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Journal of the Physical Society of Japan

Geometry-Induced Phase Transition in the ±JIsing Model

Hidetoshi Nishimori

Published Oct 15, 1986• 86 citations
Publisher

Abstract

We present an argument for the existence of a geometry-induced phase transition in the ± J Ising model on an arbitrary lattice ( d ≥2) with asymmetric probability of ferromagnetic and antiferromagnetic bonds. This transition manifests itself as a vertical phase boundary separating, possibly, a ferromagnetic and a non-ferromagnetic (paramagnetic or spin glass) phases.

DOI
10.1143/jpsj.55.3305

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Journal of the Physical Society of Japan•Apr 15, 1998•27 citations
We present an argument for the existence of a geometry-induced phase transition in the ± J Ising model on an arbitrary lattice ( d ≥2) with asymmetric probability of ferromagnetic and antiferromagnetic bonds. This transition manifests itself as a vertical phase boundary separating, possibly, a ferromagnetic and a non-ferromagnetic (paramagnetic or spin glass) phases.
The spin-1/2 quantum Heisenberg spin-glass system is studied in all spatial dimensions d by renormalization-group theory. Strongly asymmetric phase diagrams in temperature and antiferromagnetic bond probability p are obtained in dimensions d>or=3. The asymmetry at high temperatures approaching the pure ferromagnetic and antiferromagnetic systems disappears as d is increased. However, the asymmetry at low but finite temperatures remains in all dimensions, with the antiferromagnetic phase receding from the ferromagnetic phase. A finite-temperature second-order phase boundary directly between the ferromagnetic and antiferromagnetic phases occurs in d>or=6, resulting in a new multicritical point. In d=3, 4, 5, a paramagnetic phase reaching zero temperature intervenes asymmetrically between the ferromagnetic and reentrant antiferromagnetic phases. There is no spin-glass phase in any dimension.
The +/-J Ising spin glass [probabilities p and (1-p) associated with ferromagnetic and antiferromagnetic couplings, respectively] is studied by applying a real-space renormalization-group technique on a hierarchical lattice that approaches the square lattice. Within such a procedure, there is no spin-glass phase and only two finite-temperature phases are found, namely, the paramagnetic and ferromagnetic ones. In spite of a reasonably small computational effort, an accurate paramagnetic-ferromagnetic boundary is presented: the estimate for the slope at p=1 is in very good agreement with the well-known exact result, whereas the coordinates of the Nishimori point are determined within a high precision. Below the Nishimori point, such a boundary is not strictly vertical-contrary to the usual belief-in such a way that a small reentrance is found at low temperatures.
By using frustration-preserving hard-spin mean-field theory, we investigated the phase-transition dynamics in the three-dimensional field-free ± J Ising spin-glass model. As the temperature T is decreased from paramagnetic phase at high temperatures, with a rate ω=-dT/dt in time t, the critical temperature depends on the cooling rate through a clear power law ω^{a}. With increasing antiferromagnetic bond fraction p, the exponent a increases for the transition into the ferromagnetic case for p<p_{c} and decreases for the transition into the spin-glass phase for p>p_{c}, signaling the ferromagnetic - spin-glass phase transition at p_{c}≈0.22. The relaxation time is also investigated in the adiabatic case ω=0 and the dynamic exponent zν is found to increase with increasing p.
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.
The ferromagnetic transition of ± J -Ising spin-glass model on square lattice is studied by using Monte Carlo simulation. At ferromagnetic-bond density p =0.96 where the system shows ferromagnetic transition, the spontaneous magnetization, the ratio < m 4 >/ < m 2 > 2 and the non-equilibrium relaxation (NER) are studied. The transition is estimated to be at K c =0.5100 (1). The analysis by NER gives the most reliable estimation with the least CPU time for simulations among the three methods. The value of magnetization exponent β is estimated to be 0.130(10) which coincides with that of the purely ferromagnetic Ising model ( p =1). The dynamical critical exponent z is estimated to be 2.418(23) from the NER analysis which is different from that of the pure model. The size of simulated lattices is up to 2001×2000, and the number of totally updated Ising spins is 5.5×10 15 .