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I always thought that these two terms are some kind of synonyms, meaning that if you have a self-similar or scale invariant system, you can zoom in or out as you like and you will always see the same picture (physics).

But now I have just read in the context of a lattice of spins model for example, that if the system is at its critical point and therefore scale invariant, it does not mean that it is self-similar as naively discribed in the first paragraph of this question. The author of the paper I am reading even calls it a false picture on p. 9. Later on on p. 24 he explains that poles on the positive real axis on the so called Borel plane break self-similarity because lead to tha fact that to obtain the effective action (for a $\lambda\phi^4$ bare action) the scale dependent non perturbative power correction terms have to be kept. So if scale invariance and self-similarity are not exactly the same, the explanation for breaking of scale invariance should be (slightly and subtly?) different?

Now I am confused and my question simply is: what exactly is the difference between scale invariance and self-similarity (if any ...)?

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    $\begingroup$ Extremely interesting paper! Btw, check out the last paragraph on en.wikipedia.org/wiki/Renormalon Is it disagreeing with the paper's claim on renormalons in the $\lambda \phi^4$ theory :-? (if I understand the comments on p9 and p24 correctly). $\endgroup$ – Siva May 24 '13 at 17:52
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    $\begingroup$ @Siva yeah, I have just started to read the paper, but I already like it. Hm yes the last paragraph seems to disagree with what is said in the paper. I dont know how recent the recent suggested proof for the non existance of renormalons actually is? The papers mentioned in the Wiki article are all older than the ERG paper, so maybe this proof was wrong, who knows...? About those renormalons I have not yet heard before, maybe I'll have to ask about them anyway. $\endgroup$ – Dilaton May 24 '13 at 22:09
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    $\begingroup$ I first heard about renormalons in a nice talk by Argyres, based on his work with Unsal. You can look on the arXiv for their 2012 papers. Fwiw, a little googling turns up this interesting talk by Unsal. $\endgroup$ – Siva May 24 '13 at 23:51
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    $\begingroup$ The clarification is in ref [6] --arxiv.org/abs/cond-mat/0702365. Section 1.4.2, I presume. $\endgroup$ – Kostya Jun 5 '13 at 10:20
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From the point of view of non-linear dynamics where self-similarity plays an important role if the attractor is a fractal I would say that the difference is one between continuous and discrete transformations.

A self-similar transformation like the one producing the Cantor set or the Sierpinski triangle proceeds by discrete stages. The fractal which is the limit when the number of stages $N$ tends to infinity shows self-similarity (i.e. is identical to itself) only for a discrete number of stages.

For instance when zooming on the Sierpinski triangle, one may not zoom anywhere and by any zooming factor. One has to zoom only with a factor $1/3$ and center the zoom on the symmetry axis of the triangle. So basically the number of self-similar objects is an integer and has for characteristic the self-similarity dimension which is a number $D$ such as $N = L^D$ where $N$ is the number of copies produced by changing the size by $L$.

As for scale invariance which is not so largely used, it is a statement that $f(\mu x) = \mu^D \, f(x)$ with some constant $D$. The property is continuous and true for every $x$. Fractal attractors are generally not exactly scale invariant - they have often 2 or several different scalings.

Hence from this point of view the self-similarity and scale invariance may only be identical in a discrete number of points for simple fractals which have a unique scaling factor. (I am aware that this does not adress spin lattices but it answers the question in the frame of the chaos theory.)

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The discussion p. 24 on renormalons by Rosten does not look to me related to the issue of scale invariance at all. As for the one on pp. 8-9, I think this is just a remark in passing about typical spin configurations at the critical point. Let me give precise definitions to explain what is going on in the language of probability theory.

Take say the nearest neighbor Ising model in dimension $d\ge 2$. This is a collection of probability measures $\mu_{\beta,h}$ on the space $\Omega=\{-1,1\}^{\mathbb{Z}^d}$ which depends on the two parameters given by the inverse temperature $\beta\in [0,\infty)$ and the magnetic field $h\in\mathbb{R}$. The critical case to which the "false picture" remark by Rosten applies to is the particular case $\beta=\beta_{\rm c}$ and $h=0$. The remark pertains to some geometric property $P(\sigma)$ of a spin configuration $\sigma=(\sigma_{\mathbf{x}})_{\mathbf{x}\in\mathbb{Z}^{d}}\in\Omega$. Here the property $P(\sigma)$ is something like "the configuration $\sigma$ has all these nested oceans/islands of up and down spins". The belief that Rosten disputes (following Section 1.4.2 of Delamotte as Kostya rightly pointed out) is: $P(\sigma)$ is true $\mu_{\beta_{\rm c},0}$-almost surely. This belongs to the chapter on the Strong Law of Large Numbers in probability theory.

Now the property $P(\sigma)$ definitely has a "self-similar" flavor to it but I would not at all call that scale invariance. The latter has to do with a different chapter, namely, the one on the Central Limit Theorem.

