Anomalous diffusion in aperiodic environments

Iglói, Ferenc; Turban, L.; Rieger, Heiko

doi:10.1103/physreve.59.1465

Cited by 53 publications

(54 citation statements)

References 36 publications

(69 reference statements)

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“…This analogy is more than a simple coincidence, since the 1d RW and the quantum Ising spin chain are related through an exact mapping [22], which then has the same requirement for the relevance-irrelevance conditions. In 2d, where such type of mapping does not exist, the limiting value is found to be approximately around α c ≈ 4.5, thus in the region of 2 < α < α c the broadness of disorder is relevant for the RTIM, whereas it is irrelevant for the RW.…”

Section: Discussionmentioning

confidence: 99%

Random quantum magnets with broad disorder distribution

Karevski¹,

Lin²,

Rieger³

et al. 2001

Eur. Phys. J. B

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Abstract. We study the critical behavior of Ising quantum magnets with broadly distributed random couplings (J), such that P (ln J) ∼ | ln J| −1−α , α > 1, for large | ln J| (Lévy flight statistics). For sufficiently broad distributions, α < αc, the critical behavior is controlled by a line of fixed points, where the critical exponents vary with the Lévy index, α. In one dimension, with αc = 2, we obtaind several exact results through a mapping to surviving Riemann walks. In two dimensions the varying critical exponents have been calculated by a numerical implementation of the Ma-Dasgupta-Hu renormalization group method leading to αc ≈ 4.5. Thus in the region 2 < α < αc, where the central limit theorem holds for | ln J| the broadness of the distribution is relevant for the 2d quantum Ising model. PACS. 75.50.Lk Spin glasses and other random magnets -05.30.Ch Quantum ensemble theory -75.10.Nr Spin-glass and other random models -75.40.Gb Dynamic properties (dynamic susceptibility, spin waves, spin diffusion, dynamic scaling, etc.)

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Section: Discussionmentioning

confidence: 99%

Random quantum magnets with broad disorder distribution

Karevski¹,

Lin²,

Rieger³

et al. 2001

Eur. Phys. J. B

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“…[22]. In the following we calculate the exact value of the dynamical exponent using the same strategy as for the random quantum Ising model in Ref [20,21]. Our basic observation is the fact that the eigenvalue problem of the T σ (or T τ ) matrix can be mapped through an uniter transformation to a Fokker-Planck operator, which appear in the Master equation of a Sinai diffusion, i.e.…”

Section: Griffiths Phasementioning

confidence: 99%

“…In the Griffiths phase the system is gapless, thus dynamical correlations decay with a power-law, however there is long-range-order with exponentially decaying spatial correlations. For the random quantum Ising chain dynamical correlations, both at the critical point and in the Griffiths phase have been exactly determined [20][21][22] using a mapping to the Sinai model [23], i.e. random walk in a random environment.…”

Section: Introductionmentioning

confidence: 99%

Random antiferromagnetic quantum spin chains: Exact results from scaling of rare regions

2000

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We study XY and dimerized XX spin-1/2 chains with random exchange couplings by analytical and numerical methods and scaling considerations. We extend previous investigations to dynamical properties, to surface quantities and operator profiles, and give a detailed analysis of the Griffiths phase. We present a phenomenological scaling theory of average quantities based on the scaling properties of rare regions, in which the distribution of the couplings follows a surviving random walk character. Using this theory we have obtained the complete set of critical decay exponents of the random XY and XX models, both in the volume and at the surface. The scaling results are confronted with numerical calculations based on a mapping to free fermions, which then lead to an exact correspondence with directed walks. The numerically calculated critical operator profiles on large finite systems (L ≤ 512) are found to follow conformal predictions with the decay exponents of the phenomenological scaling theory. Dynamical correlations in the critical state are in average logarithmically slow and their distribution show multi-scaling character. In the Griffiths phase, which is an extended part of the off-critical region average autocorrelations have a power-law form with a non-universal decay exponent, which is analytically calculated. We note on extensions of our work to the random antiferromagnetic XXZ chain and to higher dimensions.

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“…(11) and (13) describe the ultraslow diffusion processes. This kind of diffusion has been found, for instance, in aperiodic environments [10].…”

Section: Langevin Equation and Its Corresponding Fokker-planck Equationmentioning

confidence: 67%

Solution of Fokker-Planck equation for a broad class of drift and diffusion coefficients

2011

Phys. Rev. E

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Anomalous diffusion in aperiodic environments

Cited by 53 publications

References 36 publications

Random quantum magnets with broad disorder distribution

Random quantum magnets with broad disorder distribution

Random antiferromagnetic quantum spin chains: Exact results from scaling of rare regions

Solution of Fokker-Planck equation for a broad class of drift and diffusion coefficients

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