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The total energy of the system depends on the interactions of the parti- cles with their nearest neighbor and with any external magnetic field. The Hamiltonian is then:
H =
i
j
J
sisj
− Bsi
where J is some coupling constant, and B is the external magnetic field. The first sum is over all particles in the lattice, while the second sum is taken over the ith particle’s nearest neighbors.
3 The Solution
The Ising model has an analytic solution in one dimension, for no external B field. For a single particle:
〈E(T )〉 = −
tanh
J
β
〈M (T )〉 = 0
where the angled brackets indicate a time average. β is the usual thermo- dynamic quantity β ≡ 1 /(kB T ). Note that this system does not undergo a phase transition. There is no discontinuity in the energy or its first dervative. There is also an analytic solution in two dimensions for no external B field. Again, for a single particle:
〈E(T )〉 = − 2 Jtanh(2βJ) +
K
2 π
dK dβ
∫ (^) π
0
dφ
sin^2 φ ∆(1 + ∆)
〈M (T )〉 =
[
sinh(^14 βJ)
]− 4 )^1 /^8
T ≤ Tc 0 T > Tc
where K = 2/(cosh(2βJ)coth(2βJ)) and ∆ =
1 − K^2 sin^2 φ. Tc is the criti- cal temperature of the system, kB Tc ≈ 0. 5672925 J. This critical temperature marks a phase transition. This phase transition is second order, marked by a singularity in the specific heat cv ≡ d dET. The phase transition also marks the temperature at which the magnetization takes on a nonzero value. In this case, then, the magnetization is the order parameter for the transition. As such, it is expected to fluctuate largely when T is neat Tc. The derivations of these results can be found in the references.
4 The Simulation
There is no exact solution in three dimensions, a discussion of which is con- tained in the references. Nor is there a simple solution for the 1D and 2D models when an external field is present. Thus, computer simulations must be used to determine the properties of the model in these cases. The simulation works by mimicking the way a physical system reaches and behaves at thermal equilibrium. Initially, the spins are randomized. The simulation then does a Monte Carlo step to obtain a new lattice from the old one. The Monte Carlo step involves picking a lattice site at random. It determines the energy associated with each possible spin orientation at that specific site. Periodic boundary conditions are used when calculating the spin-spin interaction with neighboring particles. It then chooses a new spin orientation based on the Maxwell-Boltzmann distribution, much like a physical system. This repeats as many times as there are latice sites, though, since the sites are chosen at random, not every site may be chosen. This is then a single Monte Carlo step. The new lattice generated undergoes another Monte Carlo step, and so on. Eventually, the system will reach a state of thermal equilibrium.
5 Validation
The exact solutions to the 1D and 2D Ising models for no external field provide an excellent way to validate the simulation for the 3D Ising model. It is simple to use the same code for 1D, 2D, and 3D simulations, so if the code produces accurate results in 1D and 2D, it is likely to do so in 3D. Additionally, there are several limiting cases that can be checked. In the high temperature limit, the spins should be randomized, since there is enough thermal energy to flip a spin, regardless of the states of its neighbors or the strength of any external B field. At the low temperature limit, the system should eventually reach a state where all spins are aligned (with each other and any external B field, if present), which is the system’s ground state. At the high B field limit, the spins should all align with the field.
S. Istrail, Statistical mechanics, three-dimensionality and NP complete- ness. In: Proceedings of the annual ACM symposium on the theory of computing (STA), 87-96 (2000).
