- #1

William Crawford

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- 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##).

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##).

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