J. M. Dixon, J. A. Tuszynski and E. J. Carpenter- Analytical Expressions for Energies, Degeneracies and Critical Temperatures of the 2D Square and 3D Cubic Ising Models

8/3/2019 J. M. Dixon, J. A. Tuszynski and E. J. Carpenter- Analytical Expressions for Energies, Degeneracies and Critical Temp

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Analytical Expressions for Energies,

Degeneracies and Critical Temperatures of the

2D Square and 3D Cubic Ising Models

J. M. Dixon a, J. A. Tuszynski b, and E. J. Carpenter b

aDepartment of Physics, University of Warwick, Coventry CV4 7AL

bDepartment of Physics, University of Alberta, Edmonton, Alberta, T6G 2J1,

Canada

Abstract

This paper revisits the fundamental statistical properties of the crucial model incritical phenomena i.e. the Ising model, guided by knowledge of the energy valuesof the Ising Hamiltonian and aided by numerical estimation techniques. We haveobtained exact energies in 2D and 3D and nearly exact analytical forms for thedegeneracies of the distinct eigenvalues. The formulae we obtained, both for energiesand degeneracies, have an exceedingly simple analytical form and are easy to use.The resultant partition functions were utilised to determine the critical behaviourof the Ising system on cubic lattices in 2D and 3D. We obtained a logarithmicdivergence of the specific heat in 2D and 3D cases and the critical temperatureestimates provided additional confirmation of the correctness of our approach.

Key words: Ising, Hamiltonian, partition function, critical temperature

1 Introduction

The Ising model, despite its simplicity has been of crucial importance in thestudy of critical phenomena [1]. For the two-dimensional square lattice, On-sager determined the free energy analytically [2] many years ago. The 3Dcubic lattice has so far eluded analytical solution although a number of ap-proximations have been made. For example, the Bragg-Williams approxima-tion assumes no short-range order apart from that resulting from long-range

Email address: [email protected] (J. A. Tuszynski).

Preprint submitted to Elsevier Science 5 August 2004


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behaviour. It is in fact equivalent to a Curie-Weiss-like molecular field ap-proximation. The Bethe-Peierls approximation [1,3,4], although differing fromthe exact 2D-Onsager result, agrees with 1D results and does account forshort-range order. The transfer matrix method [5,6] has been very useful inrigorously studying the statistical mechanics of the Ising Model. In addition,

a whole battery of analytical and numerical methods have been used to studythe model through finite chain extrapolations [7], high-temperature series [8],two-time Green functions [9], Suzuki-Trotter mapping [10], Monte Carlo meth-ods [11,12] Bethe approaches [13], and low-temperature series expansions withtemperature grouping polynomials [14,15]. As well as being a simple modelfor a number of magnetic materials, the spin 12 anti-ferromagnet is relateddirectly to the lattice gas theory [16]. To name but a few more physical ap-plications Ising-type models have also provided crude models for some solidelectrolytes [17], described binary alloys with variable composition [18], andprovided insights into protein folding [19].

The most important feature of the Ising model is the existence of criticaltemperatures which differ depending on the dimensionality and topology ofthe spin lattice. Denoting the Boltzmann constant kB, the 1D model possessespartition function singularities only in the limit as T 0 J/kB. The 2D Isingmodel has a non-zero critical temperature, Tc, whose value can be found usingcomputer simulations and which varies from 1.52 J/kB in the honeycombstructure to 2.27 J/kB for a square lattice and 3.64 J/kB for a triangularlattice. In 3D numerical estimates are 2.70 J/kB for the two interpenetratingfcc lattices of the diamond structure and 4.51 J/kB for the cubic lattice [1].

In this paper we are interested in spin- 1

2

systems in the absence of an externalmagnetic field, and all discussion will belong to one of three cases: the 1Dlinear ring of size N, the 2D N N square toroid, and the 3D N N Ncubic hypertoroid forms of the model. These toroidal geometries with eachdimension cyclic with period N, give the finite collection of spins periodicboundary conditions that approximate the case of infinite N. In the nextsection we reiterate several important, well-known results for the 1D Isingmodel, some of which may be utilised for the 2D and 3D cases. We give allthe energies, E, in closed form and the corresponding degeneracies, D(E), ineach case.

2 The 1D Partition Function

The Hamiltonian for the Ising model may be written in the general form

H = Ji,j

szi szj , (1)

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where in eq. (1) we sum over all nearest neighbour spin-pairs, szi szj , which

are denoted by i, j in the summation. The spin sites are assumed to beequally spaced, szi represents the z-component of the i

th spin and J is acoupling constant. Periodic boundary conditions will be assumed throughout,including 2D and 3D cases. As is well known, two equivalent forms of the

partition function, Z1, in 1D are given by [1,16]

Z1 = 2N

coshN

J

4

+ sinhN

J

4

(2)

and

Z1 = 2rmr=0

NC2r exp

J

4(N 4r)

(3)

where N is both the number of sites and distinct nearest-neighbour pairs. Inthe case cited the total spin of each site is 12 which we assume throughout.Here, in 1D rm = N/2 for N even and rm = (N 1)/2 when N is odd. Aseach distinct energy may be labelled with index r, eq. (3) may also be writtenas [3]

Z1 =r

D(Er)eEr (4)

where the sum in eq. (4) is actually over distinct energies and D(Er) is thedegeneracy of each energy level. The distinct energies in 1D may be representedby [20,21]

Er = (4r N) 14J, (5)

and the degeneracy, D(Er) is

D(Er) = 2 N

C2r. (6)

For large N it may be more convenient to approximate with the Laplace-deMoivre formula [22]

NC2r 2N

2

Nexp

2N

2r N

2

2(7)

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to obtain D(E) as a distribution of energies about the central energy, namelyabout r = N/4 or E = 0 J.

