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arXiv:1304.7008

Using memory to identify phase transitions on a Cayley Tree

Javier M. Magán, Auditya Sharma

Published Apr 25, 2013• 0 citations
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Abstract

We provide a concrete and systematic connection between the statistical physics of the Ising ferromagnet on a Cayley tree, and the study of memory in exponentially expanding spaces. Memory turns out to be a clear signal of the `Bethe-Peierls' phase transition, and the average of memory divided by its standard deviation provides a clear signal of the `spin-glass' transition temperature. Numerical Monte Carlo simulations are used to make transparent the existence of the two different transition temperatures. The quantities used to spot the phase transitions with Monte Carlo could be useful when studying other systems where analytical methods don't work.

DOI
10.48550/arxiv.1304.7008

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We provide a concrete and systematic connection between the statistical physics of the Ising ferromagnet on a Cayley tree, and the study of memory in exponentially expanding spaces. Memory turns out to be a clear signal of the Bethe-Peierls' phase transition, and the average of memory divided by its standard deviation provides a clear signal of the spin-glass' transition temperature. Numerical Monte Carlo simulations are used to make transparent the existence of the two different transition temperatures. The quantities used to spot the phase transitions with Monte Carlo could be useful when studying other systems where analytical methods don't work.
Abstract The quantum Ising model with random couplings and random transverse fields on the Cayley tree is studied by real-space-renormalization in order to construct the whole set of eigenstates. The renormalization rules are analyzed via large deviations. The phase transition between the paramagnetic and the spin-glass many-body-localized phases involves the activated exponent ψ = 1 and the correlation length exponent ν = 1 . The spin-glass-ordered cluster containing N SG spins is found to be extremely sparse with respect to the total number N of spins: its size grows only logarithmically at the critical point N S G c r i t i ∝ ln ⁡ N , and it is sub-extensive N S G ∝ N θ in the finite region of the spin-glass phase where the continuously varying exponent θ remains in the interval 0 < θ < 1 .
Some recent progress in Monte Carlo simulations of spin glasses will be presented. The problem of slow dynamics at low temperatures is partially alleviated by use of the parallel tempering (replica exchange) method. A useful technique to check for equilibration (applicable only for a Gaussian distribution) will be discussed. It will be argued that a finite size scaling analysis of the scaled correlation length of the system is a good approach with which to investigate phase transitions in spin glasses. This method will be used to study two questions: (i) whether there is a phase transition in zero field in the Heisenberg spin glass in three dimensions, and (ii) whether there is phase transition in a magnetic field in an Ising spin glass, also in three dimensions.
We investigate quantum criticality in the transverse-field Ising model on finite Cayley trees and demonstrate that their extensive boundary fundamentally changes the nature of the quantum phase transition compared with the infinite, boundary-free Bethe lattice. Using tree tensor-network simulations for systems with up to N=12286N=12286N=12286 sites, we identify a boundary-driven continuous transition at hxc≈1.76h_x^c\approx1.76hxc​≈1.76, distinct from the mean-field Bethe-lattice transition at hxc≈2.23h_x^c\approx2.23hxc​≈2.23. We characterize this transition through finite-size scaling of the magnetization and magnetic susceptibility and extract a consistent set of critical exponents. By introducing next-nearest-neighbor interactions, we demonstrate the robustness of the extracted critical exponents, providing numerical argument for the Ising boundary universality class on the Cayley tree. Furthermore, we show that the physics of the boundary-free Bethe lattice can be efficiently recovered on a finite Cayley tree by fine-tuning a local boundary field. The resulting \textit{effective Bethe line}, whose existence is tied to the mean-field nature of the bulk transition, effectively screens the extensive boundary and restores both the critical point and the mean-field critical scaling of the infinite Bethe lattice.
We investigate the ferromagnetic-glassy transitions that separate the low-temperature ferromagnetic and spin-glass phases in the temperature-disorder phase diagram of three-dimensional Ising spin-glass models. For this purpose, we consider the cubic-lattice $\ifmmode\pm\else\textpm\fi{}J$ (Edwards-Anderson) Ising model with bond distribution $P(J)=p\ensuremath{\delta}(J\ensuremath{-}1)+(1\ensuremath{-}p)\ensuremath{\delta}(J+1)$, and present a numerical Monte Carlo study of the critical behavior along the line that marks the onset of ferromagnetism. The finite-size scaling analysis of the Monte Carlo data shows that the ferromagnetic-glassy transition line is slightly reentrant. As a consequence, for an interval of the disorder parameter ppp, around $p\ensuremath{\approx}0.77$, the system presents a low-temperature glassy phase, an intermediate ferromagnetic phase, and a high-temperature paramagnetic phase. Along the ferromagnetic-glassy transition line, magnetic correlations show a universal critical behavior with critical exponents $\ensuremath{\nu}=0.96(2)$ and $\ensuremath{\eta}=\ensuremath{-}0.39(2)$. The hyperscaling relation $\ensuremath{\beta}/\ensuremath{\nu}=(1+\ensuremath{\eta})/2$ is satisfied at the transitions, so that $\ensuremath{\beta}/\ensuremath{\nu}=0.305(10)$. This magnetic critical behavior represents a new universality class for ferromagnetic transitions in Ising-type disordered systems. Overlap correlations are apparently not critical and show a smooth behavior across the transition.
Monte Carlo calculations have been performed for an Ising system on a square lattice with nearest-neighbour interactions distributed according to a Gaussian with zero mean. The susceptibility is computed from the time averages of the fluctuating magnetisation and its square, and exhibits the characteristic maximum as a function of temperature. The height and position of this peak are found to depend on the 'observation time' used for the computation of the time averages. This is interpreted as evidence for the absence of a phase transition.