Real space rescaling study of spin glass behaviour in three dimensions

Southern, B. W.; Young, A. P.

doi:10.1088/0022-3719/10/12/023

Cited by 180 publications

(186 citation statements)

References 25 publications

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“…which coincides with the zero-temperature limit of the Migdal-Kadanoff RG rules for spin-glasses, obtained by exact recursions on partition functions [4][5][6][7][8][9][10][11][12][14][15][16][17].…”

Section: Exactness For the Migdal-kadanoff Approximationmentioning

confidence: 79%

One-dimensional Ising spin-glass with power-law interaction: real-space renormalization at zero temperature

Monthus¹

2014

J. Stat. Mech.

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For the one-dimensional long-ranged Ising spin-glass with random couplings decaying with the distance r as J(r) ∼ r −σ and distributed with the Lévy symmetric stable distribution of index 1 < µ ≤ 2 (including the usual Gaussian case µ = 2), we consider the region σ > 1/µ where the energy is extensive. We study two real space renormalization procedures at zero temperature, namely a simple box decimation that leads to explicit calculations, and a strong disorder decimation that can be studied numerically on large sizes. The droplet exponent governing the scaling of the renormalized couplings JL ∝ L θµ(σ) is found to be θµ(σ) = 2 µ −σ whenever the long-ranged couplings are relevant θµ(σ) ≥ −1. For the statistics of the ground state energy E GS L over disordered samples, we obtain that the droplet exponent θµ(σ) governs the leading correction to extensivity of the averaged value E GS L ≃ Le0 + L θµ (σ) e1. The characteristic scale of the fluctuations around this average is of order L 1 µ , and the rescaledGaussian distributed for µ = 2, or displays the negative power-law tail in 1/(−u) 1+µ for u → −∞ in the Lévy case 1 < µ < 2. Finally we apply the zero-temperature renormalization procedure to the related Dyson hierarchical spin-glass model where the same droplet exponent appears.

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Section: Exactness For the Migdal-kadanoff Approximationmentioning

confidence: 79%

One-dimensional Ising spin-glass with power-law interaction: real-space renormalization at zero temperature

Monthus¹

2014

J. Stat. Mech.

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“…Note that the number of bonds on hierarchical lattice after I iterations is the same as the number of sites of a d-dimensional lattice of size L = 2 I . Thermodynamic quantities are then evaluated iteratively by tracing over the spins on the highest level of the hierarchy, until the lowest level is reached and the trace over the remaining two spins is calculated [13]. This procedure generates new effective couplings, which have to be included in the recursion relations.…”

Section: Numerical Resultsmentioning

confidence: 99%

The ± J spin glass in Migdal-Kadanoff approximation

Drossel

Moore

2001

Eur. Phys. J. B

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We study the low-temperature phase of the three-dimensional ±J Ising spin glass in MigdalKadanoff approximation. At zero temperature, T = 0, the properties of the spin glass result from the ground-state degeneracy and can be elucidated using scaling arguments based on entropy. The approach to the asymptotic scaling regime is very slow, and the correct exponents are only visible beyond system sizes around 64. At T > 0, a crossover from the zero-temperature behaviour to the behaviour expected from the droplet picture occurs at length scales proportional to T −ds/2 where ds is the fractal dimension of a domain wall. Canonical droplet behaviour is not visible at any temperature for systems whose linear dimension is smaller than 16 lattice spacings, because the data are either affected by the zero-temperature behaviour or the critical point behaviour.

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“…Consider a general (b, p)-DHL with the coupling constants defined by the inflation rule given by (2). Within the real space renormalization group scheme, the decimation procedure is carried on by partial tracing on the spins introduced in the last generation.…”

Section: The Model Hamiltonian and The Aperiodic Interactionsmentioning

confidence: 99%

Multifractal properties of aperiodic Ising model on hierarchical lattices: role of the geometric fluctuations

2001

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The role of the geometric fluctuations on the multifractal properties of the local magnetization of aperiodic ferromagnetic Ising models on hierachical lattices is investigated. The geometric fluctuations are introduced by generalized Fibonacci sequences. The local magnetization is evaluated via an exact recurrent procedure encompassing a real space renormalization group decimation. The symmetries of the local magnetization patterns induced by the aperiodic couplings is found to be strongly (weakly) different, with respect to the ones of the corresponding homogeneous systems, when the geometric fluctuations are relevant (irrelevant) to change the critical properties of the system. At the criticality, the measure defined by the local magnetization is found to exhibit a non-trivial F (α) spectra being shifted to higher values of α when relevant geometric fluctuations are considered. The critical exponents are found to be related with some special points of the F (α) function and agree with previous results obtained by the quite distinct transfer matrix approach.

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Real space rescaling study of spin glass behaviour in three dimensions

Cited by 180 publications

References 25 publications

One-dimensional Ising spin-glass with power-law interaction: real-space renormalization at zero temperature

One-dimensional Ising spin-glass with power-law interaction: real-space renormalization at zero temperature

The ± J spin glass in Migdal-Kadanoff approximation

Multifractal properties of aperiodic Ising model on hierarchical lattices: role of the geometric fluctuations

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