# "Long-range" 1D Ising model with exponentially decaying interactions

Consider a 1D Ising model with infinite-range but exponentially decaying interactions. For definiteness, say it is defined by the following Hamiltonian on $$n$$ spins: \begin{align} H\left[\left(\sigma_j\right)_{0 \leq j \leq n}\right] := \sum_{0 \leq j < k < n}\rho^{|j - k|}\sigma_j\sigma_k, \qquad \left(\sigma_j\right)_{0 \leq j < n} \in \{1, -1\}^n \end{align} where $$\rho \in [0, 1]$$ controls the decay of interactions. [Ultimately, I also wish to consider, at least numerically, the case where interactions are not "transition-invariant", but let's start with that for definiteness.]

I am wondering about available analytic and numerical methods to study this model, depending on the values of $$\rho$$ and the temperature.

Two (potentially) easy limiting cases I see would be the following:

• When $$\rho = 1$$ (all-to-all Ising model), the coupling matrix has rank 1 and the thermodynamic limit $$n \to \infty$$ can be understood using e.g. the saddle-point method.
• When $$\rho \longrightarrow 0$$, one approaches the no-interaction case and it is reasonable to expect that truncating the interactions to constant distance should work; it is also plausible that the required cutoff distance will decrease with $$\rho$$. If this approximation is correct (really not sure about that!), one could solve the model using the transfer matrix method for instance.

I unfortunately could not find many explicit analytic or numerical results on this model in the literature. On the analytic side, I would be more particularly interested in "explicit" estimates for the partition function in the thermodynamic limit, which I only know about for $$\rho = 1$$ or in the limit $$\rho \longrightarrow 0$$ at the moment. On the numerical side, I would be mostly interested in rigorous estimates of the resources requirements (bond dimension) of tensor network methods, such as Matrix Product States (somehow equivalent to the transfer matrix method), to solve such systems.

Any insight, even partial, would then be much appreciated!

• Terminology note: exponentially decaying interactions are usually referred to as “short-ranged”, even if their support is infinite. People sometimes say that the interactions are “quasi-long range” if they decay as a power law with distance, or “long-range” if they approach a constant at long distances. Commented Feb 1, 2023 at 22:36
• As far as I know, most universal behavior is qualitatively very similar between systems with exponentially decaying vs. finitely-supported interactions. I’m not sure how systematically that’s been studied, though - it may just be “folk wisdom”. Commented Feb 1, 2023 at 22:39
• Thanks @tparker for these insights! It is reassuring to intuition that exponentially decaying interactions give qualitatively similar behaviour to finite-ranged ones. However, on a more quantitative basis (even just based on "folk wisdom"), do we know how the resources to solve such a system, say, using MPS, scale with rho? I assume the bond dimension will increase as as rho gets closer 1 (although in this special toy model it should be polynomial in n for rho = 1 exactly, since translation invariance allows to work in the symmetric space). But can we understand "increase" more quantitatively? Commented Feb 2, 2023 at 13:04

I'll only address some aspects of your question. It is actually not completely clear to me what answer you are expecting. In particular, it is not clear what kind of numerical approaches you are interested in, since standard Monte-Carlo simulations would work here.

The discussion below does provide references to works dealing with such a model (and much more general ones) with full mathematical rigor. So they provide some possible analytic approaches (although not easy ones, because of the very general and abstract setup they consider; the arguments should be significantly simpler when restricted to the case you're interested in).

### Some rigorous results.

Let us consider a one-dimensional Ising model with exponentially decaying interactions, that is, with a formal Hamiltonian of the form $$\mathcal{H} = -\sum_{\{i,j\}\subset\mathbb{Z}} J_{j-i} \sigma_i\sigma_j,$$ with $$J_r\leq e^{-c|r|}$$ for all $$r\in\mathbb{Z}$$, for some $$c>0$$.

It has been known for a long time that such a model does not display a phase transition at any positive temperature:

• The free energy and all correlation functions are analytic in the temperature. This was first proved by Ruelle; a stronger result (valid for interactions decaying much more slowly) was obtained by Dobrushin.
• There is a unique infinite-volume Gibbs measure at every temperature. See, for instance, Section 6.5.5 in this book. (This result is also valid for interactions decaying much more slowly).

Note that, in all these results, it is actually not important that the interaction be translation-invariant (provided you have a uniform control on the way they decay with the distance). They also apply to much more general models.

### Comparison with finite-range models

Since this was mentioned in a comment by tparker, let me say a few words about the respective behaviors of models with exponentially decaying interactions, say such that $$e^{-c|r|} \geq J_r \geq e^{-c'|r|}$$, for all $$r$$ and some $$c'>c>0$$.

It is generally considered that such systems display the same qualitative properties as their counterpart with finite-range interactions. To a large extent, this is true. But not always. For instance, the 2-point function can have very different asymptotic behaviors. Also, the correlation length can fail to be analytic in the temperature in the case of exponentially decaying interactions. All this is discussed (for a more general class of models and in any dimension) in this paper.

• Thanks for pointing to the textbook reference, I wasn't aware of it and will definitely give a look! Regarding numerical techniques, I was mostly thinking about tensor network methods, e.g. Matrix Product States; therefore, I guess the issue is how the required bond dimension would scale with $\rho$. On the more analytic side, I was also wondering about the existence of "explicit" expressions for the partition function in the thermodynamic limit. I've edited the question accordingly. Commented Feb 3, 2023 at 13:48