D(E) =4 2N

2Nexp

12N

E14

J

2

. (8)

This is Gaussian with a variance ofN(14J)2. For a large N we can convert thesum in eq. (4) to an integral. This integral then gives us the area under theD(E) versus E curve as 2N, i.e. the total number of states in the 1D chainwith N spins.

3 Energies in 1D, 2D, and 3D

We can extend the labelling of energies with r into higher dimensional cases.We again, set up our labels so that when r = 0 we obtain the minimum energy,Emin, of the system. This is simply

Emin = 14J 12 NsNNN (9)

where Ns is the total number of sites and NNN is the number of nearestneighbours for each site. Thus in 2D, Ns = N

2 and NNN = 4 whereas in 3DNs = N

3 and NNN = 6.

In 2D we may again label energies with r = 0, 1, 2, . . . , rm where here rm =N22 ifN is even and rm = N2N1 if N is odd. However, the energies arenot evenly distributed in 2D as was the case in 1D so that as r increases from0 the energy rises from Emin = 2N2 14J by a step of 2J first and then, insteps ofJ. IfN is odd, this is the complete pattern, but if N is even the stepsofJ are followed by a final jump of 2J, producing a distribution symmetricabout 0J.

In 3D, Em = 3N3 14J and for N even, rm = 32N3 6 whereas if N is oddrm =

32

(N3

N2)

3. For N even, the energy in 3D rises first by 3

J, then

2J, then in steps of J until the last two energy steps at the rm end of thedistribution where there is a jump of 2J and then, lastly, a change of 3J.If N is odd, however, the distribution of energies is not symmetrical and istruncated before the final two changes. To summarize we have:

in 1D(1) for N even,

Energy = (r 14

N) J, where rm = 12N

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(2) for N odd,Energy = (r 14N) J, where rm = 12(N 1)

in 2D(1) for N even,

Energy = (r r=0 + r=rm 12N2 + 1) J, where rm = N2 2(2) for N odd,

Energy = (r r=0 12N2 + 1) J, where rm = N2 N 1where r=ri is unity when r = ri and zero otherwise.

in 3D(1) For N even,

Energy = (r 3r=0 r=1 + r=rm1 + 3r=rm 34N3 + 3) J, where rm =(3N3/2) 6

(2) For N odd,Energy = (r 3r=0 r=1 34N3 + 3) J, where rm = 32(N3 N2) 3.

The number of distinct energies in each case is obviously just rm + 1. Theseresults are easily verified in 2D and 3D for small N with a computer. How-ever, physical reasoning which leads to them may be easily proved using thechequer-board picture as follows.

To see how the argument proceeds we consider the 2D case as an example.The N N arrangement of spins may be regarded as N2 spins or, taking theperiodic boundary conditions into account, 2N2 interaction pairs or bonds.Each of these bonds has two states which we may term aligned or antialignedwith energies 1

4J and +1

4J, respectively. The energy of a state with n+

aligned pairs and n = 2N2

n+ antialigned pairs is therefore given by

E = (n n+) J/4 = [n (2N2 n)] J/4 = Emin + n J/2 (10)

where Emin = 12N2 J and corresponds to the minimum possible energy,obviously when there are no antialigned pairs, i.e. either all spins are up or allspins are down.

With periodic boundary conditions it is easy to establish that each of the N

rows and N columns must contain an even number of antialigned pairs. Fromthis fact we can deduce that no states exist with n+ greater than 2N(N 1)when N is odd and 2N2 when N is even. To see that states at these limits doexist we have illustrated in Fig. 1A cases when N = 3, 5, and 7 and in Fig. 1Bcases with N = 4, 6, and 8. The direction of the spin component is denoted byan arrow at each of the sites. When N is even, each spin component is oppositeto its four nearest neighbours so that a chequer-board pattern emerges. Ascan be seen in Fig. 1A this is almost true for N odd although some mismatch

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occurs. Thus the maximum energy, Emax, is given by

Emax =

+2N(N 1) 14J when N is odd+2N2 1

4J when N is even

(11)

To produce these chequer-board patterns one can begin with an NN latticein 2D with one spin up in the top left-hand corner. Each of the four nearestneighbours (using periodic boundary conditions) is now given a down spin.One now proceeds across the top row repeating the procedure, i.e. the nextspin along the top row is spin-down so each of its four neighbours is given upspins. The procedure continues to the second row after the first is completed,etc. When N is even we always end up with all spin pairs antialigned so theenergy is a maximum, i.e. 1

2N2 J. When N is odd, one finds that one pair

in one row (or column) cannot be made antialigned so the end result is the

maximum energy of12N(N 1) J.

