Two-dimensional<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>X</mml:mi><mml:mi>Y</mml:mi></mml:mrow></mml:math>and clock models studied via the dynamics generated by rough surfaces

Brito, Alexandre Faissal; Redinz, José Arnaldo; Plascak, J. A.

doi:10.1103/physreve.81.031130

Cited by 28 publications

(36 citation statements)

References 21 publications

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“…( 6), T 1 and T 2 in the thermodynamic limit are evaluated to be T 1 = 0.410 and T 2 = 0.921. These values are consistent with those evaluated in previous studies (see Table II) [39,40]. Here, the values of b and c obtained by fitting are (b, c) = (0.182, −0.104) for T 1 and (b, c) = (0.049, 0.016) for T 2 .…”

Section: Methods and Resultssupporting

confidence: 92%

Machine-learning detection of the Berezinskii-Kosterlitz-Thouless transitions in the q-state clock models

Miyajima,

Murata,

Tanaka

et al. 2021

Preprint

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We demonstrate that a machine learning technique with a simple feedforward neural network can sensitively detect two successive phase transitions associated with the Berezinskii-Kosterlitz-Thouless (BKT) phase in q-state clock models simultaneously by analyzing the weight matrix components connecting the hidden and output layers. We find that the method requires only a data set of the raw spatial spin configurations for the learning procedure. This data set is generated by Monte-Carlo thermalizations at selected temperatures. Neither prior knowledge of, for example, the transition temperatures, number of phases, and order parameters nor processed data sets of, for example, the vortex configurations, histograms of spin orientations, and correlation functions produced from the original spin-configuration data are needed, in contrast with most of previously proposed machine learning methods based on supervised learning. Our neural network evaluates the transition temperatures as T2/J = 0.921 and T1/J = 0.410 for the paramagnetic-to-BKT transition and BKT-to-ferromagnetic transition in the eight-state clock model on a square lattice. Both critical temperatures agree well with those evaluated in the previous numerical studies.

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Section: Methods and Resultssupporting

confidence: 92%

Machine-learning detection of the Berezinskii-Kosterlitz-Thouless transitions in the q-state clock models

Miyajima,

Murata,

Tanaka

et al. 2021

Preprint

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“…As a last application, we study the permutation measures and applied to Ising surfaces [40] , [41] . These surfaces are obtained by accumulating the lattice spin values of the Ising model defined by the Hamiltonian.…”

Section: Resultsmentioning

confidence: 99%

Complexity-Entropy Causality Plane as a Complexity Measure for Two-Dimensional Patterns

et al. 2012

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Complexity measures are essential to understand complex systems and there are numerous definitions to analyze one-dimensional data. However, extensions of these approaches to two or higher-dimensional data, such as images, are much less common. Here, we reduce this gap by applying the ideas of the permutation entropy combined with a relative entropic index. We build up a numerical procedure that can be easily implemented to evaluate the complexity of two or higher-dimensional patterns. We work out this method in different scenarios where numerical experiments and empirical data were taken into account. Specifically, we have applied the method to fractal landscapes generated numerically where we compare our measures with the Hurst exponent; liquid crystal textures where nematic-isotropic-nematic phase transitions were properly identified; 12 characteristic textures of liquid crystals where the different values show that the method can distinguish different phases; and Ising surfaces where our method identified the critical temperature and also proved to be stable.

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“…This result reflects the continuum situation in the Maes and Shlosman program [18]. As a consequence, S L+αK t = S L t S αK t , and it is not immediate that the joint dynamics also rotates the discrete Gibbs measures in the sense of Proposition 1.5, part (2). To see that this is nevertheless true one has to follow the same arguments as in Section 3.2 and notice |Γ joint | e ̺ < ∞.…”

Section: Translation-invariant Invariant Measures Are Gibbs Measuresmentioning

confidence: 72%

“…We are now in the position to finish the proof of Proposition 1.4, part (2). This is a standard argument from [17] using translation-invariance and explicit control over boundary terms, applied to the q-state model.…”

Section: Translation-invariant Invariant Measures Are Gibbs Measuresmentioning

confidence: 94%

A class of nonergodic interacting particle systems with unique invariant measure

Jahnel¹,

Külske²

2014

Ann. Appl. Probab.

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Dedicated to A. van Enter on the occasion of his 60th birthdayWe consider a class of discrete q-state spin models defined in terms of a translation-invariant quasilocal specification with discrete clock-rotation invariance which have extremal Gibbs measures µ ′ ϕ labeled by the uncountably many values of ϕ in the one-dimensional sphere (introduced by van Enter, Opoku, Külske [J. Phys. A 44 (2011) 475002, 11]). In the present paper we construct an associated Markov jump process with quasilocal rates whose semigroup (St) t≥0 acts by a continuous rotation St(µ ′ ϕ ) = µ ′ ϕ+t . As a consequence our construction provides examples of interacting particle systems with unique translation-invariant invariant measure, which is not long-time limit of all starting measures, answering an old question (compare Liggett [Interacting Particle Systems (1985) Springer], question four, Chapter one). The construction of this particle system is inspired by recent conjectures of Maes and Shlosman about the intermediate temperature regime of the nearest-neighbor clock model. We define our generator of the interacting particle system as a (noncommuting) sum of the rotation part and a Glauber part.Technically the paper rests on the control of the spread of weak nonlocalities and relative entropy-methods, both in equilibrium and dynamically, based on Dobrushin-uniqueness bounds for conditional measures.

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Two-dimensionalXYand clock models studied via the dynamics generated by rough surfaces

Cited by 28 publications

References 21 publications

Machine-learning detection of the Berezinskii-Kosterlitz-Thouless transitions in the q-state clock models

Machine-learning detection of the Berezinskii-Kosterlitz-Thouless transitions in the q-state clock models

Complexity-Entropy Causality Plane as a Complexity Measure for Two-Dimensional Patterns

A class of nonergodic interacting particle systems with unique invariant measure

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