Love thy neighbor, and everyone else too

Suppose, as before, a 1d lattice of \(N\) particles, each with a spin \(\sigma_i\in\{-1,+1\}\).

In that problem, particles could interact with their nearest neighbors, but no further. As a result, no long-range order is possible above a temperature of absolute zero.

Here, consider the case where particles can interact with particles infinitely far down the lattice with the energy

\[E(\sigma_i, \sigma_j) = -\frac{\sigma_i\cdot\sigma_j}{\left| i-j\right| ^{\gamma_r}}\]

What is the largest value of the exponent \(\gamma_r\) for which the lattice will prefer an ordered state at some non-zero temperature, \(T_c\), in the limit \(N\rightarrow \infty\)?


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