From the chequer-board pattern we consider the process whereby we progressin steps towards the Emin state with all the spins aligned taking the N evencase as an example. As N is even there are 12N

2 sites which have an up spinon them. If any up spin is reversed the four surrounding spins together withthe reversed up spin will switch from antialigned to aligned, producing a shiftin energy of 2J. Each of the 1

2N2 up spins can be reversed progressively

(changing the energy by 2Jat each step) until the entire region is spin down(spins down on every site). Alternatively we can start with all spins down onevery site, i.e. a state of lowest energy. If we now introduce an adjacent pair of

spin-up sites we increase the energy to Emin + 3J. If we build up a chequer-board pattern avoiding this defect pair we can generate a second similar seriesof energies ranging from Emin + 3J to Emax 3J, the separations withinthis series of energies again being 2J. This second series generates energiesbetween the energies of the first series. Thus the first series for N = 4 fromminimum to maximum is:

8J, 6J, 4J, 2J, 0J, +2J, +4J, +6J, +8J

whereas the second series is

5J, 3J, J, +J, +3J, +5JAs the transitions between the aligned and antialigned states carry an energyshift of 12J and each system must contain an even number of antialignedpairs the smallest possible energy changes are J. We have just proven theexistence of states with all possible energies respecting this J shift betweenstates between Emin and Emaxwith the exceptions ofEmin + J and Emax J. The first of these energies, i.e. Emin + J, corresponds to a system with

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two antialigned spin pairs. Such a state requires the two pairs in the samecolumn for otherwise there would be an odd number of antialigned pairs intwo columns contrary to the above discussion. However, there is no way toplace the two spins in the same column without placing them in two differentrowsrows with an odd number of antialigned pairs. Hence no such state can

exist. An even N, 2D arrangement of spins has an energy, E, and if we considerthe mirror arrangement where the spin components on alternating sites arereversed, it is obvious that this new arrangement will have the mirror energyE. The EmaxE J can be shown to be equivalent to the Emin + J case bythis mirror symmetry argument.

For a three-dimensional system, an analogous argument may be used. We againconsider the case when N is even for simplicity, for example, to go from a spindown state to a 3D chequer-board state, the steps involve energy differencesof 3J. The two-spin defect then has an energy shift of 5J. A third seriescan be constructed starting from 7

Jtriplet defects. When the argument is

carried through we find that we have produced three missing energies at eachend of the range, i.e. Emin + J, Emin + 2J, Emin + 4J where Emin hereagain corresponds to the states with all spin components down or all spincomponents up.

When N is odd the possible energy states are very similar near the minimum,with the same patterns of gaps as discussed above. In contrast, the inabilityto form a perfect chequer-board pattern means that the series are truncatedat Emax =

12N(N 1) J as discussed above. The defects associated with

this truncation can overlap the series generation defects so that no states areinaccessible. Accordingly only the low-energy gaps exist.

4 Variances of Energy Degeneracy Distributions in 2D and 3D

Since each model has a fixed number of two-state spins, it may be convenientlyrepresented on a digital computer by an integer of the same number of bits.The energy degeneracies of systems may then be readily studied by examininga randomly chosen subsample of all possible states. It has been observed thatsome pseudo-random number generator routines produce results with subtle

correlations that affect the results of some algorithms related to the Isingmodel [23]. We have tried to guard against such an effect by comparing withdifferent random sources [24,25] and have noted that the changes in our re-sults for the variance of the distributions using the different generators werenegligible. The data we present here used the default on our computer, anon-linear additive feedback routine in the standard C language library.

We calculated variances of the energies, 2, from samples of 106 states for 2D

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and 3D on a range of small Ns. No attempt was made to prevent the repeatsampling of the the same state, allowing a uniform sampling across all thecases even when the system has fewer than 106 states. We see 2 = 2N2(1

4J)2

in 2D and 2 = 3N3(14J)2 in 3D in the sampling results. (Also note that2 = N(1

4J)2 in 1D, see Sec. 2.) As these trends were first observed on some

small systems where exact variances are known, we believe them to be exact.

In Table 1 we show the energy variances, 2, and a comparison between themand 2N2 or 3N3 in 2D or 3D, respectively. We display also a percentage error.Note that the cases with N = 2 are exceptions, having variances twice what wewould otherwise expect. Similar considerations reveal that the energy meansare just 0 J in all cases.

5 Energy Degeneracy Distributions in 2D

One might naively think that the distribution in 2D would follow a Gaussianpattern over the whole energy range. The energy degeneracies would then be

D(E) =4 2N2

22exp1

22(E 2)2

(12)

Indeed if one plots 2D data for D(E) versus E, the actual degeneracies do ap-pear to fit such a distribution quite well, but this picture is somewhat maskedby the very flat wings or edges of D(E) versus E curves (see as an exam-ple Fig. 2 where |E| > 50). There is an important reason why D(E) curvescannot be Gaussian. If they were, the distributions in 2D and 3D would onlydiffer from 1D by the variance, 2, varying with N as described above. In 1Dit is well known that the only critical temperature is Tc = 0 J/kB so if theGaussian hypothesis were correct, the 2D and 3D cases too would only haveTc = 0 J/kB as a critical temperature. This is manifestly false since, the crit-ical temperature in 2D is known to be Tc = 2.27 J/kB [3]. In 3D the criticaltemperature is even higher. This crisis is avoided if instead of D(E) versus E,

where the character of the curve in the flat wings is hidden, we plot ln D(E)versus Ethe discrepancy between actual degeneracies and the parabolic equiv-alent of the Gaussian becomes apparent. In Fig. 3 we have plotted a specificcase as an example. The difference between parabolic and Gaussian behaviouris not small and, as we see later, the critical behaviour of 2D and 3D Isingmodels arises from the specific nature of the wings of the D(E) distribution.It is therefore crucial that the description of these regions of the distributionbe accurately given.

