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I'm trying run and experiment with Monte Carlo Simulations of 2D lattice Ising Model and Classical Heisenberg Model.

I've made a brief research on both models and I saw that main differences of these two models are:

  1. In Ising Model, spins of each vertice may only be $\sigma_i=\{-1, 1\}$ but in Classical Heisenberg Model spins are 3-component unit vectors $\vec{\sigma_i}\in R^3$ and $|\vec{\sigma_i}|=1$.
  2. Hamiltonian of Ising Model is $H=\sum_{<i,j>}{J_{i,j}\sigma_i\sigma_j}$ and of Classical Heisenberg Model is $H=\sum_{<i,j>}{J_{i,j}\vec{\sigma_i}\cdot\vec{\sigma_j}}$ where $<i,j>$ represents closest neighbors of spin i.

Question 1: Is there any main differences am I missing?

Question 2: In Ising Model we can calculate the magnetization of system using $M=\frac{1}{N}\sum_{i}{\sigma_i}$ formula. How do we calcualte magnetization of system in Classical Heisenberg Model?

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    $\begingroup$ 1. No, these are the main differences at the level of the definitions of these models (but they differ considerably at the level of their properties). 2. It's the same definition (the empirical average of the spins $\vec{\sigma}_i$). $\endgroup$
    – Yvan Velenik
    Commented May 20, 2023 at 9:27
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    $\begingroup$ Yes, this is correct. $\endgroup$
    – Yvan Velenik
    Commented May 20, 2023 at 9:53
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    $\begingroup$ I don't think that my answers above deserve more than being comments. But I am glad they helped you. I have used various Monte-Carlo simulation schemes in the past, yes, but as an amateur (usually to prepare illustrations for my papers or my book); my actual research is in mathematical physics. $\endgroup$
    – Yvan Velenik
    Commented May 20, 2023 at 10:11
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    $\begingroup$ There are infinitely many ways of setting up a reversible Markov chain with the Ising distribution as stationary measure. They may practically differ in efficiency (rejection rate, etc). You should probably ask another question on this issue (with an explicit description of the actual algorithms). $\endgroup$
    – Yvan Velenik
    Commented May 20, 2023 at 10:23
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    $\begingroup$ If you put a "statistical-mechanics" and/or "Ising-model" tag to your question, then I'll see it (they are among the tags I follow). $\endgroup$
    – Yvan Velenik
    Commented May 20, 2023 at 10:34

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