# Finite range 1D Ising model vs. infinite range Ising model

Ising model is defiend as $$\mathcal{H}=-H\sum_i S_i -\frac{1}{2}\sum_{i,j}J_{ij}S_i,S_j$$ In 1D we assume that indices $$i,j$$ are integers, $$i,j\in\mathbb{Z}$$, and that the coupling depends only on the distance between the spins: $$J_{ij} = J(|i-j|).$$ Then (at least for the nearest neighbor coupling) the model does not exhibit a phase transition at finite temperature.

On the other hand, if we consider that $$J_{ij}=J$$ is a constant, we have the infinite-range Ising model, which can be solved by mean field theory and exhibits a phase transition at finite temperature. However, this formally falls under the 1D case described above.

Question: Is this contradiction appears only in infinite range limit (i.e., there is no phase transition in a finite range 1D Ising model) OR should finite range Ising model be considered as different from 1D?

Auxhiliary question: what really determines formally the dimensionality of the model? The number of nearest neighbors?

Background: what really interests me is a possibility of a phase transition in an Ising/Potts model with random couplings (one-dimensional in real space), but I would like to understand well the basics.

Keeping with your notations, let us consider the Ising model on $$\mathbb{Z}$$ with coupling constants $$J_{ij} = |i-j|^{-\alpha}$$, with $$\alpha > 1$$ (otherwise the model is ill-defined; note that the mean-field model is pathological in that the interaction is dependent on the system size, $$J_{ij} = J/N$$ for a chain of length $$N$$).
Then, one can prove that this model displays long-range order at low temperatures if and only if $$\alpha\leq 2$$.
The fact that there is uniqueness in one dimension when the interaction decays faster than $$r^{-2}$$ is pretty general; see this answer for more information on the existence/absence of phase transitions in one-dimensional systems. In particular, any one-dimensional model with finite-range, translation-invariant interactions (and bounded spins) is disordered at all temperatures.
• By finite range you mean that $J_{ij}=0$ if $|i-j|$ is greater then some cutoff? Commented May 6, 2021 at 11:08
• What I really have in mind is genetic sequence: it is physically linear, with every position taking 4 or 20 values (depending on whether we work on nucleotide or amino-acid level). There may be interactions between positions that are very far one from another - they do not encessarily decay with distance, but one could say that $J_{ij}$ is sparse. There is also random field at every position. Commented May 6, 2021 at 11:28