Multicritical points for spin-glass models on hierarchical lattices

Ohzeki, Masayuki; Nishimori, Hidetoshi; Berker, A. Nihat

doi:10.1103/physreve.77.061116

Cited by 46 publications

(81 citation statements)

References 25 publications

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“…Since the series of work [3][4][5][6] revealed that the duality is capable to estimate the approximate values of the critical points even for such complicated systems, the duality has been developed as one of the rare successful approaches to finite-dimensional spin glasses, where very little analytical work exists. More precise analysis of the locations of the critical points in spin glasses has been performed by employing a real-space renormalization technique, i.e., the partial summation of the degree of freedom [7,8].…”

Section: Introductionmentioning

confidence: 99%

Duality analysis on random planar lattices

Ohzeki

Fujii

2012

Phys. Rev. E

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The conventional duality analysis is employed to identify a location of a critical point on a uniform lattice without any disorder in its structure. In the present study, we deal with the random planar lattice, which consists of the randomized structure based on the square lattice. We introduce the uniformly random modification by the bond dilution and contraction on a part of the unit square. The random planar lattice includes the triangular and hexagonal lattices in extreme cases of a parameter to control the structure. The duality analysis in a modern fashion with real-space renormalization is found to be available for estimating the location of the critical points with wide range of the randomness parameter. As a simple testbed, we demonstrate that our method indeed gives several critical points for the cases of the Ising and Potts models, and the bond-percolation thresholds on the random planar lattice. Our method leads to not only such an extension of the duality analyses on the classical statistical mechanics but also a fascinating result associated with optimal error thresholds for a class of quantum error correction code, the surface code on the random planar lattice, which known as a skillful technique to protect the quantum state.

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Section: Introductionmentioning

confidence: 99%

Duality analysis on random planar lattices

Ohzeki

Fujii

2012

Phys. Rev. E

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“…Similar previous studies, on other spin-glass systems, are reported in Refs. [12,13,[40][41][42][43][44][45][46][47]. Figure 1 shows a calculated sequence of phase diagram cross sections for the left-chiral (c = 0) (top row) and quenched random left-and right-chiral (c = 0.5) (bottom row) systems with, in both cases, quenched random ferromagnetic and antiferromagnetic interactions.…”

Section: Renormalization-group Method: Migdal-kadanoff Approximamentioning

confidence: 99%

Phase transitions between different spin-glass phases and between different chaoses in quenched random chiral systems

Çağlar

Berker

2017

Phys. Rev. E

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The left-right chiral and ferromagnetic-antiferromagnetic double-spin-glass clock model, with the crucially even number of states q = 4 and in three dimensions d = 3, has been studied by renormalization-group theory. We find, for the first time to our knowledge, four spin-glass phases, including conventional, chiral, and quadrupolar spin-glass phases, and phase transitions between spin-glass phases. The chaoses, in the different spin-glass phases and in the phase transitions of the spin-glass phases with the other spin-glass phases, with the non-spin-glass ordered phases, and with the disordered phase, are determined and quantified by Lyapunov exponents. It is seen that the chiral spin-glass phase is the most chaotic spin-glass phase. The calculated phase diagram is also otherwise very rich, including regular and temperature-inverted devil's staircases and reentrances.

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“…To account for this, a renormalisation inspired expansion was introduced in [11,20] to account for corrections. While most natural for hierarchical lattices [20], it has been extended to square lattices and achieves an even tighter match with previous numerical results using only a first order correction [11]. This technique proceeds by replacing the term e J H in x H 0 (and, similarly, the term e J H + e −J H in x * H 0 ) with an equivalent effective weight arising from an averaging effect over several neighbouring spins.…”

Section: B Replica Methodsmentioning

confidence: 99%

Implications of ignorance for quantum-error-correction thresholds

Kay

2014

Phys. Rev. A

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Quantum error correcting codes have a distance parameter, conveying the minimum number of single spin errors that could cause error correction to fail. However, the success thresholds of finite per-qubit error rate that have been proven for the likes of the Toric code require them to work well beyond this limit. We argue that without the assumption of being below the distance limit, the success of error correction is not only contingent on the noise model, but what the noise model is believed to be. Any discrepancy must adversely affect the threshold rate, and risks invalidating existing threshold theorems. We prove that for the 2D Toric code, suitable thresholds still exist by utilising a mapping to the 2D random bond Ising model.

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Multicritical points for spin-glass models on hierarchical lattices

Cited by 46 publications

References 25 publications

Duality analysis on random planar lattices

Duality analysis on random planar lattices

Phase transitions between different spin-glass phases and between different chaoses in quenched random chiral systems

Implications of ignorance for quantum-error-correction thresholds

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