What is the Ground State of the 1D Ising Model with Neighbor Interactions?

[mahblah]

Homework Statement

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 $s_i = \pm 1$

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 interested about solution, but i want to know what i must do---

thanks all,
mahblah.

 

[mark726]

Hi mahblah,

your goal is to understand and explain natural phenomena through observation and experimentation. In this case, you are studying the one-dimensional Ising model, which is a mathematical model that simulates the behavior of magnetic materials. Your goal is to find the ground state, which is the most stable configuration of the system at a temperature of 0 Kelvin.

To do this, you can use the partition function, as you have mentioned. The partition function is a mathematical tool that helps us calculate the probability of a particular state occurring in a system. In this case, the state we are interested in is the ground state.

To find the ground state, you can use the partition function to calculate the average energy of the system at T=0. This can be done by summing over all possible configurations of the system and calculating the energy for each configuration. The configuration with the lowest energy will be the ground state.

In addition to the partition function, you can also use other techniques such as mean-field theory or Monte Carlo simulations to find the ground state of the Ising model.

I hope this helps guide you in your research.