Partition Function Zeros and the Ising Model in a Field

Todd

Supervised by Dr Iwan Jensen University of Melbourne

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Introduction The Ising model is a mathematical model of ferromagnetism in statistical mechanics. It was first studied by Ernst Ising in the early 1920s to analyse phase transitions in ferromagnets. In this study we consider the 2-dimensional square lattice. We aim to better understand phase transitions in ferromagnets, and to predict a ferromagnet’s potential for a phase transition.

Background Ferromagnetism is a mechanism by which certain materials, such as iron, can form permanent magnets.

In a lattice of iron atoms, each iron atom has 4 unpaired electrons. Each electron has a magnetic dipole moment that arises from its quantum mechanical spin. So we can think of each iron atom as a tiny magnet.

The magnetic moments of each iron atom spontaneously align and combine to create a strong magnetic field. However, as the temperature increases, we find that beyond a certain point, the lattice loses its magnetisation.

Figure 1 - Magnetisation of a ferromagnet by temperature (1)

This change from magnetised to non-magnetised is called a phase transition. A phase transition is where a small change in a parameter such as temperature or pressure leads to a large-scale, qualitative change in the state of the system.

The Ising Model is used to analyse these critical points of phase transition.

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Ising Model The starting point for the Ising Model is a lattice. In this project, we consider a two dimensional square lattice. In this case, each red dot represents an electron (or an iron atom). Each black line is a bond. We say two electrons are nearest neighbours if there is a bond connecting them.

For added simplification, we consider that the lattice “wraps-around”. Each electron in the top row has an additional bond which wraps around and connects to the corresponding electron in the bottom row. Similarly, we have extra bonds from left to right. This negates the difference between “boundary” and “interior” electrons.

In the thermodynamic limit, in which the size of the lattice tends to infinity, the percentage of electrons on the boundary approaches zero. So although the wrap-around model introduces physically unrealistic long-range interactions, we assume that it affects a “negligibly small” percentage of electrons and thus will not affect the overall behaviour of the system. (2)

We assume that each electron can only spin either up or down, which reflects the direction of its magnetic dipole moment. If it spins up we assign it a value of plus one and if it spins down we assign it a value of minus one.

The Hamiltionian is a function that gives the total energy of the system. In the Ising Model it is given by:

E is a parameter associated with nearest neighbour interactions. In the case of a ferromagnet, it is positive. J is a parameter associated with an external magnetic field.

The first sum is over all bonds. A configuration in which nearest neighbours have the same spin has a lower total energy and is therefore more stable than one in which they are not. The second sum is over all electrons. It is more stable for an electron to align with the external magnetic field.

We assume that all other contributions to the total energy are negligible or non-consequential.

Figure 2 - Representation of square lattice

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Partition Function The partition function describes the statistical properties of a system in thermodynamic equilibrium (3). Most of the thermodynamic functions can be derived from the partition function. The partition function is given by:

Where β = 1/kT. k is the Boltzmann constant, and T is the temperature in absolute degrees. We sum over every possible configuration, σ, of the system.

The partition function is related to the probability of the system being in a certain configuration, σ, by:

Historically, Gibbs mathematically described a phase transition as a point of non-analyticity in a thermodynamic function(4). However, physical systems are finite, and therefore their thermodynamic functions are manifestly analytic (such as in figure 1). So intuitively Gibbs’ definition implies that phase transitions do no occur in physical systems.

However, if we generalise the parameters (such as the temperature) of the thermodynamic functions to complex values, we find that there do exist singularities for these thermodynamic functions. Yang and Lee proposed that as the system gets larger and larger, the complex singularities of its thermodynamic functions approach the real (physical) domain of its thermodynamic functions. Phase transitions arise from this.

Calculating Partition Function Zeros First of all, for a given lattice size, we consider every possible configuration. For a lattice with n sites, there are 2n possible configurations. We then map each possible configuration to a matrix. Each spin up site maps to a one, and each spin down to a zero. For a 2 by 2 lattice, the matrix representation of each possible configuration is shown below.