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We have fitted exact energy degeneracies in 2D to the form (see Figs. 4, 5, 6,and 7)

D(E) = exp( + E2 + E4) (13)

by least-squares fitting the curve +E2+E4 to ln D(E) for N = 4, 6, 8, 10,16, and 32. We obtained the parameters , , and displayed in Table 2 whichgave good fits to the data. The parameter remains close to the logarithm ofthe premultiplicative factor of the Gaussian in eq. (12), namely

= ln

2 2N2

N

. (14)

In Table 3 we have compared values of with together with their relativepercentage difference, (

)/

100 %. It is seen that this difference appears

to tend to a very small and constant value as N increases and that for verylarge N, scales as N2, the size of the 2D Ising array. In Table 4 we presentestimates of our errors in fitting the quartic ln D(E) versus E in terms of theroot mean square of the differences between the fitted and actual degeneracies.

The parameters and we found to vary with N in the following way, fittingto the data from N = 4, 6, 8, 10, 16, and 32,

(N) = (0.53856 + 2.16001N)2 (J/4)2

(N) = +(

1.01841 + 2.09255N)6 (

J/4)4.

(15)

It would appear that there is a trend in the behaviour as function ofN in that scales as N2 and as N6 for large N.

6 Energy Degeneracies in 3D

In 3D we attempted to fit the ln D(E) versus E data with a quartic energy

dependence as in 2D and eq. (13). However, it was soon established that anadditional power in the polynomial function of energy greatly improved thefits. That is, we performed a least squares fitting to expressions of the form

ln D(E) = + E2 + E4 + E6. (16)

In Fig. 8 we illustrate an attempted fitting of the 3D N = 4 degeneracies toa quartic ln D(E) form and, in Fig. 9, we compare it with a fit to the sixth

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power expression of eq. (16). We notice that in Fig. 8 the fit is very poor in thecentre of the distribution where the term in should dominate (i.e. the pureGaussian). Some two-thirds of the way down the distributions peak the fit isalso not good and a similar situation holds in the tails. However, in Fig. 9, thelast two areas fit well and the centre portion is much improved. In Table 5 we

present the best-fit values of , , , and in 3D for N = 3, 4, and 6 and inTable 6 differences between and for the same values of N. In Table 7 wedisplay the RMS value of the degerenacy differences between the model andthe data for the case N = 6 in 3D.

Before such fittings can take place the logarithm of the actual degeneracies andenergies must be found. This is difficult because as N increases the number ofspin states increases dramatically, e.g. for N = 6 the total number of states is26

3 1065. We have developed a procedure to overcome this problem withinreasonable computer time which entails the use of what we call ns curves.In this procedure we consider changes to the Ising 3D system where all the

spins are initially up. Then we sample the states with a fixed number, ns ofspins turned down. Let us suppose the number we sample, for example, is100 000. Of these, those with a given energy will be a given fraction of thetotal sampled, i.e. 100 000. For a fixed ns, for each energy a fraction of thetotal number of states may be determined for that particular energy. In 3D

the total number of states with ns spins turned down will be2N

3

Cns. Eachfraction for each energy obtained for a fixed ns is then scaled by this factor.We expect that, provided the number of states sampled is large, we shouldobtain reasonable agreement with the actual degeneracies, D(E). In fact, wehave compared with exact degeneracies in 1D, 2D, and 3D where the ns curve

procedure works extremely well. Not only this but the area under each curveis automatically, by construction, exact giving the correct number of statesand it was also shown that the variance is at least 99% of what it should bein 2D and 3D. The shape of a given ln D(E) versus E curve for a fixed ns isillustrated in Fig. 10 and a complete 3D ln D(E) versus E curve, obtained bythis method, is shown in Fig. 11 for N = 6. When ns curves are approximatelyparabolic, e.g. when N = 4 each curve, up to 11 spins turned-over, takes theform A(E B)2 + C where

A1 = 9.21603 + 13.3109ns,B = 30.4169 + 9.06772ns,

C = 9.13203 + 1.87626ns.

(17)

We see that parameters B and C vary linearly with ns as does A1. Interest-

ingly, these ns curves have mean energies E(ns), which depend on the numberof spins turned down and the size N of the system, in the following ways:

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in 1D for N > 2

E(ns) =4

N 1

N

2

2

ns N2

2 N (18)

in 2D we have checked numerically for N = 3, 4, and 5 that

E(ns) =8

N2 1

N2

2

2

ns N2

2

2 2N2 (19) in 3D for N = 4 we have verified that

E(ns) =12

N3 1

N3

2

2

ns N3

2

2 3N3 (20)In eqs. (18), (19), and (20) we see a striking similarity of form when the dimen-

sion is increased from 1D. The factor outside the square brackets is merely 4the dimension (1, 2, or 3) divided by the size of the spin system minus unity.The first term inside each square bracket is the square of the size dividedby two and the second factor is the square of the difference between ns andthe size divided by two. The last factor in each case is clearly the number ofspin pair interactions. Furthermore, for ns = N/2 as N , E 1 in 1D,E 2 in 2D, and E 3 in 3D. This is readily seen, for example in 1D, byrearranging E(ns) in eq. (18) as

