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Ising chain hamiltonian
As discussed in Ising Model page, the Hamiltonian consists of 2 components: interaction with neighboring spins and interaction with external magnetic field.

$$H=-\sum_{i,j}J\sigma_i \sigma_j -\sum_ih\sigma_i$$

To simplify the problem, we have made a reasonable assumption that a spin interacts only with the closest neighbors and the interaction amplitude $$J $$ is constant. From energy considerations, one would expect that spins will try to align with each other (for $$J>0$$) and will be pointing parallel to external field $$h$$. In next few section we will elaborate how the system's net magnetization varies with temperature and interaction constants for both 1D and 2D chains.

One-dimensional spin chain
The magnetization of 1D Ising spin chain can be directly calculated from system's Hemholtz Free energy :

$$m=\frac{M}{N}=-\frac{1}{N}\frac{\partial F}{\partial h}=\frac{e^{\beta J}\sinh{(\beta h)}}{\sqrt{e^{-2\beta J}+e^{2\beta J}\sinh^2{(\beta h)}}}$$

Magnetization values for different $$\beta J$$ and $$\beta h$$ are provided in Fig.1. It is easy to see that there is no spontaneous magnetization in 1D chain (i.e. $$m=0 $$ at $$h=0$$) and external magnetic field is always required to magnetize the system. Apart from that, one can notice that it is significantly harder to magnetize the spin chain at negative $$\beta J$$. This is in compliance with understanding of anti-ferromagnetic interaction - at negative $$\beta J$$, neighboring spins would prefer to align in opposite directions and considerable amount of external magnetic field is required to break this order. Increasing $$\beta J$$ above 0 would impose ferromagnetic order on a system, where now neighboring spins would prefer to point in the same direction. This will make magnetization process much easier and, thereby, very small external field is required to completely align the system. This is in agreement with magnetization plot becoming steeper as $$\beta J$$ increases.

Zero field
The net magnetization of 2D Ising spin chain in the absence of external magnetic field can be solved exactly. The exact functional form for such system was first guessed by L. Onsager in 1948:

$$ m=(1-[\sinh{(2\beta J)}]^{-4})^{\frac{1}{8}}$$

Few years later, exact derivation was provided by different theoreticians using a limiting process of transfer matrix eigenvalues. The key difference from 1D case is that 2D spin chain can be magnetized even without external magnetic field given that spin interaction is sufficient enough. As one can see in Fig.2, there is indeed a spontaneous magnetization in the absence of magnetic field at $$\beta J\approx0.44$$



Non-zero field
There is no analytical solution for 2D Ising spin chain in the presence of magnetic field. However, there is an approximate self-consistent solution obtained using mean-field approach, which is is given by:

$$m=\tanh{(\beta h+4\beta J m)}$$

To investigate this spin system, we will be running Metropolis Monte Carlo simulations as discussed in the main Ising Model page. The procedure is as following:


 * Construct $$100\times100$$ matrix of randomly directed spins (up/down). Calculate the total energy
 * Choose a random site and flip its spin. Calculate the new energy
 * If $$E_{new}E_{old}$$, keep the current state with $$e^{-\beta(E_{new}-E_{old})}$$ probability
 * Iterate $$10^7$$ times2D h plot.png

Fig.3 shows the Monte Carlo simulation histogram for 2D spin chain with $$\beta J=0.1$$ at different $$\beta h$$. It is easy to observe that the simulation converges faster at higher fields which results in more sharp peaks. To check the validity of magnetization data, it is then compared to theoretical values from self-consistent equation in Fig.4. Error bars represent the standard deviation of corresponding histogram. There is a clear match between the two methods, which implies that Metropolis Monte Carlo can evaluate the properties of 2D Ising spin chains to a good precision.