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APS

Global Random-Field Spin-Glass Phase Diagrams in Two and Three Dimensions

Gabriele Migliorini, A. Nihat Berker

Published Mar 1, 1997• 0 citations
OpenAlex

Abstract

An Ising model including field randomness and $\ifmmode\pm\else\textpm\fi{}J$ bond randomness is studied by renormalization-group theory in spatial dimensions $d=2$ and 3. In $d=3,$ with field randomness but no $\ifmmode\pm\else\textpm\fi{}J$ randomness, ferromagnetic and paramagnetic phases occur with no intervening spin-glass phase. Also in $d=3,$ at sufficient $\ifmmode\pm\else\textpm\fi{}J$ randomness, a spin-glass phase occurs, but is replaced by the paramagnetic phase for any nonzero random field, also implying its disappearance for any nonzero uniform field. In $d=2,$ no ferromagnetic phase under random fields and no spin-glass phase occur. Global phase diagrams uniting the random-field and spin-glass problems are evaluated. The strong violation of critical phenomena universality, previously found adjoining random-bond tricriticality in $d=3,$ is seen here in $d=2.$

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An Ising model including field randomness and \ifmmode±\else\textpm\fiJ\ifmmode\pm\else\textpm\fi{}J\ifmmode±\else\textpm\fiJ bond randomness is studied by renormalization-group theory in spatial dimensions d=2d=2d=2 and 3. In d=3,d=3,d=3, with field randomness but no \ifmmode±\else\textpm\fiJ\ifmmode\pm\else\textpm\fi{}J\ifmmode±\else\textpm\fiJ randomness, ferromagnetic and paramagnetic phases occur with no intervening spin-glass phase. Also in d=3,d=3,d=3, at sufficient \ifmmode±\else\textpm\fiJ\ifmmode\pm\else\textpm\fi{}J\ifmmode±\else\textpm\fiJ randomness, a spin-glass phase occurs, but is replaced by the paramagnetic phase for any nonzero random field, also implying its disappearance for any nonzero uniform field. In d=2,d=2,d=2, no ferromagnetic phase under random fields and no spin-glass phase occur. Global phase diagrams uniting the random-field and spin-glass problems are evaluated. The strong violation of critical phenomena universality, previously found adjoining random-bond tricriticality in d=3,d=3,d=3, is seen here in d=2.d=2.d=2.
An Ising spin glass of random bonds with the Gaussian and two-peaked distributions in two- and three-dimensions is investigated with the cumulant expansion and two-step decimation transformations. It is found that the spin glass state exists in two- and three-dimensions and that the ferromagnetic phase shrinks to that of the pure system in two-dimension for the two-peaked distribution. The specific heat exponent at the spin glass transition point is found to be -1.08 and -0.471 for the square and simple cubic lattices, respectively. Also the effect of bond correlations on the spin glass state is considered.
The transverse-field Ising models with random exchange interactions in finite dimensions are investigated by means of a real-space renormalization-group method. The scheme yields the exact values of the critical point and critical exponent $\ensuremath{\nu}$ in one dimension and some previous results in the case of random ferromagnetic interactions are reproduced in two and three dimensions. We apply the scheme to spin glasses in transverse fields in two and three dimensions, which have not been analyzed very extensively. The phase diagrams and the critical exponent $\ensuremath{\nu}$ are obtained and evidence for the existence of an infinite-randomness fixed point in these models is found.
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.
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The four-dimensional $\ifmmode\pm\else\textpm\fi{}J$ random-bond Ising model is studied using ground-state calculations. System sizes up to N=64{N=6}^{4}N=64 spins are considered. Here it is found that the ferromagnetic--spin-glass transition occurs at a critical concentration pc=0.28(1){p}_{c}=0.28(1)pc​=0.28(1) of the antiferromagnetic bonds, which is comparable to values found previously by high-temperature series expansions. The transition is characterized by a correlation-length exponent $\ensuremath{\nu}=1.0(1)$ and an order-parameter exponent $\ensuremath{\beta}=0.4(1).$ Thus, this transition is in a different universality class from four-dimensional bond percolation, where $\ensuremath{\nu}=0.678(50)$ and $\ensuremath{\beta}=0.639(20).$