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Annalen der Physik

Critical and off-critical singularities in disordered quantum magnets

Heiko Rieger

Published Nov 1, 1998• 2 citations
Publisher

Abstract

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.

DOI
10.1002/(sici)1521-3889(199811)7:5/6<564::aid-andp564>3.0.co;2-u

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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.
Abstract 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.
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 study the effect of spatial correlations on quenched disorder in random quantum magnets at and near a quantum critical point. In random transverse-field Ising systems disorder correlations that decay algebraically with an exponent $\ensuremath{\rho}$ change the universality class of the transition for small enough $\ensuremath{\rho}$, and off-critical Griffiths-McCoy singularities are enhanced. We present exact results for 1D utilizing a mapping to fractional Brownian motion and generalize the predictions for the critical exponents and the generalized dynamical exponent in the Griffiths phase to $d\ensuremath{\ge}2$.
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.
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 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.