Figure 3 – Matrix representations of all configurations for a 2 by 2 lattice

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There are 8 bonds in a 2 by 2 lattice. For each configuration, we count the number of ones and the number of 0-1 bonds. This can be summarized in the table below.

Multiplicity Number of ones Number of 0 – 1 bonds

1 0 0

4 1 4

4 2 4

2 2 8

4 3 4

1 4 0

Relating back to the Hamiltionian, the sum over all bonds is: 2*(n – number of 0-1 bonds) , where n is the total number of lattice sites.

The sum over all lattice sites is: n – 2*(number of ones) In this way we can calculate the total energy of each configuration. We can then calculate the partition function. In this case it is given by: Z = e8Eβ e4Jβ+4e2Jβ+4+2e-8Eβ+4e-2Jβ+e8Eβ e-4Jβ

We substitute for u= e-2Eβ and z= e-2Jβ, where u and z are Boltzmann weights. In this way we can solve for the partition function zeros as a polynomial in two variables. Since we are solving for the zeros of the partition function, this simplifies to:

0=1+4u4z+4u4z2+2u8z2+4u4z3+z4

This is simply equal to multiplicity*unumber of 0-1 bonds *znumber of ones, summed over every row in the table.

However, as the lattice becomes larger and larger, it becomes impractical to consider every configuration in order to calculate the partition function.

Transfer Matrix Method A much more efficient way to calculate the partition function is by the transfer matrix method.

To calculate the partition function by this method, first we consider one column in the lattice. We then consider each possible configuration for this column. For a length 2 column, these are shown below:

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a b c d

We then create the transfer matrix, M. This is a square matrix of length 2column length.

M =

To calculate each element, we first take the column on the left side of the arrow. Then we add the column on the right side to the initial column. We consider all new bonds created, and all new lattice sites. Here the red line separates the two columns, and the blue lines indicate new bonds created.

Each element is given by:

unumber of new 0-1 bondsznumber of new ones

So, a→a=1 and b→d=uz2

Since interactions don’t extend beyond one column, this matrix gives us the entropy contribution of adding any column to any other column. In this case, M is given by: M = If we then square this matrix, we get the entropy contribution of adding two columns. The top left element for example is:

M211 = (a→a)(a→a) + (a→b)(b→a) + (a→c)(c→a) + (a→d)(d→a)

So this element gives us the entropy contribution of adding any of the four columns to ‘a’, then also adding a column ‘a’ to the first column.

a→a a→b a→c a→d

b→a b→b b→c

c→a

b→d

c→b c→c c→d

d→d d→c d→a d→b

a→a b→d

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Lastly, since our lattice wraps around, the final column must be equal to the column which was originally added to. This means that the only physical contributions to the total entropy lie along the diagonal of the matrix.

So, for example, the entropy of a 2 by 3 lattice is given by:

Z = Trace(M3)

The Project In this project, we consider lattice sizes with (width * length) of (L * L+1), where L is an integer greater than 2. This is because square lattices with an odd number of lattice sites do not have a configuration which is non-magnetised.

We fix the u-value in the partition function, and solve for all complex zeros in the z-plane. Clearly, these are singularities of the partition function. In this case, we consider u-values along the line 5+i between zero and the unit circle.

Where |u|=0, all zeros lie on the unit circle in the z-plane. The Lee-Yang theorem states that for all physical values of u, between 0 and 1 on the real line, every complex zero lies on the unit circle in the z-plane.

As the magnitude of u increases along the line 5+i , we see that beyond a point of around 0.45 the zeros break away from the unit circle and form a spiral.

Figure 4 – Partition function zeros in the z-plane for an 8 by 9 lattice where |u|=0

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|u|=0.418 |u|=0.438

|u|=0.459 |u|=0.479

We define a function to give us the maximum distance of any zero from the unit circle: distancemax = max(||xi|-1|) , for all i where xi is the value of a zero in the z-plane We plot the ‘distancemax’ against the magnitude of u, along with its discrete derivate.

