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I have been running around the block trying to find answers for this question, and I keep running into caveats. So, I just want to write down the list of things I want to know: Given that the order parameter is magnetization,

Consider the simple two-spin-state Ising model. Where the Hamiltonian is defined by $$ \mathcal{H} = J\sum _{i,j} s_i s_j + B \sum s_i. $$

  1. The 1D Ising model shows no phase transitions as temperature changes: there are no divergences to look out for.

  2. If we are not present in an external field $(B=0)$, will the 2D Ising model will a second-order/continuous phase transition as temperature passes over critical point ($T_c$)?

  3. In the presence of an external field $(B\neq 0)$, will the 2D Ising model show a first-order phase transition as temperature passes over critical point ($T_c$)?

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  • $\begingroup$ That's too many questions at once. I'd suggest you at least cut #3. (It is not even clear what you mean by "bottleneck".) -- Also: 2D classical? Quantum? As a function of the temperature? Or also of the field? $\endgroup$ Jan 25 at 20:41
  • $\begingroup$ @NorbertSchuch, I have added some additions. Hope they help. $\endgroup$
    – megamence
    Jan 25 at 20:48

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For your 1, suppose we are on a 2D square lattice and $J<0$ (so ferromagnetic interaction), then yes there is a continuous transition at a critical temperature between ferromagnetic and paramagnetic phases. In fact the model was solved exactly by Onsager, whose main motivation was to prove the existence of a phase transition. The lattice geometry is not crucial: the same thing happens on triangular lattice for example. But the sign of $J$ does matter sometimes.

For your 2, the critical point is located at $T=T_c, B=0$. For finite $B$, there is no singularity in free energy anymore. However, there is indeed a first-order transition as you tune $B$ from say positive to negative below the critical temperature crossing $B=0$. This first-order line terminates at the critical point.

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  • $\begingroup$ Can be the statement about first order transition captured from mean-field analysis? $\endgroup$ Jan 25 at 22:02
  • $\begingroup$ @ArtemAlexandrov Yes, the mean-field analysis (aka Landau theory) can capture both 1 and 2 qualitatively. Quantitative aspects of the critical point are beyond mean-field in dimensions < 4. $\endgroup$
    – Meng Cheng
    Jan 25 at 23:01
  • $\begingroup$ The lattice geometry is actually quite crucial. For instance, the graph has to be isoradial: arxiv.org/abs/1204.0505 $\endgroup$
    – PeaBrane
    Jan 28 at 2:42
  • $\begingroup$ For ferromagnetic Ising model on Bravais lattice (which is what I meant by lattice), I think the phase diagram (qualitatively) and critical theory does not really depend on geometry. The situation could be different for other types of stat mech model. $\endgroup$
    – Meng Cheng
    Jan 28 at 3:01

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