E(ns) =N

N

1 4

N

1 ns N

2 2

(21)

and similarly for 2D and 3D. A fortuitously exact approximation to thesemay be obtained if we consider a specific spin pair. There are four distinctpossibilities which may arise. If the first spin is up, the second can be down,written , or up, . Similarly, if the first spin is down, the second may beup, , or down . Let us write the probabilities of the four possibilities asP, P, P, and P. In 1D we may express P by

P =nsN

ns 1N

1

(22)

where ns/N is the probability that the first spin is down and (ns 1)/(N 1)is the probability that the second is down given that the first spin is downsince there is one less to be down for the second. Similarly,

P =N ns

N N ns 1

N 1 , (23)

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P =N ns

N ns

N 1 , and (24)

P =nsN

N nsN 1 . (25)

The energies of the and cases are J/4 and conversely, both and states are +J/4. If we navely multiply the mean energy of the single spinpair by the 12NNNNs spin pairs we can estimate for the mean energy, E, of anns curve,

E =

NNN2

Ns

(P + P P P)( 14J) (26)

which gives exactly eq. (18). The argument is easily generalised in 2D by re-placing the number of spins, N, in the probabilities with N2. Similarly in 3D,N is replaced by N3 in the four probabilities. Eqs. (19) and (20) may also beproved in this way.

7 Critical Behaviour

The aim in this section is to show that critical behaviour and critical tempera-tures are consistent with the picture we have so far in terms of the degeneracies,D(E), in 2D and 3D. Our starting point is the standard expression for specificheat, CV, in terms of a statistical average of the square of the energy, namely

CV =1

kBT2

E2

(27)

where

E2

=

E2D(E) eEdE

D(E) eEdE, (28)

= (kB

T)1 and T the temperature.

The integrals are over the entire energy distributions, but this infinite rangecan be truncated to the range of energies with non-zero degeneracies, that isbetween 2N2 J/4 and +2N2 J/4 in 2D, 3N3 J/4 and +3N3 J/4 in 3D.If we denote the minimum allowed energy of the distribution by E0 then

CV =1

kBT2

ln I() (29)

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where

I() = 2

E00

exp(E E2 + E4 E6)dE, (30)

and , , and are the moduli of the , , and , respectively. Additionally,this allows us to avoid confusion with = (kBT)

1. We also define a smallparameter, , to denote proximity to the critical temperature by

=T TckBT2c

(31)

so that

= c + kBTc2

. . . (32)

where c = (kBTc)1, Tc being the critical temperature. We write E = E0 + x

where x is a shift in energy away from E0, which we suppose is close to thecritical energy. For simplicity we drop (kBT

2)1 in eq. (29) since it is commonto all the integrals and does not contribute to criticality because it only appearsin our expressions as common factors within ratios. We then define

P =1

I()

I()

. (33)

The component

E0 E20 + E40 E60 (34)

and cancel between the numerator and denominator of P and we find

P 0(E

20 + 2E0x + x

2) exp[( 2kBTc)x x2]dx

0 exp[( 2kBTc)x x2]dx

(35)

where = 6E20 + 15E40 . We have restricted the integration range toE0 = E0 < x < E0 so that it is close to the energy region where criticalityoccurs, and used the fact that by definition

+ 2E0 4E30 + 6E50 = c + 2kBTc + 2E0 4E30 + 6E50 = + 2kBTc. (36)

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Here, is assumed small and close to E0. Thus P may be written as

P = E20 2E0I1I0

I2I0

(37)

where

Ik =

0

xk exp[x x2( 6E20 + 15E40)]dx, (38)

and the small terms in 2 have been dropped. We may examine each ofI0, I1,and I2 in turn and drop terms in x

2 and x2 as they are much smaller thanthe term in x2. After elementary integration we find

I1 = 12 exp

2

4exp + 2

2 exp2

4 . (39)

For small and 0 this is finite. In a similar way, I2 takes the form

I2 =1

exp

2

4

L2L1

Z2

Z

3/2+

2

42

eZ

2

dZ (40)

where we have changed variables to Z =

(x 1

2/) and defined L1 =1

2/

and L2 =

( + 12/). The last two terms of eq. (40) clearly 0 as 0 and the first is finite when = 0. Lastly, we examine I0 which becomes

I0 =1

exp

+2/4

erf+

exp2

exp

2/4

. (41)

In the thermodynamic limit as N , 0, so when 0, I0 maybecome close to zero and we have critical behaviour. Restoring the minus sign

taken out of eq. (29), dividing by = x, defining A =

and B =

,

and assuming that exp(+2/4) is close to one, we define F1 = I0/ whichbecomes

F1 =1

B2

exp

x24B2

exp

A2

erf(A)

Bx(42)

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which we compare with F2 = (ln )/ in Fig. 12. Clearly F1 and F2 becomevirtually identical as 0 confirming the possibility of the logarithmic sin-gularity in the specific heat via the ln dependence in F1 close to criticality in2D which is well known. However, as we have determined in a number of 2Dcases and provided a possible formula for higher values ofN, namely eq. (15),

so in principle, B is known in eq. (42). Hence, by fitting carefully to (ln )/,we may determine which indicates how far the critical energy is away fromthe critical region.

