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arXiv:cond-mat/9512162 · Physical review. B, Condensed matter

Griffiths singularities in the disordered phase of a quantum Ising spin glass

Heiko Rieger, A. P. Young

Published Aug 1, 1996• 137 citations
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Abstract

We study a model for a quantum Ising spin glass in two space dimensions by Monte Carlo simulations. In the disordered phase at T=0, rare strongly correlated regions give rise to strong Griffiths singularities, as originally found by McCoy for a one-dimensional model. We find that there are power law distributions of the local susceptibility and local nonlinear susceptibility, which are characterized by a smoothly varying dynamical exponent z. Over a range of the disordered phase near the quantum transition, the local nonlinear susceptibility diverges. The local susceptibility does not diverge in the disordered phase but does diverge at the critical point. Approaching the critical point from the disordered phase, the limiting value of z seems to equal its value precisely at criticality, even though the physics of these two cases seems rather different. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.3328

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We study a model for a quantum Ising spin glass in two space dimensions by Monte Carlo simulations. In the disordered phase at T=0, rare strongly correlated regions give rise to strong Griffiths singularities, as originally found by McCoy for a one-dimensional model. We find that there are power law distributions of the local susceptibility and local nonlinear susceptibility, which are characterized by a smoothly varying dynamical exponent z. Over a range of the disordered phase near the quantum transition, the local nonlinear susceptibility diverges. The local susceptibility does not diverge in the disordered phase but does diverge at the critical point. Approaching the critical point from the disordered phase, the limiting value of z seems to equal its value precisely at criticality, even though the physics of these two cases seems rather different. \textcopyright{} 1996 The American Physical Society.
We report a Monte Carlo study of the effects of fluctuations in the bond distribution of Ising spin glasses in a transverse magnetic field, in the paramagnetic phase in the T\ensuremath{\rightarrow}0 limit. Rare, strong fluctuations give rise to Griffiths singularities, which can dominate the zero-temperature behavior of these quantum systems, as originally demonstrated by McCoy for one-dimensional (d=1) systems. Our simulations are done on a square lattice in d=2 and a cubic lattice in d=3, for a Gaussian distribution of nearest neighbor (only) bonds. In d=2, where the linear susceptibility was found to diverge at the critical transverse field strength ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}$ for the order-disorder phase transition at T=0, the average nonlinear susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathrm{nl}}$ diverges in the paramagnetic phase for \ensuremath{\Gamma} well above ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}$, as is also demonstrated in the accompanying paper by Rieger and Young. In d=3, the linear susceptibility remains finite at ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}$, and while Griffiths singularity effects are certainly observable in the paramagnetic phase, the nonlinear susceptibility appears to diverge only rather close to ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{c}}$. These results show that Griffiths singularities remain persistent in dimensions above one (where they are known to be strong), though their magnitude decreases monotonically with increasing dimensionality (there being no Griffiths singularities in the limit of infinite dimensionality). \textcopyright{} 1996 The American Physical Society.
We study the ±J transverse-field Ising spin-glass model at zero temperature on d-dimensional hypercubic lattices and in the Sherrington-Kirkpatrick (SK) model, by series expansions around the strong-field limit. In the SK model and in high dimensions our calculated critical properties are in excellent agreement with the exact mean-field results, surprisingly even down to dimension d=6, which is below the upper critical dimension of d=8. In contrast, at lower dimensions we find a rich singular behavior consisting of critical and Griffiths-McCoy singularities. The divergence of the equal-time structure factor allows us to locate the critical coupling where the correlation length diverges, implying the onset of a thermodynamic phase transition. We find that the spin-glass susceptibility as well as various power moments of the local susceptibility become singular in the paramagnetic phase before the critical point. Griffiths-McCoy singularities are very strong in two dimensions but decrease rapidly as the dimension increases. We present evidence that high enough powers of the local susceptibility may become singular at the pure-system critical point.
We study the quantum phase transition in the two-dimensional random Ising model in a transverse field by Monte Carlo simulations. We find results similar to those known analytically in one dimension. At the critical point the dynamical exponent is infinite and the typical correlation function decays with a stretched exponential dependence on distance. Away from the critical point there are Griffiths-McCoy singularities, characterized by a single continuously varying exponent, ${z}^{\ensuremath{'}}$, which diverges at the critical point, as in one dimension. Consequently, the zero temperature susceptibility diverges for a range of parameters about the transition.
We study disordered systems at and near quantum phase transitions, which are transitions at zero temperature that are driven by quantum instead of thermal fluctuations. Special emphasis is put on the so-called Griffiths-McCoy regions surrounding these transitions in magnetic systems, where various susceptibilities are found to be divergent for T → 0 without the system being critical. We present a phenomenological picture for uncorrelated and weakly or strongly correlated disorder and check it for various models. In higher dimensions the phenomena seem to be connected to a percolation transition.
For the two dimensional random bond disordered Ising ferromagnet, we measured bulk data of the magnetic susceptibility (χχχ) and correlation length (ξξξ) up to ξ≃536ξ\simeq 536ξ≃536, with the use of a novel finite size scaling Monte Carlo technique. Our data are exclusively consistent with normal power-law critical behaviors with only one singular point at criticality, disproving the existence of Griffiths singularity even in an extremely deep scaling region. The critical exponents of χχχ and ξξξ increase continuously with the strength of disorder.
We employ scaling arguments and optimal fluctuation theory to establish a general relation between quantum Griffiths singularities and the Harris criterion for quantum phase transitions in disordered systems. If a clean critical point violates the Harris criterion, it is destabilized by weak disorder. At the same time, the Griffiths dynamical exponent z' diverges upon approaching the transition, suggesting unconventional critical behavior. In contrast, if the Harris criterion is fulfilled, power-law Griffiths singularities can coexist with clean critical behavior, but z' saturates at a finite value. We present applications of our theory to a variety of systems including quantum spin chains, classical reaction-diffusion systems and metallic magnets, and we discuss modifications for transitions above the upper critical dimension. Based on these results we propose a unified classification of phase transitions in disordered systems.