Figure 5 – Partition function zeros in the z-plane for an 8 by 9 lattice for varying magnitudes of u

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‘distancemax’ by |u| Discrete derivative In the derivative there is a discontinuity at around |u|=0.45. If we add more points and zoom in on this discontinuity we find that it does not smooth out. We aim to narrow down on this point and determine if it converges to any value in the thermodynamic limit.

Narrowing in on the Discontinuity We first consider a list of magnitudes for u in increasing order around the point 0.45. For example (0.40,0.41,...,0.49,0.50). We then plot them by the following function:

)

))

)

)

The end result is shown to the right. The maximum value in the transformed graph occurs immediately after the non-analyticity.

Here we take the radius (r1) of maximum value in the transformed graph, as well as the radii for the closest two points to its left (r2 and r3, where r2>r3).

We add the following values to the list of magnitudes for u (such that the list retains its increasing order):

, for all values of a in the list, where b is the value immediately following a.

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We then plot this new list by the same function, and repeat the process of adding two new points. By making use of this ‘log’ transformation, it is simple to automate this process. In addition, since we only add two points around the non-analyticity per cycle, this method minimises computation time.

Below are the resulting graphs after 35 cycles for an 8 by 9 lattice.

One last consideration is that for lattices larger than a size of 10 by 11, it can become extremely time consuming (or not possible) to calculate the zeros using a program such as Maple. In these cases we use an external program, unisolve(5). This program is capable of solving for the zeros of a 16 by 17 lattice in only a few seconds.

Results The points of non-analyticity for varying lattices sizes are given in the table below. Width of lattice

|u| at non-analyticity Width of lattice |u| at non-analyticity

4 0.48941 11 0.461148 5 0.49143 12 0.458763 6 0.485953 13 0.456809 7 0.47994 14 0.455187 8 0.47241 15 0.453837 9 0.467667 16 0.452703 10 0.464066

Figure 6 – Left, ‘distancemax’ by 100*|u|. Right, ‘log’ transformation by 100*|u|

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We then fit a least squares exponential curve to the data, to make a prediction of what the |u| value might converge to in the thermodynamic limit.

Here, |u| 0.44903, in the thermodynamic limit.

Possibilities for Further Research Of course, it would be interesting to consider the same project for different arguments of u. We would see if this approaches any value as the argument of u tends to 0.

An additional area of research is to understand why the zeros form the shape that they do as |u| increases, and to see if there is something universal to the nature of this movement.

Acknowledgments Huge thanks to my supervisor, Dr Iwan Jensen. I did not have any programming experience before this project, but he was always patient each of the many times I was having trouble. Lastly, thank you to AMSI and the University of Melbourne for this fantastic opportunity.

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

7 9 11 13 15 17

Figure 7 – Least squares exponential curve for width of lattice by |u| of the non-analyticity

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References 1. ‘Monte Carlo Simulation of the Site-Diluted Ising Model’, http://quantumtheory.physik.unibas.ch/people/bruder/Semesterprojekte2007/p3/isingResults.html, Website accessed 20/2/2016 2. Cipra, BA, 1987, ‘An Introduction to the Ising Model’, http://www.ww.amc12.org/sites/default/files/pdf/upload_library/22/Hasse/00029890.di991727.99p0087h.pdf, Website accessed 20/2/2016 3. Wikipedia, ‘Partition Function (Statistical Mechanics)’, https://en.wikipedia.org/wiki/Partition_function_%28statistical_mechanics%29, Website accessed 20/2/2016 4. Biskup, M, Borgs, C et al, 2004, ‘Partition Function Zeros at First-Order Phase Transitions: A General Analysis*’ http://research.microsoft.com/en-us/um/people/borgs/papers/zeroscmp.pdf, Website accessed 20/2/2016 5. ‘MPSolve home page’, http://numpi.dm.unipi.it/mpsolve-2.2/, Website accessed 20/2/2016