To find specific critical temperatures we go back to eq. (30). We may considercritical behaviour to occur outside the central Gaussian part of the D(E)distribution so we divide I() into two parts, one near the centre of the distri-bution, which we designate as noncritical and write INC, and the other nearto the tails we express by IC. Thus

I = INC + IC (43)

where

INC = 2

E0/20

ef(E) dE, (44)

and

IC = 2

E0

E0/2

ef(E) dE, (45)

and

f(E) = E E2 + E4 E6. (46)

In eq. (44) we have tentatively assumed that the critical region is somewherebetween E0 and E0/2 and we show this later with specific calculations. Ex-panding f(E) about E0 in a Taylor series and retaining only terms up to order

E E0 we readily find

IC 2C(E0)f(E0)

exp(E0f

(E0)) exp

1

2E0f

(E0)

(47)

where

C(E0) = exp [f(E0) E0f(E0)] (48)

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and f(E) indicates a derivative with respect to energy. Hence the factor

ln IC() in eq.(29) with I() IC(), since we are only interested inthe particular temperature of criticality, becomes

ln IC =

ln C(E0) ln f(E0) + ln eE0f(E0) 1 e12 E0f(E0)=

f(E0) E0f(E0) ln f(E0) + E0f(E0) + ln

1 e12 E0f(E0)

(49)

After a little algebra eq. (49) reduces to

ln IC = E20 +

2E0f(E0)

+E20 exp[

12 E0f

(E0)]

1 exp[12

E0f(E0)](50)

The condition for criticality from eq. (50) is clearly f

(E0) = 0. We may nowconsider the 2D and 3D cases in turn once the asymptotic behaviour of severalkey parameters has been established.

According to eq. (15) N2 and N6 in the 2D case. One can lookupon this in a different way, in that in 2D each term of + E2 + E4 scaleswith large N in the same way, namely as E scales as N2. (The extrema of thedistributions are 2N2 J/4 for even N.)

In 3D, the corresponding expression, + E2 + E4 + E6, has terms thatscale as N3 for large N, with N3, N9, and N15.

In the 2D case = 0, i.e. there is no sixth power of energy appearing in theexponent of D(E), so that

f(E) = E2 + E4 E (51)

and hence

= C = | 2E + 4E3|= value of at criticality

(52)

For very large N we may use the relations in eq. (15) for and so that

kBTC = 1C = | 2E + 4E3|1 (53)

when E is close to 2N2.

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Evaluating the condition ddE(kBTc) = 0 with E =

/(6) in 2D yields,

taking the values of and for the largest N from Table 2, kBTc = 2.07 J,a value very close to the Onsager result. Using eq. (15) in the same contextgives Tc = 2.02 J/kB which is independent of N, as it should be.

Our result in eq. (53) compares favourably with the celebrated Onsager result(within 10 % of the exact result) in spite of relying on a very simple expansion.Undoubtedly carrying the procedure to a higher order would produce a stillcloser result. Moreover, we have obtained a logarithmic singularity for thespecific heat in a rather straightforward manner.

For 3D, the corresponding procedure yields the condition

kBTc = (2E + 4E3 + 6E5)1 (54)

with

E =12

1442 240

60(55)

Making appropriate substitutions, we find using our method that for large N,the critical temperature in 3D is 4.11 J/kB.

8 Discussion and Summary

The Ising model has provided a rich source for the study of phase transitionsand critical phenomena over close to a century. Its apparent simplicity andelegance belied a minefield of mathematical challenges. Its three-dimensionalimplementation has not been solved analytically to this day. In this paper wehave attempted to find manageable analytical expressions for energies, degen-eracies, and partition functions of Ising lattices of arbitrary size in two andthree dimensions. We have only considered square lattices in two dimensionsand cubic lattices in three dimensions assuming periodic boundary conditionsin both cases. The formulas for the energies we have obtained are exact and

expressed as multiples ofJ. When the size of the lattice, N, is even the dis-tribution of energies is symmetrical about its maximum, but is not Gaussianexcept for the central portion. We find in 2D that a quartic energy dependenceenters the exponent with the opposite sign to the quadratic term. The energiesdiffer initially by 2J and then in steps of J until the other tail end, wherethe energy changes by 2J again. For N odd, the distribution is not symmet-rical but nevertheless still contains a positive quartic energy component inthe associated exponential. Energies on the low energy side of the distribution

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initially change by 2J units and thereafter by J (with no subsequent dropon the high energy side of 2J).

In three dimensions with N even (N3 sites) the degeneracy distribution isagain symmetrical. However, a proper fit requires the addition of a negative

sixth power exponent in addition to a quartic and quadratic energy depen-dence. Thus the signs of the contributions alternate, beginning with a negativequadratic. Furthermore, on the lower energy tail energies change by 3J first,then 2Jand then subsequently change in Junits until, at the high energy tailthere are changes of 2J units and 3J units. However, if N is odd in 3D thedistribution is again asymmetric, with the low energy tail changes the sameas for N even, but no large gaps on the high energy side of the curve.

In principle our formula for energy degeneracy distributions allows us to cal-culate the partition function of an arbitrary size lattice is 2 and 3 dimensions.This may require the use of non-Gaussian integration techniques [26] involving

integrals of the type

Ik (a, b) =

0

x2k1 exp(ax4k bx2k) dx

= (2k)1(2a)/2()exp

b2

8a

D

b2a

(56)

where D are parabolic cylinder functions. This may be applied to calculate

E2

of eq. (27) with the general expression for

D(E) in eq. (16) together witha convergent polynomial series expansion for the sixth power term in theexponential. Hence, for an arbitrary n-th moment of the distribution,

En =

En exp(E E2 + E4 E6) dEexp(E E2 + E4 E6) dE , (57)

we can follow the methodology outlined in Tuszynski et al. [26].

