Renormalization flow of the hierarchical Anderson model at weak disorder

F. L. Metz, L. Leuzzi, and G. Parisi

Phys. Rev. B 89, 064201 – Published 4 February 2014

Abstract

We study the flow of the renormalized model parameters obtained from a sequence of simple transformations of the 1D Anderson model with long-range hierarchical hopping. Combining numerical results with a perturbative approach for the flow equations, we identify three qualitatively different regimes at weak disorder. For a sufficiently fast decay of the hopping energy, the Cauchy distribution is the only stable fixed point of the flow equations, whereas for sufficiently slowly decaying hopping energy the renormalized parameters flow to a $δ$ -peak fixed-point distribution. In an intermediate range of the hopping decay, both fixed-point distributions are stable and the stationary solution is determined by the initial configuration of the random parameters. We present results for the critical decay of the hopping energy separating the different regimes.

Received 5 November 2013

DOI:https://doi.org/10.1103/PhysRevB.89.064201

Authors & Affiliations

F. L. Metz^1,*, L. Leuzzi^1,2, and G. Parisi^1,2,3

¹Dipartimento di Fisica, Università “La Sapienza”, Piazzale A. Moro 2, I-00185 Rome, Italy
²IPCF-CNR, UOS Roma “Kerberos”, Università “La Sapienza”, Piazzale A. Moro 2, I-00185 Rome, Italy
³INFN, Piazzale A. Moro 2, 00185 Rome, Italy

^*Corresponding author: fmetzfmetz@gmail.com

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Vol. 89, Iss. 6 — 1 February 2014

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Images

Figure 1
Numerical results for the flow of the distribution $P^{(ℓ)} (μ)$ of the renormalized random potentials (taken away their mean value) for $α = 1.25$ and $E = E_{\infty}^{pure}$ . The distribution $p (ɛ)$ has a Gaussian form, with mean zero and standard deviation $W = 10^{- 2}$ . The solid lines show Gaussian distributions with mean zero and standard deviations given by Eq. (17). The numerical data have been obtained through the population dynamics algorithm with $N_{p} = 10^{7}$ (see the main text).
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Figure 2
Numerical results for the flow of the distribution $P^{(ℓ)} (μ)$ of the renormalized random potentials for $α = 2.25$ and $E = E_{\infty}^{pure}$ . The distribution $p (ɛ)$ has a Gaussian form, with mean zero and standard deviation $W = 10^{- 2}$ . The solid line depicts a Cauchy distribution with parameters taken from a fitting of the data for $ℓ = 15$ . The numerical data have been obtained through the population dynamics algorithm with $N_{p} = 10^{7}$ (see the main text).
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Figure 3
Numerical results for the flow of the standard deviation of $P^{(ℓ)} (μ)$ obtained from the population dynamics algorithm for $N_{p} = 10^{7}$ , $E = E_{\infty}^{pure}$ , and two values of $α$ . The distribution $p (ɛ)$ has a Gaussian form, with mean zero and standard deviation $W = 10^{- 2}$ . The black solid line is the analytical result of Eq. (17), while the red dashed line is just a guide.
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Figure 4
Average fraction of runs of the population dynamics algorithm for which the standard deviation of $P^{(ℓ)} (μ)$ is given by Eq. (17). The fraction $F_{S}$ is calculated using $S = 50$ independent runs and $F$ is computed by averaging $F_{S}$ over five independent data sets. The initial configuration $μ_{1}^{(0)}, \dots, μ_{N_{p}}^{(0)}$ is drawn from a Gaussian distribution with mean $E_{0}^{pure} - E$ and standard deviation $W = 10^{- 2}$ . We have that $E = E_{\infty}^{pure}$ , and the values of $α$ are indicated on the figure.
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Figure 5
Numerical results for the flow of the standard deviation of $P^{(ℓ)} (μ)$ obtained from the population dynamics algorithm for $N_{p} = 10^{7}$ , $E = E_{\infty}^{pure}$ , and $α = 2.25$ . The distribution $p (ɛ)$ has a Gaussian form, with mean zero and standard deviation $W = 10^{- 2}$ . The value of $ℓ$ where the Cauchy fixed-point distribution becomes unstable is denoted by $ℓ_{c}$ . The red dashed line is just a guide.
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Figure 6
Average value $L_{c} = \bar{ℓ_{c}}$ (see Fig. 5) as a function of the population size $N_{p}$ for $α = 2.25$ and $E = E_{\infty}^{pure}$ . The distribution $p (ɛ)$ has a Gaussian form, with mean zero and standard deviation $W = 10^{- 2}$ . The average $L_{c}$ is computed using 50 independent runs of the population dynamics algorithm. The solid line is the best-fit $L_{c} = a + b ln N_{p}$ of the data, with parameters $a = 11.1 (1.3)$ and $b = 2.14 (9)$ .
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