Abstract
The Ising model is used to demonstrate how to proceed from a detailed system analysis to a computer simulation of the physics involved. The model itself describes an n-dimensional spin 1∕2 lattice which can undergo phase transitions from a ferromagnetic/antiferromagnetic (ordered spins) to a paramagnetic state (unordered spins) with increasing system temperature. The model can be solved analytically in one and two dimensions and the corresponding analysis is presented here. The purpose of a computer simulation of the Ising model will be the calculation of expectation values of certain observables as a function of temperature. The Metropolis algorithm is employed to generate randomly a sequence of modifications of spin configurations which will then be used to measure the observables of interest. Important problems of the simulation like initialization, thermalization, finite size effects, measurement of observables, and the prevention of correlations between subsequent spin configurations are discussed in detail.
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Notes
- 1.
For a short introduction to phase transitions in general please consult Appendix F.
- 2.
- 3.
In this discussion we regard the spin as a classical quantity. In the quantum mechanic case one has to replace the vectors by vector operators S i .
- 4.
We note in passing that the Hamiltonian (15.4) is invariant under a spin flip of all spins if h = 0 (\(\mathbb{Z}_{2}\) symmetry). This symmetry is broken if h ≠ 0, i.e. the spins align with the external field h.
- 5.
We note that \(H \propto \mu \cdot B\) where B is the magnetic field and μ is the magnetic moment. Furthermore, μ can be expressed as \(\mu = -\mu _{B}gS/\hslash = -\mu _{B}g\sigma /2\), where μ B is the Bohr magneton, g is the Landé g-factor and \(\sigma\) is the vector of Pauli matrices. The sign is convention.
- 6.
In particular we assume ergodicity of the system as will be explained in Chap. 16
- 7.
\(\left \langle E\right \rangle\) is also referred to as internal energy U.
- 8.
We transform
$$\displaystyle{ \lambda _{1}^{N} +\lambda _{ 2}^{N} =\lambda _{ 1}^{N}\left [1 + \left (\frac{\lambda _{2}} {\lambda _{1}}\right )^{N}\right ]\;, }$$and use that
$$\displaystyle{ \left (\frac{\lambda _{2}} {\lambda _{1}}\right )^{N} \rightarrow 0,\qquad \text{as}\qquad N \rightarrow \infty \;. }$$ - 9.
In particular \(\text{var}\left (E\right ) = \left \langle E^{2}\right \rangle -\left \langle E\right \rangle ^{2}\) is to be determined and only the second term is already known. The first term, \(\left \langle E^{2}\right \rangle\), is then estimated with the help of
$$\displaystyle{ \left \langle E^{2}\right \rangle = \frac{1} {M}\sum _{i=1}^{M}E_{ i}^{2}\;. }$$ - 10.
Periodic boundary conditions in two dimensions imply that
$$\displaystyle{ \sigma _{N+1,j} =\sigma _{1,j}\qquad \text{ and }\qquad \sigma _{i,N+1} =\sigma _{i,1}\;, }$$for all i, j.
- 11.
In the following we will refer to the notation (i), \(i = 1, 2,\ldots,N^{2}\) as the single-index notation while the notation (i, j), \(i,j = 1, 2,\ldots,N\) will be referred to as the double-index notation.
- 12.
The number of configurations discarded is referred to as the thermalization length.
- 13.
A migration through all lattice sites is referred to as a sweep.
- 14.
We note from Eq. (15.47) that we have to perform four times as many measurements in order to reduce the error by a factor 2.
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Stickler, B.A., Schachinger, E. (2016). The Ising Model. In: Basic Concepts in Computational Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-27265-8_15
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