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arXiv:cond-mat/0607411 · Physical Review E

Probing the tails of the ground-state energy distribution for the directed polymer in a random medium of dimension d=1,2,3 via a Monte Carlo procedure in the disorder

Cécile Monthus, Thomas Garel

Published Nov 13, 2006• 29 citations
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Abstract

In order to probe with high precision the tails of the ground-state energy distribution of disordered spin systems, K\"orner, Katzgraber, and Hartmann have recently proposed an importance-sampling Monte Carlo Markov chain in the disorder. In this paper, we combine their Monte Carlo procedure in the disorder with exact transfer matrix calculations in each sample to measure the negative tail of ground-state energy distribution ${P}_{d}({E}_{0})$ for the directed polymer in a random medium of dimension $d=1,2,3$. In $d=1$, we check the validity of the algorithm by a direct comparison with the exact result, namely, the Tracy-Widom distribution. In dimensions $d=2$ and $d=3$, we measure the negative tail up to ten standard deviations, which correspond to probabilities of order ${P}_{d}({E}_{0})\ensuremath{\sim}{10}^{\ensuremath{-}22}$. Our results are in agreement with Zhang's argument, stating that the negative tail exponent $\ensuremath{\eta}(d)$ of the asymptotic behavior $\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{P}_{d}({E}_{0})\ensuremath{\sim}\ensuremath{-}{\ensuremath{\mid}{E}_{0}\ensuremath{\mid}}^{\ensuremath{\eta}(d)}$ as ${E}_{0}\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ is directly related to the fluctuation exponent $\ensuremath{\theta}(d)$ [which governs the fluctuations $\ensuremath{\Delta}{E}_{0}(L)\ensuremath{\sim}{L}^{\ensuremath{\theta}(d)}$ of the ground-state energy ${E}_{0}$ for polymers of length $L$] via the simple formula $\ensuremath{\eta}(d)=1∕[1\ensuremath{-}\ensuremath{\theta}(d)]$. Throughout the paper, we comment on the similarities and differences with spin glasses.

