Ground state of the one-dimensional spin-1/2 Ising model

[William Crawford]

TL;DR Summary

How to derive the low energy ground state for the one-dimensional spin-1/2 Ising model on either a periodic or an infinite chain.

Hi,

I know that the ground state of the spin-1/2 Ising model is the ordered phase (either all spin up or all spin down). But how do I actually go about deriving this from say the one-dimensional spin hamiltonian itself, without having to solve system i.e. finding the partition function? $$ \mathcal{H} = -J\sum_n s_{n}s_{n+1}, \qquad s_n=\pm1 $$ I've tried computing the derivative of $\mathcal{H}$ w.r.t. the spin variable $s_i$, but this leaves me with the trivial difference equation $s_n + s_{n+1} = 0$ yielding the high energy solution $s_n = (-1)^ns_0$ and not the low energy solution that I was searching for (assuming $J>0$).

 

[William Crawford]

Never mind, I've solved it myself. Simply using that
$$ \min_{\lbrace s_n\rbrace}\mathcal{H} = \sum_n\min\left(-Js_ns_{n+1}\right). $$