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Journal of the Korean Physical Society

Localization of directed polymers in random media due to a columnar defect

Jong-Dae Lee, J. M. Kim, Jae Hwan Lee

Published Apr 1, 2015• 1 citation
Publisher

Abstract

We investigate the localization of directed polymers in random media due to the presence of an attractive columnar defect at the center of a two-dimensional substrate. If the defect’s strength is too weak to affect the polymers, the localization length of the polymers exhibits a power-law behavior as a function of the polymer length, as in the case of no defect. When the defect’s strength is greater than a critical value, the localization length approaches a finite value, thereby yielding a localization length exponent (or liberation exponent) ν ⊥ = 1.8004(31). The correlation length, which is perpendicular to the localization length, is defined as the distance over which the polymer is localized by the defect. The correlation length exponent ν ‖ = 2.881(5) is estimated from the data collapse via the scaling relation ζ = ν ⊥/ν ‖, where ζ = 5/8 represents the wandering exponent. In addition, we measure the number of times that the optimal path passes through the defect, because that number increases as the defect’s strength increases. This measurement yields a new critical exponent λ = 0.59.

DOI
10.3938/jkps.66.1207

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We investigate the localization of directed polymers in random media due to the presence of an attractive columnar defect at the center of a two-dimensional substrate. If the defect’s strength is too weak to affect the polymers, the localization length of the polymers exhibits a power-law behavior as a function of the polymer length, as in the case of no defect. When the defect’s strength is greater than a critical value, the localization length approaches a finite value, thereby yielding a localization length exponent (or liberation exponent) ν ⊥ = 1.8004(31). The correlation length, which is perpendicular to the localization length, is defined as the distance over which the polymer is localized by the defect. The correlation length exponent ν ‖ = 2.881(5) is estimated from the data collapse via the scaling relation ζ = ν ⊥/ν ‖, where ζ = 5/8 represents the wandering exponent. In addition, we measure the number of times that the optimal path passes through the defect, because that number increases as the defect’s strength increases. This measurement yields a new critical exponent λ = 0.59.
We analyze, via Imry-Ma scaling arguments, the strong disorder phases that exist in low dimensions at all temperatures for directed polymers and interfaces in random media. For the uncorrelated Gaussian disorder, we obtain that the optimal strategy for the polymer in dimension 1+d with 0<d<2 involves at the same time (i) a confinement in a favorable tube of radius R(S) approximately L (nu(S) ) with nu(S) =1/(4-d)<1/2 (ii) a superdiffusive behavior R approximately Lnu with nu=(3-d)/(4-d)>1/2 for the wandering of the best favorable tube available. The corresponding free energy then scales as F approximately Lomega with omega=2nu-1 and the left tail of the probability distribution involves a stretched exponential of exponent eta=(4-d)/2. These results generalize the well known exact exponents nu=2/3, omega=1/3, and eta=3/2 in d=1, where the subleading transverse length R(S) approximately L(1/3) is known as the typical distance between two replicas in the Bethe ansatz wave function. We then extend our approach to correlated disorder in transverse directions with exponent alpha and/or to manifolds in dimension D+d= d(t) with 0<D<2. The strategy of being both confined and superdiffusive is still optimal for decaying correlations ( alpha<0 ), whereas it is not for growing correlations ( alpha>0 ). In particular, for an interface of dimension ( d(t) -1) in a space of total dimension 5/3< d(t) <3 with random-bond disorder, our approach yields the confinement exponent nu(S) =( d(t) -1)(3- d(t) )/(5 d(t) -7). Finally, we study the exponents in the presence of an algebraic tail 1/ V1+micro in the disorder distribution, and obtain various regimes in the (micro,d) plane.
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For the quantum Ising model with ferromagnetic random couplings J i , j > 0 and random transverse fields h i > 0 at zero temperature in finite dimensions d > 1, we consider the lowest order contributions in perturbation theory in ( J i , j / h i ) to obtain some information on the statistics of various observables in the disordered phase. We find that the two-point correlation scales as , where ξ typ is the typical correlation length, u is a random variable and ω coincides with the droplet exponent ω DP ( D = d − 1) of the directed polymer with D = ( d − 1) transverse directions. Our main conclusions are as follows. (i) Whenever ω > 0, the quantum model is governed by an infinite-disorder fixed point: there are two distinct correlation length exponents related by ν typ = (1 − ω)ν av ; the distribution of the local susceptibility χ loc presents the power-law tail P (χ loc ) ∼ 1/χ 1 + μ loc , where μ vanishes as ξ −ω av so that the averaged local susceptibility diverges in a finite neighborhood 0 < μ < 1 before criticality (Griffiths phase); the dynamical exponent z diverges near criticality as z = d /μ ∼ ξ ω av . (ii) In dimensions d ⩽ 3, any infinitesimal disorder flows toward this infinite-disorder fixed point with ω( d ) > 0 (for instance ω( d = 2) = 1/3 and ω( d = 3) ∼ 0.24). (iii) In finite dimensions d > 3, a finite disorder strength is necessary to flow toward the infinite-disorder fixed point with ω( d ) > 0 (for instance ω( d = 4) ≃ 0.19), whereas a finite-disorder fixed point remains possible for a small enough disorder strength. For the Cayley tree of effective dimension d = ∞, where ω = 0, we discuss the similarities and differences with the case of finite dimensions.
We consider the model of the directed polymer in a random medium of dimension 1+31+31+3, and investigate its multifractal properties at the localization-delocalization transition. In close analogy with models of the quantum Anderson localization transition, where the multifractality of critical wavefunctions is well established, we analyze the statistics of the position weights ${w}_{L}(\stackrel{P\vec}{r})$ of the endpoint of the polymer of length LLL via the moments ${Y}_{q}(L)={\ensuremath{\sum}}_{\stackrel{P\vec}{r}}\phantom{\rule{0.2em}{0ex}}{[{w}_{L}(\stackrel{P\vec}{r})]}^{q}$. We measure the generalized exponents $\ensuremath{\tau}(q)$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}(q)$ governing the decay of the typical values ${Y}_{q}^{\mathrm{typ}}(L)={e}^{\overline{\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{Y}_{q}(L)}}\ensuremath{\sim}{L}^{\ensuremath{-}\ensuremath{\tau}(q)}$ and disorder-averaged values $\overline{{Y}_{q}(L)}\ensuremath{\sim}{L}^{\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}(q)}$, respectively. To understand the difference between these exponents, $\ensuremath{\tau}(q)\ensuremath{\ne}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}(q)$ above some threshold $q>{q}_{c}\ensuremath{\sim}2$, we compute the probability distributions of y=Yq(L)∕Yqtyp(L)y={Y}_{q}(L)∕{Y}_{q}^{\mathrm{typ}}(L)y=Yq​(L)∕Yqtyp​(L) over the samples: We find that these distributions becomes scale invariant with a power-law tail 1∕y1+xq1∕{y}^{1+{x}_{q}}1∕y1+xq​. These results thus correspond to the Evers-Mirlin scenario [Phys. Rev. Lett. 84, 3690 (2000)] for the statistics of inverse participation ratios at the Anderson localization transitions. Finally, the finite-size scaling analysis in the critical region yields the correlation length exponent $\ensuremath{\nu}\ensuremath{\sim}2$.
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