Let me just work in the single phase region $\beta\le \beta_{\rm c}$. Then the general picture is that for any $\beta, h$ you can find a quantity $[\phi]_{\beta,h}$ for which the following happens. Pick some fixed integer $L>1$. For each integer $n\ge0$ define a probability measure $\nu_{\beta,h,L,n}$ on the space of temperate distributions $S'(\mathbb{R}^d)$ as the law of the random distribution $\phi_n(x)$ given by $$ \phi_{n}(x)=L^{-n(d-[\phi]_{\beta,h})}\sum_{\mathbf{y}\in\mathbb{Z}^d} (\sigma_{\mathbb{y}}-\langle \sigma_{\mathbb{y}}\rangle)\ \delta^d(x-L^{-n}\mathbf{y}) $$ with $\sigma$ sampled according to the lattice measure $\mu_{\beta,h}$, and where the magnetization $\langle \sigma_{\mathbb{y}}\rangle$ is the expectation of a single spin for that measure. Then the $\phi_n$ should converge in (probability) distribution to a random (Schwartz) distribution with law $\nu_{\beta,h,L,\infty}$. Just from the existence and uniqueness of this limit, it follows that the limit law or scaling limit $\nu_{\beta,h,L,\infty}$ is scale invariant with respect to the group $L^{\mathbb{Z}}$, i.e., rescaling by powers of $L$ (also the linear size of a block in a block-spin RG procedure). This is a discrete scale invariance property (related to Stan's answer). In fact here one expects this to hold for any $L$ and thus $\nu_{\beta,h,L,\infty}$ should be invariant by the full multiplicative group $(0,\infty)$. In other words it should have the continuous scale invariance property. BTW, the difference between discrete and continuous scale invariances is discussed for instance in this review by Sornette. In our situation continuous scale invariance means in particular that say the two point function satisfies $$ \langle\phi(\lambda x)\phi(\lambda y)\rangle=\lambda^{-2[\phi]_{\beta,h}}\langle\phi(x)\phi(y)\rangle $$ for all $\lambda\in (0,\infty)$.

Now the key question is: how to pick the crucial quantity $[\phi]_{\beta,h}$ (the scaling dimension of the spin field) needed to see the scale invariance?

For $d=2$, one has $[\phi]_{\beta_{\rm c},0}=\frac{1}{8}$ and the limit $\nu$ is non-Gaussian: the $m=3$ unitary minimal conformal field theory.

For $d\ge 4$, one has $[\phi]_{\beta_{\rm c},0}=\frac{d-2}{2}$ and the limit is the massless Gaussian field.

The most interesting situation is $d=3$ where according to best present estimates $[\phi]_{\beta_{\rm c},0}\simeq 0.5181489$ (see this article).

Now in any dimension and when $(\beta,h)\neq (\beta_{\rm c},0)$ one has $[\phi]_{\beta_{\rm c},0}=\frac{d}{2}$. In this case the limit is called Gaussian white noise. The two point function should be homogeneous of degree $-d$. This might suggest that it should be given by $$ \langle\phi(x)\phi(y)\rangle\sim \frac{1}{|x-y|^d} $$ but that is wrong. The unique rotationally invariant element of $S'(\mathbb{R}^d)$ (up to multiplicative constant) is $\delta^d(x)$. One is really in a central limit regime where one divides a sum of $N=L^{nd}$ real random variables by $\sqrt{N}$. Finally, if instead of the right guess for $[\phi]_{\beta,h}$ one uses some number $\alpha>0$ and writes $$ \phi_{n}(x)=L^{-n(d-\alpha)}\sum_{\mathbf{y}\in\mathbb{Z}^d} (\sigma_{\mathbb{y}}-\langle \sigma_{\mathbb{y}}\rangle)\ \delta^d(x-L^{-n}\mathbf{y}) $$ then the following happens. If $\alpha<[\phi]_{\beta,h}$, there is too much suppression and the $\phi_n$ converge in distribution to the (non random) identically zero field. If $\alpha>[\phi]_{\beta,h}$ there is loss of tightness, i.e., probability mass escaping to infinity and the limit of the $\phi_n$ does not exist. Again, this is like the central limit theorem if you don't divide by the correct power of $N$, namely, $\sqrt{N}$.

BTW, pretty much everything I said above is substantiated by rigorous mathematical proofs. For instance, the case $(\beta,h)\neq(\beta_{\rm c},0)$ with convergence to white noise follows from a general result by Newman for FKG systems. For $d=2$ and $(\beta,h)=(\beta_{\rm c},0)$, this is in the recent work of Chelkak, Hongler and Izyurov, Dubédat, as well as Garban and Newman. Of course, the main open case is $d=3$ and $(\beta,h)=(\beta_{\rm c},0)$, i.e., the infamous 3D Ising CFT. As far as I know, the best rigorous result so far (see these recent lectures by Duminil-Copin) is that at $(\beta,h)=(\beta_{\rm c},0)$ the two-point function satisfies $$ \frac{c_2}{|\mathbf{x}-\mathbf{y}|^{2}}\le \langle \sigma_{\mathbf{x}}\sigma_{\mathbf{y}}\rangle\le \frac{c_1}{|\mathbf{x}-\mathbf{y}|} $$ for come positive constants $c_1,c_2$, when the actual conjecture is more like $$ \langle \sigma_{\mathbf{x}}\sigma_{\mathbf{y}}\rangle \sim \frac{1}{|\mathbf{x}-\mathbf{y}|^{1.0362978}}\ . $$

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