2
$\begingroup$

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 ($T_c$)?

$\endgroup$
2
  • $\begingroup$ 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, 2022 at 20:41
  • $\begingroup$ @NorbertSchuch, I have added some additions. Hope they help. $\endgroup$
    – megamence
    Jan 25, 2022 at 20:48

1 Answer 1

4
$\begingroup$

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.

$\endgroup$
4
  • $\begingroup$ Can be the statement about first order transition captured from mean-field analysis? $\endgroup$ Jan 25, 2022 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, 2022 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, 2022 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, 2022 at 3:01

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.