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

Çağlar, Tolga; Berker, A. Nihat

doi:10.1103/physreve.96.032103

Cited by 13 publications

(6 citation statements)

References 83 publications

(118 reference statements)

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“…Due to the presence of the external field that does not allow a Fourier expansion around the uniform solution η i→j (θ i ) = 1/2π as in Ref. [21], we follow the second approach, dividing the [0, 2π) interval into Q equal bins and hence moving from the XY model to the Q-state clock model [21,37,[39][40][41][42][43].…”

Section: Rs Solution Via Belief Propagation Bp Equations and The Popu...mentioning

confidence: 99%

The random field XY model on sparse random graphs shows replica symmetry breaking and marginally stable ferromagnetism

Lupo

Parisi

Ricci-Tersenghi

2019

J. Phys. A: Math. Theor.

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The ferromagnetic XY model on sparse random graphs in a randomly oriented field is analyzed via the belief propagation algorithm. At variance with the fully connected case and with the random field Ising model on the same topology, we find strong evidences of a tiny region with Replica Symmetry Breaking (RSB) in the limit of very low temperatures. This RSB phase is robust against different choices of the external field direction, while it rapidly vanishes when increasing the graph mean degree, the temperature or the directional bias in the external field. The crucial ingredients to have such a RSB phase seem to be the continuous nature of vector spins, mostly preserved by the O(2)-invariant random field, and the strong spatial heterogeneity, due to graph sparsity. We also uncover that the ferromagnetic phase can be marginally stable despite the presence of the random field. Finally, we study the proper correlation functions approaching the critical points to identify the ones that become more critical.arXiv:1902.07132v2 [cond-mat.dis-nn]

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Section: Rs Solution Via Belief Propagation Bp Equations and The Popu...mentioning

confidence: 99%

The random field XY model on sparse random graphs shows replica symmetry breaking and marginally stable ferromagnetism

Lupo

Parisi

Ricci-Tersenghi

2019

J. Phys. A: Math. Theor.

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“…The most effective approach [24] is to discretize such an interval in Q bins of width 2π/Q each. The resulting model is no longer endowed with the O(2) continuous symmetry, but with the discrete Z Q symmetry, and it is known as the Q-state clock model [22,24,[41][42][43][44][45].…”

Section: The Xy Model On Sparse Graphsmentioning

confidence: 99%

Comparison of Gabay–Toulouse and de Almeida–Thouless instabilities for the spin-glass XY model in a field on sparse random graphs

Lupo

Ricci-Tersenghi

2018

Phys. Rev. B

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Vector spin glasses are known to show two different kinds of phase transitions in presence of an external field: the so-called de Almeida-Thouless and Gabay-Toulouse lines. While the former has been studied to some extent on several topologies (fully connected, random graphs, finitedimensional lattices, chains with long-range interactions), the latter has been studied only in fully connected models, which however are known to show some unphysical behaviors (e. g. the divergence of these critical lines in the zero-temperature limit). Here we compute analytically both these critical lines for XY spin glasses on random regular graphs. We discuss the different nature of these phase transitions and the dependence of the critical behavior on the field distribution. We also study the crossover between the two different critical behaviors, by suitably tuning the field distribution.

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“…The spatially and temporally heterogeneous units, and the coupled interaction among units results in the phenomenon that a microscopic change at one scale has implications ramifying across scales throughout the system [69,70]. Therefore, in biotic systems, the coarse-grained variables are not coarse-grained characterization of a group of units whose higher order coupling could be considered as irrelevant fluctuations and be safely discarded (e.g., field in physics [58][59][60]122]), but are representation of the environ-ment computed by the system itself to capture coarse-grained regularities.…”

Section: Self-organization Of Dnns Through a Feedback-control Loop Co...mentioning

confidence: 99%

“…This forward-backward propagation composes one iteration of the loop. coarse-grain over the symmetric behaviors of constituent units in the Ising model (through statistical field theory, e.g., techniques such as renormalization groups [58][59][60]122]).…”

