# 1D Ising ground state

1. Jun 27, 2011

### mahblah

1. The problem statement, all variables and given/known data
Find the ground state (stable configuration at T = 0) of the one-dimensional ising model with first and second neighbour intercations:

$H = -J_1 \sum_{i} s_i s_{i+1} -J_2 \sum_{i} s_i s_{i+2}$

where $s_i = \pm 1$

3. The attempt at a solution

I really don't know what i should do.. i don't know what i must FIND, this is my problem!
maybe i must find the Partition function and calculate the average magnetization ? so i say:

$Z = \sum_{s} \exp{[-\beta H]} = (2 \cosh(\beta J_1) \cosh(\beta J_2))^N$

but seems that:

$<s>_{T=0} = \frac{ \sum_{s} \exp{[-\beta H]} \sum_{i}s_i } {\sum_{s} \exp{[-\beta H]}} =0$

i'm not intrested about solution, but i want to know what i must do---

thanks all,
mahblah.

Last edited: Jun 27, 2011