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The localisation transition temperatureTc

equals the replica pair transition tempera- tureT2for SDLs

The direct way to search for a relation to a binding problem is to study the associated binding problem. We have checked for the SDL via numerical transfer matrix calculations that the localisation temperature in disorder Tc

equals the transition temperature T2for replica pair binding, Tc =T2. As

stated before, we use the same transfer matrix algorithm for the replica pair system with a short-range binding potential by exploiting that the binding of two SDLs can be rewritten as a binding problem of one effective SDL in an external potential using relative coordinates [82]. This SDL has a bending stiffness ofκ0=κ_/2and the “potential energy” we are interested in (cf. eq.

(6)) for the replica pair binding is−βg2 Rdx δ(z). For the interpretation of

the simulation results one has to keep in mind that the energy functional is temperature dependent and, therefore, derivatives of the free energy with respect to the inverse temperature β are not given directly by cumulants of the internal energy. Using the notations

Hel= κ Z

L1/ν_(T c−T)/Tc −1 0 1 2 3 4 Epot L − 1/ ν −20 −10 0 L

Figure 25: Finite size scaling for the SDL

adsorption problem that corresponds to the two-replica binding, see text for de- tailed explanation. The scaling uses Tc=

1.44and ν = 2. As only one sample is needed for the calculation we used larger system lengths L = 100 (green, Greek crosses),200 (blue, saltires),. . . ,500 mauve, full squares).

as before for the elastic part of the energy and

Hbind= −g2 Z

dx δ(z)

for the reduced87binding energy the partition function is given by 87_{Reduced in the sense that factors of T}

are missing

Z= Z

Dz e−βHel−β2Hbind.

As always the free energy is computed via F= −β−1ln Zimplying that €(βF)

€β = hHel+2βHbindi =U+βhHbindi

where U is the total internal energy (treating βHbindas a potential). Thus, the

derivative of the difference of the free energies with and without the adsorption term with respect to the inverse temperature, which (assuming hyperscaling) should give the divergent correlation length βδF=β(Fg−F0) ∼L(Tc−T)ν,

is close to criticality identical to the “potential energy” Epot= −βHbind and

usual finite size scaling should be applicable, as is shown in Fig. 25. We retain the known correlation length exponent ν =2for the adsorption problem [82]. We see matching curves for the used system sizes L=100, 200 . . . 500 around T−Tc ≈0 for T2 = 1.44, which equals Tc ≈ 1.4for the SDL in

disorder.

Pair overlap as order parameter

More importantly, we identify a binding-related order parameter for the localisation transition. For DLs, the disorder-averaged overlap

q= lim L→∞ 1 L L Z 0 dx δ(z1(x) −z2(x))

of two replicas has been proposed as order parameter [44, 45]. Up to now, it has been numerically unfeasible to verify this order parameter for DLs in d>2dimensions where a localisation transition exists because the relevant 2d-dimensional two replica phase space is too large. For SDLs, on the other hand, the transition is numerically accessible already in 1+1dimensions and we show that the overlap q is indeed a valid order parameter using an adaptation of the transfer matrix technique from Ref. 44, see Fig. 26.

This involves simulating two interacting SDLs, therefore88 we have to 88This is very similar to how energies are computed, see Ref. 4, but involves twice the number of degrees of freedom

compute the full overlap-distribution. That is we have to keep track for every point in the four-dimensional two-line phase-space (two displacements

and two tangent orientations) how much overlap weight has been accumu- lated by all paths ending there. This is done by iteratively computing the overlap-distribution QL(z1,v1; z2, v2), whose transfer matrix equation reads

QL ∝

∑

u1,u2 |u1−v1|≤∆v |u2−v2|≤∆v e−β∆EL(u1−v1,z1)−β∆EL(u2−v2,z2)_[QL−1(z 1−u1,u; z2−u2,u2) +wL(z1−u1,u1)wL(z2−u2,u2)δz1,z2].

The first term in the brackets represents the contribution from shorter lengths and the second term gives the current overlap at length L. We omitted the normalisation factor which is given by the square of the normalisation factor used in the normalisation of the weights. Using this we can directly compute the overlap at a given length as

q(L) = 1

Lz

∑

₁,z2

v1,v2

QL(z1,v1; z2,v2).

As this clearly is associated with a greater amount of computational effort, we only use lengths up to L=30and 103samples.

Figure 26: Two plots regarding the overlap

order parameter q. Large: We estimate the value of q at infinite length, q∞, from finite

lengths using a fit that accounts for the first correction in L−1

qT(L) =a(T)/L+q∞(T).

In the plot we show q∞ as a func-

tion of the temperature T. Inset: Double- logarithmic plot of the overlap q (at the largest length considered, i.e. L = 30) versus Tc−T(with Tc=1.44), the solid

line is given by q ∼ (Tc−T)−β 0 with β0≈ −1.36. T 0 4 6 8 10 0 0.2 0.4 0.6 0.8 1 q∞ Tc−T 10−2 10−0 10−2 100 10−1 q

For DLs, it has been found[45] that the overlap at criticality decays as q∼ LΣ with Σ_{= −2ζ = −(1+ω)}in d₌ 3. This has been extended to

finite temperatures yielding q∼ |T−Tc|−νΣ. Indeed, we find a qualitatively

similar behaviour q∼ |T−Tc|−β

0

with an exponent β0 _{≈ 1.3−1.4. Our best estimate for Σ is Σ ≈ −0.75,}

cf. Fig. 27. Because of small simulation lengths L we do not conclude this deviation to be a definite statement against the renormalisation group results presented in Ref. 45, but, interestingly, this would, unlike the DL renormal- isation group result Σ= −(1+ω) < −1, suggest[20] that two SDLs in a

random potential are certain to meet eventually.

At this point it is noteworthy that our own renormalisation group analysis from before suggested that the effective disorder correlator R transforms under a scale-change x→bxlike R→b−4ε+4ζR. This disorder correlator captures the effective disorder energy per length and it is therefore reasonable to assume that q should show identical scaling behaviour which would imply

which is not directly dependent on the type of elasticity considered. By this reasoning, we can give a “scale-change” estimate for Σ

Σ|_SC ≈ −0.628

which is close (considering the small system sizes) to the numerical value

Σ ≈ −0.75. For the correlation length exponent ν we find values ν ≈ 2

compatible with the corresponding problem of DLs [33, 36, 39]; such that our present results deviate from β0₌

νΣ. Nevertheless, the connection between

DLs and SDLs provides the first system to test the proposed order parame- ter in a localisation transition numerically and to determine the otherwise inaccessible exponents β0 _{or Σ.} (L/ξ)1/ν _{= (T} c−T)L1/ν −40 20 0 q/ ξ Σ 10−2 100 102 104

Figure 27: Finite size scaling (L =

15, . . . ,30) for the overlap order parame- ter. Our best results are Σ ≈ 0.75for the exponent related to the decay of the order parameter, and ν≈2for the corre- lation length exponent. As before, we used Tc=1.44.

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