Section: Self-organization Of Dnns Through a Feedback-control Loop Co...mentioning

confidence: 99%

“…By discussing the epistemology of physics, where the renormalizibility of physical systems [58][59][60] is an emergent orga-nizing principle founding on conservative symmetries [3,61] in which the microscopic details are unknowable or irrelevant [34], we motivate that we might need to investigate mesoscopic organizing principles of organized complex systems [40] that are stable [62] from an informational perspective [46,49,51,63,64]. The proliferation of adjacent states [7, p. 263] with similar free energy (i.e., conservative symmetric states) in DNA macromolecules [50], and the transitions among those states by the breaking conservative symmetries (induced by spontaneous symmetry breaking in quantum field theories or deterministic chaos [61,65]) make us speculate that the lack of conservative symmetries in DNNs [66,67] is a feature looked in a backward perspective.…”

Section: Introductionmentioning

confidence: 99%

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Complexity from Adaptive-Symmetries Breaking: Global Minima in the Statistical Mechanics of Deep Neural Networks

M.¹

2022

Preprint

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Background: The scientific understanding of complex systems and deep neural networks (DNNs) are among the unsolved important problems of science; and DNNs are evidently complex systems. Meanwhile, conservative symmetry arguably is the most important concept of physics, and P.W. Anderson, Nobel Laureate in physics, speculated that increasingly sophisticated broken symmetry in many-body systems correlates with increasing complexity and functional specialization. Furthermore, in complex systems such as DNA molecules, different nucleotide sequences consist of different weak bonds with similar free energy; and energy fluctuations would break the symmetries that conserve the free energy of the nucleotide sequences, which selected by the environment would lead to organisms with different phenotypes.Purpose: When the molecule is very large, we might speculate that statistically the system poses in a state that would be of equal probability to transit to a large number of adjacent possible states; that is, an adaptive symmetry whose breaking is selected by the feedback signals from the environment. In physics, quantitative changes would accumulate into qualitative revolution where previous paradoxical behaviors are reconciled under a new paradigm of higher dimensionality (e.g., wave-particle duality in quantum physics). This emergence of adaptive symmetry and complexity might be speculated as accumulation of sophistication and quantity of conservative symmetries that lead to a paradigm shift, which might clarify the behaviors of DNNs.Results: In this work, theoretically and experimentally, we characterize the optimization process of a DNN system as an extended symmetry-breaking process where novel examples are informational perturbations to the system that breaks adaptive symmetries. One particular finding is that a hierarchically large DNN would have a large reservoir of adaptive symmetries, and when the information capacity of the reservoir exceeds the complexity of the dataset, the system could absorb all perturbations of the examples and self-organize into a functional structure of zero training errors measured by a certain surrogate risk. In this diachronically extended process, complexity emerges from quantitative accumulation of adaptive-symmetries breaking.Method: More specifically, this process is characterized by a statistical-mechanical model that could be appreciated as a generalization of statistics physics to the DNN organized complex system, and characterizes regularities in higher dimensionality. The model consists of three constitutes that could be appreciated as the counterparts of Boltzmann distribution, Ising model, and conservative symmetry, respectively: (1) a stochastic definition/interpretation of DNNs that is a multilayer probabilistic graphical model, (2) a formalism of circuits that perform biological computation, (3) a circuit symmetry from which self-similarity between the microscopic and the macroscopic adaptability manifests. The model is analyzed with a method referred as the statistical as...

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

Cited by 13 publications

References 83 publications

The random field XY model on sparse random graphs shows replica symmetry breaking and marginally stable ferromagnetism

The random field XY model on sparse random graphs shows replica symmetry breaking and marginally stable ferromagnetism

Comparison of Gabay–Toulouse and de Almeida–Thouless instabilities for the spin-glass XY model in a field on sparse random graphs

Complexity from Adaptive-Symmetries Breaking: Global Minima in the Statistical Mechanics of Deep Neural Networks

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