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First question: Wiki says Ising Model higher than four dimensions can be described by mean field theory. What is the reason for this? Does this mean there is no phase transition for higher dimensions than four? I can see fluctuations become less important in higher dimensions since there are many "neighbors" to stabilize the interactions but I'm looking for a more quantitative argument.

Second question: The transverse Ising model $$H = -J\sum_{ij} \sigma^z_i \sigma^z_j-h \sum_i\sigma^x_i$$

is the quantum correspondence of Ising model. Does the statement above hold for this model?

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The criterion for determining whether mean-field theory is good or not is the Ginzburg criterion. You can estimate how accurate mean-field theory is by computing the leading corrections to it. In particular, the exact Ising model is $$ H = -J \sum_{\langle i j \rangle} s_i s_j, $$ and mean-field theory is done by taking $$ s_i s_j = M^2 + M (s_i + s_j) + (s_i - M) (s_j - M), $$ where $M = \langle s_i \rangle$, and then approximating the last term as zero, $$ (s_i - M) (s_j - M) \approx 0. $$ One way to determine that mean-field theory is failing is to calculate the expectation value of this omitted term within mean-field theory and compare it to the magnitude to $M^2$ near the critical point. If that ratio is large, then the term you omitted will cause a large correction, so you're not justified in omitting it. One can show that if the system has correlation length $\xi$, then $$ \langle (s_i - M) (s_j - M) \rangle \propto \xi^{2-d}. $$ Meanwhile, within mean-field theory, $M^2 \propto \xi^{-2}$, so the Ginzburg criterion says that mean-field theory is accurate if $$ \xi^{2-d} \ll \xi^{-2}. $$ This means that, close enough to the phase transition ($\xi$ large enough), mean-field theory is asymptotically correct for $d>4$ and incorrect for $d<4$. (As an aside, if we considered longer-range interactions, the range of the interaction alters this criterion.)

Does this mean there is no phase transition for higher dimensions than four?

No, this means that there is always a phase transition for $d>4$ (mean-field theory predicts a phase transition).

Second question: The transverse Ising model [...] is the quantum correspondence of Ising model. Does the statement above hold for this model?

The $d$-dimensional transverse-field Ising model (TFIM) actually corresponds to a $d+1$-dimensional classical finite-temperature Ising model. So the upper-critical dimension is $3$ for the TFIM.

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  • $\begingroup$ It's a while ago but may I ask a follow-up to this? what is the definition of long-range interactions? It changes the criterion because it changes the $t$ dependence of $\xi$? $\endgroup$ – Histoscienology Feb 18 '20 at 23:41
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    $\begingroup$ @Histoscienology There is a nice discussion of this on Cardy's textbook Scaling and Renormalization in Statistical Physics. As shown in his chapter on mean-field theory, if one has an Ising model of the form $H = -\frac{1}{2}\sum_{r,r'} J(r - r') s_r s_{r'}$, one can define the range of an interaction as $R = \sum_r r^2 J(r)/\sum_{r'}J(r')$, and then the Ginzburg criterion becomes $\xi^{4 - d} \ll R^4$. $\endgroup$ – Seth Whitsitt Feb 19 '20 at 23:20
  • $\begingroup$ If $R$ is quite large, in practice mean-field theory is ok even if $d<4$, since one can never tune close enough to criticality to make $\xi$ large enough to violate this. Note that the quantity $R$ diverges if $J(r)$ decays slower than $J(r) \sim 1/r^{\alpha}$ with $\alpha \leq d +2$. In this case, the Ginzburg criterion is radically altered, and one gets new "long-range Ising" fixed points. Section 4.3 in Cardy's textbook discusses this case briefly (and I can point you to the relevant literature if you are interested). $\endgroup$ – Seth Whitsitt Feb 19 '20 at 23:23
  • $\begingroup$ Sorry, there is a typo in my first response. What I defined to be $R$ should really be $R^2$. $\endgroup$ – Seth Whitsitt Feb 19 '20 at 23:33

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