DOI
10.1103/physreve.74.051109

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In order to probe with high precision the tails of the ground-state energy distribution of disordered spin systems, K"orner, Katzgraber, and Hartmann have recently proposed an importance-sampling Monte Carlo Markov chain in the disorder. In this paper, we combine their Monte Carlo procedure in the disorder with exact transfer matrix calculations in each sample to measure the negative tail of ground-state energy distribution Pd(E0){P}_{d}({E}_{0})Pd​(E0​) for the directed polymer in a random medium of dimension d=1,2,3d=1,2,3d=1,2,3. In d=1d=1d=1, we check the validity of the algorithm by a direct comparison with the exact result, namely, the Tracy-Widom distribution. In dimensions d=2d=2d=2 and d=3d=3d=3, we measure the negative tail up to ten standard deviations, which correspond to probabilities of order Pd(E0)\ensuremath∼10\ensuremath−22{P}_{d}({E}_{0})\ensuremath{\sim}{10}^{\ensuremath{-}22}Pd​(E0​)\ensuremath∼10\ensuremath−22. Our results are in agreement with Zhang's argument, stating that the negative tail exponent \ensuremathη(d)\ensuremath{\eta}(d)\ensuremathη(d) of the asymptotic behavior lnPd(E0)\ensuremath∼\ensuremath−\ensuremath∣E0\ensuremath∣\ensuremathη(d)\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{P}_{d}({E}_{0})\ensuremath{\sim}\ensuremath{-}{\ensuremath{\mid}{E}_{0}\ensuremath{\mid}}^{\ensuremath{\eta}(d)}lnPd​(E0​)\ensuremath∼\ensuremath−\ensuremath∣E0​\ensuremath∣\ensuremathη(d) as E0\ensuremath→\ensuremath−\ensuremath∞{E}_{0}\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}E0​\ensuremath→\ensuremath−\ensuremath∞ is directly related to the fluctuation exponent \ensuremathθ(d)\ensuremath{\theta}(d)\ensuremathθ(d) [which governs the fluctuations \ensuremathΔE0(L)\ensuremath∼L\ensuremathθ(d)\ensuremath{\Delta}{E}_{0}(L)\ensuremath{\sim}{L}^{\ensuremath{\theta}(d)}\ensuremathΔE0​(L)\ensuremath∼L\ensuremathθ(d) of the ground-state energy E0{E}_{0}E0​ for polymers of length LLL] via the simple formula \ensuremathη(d)=1∕[1\ensuremath−\ensuremathθ(d)]\ensuremath{\eta}(d)=1∕[1\ensuremath{-}\ensuremath{\theta}(d)]\ensuremathη(d)=1∕[1\ensuremath−\ensuremathθ(d)]. Throughout the paper, we comment on the similarities and differences with spin glasses.
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.
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 $\ensuremath{\nu}\ensuremath{\sim}2$.
A two-dimensional (2D) spin-1/2 antiferromagnetic Heisenberg model with a specific kind of quenched disorder is investigated, using the first-principles nonperturbative quantum Monte Carlo calculations (QMCs). The employed disorder distribution has a tunable parameter ppp which can be considered as a measure of randomness (p=0p=0p=0 correponds to the clean model). Through large-scale QMCs, the dynamic critical exponents zzz, the ground-state energy densities E0{E}_{0}E0​, and the Wilson ratios WWW of various ppp are determined with high precision. Interestingly, we find that the ppp dependencies of zzz and WWW are likely to be complementary to each other. For instance, while the zzz values of $0.4\ensuremath{\le}p\ensuremath{\le}0.9$ match well among themselves and are statistically different from that of p=0p=0p=0, the WWW values for p<0.7p<0.7p<0.7 are in reasonably good agreement with $W\ensuremath{\sim}0.1243$ of the clean case. Surprisingly, our study indicates that a threshold of randomness, pW{p}_{W}pW​, associated with WWW exists. In particular, beyond this threshold the magnitude of WWW grows with ppp. This is somehow counterintuitive since one expects the spin correlations should diminish accordingly. Similarly, there is a threshold pz{p}_{z}pz​ related to zzz after which a constant value is obtained for zzz. The results presented here are not only interesting from a theoretical perspective but also can serve as benchmarks for future related studies.
For Anderson localization models, there exists an exact real-space renormalization procedure at fixed energy which preserves the Green's functions of the remaining sites [H. Aoki, J. Phys. C 13, 3369 (1980)]. Using this procedure for the Anderson tight-binding model in dimensions d=2,3d=2,3d=2,3, we study numerically the statistical properties of the renormalized on-site energies ϵϵϵ and of the renormalized hoppings VVV as a function of the linear size LLL. We find that the renormalized on-site energies ϵϵϵ remain finite in the localized phase in d=2,3d=2,3d=2,3 and at criticality (d=3)(d=3)(d=3), with a finite density at ϵ=0ϵ=0ϵ=0 and a power-law decay 1/ϵ21/{ϵ}^{2}1/ϵ2 at large ∣ϵ∣|ϵ|∣ϵ∣. For the renormalized hoppings in the localized phase, we find: $\text{ln}\text{ }{V}_{L}\ensuremath{\simeq}\ensuremath{-}\frac{L}{{\ensuremath{\xi}}_{loc}}+{L}^{\ensuremath{\omega}}u$, where ${\ensuremath{\xi}}_{loc}$ is the localization length and uuu a random variable of order one. The exponent $\ensuremath{\omega}$ is the droplet exponent characterizing the strong disorder phase of the directed polymer in a random medium of dimension $1+(d\ensuremath{-}1)$, with $\ensuremath{\omega}(d=2)=1/3$ and $\ensuremath{\omega}(d=3)\ensuremath{\simeq}0.24$. At criticality (d=3)(d=3)(d=3), the statistics of renormalized hoppings VVV is multifractal, in direct correspondence with the multifractality of individual eigenstates and of two-point transmissions. In particular, we measure ${\ensuremath{\rho}}_{typ}\ensuremath{\simeq}1$ for the exponent governing the typical decay $\overline{\text{ln}\text{ }{V}_{L}}\ensuremath{\simeq}\ensuremath{-}{\ensuremath{\rho}}_{typ}\text{ }\text{ln}\text{ }L$, in agreement with previous numerical measures of ${\ensuremath{\alpha}}_{typ}=d+{\ensuremath{\rho}}_{typ}\ensuremath{\simeq}4$ for the singularity spectrum $f(\ensuremath{\alpha})$ of individual eigenfunctions. We also present numerical results concerning critical surface properties.
Extreme-value distributions are studied in the context of a broad range of problems, from the equilibrium properties of low-temperature disordered systems to the occurrence of natural disasters. Our focus here is on the ground-state energy distribution of disordered many-body quantum systems. We derive an analytical expression that, upon tuning a parameter, reproduces with high accuracy the ground-state energy distribution of the systems that we consider. For some models, it agrees with the Tracy-Widom distribution obtained from Gaussian random matrices. They include transverse Ising models, the Sachdev-Ye model, and a randomized version of the PXP model. For other systems, such as Bose-Hubbard models with random couplings and the disordered spin-1/2 Heisenberg chain used to investigate many-body localization, the shapes are at odds with the Tracy-Widom distribution. Our analytical expression captures all of these distributions, thus playing a role to the lowest energy level similar to that played by the Brody distribution to the bulk of the spectrum.
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