Loop expansion around the Bethe solution for the random magnetic field Ising ferromagnets at zero temperature

Angelini, Maria Chiara; Lucibello, Carlo; Parisi, G.; Ricci-Tersenghi, Federico; Rizzo, Tommaso

doi:10.1073/pnas.1909872117

Cited by 10 publications

(13 citation statements)

References 37 publications

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“…Recently, ref. [100] proposed to expand the RFIM around an exact solution on the "Bethe lattice" (an infinite tree without loops with all vertices equivalent and having coordination number 2d, like for the cubic lattice in d dimension). While this approach is very different from the traditional one, their calculations were consistent with dimensional reduction in d close to 6.…”

Section: Jhep03(2021)219mentioning

confidence: 99%

Random field Ising model and Parisi-Sourlas supersymmetry. Part II. Renormalization group

Kaviraj

Rychkov

Trevisani

2021

J. High Energ. Phys.

104

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We revisit perturbative RG analysis in the replicated Landau-Ginzburg description of the Random Field Ising Model near the upper critical dimension 6. Working in a field basis with manifest vicinity to a weakly-coupled Parisi-Sourlas supersymmetric fixed point (Cardy, 1985), we look for interactions which may destabilize the SUSY RG flow and lead to the loss of dimensional reduction. This problem is reduced to studying the anomalous dimensions of “leaders” — lowest dimension parts of Sn-invariant perturbations in the Cardy basis. Leader operators are classified as non-susy-writable, susy-writable or susy-null depending on their symmetry. Susy-writable leaders are additionally classified as belonging to superprimary multiplets transforming in particular OSp(d|2) representations. We enumerate all leaders up to 6d dimension ∆ = 12, and compute their perturbative anomalous dimensions (up to two loops). We thus identify two perturbations (with susy- null and non-susy-writable leaders) becoming relevant below a critical dimension dc ≈ 4.2 - 4.7. This supports the scenario that the SUSY fixed point exists for all 3 < d ⩽ 6, but becomes unstable for d < dc.

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Section: Jhep03(2021)219mentioning

confidence: 99%

Random field Ising model and Parisi-Sourlas supersymmetry. Part II. Renormalization group

Kaviraj

Rychkov

Trevisani

2021

J. High Energ. Phys.

104

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“…Since we are at T = 0, the connected correlation function in Eq. ( 17) is ill-defined; therefore, we work with its rescaled version that we called response function [25]…”

Section: Spin Glass Models On the Bethe Latticementioning

confidence: 99%

“…The expansion around the BL solution has the same advantages as standard field-theoretical loop expansions, but has a larger range of applicability, as it can be used for any problem that displays a continuous phase transition on the BL. Recent applications include the Random Field Ising model (RFIM) at zero temperature [25], the bootstrap percolation [26] and the glass crossover [27]. It has also been applied to the SG in a field in the limit of high connectivity for T > 0 [28], showing that in such a limit the expansion is completely equivalent to the standard expansion around the MF-FC solution [11,13].…”

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confidence: 99%

“…The term N (p, L) has already been computed [25] and it reads respectively for the quartic and cubic loops…”

mentioning

confidence: 99%

“…However, the above sums over L A and L B are divergent respectively for D ≤ 6, D ≤ 8 and D ≤ 8 and thus the Ginzburg criterion tells us that the critical exponents cannot be those of the Gaussian theory below D U = 8. The same argument applied to the quartic loop would give a critical dimension equal to 6 (the diagram indeed appears in the computation of the connected correlation of the RFIM [25]) and allows to neglect the quartic loop with respect to the cubic one.…”

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confidence: 99%

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Unexpected upper critical dimension for spin glass models in a field predicted by the loop expansion around the Bethe solution at zero temperature

Angelini,

Lucibello,

Parisi

et al. 2021

Preprint

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The spin-glass transition in a field in finite dimension is analyzed directly at zero temperature using a perturbative loop expansion around the Bethe lattice solution. The loop expansion is generated by the M -layer construction whose first diagrams are evaluated numerically and analytically. The Ginzburg criterion, from both the paramagnetic and spin-glass phase, reveals that the upper critical dimension below which mean-field theory fails is DU = 8, at variance with the classic results DU = 6 yielded by finite-temperature replica field theory. The different outcome follows from two crucial properties: finite-connectivity of the lattice and zero temperature. They both also lead to specific features of the replica-symmetry-breaking phase.

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Bethe M-layer construction on the Ising model

Angelini,

Palazzi,

Parisi

et al. 2024

J. Stat. Mech.

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In statistical physics, one of the standard methods to study second order phase transitions is the renormalization group that usually leads to an expansion around the corresponding fully connected solution. Unfortunately, often in disordered models, some important finite dimensional second-order phase transitions are qualitatively different or absent in the corresponding fully connected model: in such cases the standard expansion fails. Recently, a new method, the M-layer one, has been introduced that performs an expansion around a different soluble mean field model: the Bethe lattice one. This new method has been already used to compute the upper critical dimension D U of different disordered systems such as the Random Field Ising model or the Spin glass model with field. If then one wants to go beyond and construct an expansion around D U to understand how critical quantities get renormalized, the actual computation of all the numerical factors is needed. This next step has still not been performed, being technically more involved. In this paper we perform this computation for the ferromagnetic Ising model without quenched disorder, in finite dimensions: we show that, at one-loop order inside the M-layer approach, we recover the continuum quartic field theory and we are able to identify the coupling constant g and the other parameters of the theory, as a function of macroscopic and microscopic details of the model such as the lattice spacing, the physical lattice dimension and the temperature. This is a fundamental step that will help in applying in the future the same techniques to more complicated systems, for which the standard field theoretical approach is impracticable.

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Loop expansion around the Bethe solution for the random magnetic field Ising ferromagnets at zero temperature

Cited by 10 publications

References 37 publications

Random field Ising model and Parisi-Sourlas supersymmetry. Part II. Renormalization group

Random field Ising model and Parisi-Sourlas supersymmetry. Part II. Renormalization group

Unexpected upper critical dimension for spin glass models in a field predicted by the loop expansion around the Bethe solution at zero temperature

Bethe M-layer construction on the Ising model

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