In addition to a general method this paper has provided a specific test of

the methods validity, especially in the particularly challenging area in thevicinity of the critical point. Using the relatively simple evaluation techniqueof incorporating only the tail end of the density function in the central regime ithas been demonstrated that the 2D specific heat, CV, develops a logarithmicdivergence with respect to the reduced critical temperature. Moreover thefirst estimate of the critical temperature, in this case, was within 10 % of theexact value and the inclusion of further terms would undoubted improve theagreement.

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9 Acknowledgments

This project was supported by funding from NSERC and MITACS. One ofthe authors (JMD) expresses his gratitude for the hospitality and kindness of

the faculty and staff of the Department of Physics at the University of Albertaduring his sabbatical leave in Edmonton.

References

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[2] L. Onsager, Crystal statistics. I. A two-dimensional model with an order-

disorder transition, Phys. Rev. 65 (34) (1944) 117149.

[3] K. Huang, Statistical Mechanics, Wiley, New York, 1987.

[4] J. M. Dixon, J. A. Tuszynski, P. Clarkson, From Nonlinearity to Coherance,Universal Features of Nonlinear Beahaviour in Many-Body Physics, ClarentonPress, 1997.

[5] H. A. Kramers, G. H. Wannier, Statistics of the two-dimensional ferromagnet.Part I, Phys. Rev. 60 (3) (1941) 252262.

[6] R. Kubo, An analytic method in statistical mechanics, Busserion Kenkyu 1(1943) 113, in Japanese.

[7] J. C. Bonner, M. E. Fisher, Linear magnetic chains with anisotropic coupling,Phys. Rev. 135 (3A) (1964) A640A658.

[8] G. A. Baker Jr., G. S. Rushbrooke, H. E. Gilbert, High-temperature seriesexpansions for the spin-12 Heisenberg model by the method of irreduciblerepresentations of the symmetric group, Phys. Rev 135 (5A) (1964) A1272A1277.

[9] J. Kondo, K. Yamaji, Greens-function formalism of the one-dimensionalHeisenberg spin system, Prog. Theor. Phys. 47 (3) (1972) 807818.

[10] J. J. Cullen, D. P. Landau, Monte Carlo studies of one-dimensional quantumHeisenberg and XY models, Phys. Rev. B 27 (1) (1983) 297313.

[11] J. W. Lyklema, Monte Carlo study of the one-dimensional quantum Heisenbergferromagnet near T = 0, Phys. Rev. B 27 (5) (1983) 31083110.

[12] M. Marcu, J. Muller, F.-K. Schmatzer, Quantum Monte Carlo simulation of theone-dimensional spin-S xxz model. II. high precision calculations for S = 12 , J.Phys. A 18 (16) (1985) 31893203.

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[13] P. Schlottmann, Low-temperature behavior of the S = 12 ferromagneticHeisenberg chain, Phys. Rev. B 33 (7) (1986) 48804886.

[14] M. Takahashi, M. Yamada, Critical behavior of spin-12 one-dimensionalHeisenberg ferromagnet at low temperatures, J. Phys. Soc. Japan 55 (6) (1986)20242036.

[15] C. K. Majumdar, I. Ramarao, Critical field and low-temperature critical indicesof the ferromagnetic Ising model, Phys. Rev. B 22 (7) (1980) 32883293.

[16] H. Stanley, Intrudaction to Phase Trasitions and Critical Phenomena,Clarendon Press, Oxford, 1971.

[17] G. D. Mahan, W. L. Roth (Eds.), Supersonic Conductors, Plenum Press, NewYork, 1976.

[18] A. Bienenstock, J. Lewis, Order-disorder of nonstoichiometric binary alloys andIsing antiferromagnets in magnetic fields, Phys. Rev. 160 (2) (1967) 393403.

[19] V. Munoz, What can we learn about protein folding from Ising-like models?,Curr. Opin. Struct. Biol. 11 (2) (2001) 212216.

[20] J. A. Tuszynski, J. M. Dixon, An algorithm to obtain exact eigenvalues andeigenstates of the arbitrary spin Ising Hamiltonian in d-dimensions, Phys. Lett.A 283 (5-6) (2001) 300308.

[21] J. M. Dixon, J. A. Tuszynski, M. L. A. Nip, Exact eigenvalues of the IsingHamiltonian in one-, two- and three-dimensions in the absence of a magneticfield, Physica A 289 (1-2) (2001) 137156.

[22] J. Spanier, K. B. Oldham, An Atlas of Functions, Hemisphere Publishing

Corporation, New York, 1987.

[23] H. G. Ballesteros, V. Martn-Mayor, Test for random number generators:Schwinger-Dyson equations for the Ising model, Phys. Rev. E 58 (5) (1998)67876791.

[24] The RAND Corporation, A Million Random Digits with 100000 NormalDeviates, 1st Edition, The Free Press, Glencoe, Illinois, 1955,http://www.rand.org/publications/classics/randomdigits/.

[25] G. Marsaglia, The Marsaglia Random Number CDROM including the DiehardBattery of Tests of Randomness, 1995, http://stat.fsu.edu/pub/diehard/.

[26] J. A. Tuszynski, M. J. Clouter, H. Kieffe, Non-Gaussian models for criticalfluctuations, Phys. Rev. B 33 (5) (1986) 34233435.

