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Hamiltonian and Partition Function

We have N spins $\sigma_i$, i=1, 2,... on a lattice with coordinate number z. Each spin can be either up or down[*]:

\sigma_i = 
 1,& \text{``up''}\  -1,& \text{``down''}



Each spin interacts with external field H:

E_1 = -H\sum_i \sigma_i 

Each pair of neighbors interacts:
Parallel spins are favored--energy -J, J>0
Antiparallel spins are discouraged--energy +J
This can be written as $-J\sigma_1\sigma_2$. The total interaction:

E_0 = -\frac{J}{2}\sum_{(i,j)}\sigma_i\sigma_j

Here (i,j) means sum over all pairs of nearest neighbors
 E = -H\sum_i \sigma_i -\frac{J}{2}\sum_{(i,j)}\sigma_i\sigma_j\end{displaymath} (1)

Partition Function and Thermodynamic Variables

Microscopic state of the system--combination of all spins:

\{\sigma\} = \{\sigma_1,\sigma_2,\dots\}\end{displaymath}

Partition function:

Z_N = \sum_{\{\sigma\}} \exp\bigl(-\beta E(\{\sigma\})\bigr)\end{displaymath}

We don't have momenta here--this is actually a configuration integral rather than partition function! Free energy:

A = -kT\ln Z_N\end{displaymath}

Average magnetization (H is conjugated to $N\sigma$!)

M(H,T) = \left\langle \sigma\right\rangle = -\frac{1}{N}\frac{\partial A(H,T) }{\partial H}\end{displaymath}


\chi = \frac{\partial M(H,T) }{\partial H} = -\frac{1}{N}\frac{\partial^2 A(H,T)
 }{\partial H^2} \ge 0\end{displaymath}


E = A - T \frac{\partial A(H,T) }{\partial T} = -T^2 \frac{\partial
 A(H,T)/T }{\partial T} \end{displaymath}

Specific heat:

C = \frac{\partial E(H,T) }{\partial T}\end{displaymath}

© 1997 Boris Veytsman and Michael Kotelyanskii
Mon Oct 13 22:07:20 EDT 1997