The Depinning Transition in Presence of Disorder: A Toy Model

Abstract

We introduce a toy model, which represents a simplified version of the problem of the depinning transition in the limit of strong disorder. This toy model can be formulated as a simple renormalization transformation for the probability distribution of a single real variable. For this toy model, the critical line is known exactly in one particular case and it can be calculated perturbatively in the general case. One can also show that, at the transition, there is no fixed distribution accessible by renormalization which corresponds to a disordered fixed point. Instead, both our numerical and analytic approaches indicate a transition of infinite order (of the Berezinskii–Kosterlitz–Thouless type). We give numerical evidence that this infinite order transition persists for the problem of the depinning transition with disorder on the hierarchical lattice.

Appendices

Appendix 1: The Large $a$ Limit of the Toy Model (6)

In this appendix we try to explain the $1/a$ dependence of $\lambda _c$ that we observed for large $a$ in Table 1.

Consider a distribution $P_0(X)$ characterized by its first two moments

$$\begin{aligned} \int P_0(X) X d X = \mu ; \int P_0(X) X^2 d X = \mu ^2 + \sigma . \end{aligned}$$

The parameters $\mu$ and $\sigma$ change if one varies the initial distribution $P_0(X)$. Let us call $\mu _c(a, \sigma )$ the critical value of $\mu$ at the depinning transition (8).

By the simple change of scale $a\rightarrow a', \mu \rightarrow a' \mu / a, \sigma \rightarrow a'^2 \sigma / a^2$ it is clear that $\mu _c(a,\sigma )$ should be of the form

$$\begin{aligned} \mu _c(a, \sigma ) = a \ F\left( {\sigma \over a^2} \right) . \end{aligned}$$

On the other hand, if $a$ is very large compared to $\mu$ and $\sqrt{\sigma }$, for many iterations of the renormalization, the distribution is renormalized as if $a$ was infinite: after $n$ steps, one has $\mu _n \simeq 2^n \mu$ and $\sigma _n \simeq 2^n \sigma$. Therefore for $\mu \ll a$ and $\sigma \ll a^2$ one has

$$\begin{aligned} \mu _c(a, \sigma ) \simeq \sigma G(a). \end{aligned}$$

Combining these two equations leads to the result that

$$\begin{aligned} \mu _c(a,\sigma ) \sim {\sigma \over a}. \end{aligned}$$

For the particular case of the binary distribution (9) one has $\mu = 2 \lambda -1$ and therefore $\lambda _c -1/2 \sim 1/a$ as observed in Table 1.

Appendix 2: Absence of Critical Fixed Distributions of the Map (22)

In this appendix, we show for $a$ an integer, that the only fixed distributions accessible by the renormalization group (19) are $H(z)=0$ (which corresponds to the pinned phase), $H(z)=1$ (which corresponds to the unpinned phase) and $H(z)=z^a$ (which corresponds to the critical point of the pure system).

We have seen that in terms of the generating function $H_n(z)$, the renormalization transformation (6) can be written as

$$\begin{aligned} H_{n+1}(z) = {H_n^2(z) - Q_n(z) \over z^a} + Q_n(1) \end{aligned}$$

(49)

where $Q_n(z)$ is the polynomial of degree $a-1$ obtained by keeping the first $a$ coefficients of the expansion of $H_n^2(z)$ around $z=0$ :

$$\begin{aligned} \text {If } \quad H_n^2(z) = \sum _{k=0}^\infty \beta _n^{(k)} \ z^k \quad \quad \text {then } \quad Q_n(z) = \sum _{k=0}^{a-1} \beta _n^{(k)} \ z^k. \end{aligned}$$

(50)

It is clear that if the $p_n^{(k)}$’s in (19, 21) are non-negative then the $\beta ^{(k)}$’s are also non-negative. If one looks for a fixed distribution (i.e. for a fixed point $H_*(z)$ of the map (22)) one gets

$$\begin{aligned} H_*(z) = {z^a \pm \sqrt{\Delta (z)} \over 2} \end{aligned}$$

(51)

with

$$\begin{aligned} \Delta (z)= z^{2 a} + 4 Q_*(z) - 4 z^a Q_*(1). \end{aligned}$$

(52)

Let us first observe that $\Delta (z)$ cannot vanish on the positive real axis:

• For $z \le 1$, by writing

  $$\begin{aligned} \Delta (z) = z^{2 a} + 4 \sum _{k=0}^{a-1} \beta _*^{(k)} (z^k - z^a) \end{aligned}$$

  it is obvious that $\Delta (z) >0$ .