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Fig. 1. Chequer-board patterns. spin site; spin up; spin down. A) N = 3, 5, 7B) N = 4, 6, 8

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Fig. 2. A 2D plot ofD(E), in units of 51017, versus E for N = 8 showing a centralGaussian shape with very flat wings. () actual data; () Gaussian fit.

Fig. 3. A 2D plot of ln D(E) versus E for N = 8. () actual data; () centralparabola obtained if D(E) versus E where Gaussian.

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Fig. 4. A 2D fit to data points for a plot of ln D(E) versus E for N = 6. () actualdata; () least squares fit.


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Fig. 8. A least squares fit (line) of 3D degeneracy data (dots) to the quartic expres-sion ln D(E) = + E2 + E4 versus E for N = 4 showing poor fit at the centre.( = 41.0343, = 0.00172779, = +1.873 108)

Fig. 9. A least squares fit (line) of 3D degeneracy data (dots) to the sixth powerform ln D(E) = + E2 + E4 + E6 versus E for N = 4 . ( = 41.7412, = 0.00221899, = +5.87478 108, = 7.92728 1013)

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Fig. 10. An illustration of the ns-curve method for N = 4 in 3D. We show portionsof the 3 curves for ns = 5, 7, and 11.

Fig. 11. A complete 3D fit (line) to data points (dots) for ln D(E) versus E, usingthe ns-curve method, for N=6.

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Fig. 12. A) plot of the function F1, see eq. (42), versus x and the function F2 = ln x/xfor A = 45.2446, B = 0.22604. B) As in part A, but with different scales on thecoordinate axes.

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N 2 Dimensions 3 Dimensions

2 2N Error % 2 2N Error %

2 16.011 8 100.13 47.884 24 99.52

3 18.000 18 0.0014 81.056 81 0.069

4 31.964 32 0.11 192.070 192 0.0365 49.974 50 0.053 375.318 375 0.0856 72.040 72 0.055 646.902 648 0.177 98.093 98 0.095 1 030.044 1 029 0.10

8 128.104 128 0.081 1 535.339 1 536 0.0439 162.144 162 0.089 2 186.985 2 187 0.000 67

10 200.093 200 0.047 3 003.016 3 000 0.10

11 241.877 242 0.051 3 990.177 3 993 0.07112 287.871 288 0.045 5 169.494 5 184 0.2813 337.266 338 0.21 6 577.363 6 591 0.2114 391.520 392 0.12 8 224.661 8 232 0.08915 450.087 450 0.019 10 1 21.187 10 1 25 0.03816 513.271 512 0.25 12 309.085 12 288 0.17

17 579.738 578 0.30 14 734.258 14 739 0.03218 646.571 648

0.22 17 494.118 17496

0.011

19 720.995 722 0.14 20 588.168 20 577 0.054Table 1Calculations of approximate variances for the degeneracy distributions in 2D and

3D for N = 2 through 19. Percentage errors are between 2 and 2N2 in 2D and w2

and 3N3 in 3D. Each estimate is based on 106 random samples.

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N

4 9.71229 1.56078102 6.72956106

6 23.0990 6.52493103 4.37475107

8 42.0658 3.55286103 6.5596810810 66.5968 2.23865103 1.57079108

16 173.408 8.59737104 8.410911010

32 700.841 2.12942104 1.221921011Table 2Values of the parameters , and in ln D(E) = + E2 + E4 for the fit to the

ln D(E) distribution in 2D for N = 4, 6, 8, 10, 16, and 32.

N 100%4 9.71229 9.82484 1.14556

6 23.0990 23.2823 0.78729

8 42.0658 42.4027 0.794637

10 66.5968 67.1329 0.798561

16 173.408 174.794 0.792834

32 700.841 706.438 0.792225

Table 3

Comparison of with = ln(2N2

+1/N) in eq. (14) for N = 4, 6, 8, 10, 16, and32 in 2D.

N RMS Total Degeneracy Difference

4 0.0656 9.71229

6 0.0403 23.0990

8 0.0420 42.0658

10 0.0500 66.5968

16 0.0791 173.408

32 0.1609 700.841

Table 4Estimate of errors to fitting ln D(E) versus E in 2D. Root-mean square (RMS)

where RMS =

i(xi xi)1/22

/N0. xi fitted log degeneracy, xi data logdegeneracy, N0 total number of data points.

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N

3 16.8377 6.02093 103

+1.12708 106

8.92867 1011

4 41.7412 2.21899 103 +5.87478 108 7.92728 1013

6 145.659 6.49072 104 +1.49938 109 1.85000 1015

6 145.112 6.3841 104 +1.44836 109 1.77912 1015Table 5Parameters , , , and to fit D(E) = + E2 + E4 + E6 to ln D(E) function

in 3D for N = 3, 4, and 6. Two similar fits are given for N = 6.

N 100%3 16.8377 16.9851 0.875416

4 41.7412 42.2000 1.09915

6 145.659 146.950 0.886317

Table 6Values of and in 3D for N = 3, 4, and 6 with percentage errors.

N RMS Total Degeneracy Difference

6 0.00670 145.659

Table 7Estimate of errors to fitting ln D(E) versus E in 3D for N=6. Root-mean square

(RMS) where RMS =

i(xi xi)1/22

/N0. xi fitted log degeneracy, xi datalog degeneracy, N0 total number of data points.

Documents

J. M. Dixon, J. A. Tuszynski and E. J. Carpenter- Analytical Expressions for Energies, Degeneracies and Critical Temperatures of the 2D Square and 3D Cubic Ising Models