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→Peierls droplets: Specified that it is the *average* magnetization which is 0 |
→Definition: The Ising Hamiltonian is an example of a pseudo-Boolean function |
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<math display="block">H(\sigma) = -\sum_{\langle ij\rangle} J_{ij} \sigma_i \sigma_j - \mu \sum_j h_j \sigma_j,</math>
where the first sum is over pairs of adjacent spins (every pair is counted once). The notation <math>\langle ij\rangle</math> indicates that sites <math>i</math> and <math>j</math> are nearest neighbors. The [[magnetic moment]] is given by <math>\mu</math>. Note that the sign in the second term of the Hamiltonian above should actually be positive because the electron's magnetic moment is antiparallel to its spin, but the negative term is used conventionally.<ref>See {{harvtxt|Baierlein|1999}}, Chapter 16.</ref> The Ising Hamiltonian is an example of a [[pseudo-Boolean function]]; tools from the [[analysis of Boolean functions]] can be applied to describe and study it.
The ''configuration probability'' is given by the [[Boltzmann distribution]] with [[inverse temperature]] <math>\beta\geq0</math>: <math display="block">P_\beta(\sigma) = \frac{e^{-\beta H(\sigma)}}{Z_\beta},</math>
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