• For $z \ge 1$, using the fact that

  $$\begin{aligned} \sum _{k=-a}^\infty p^{(k]}=1= \sum _{k=0}^\infty \beta ^{(k)} \end{aligned}$$

  one can rewrite $\Delta (z)$ as

  $$\begin{aligned} \Delta (z)= z^{2 a} \sum _{k=a}^\infty \beta _*^{(k)} + \sum _{k=0}^{a-1} \beta _*^{(k)}(z^a-2)^2 + 4 \sum _{k=0}^{a-1} \beta _*^{(k)}( z^k - 1) \end{aligned}$$

and in this case too, $\Delta (z)$ is strictly positive on the whole positive axis. We are now going to prove that $\sqrt{\Delta (z)}$ has to be a polynomial.

If $\sqrt{\Delta (z)}$ was not a polynomial, then, because $\Delta (z)$ does not vanish on the positive real axis, the singularity of $H_*(z)$ closest to $z=0$ would not be on the positive real axis. It is however known (Pringsheim’s theorem [20]) that a series with positive coefficients has a singularity at the intersection of the positive real axis and its circle of convergence of the series. Therefore, the coefficients of $H_*(z)$ would not be all positive.

Therefore $\sqrt{\Delta (z)}$ has to be a polynomial. From (52) and the fact that $Q_*(z)$ is of degree $a-1$, the only possibilities are $Q_*(z)=Q_*(1)^2$ which implies either $Q_*(z)=0$ or $Q_*(z)=1$. Thus (52) implies that $\Delta (z) = z^{2 a}$ or $\Delta (z) = (z^{ a}-2)^2$ which gives $H_*(z)=0$, $z^a$ or $1$.

Appendix 3: The Extra Critical Fixed Point of the Truncated Transformation (27)

In this appendix we show that if one truncates the transformation as in (6) by (27), there appears a new critical fixed distribution. This new fixed point disappears in the $a' \rightarrow \infty$ limit.

For the sake of simplicity, let us limit the discussion to the case $a=1$ and to a large integer value of $a'$. For integer $a$ and $a'$ the transformation (19) remains the same for $k < a'$. The only change is for $k=a'$ where it becomes

$$\begin{aligned} p_{n+1}^{(a')} = \sum _{q_1=0}^{a'} \quad \sum _{q_2=a'-q_1}^{a'} \quad p_n^{(q_1)} \ p_n^{(q_2)}. \end{aligned}$$

(53)

Because the transformation for $k < a'$ does not depend on $a'$, the recursion relation of the generating function $H_n(z)$

$$\begin{aligned} H_n(z) = \sum _{k=-a}^{a'} p_n^{(k)} \ z^{k+a} \end{aligned}$$

(54)

is the same as (24) up to terms of order $z^{a+a'-1}$

$$\begin{aligned} H_{n+1}(z)= {H_n(z)^2 - H_n(0)^2 \over z}+ H_n(0)^2 + O\left( z^{a+a'-1} \right) . \end{aligned}$$

(55)

Therefore a fixed point of (55) should be of the form

$$\begin{aligned} H_*(z) = {z \over 2} + H(0)\sqrt{1 - z + {z^2 \over 4 H(0)^2}} + O\left( z^{a+a'} \right) . \end{aligned}$$

For $k < a+a'-1$ (here $a=1$) one gets for the weights of the fixed distribution

$$\begin{aligned} p_*^{(k)} = {1 \over 2 \pi i} \oint {dz \over z^{k+2}} \left[ {z \over 2} + H(0) \sqrt{1 - z + {z^2 \over 4 H(0)^2}}\right] \end{aligned}$$

where the integration contour is a small circle around the origin.

For $H(0)$ close to $1$ this gives

$$\begin{aligned} p_*^{(k)} \simeq {1 \over 2^{k-1}} {1- H(0) \over \pi } \int \limits _0^1 dy \sqrt{1 - y^2} \ \cos \left( k \sqrt{2 (1 - H(0))} y \right) . \end{aligned}$$

If $q$ is the first zero of $\int \limits _0^1 dy \sqrt{1 - y^2} \ \cos ( q y )$ all the $p_*^{(k)}$ are positive as long as $k < q/\sqrt{2 (1- H(0)}$. Adjusting the boundary condition at $k=a'$ selects one particular value $H(0)$ which should be such that

$$\begin{aligned} 1- H(0) \simeq {q^2 \over 2 a'^2}. \end{aligned}$$

In the limit $a' \rightarrow \infty$, obviously $H(0) \rightarrow 1$ and this extra fixed point merges with the fixed point $H_*(z)=1$.