magnetization in the presence of the magnetic field is suggested. The critical temperatures are deduced for N = 16 and N =36 using graph theoretical procedures while the magnetization critical exponent δ is computed from the magnetization values.