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arXiv:cond-mat/0009007 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Dynamic scaling in diluted systems: Deactivation through thermal dilution

Manuel I. Marqués, Julio A. Gonzalo

Published Apr 18, 2001• 0 citations
arXivHTMLPDF

Abstract

Activated scaling is confirmed to hold in transverse field-induced phase transitions of randomly diluted Ising systems. Quantum Monte Carlo calculations have been made not just at the percolation threshold (pc) but also well below and above it. We follow the evolution of the activated scaling at zero temperature in the phase transition from ferromagnetic to quantum Griffiths phase (p>pc) at the phase boundary (p=pc) and for transitions inside the nonferromagnetic quantum Griffiths phase (p<pc). A novel deactivation phenomenon inside the nonferromagnetic Griffiths-McCoy phase (p<pc) is observed using a thermal (in contrast to random) dilution of the system.

DOI
10.1103/physreve.63.056114

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Activated scaling is confirmed to hold in transverse field-induced phase transitions of randomly diluted Ising systems. Quantum Monte Carlo calculations have been made not just at the percolation threshold (pc) but also well below and above it. We follow the evolution of the activated scaling at zero temperature in the phase transition from ferromagnetic to quantum Griffiths phase (p>pc) at the phase boundary (p=pc) and for transitions inside the nonferromagnetic quantum Griffiths phase (p<pc). A novel deactivation phenomenon inside the nonferromagnetic Griffiths-McCoy phase (p<pc) is observed using a thermal (in contrast to random) dilution of the system.
We study the transverse-field Ising model on a square lattice with bond- and site-dilution at zero temperature by stochastic series expansion quantum Monte Carlo simulations. Tuning the transverse field hhh and the dilution ppp, the quantum phase diagram of both models is explored. Both quantum phase diagrams show long-range order for small hhh and small ppp. The ordered phase of each is separated from the disordered (quantum) Griffiths phase by second-order phase transitions on two critical lines touching at a multi-critical point. Using Binder ratios we locate quantum critical points with high accuracy. The order-parameter critical exponent βββ and the average correlation-length exponent νavν_{\mathrm{av}}νav​ are determined along the critical lines and at the multi-critical points via finite-size scaling. We find three internally consistent sets of critical exponents and compare them with potentially connected universality classes. The quantum Griffiths phase in the vicinity of the phase transition lines is analyzed through the local susceptibility. Our results indicate that activated scaling occurs not only at the percolation transition, but also at the phase transition line for ppp smaller than the percolation threshold pcp_cpc​.
When a quantum many-particle system exists on a randomly diluted lattice, its intrinsic thermal and quantum fluctuations coexist with geometric fluctuations due to percolation. In this paper, we explore how the interplay of these fluctuations influences the phase transition at the percolation threshold. While it is well known that thermal fluctuations generically destroy long-range order on the critical percolation cluster, the effects of quantum fluctuations are more subtle. In diluted quantum magnets with and without dissipation, this leads to novel universality classes for the zero-temperature percolation quantum phase transition. Observables involving dynamical correlations display nonclassical scaling behavior that can nonetheless be determined exactly in two dimensions.
Using Monte Carlo techniques, we study the decay of magnetization in diluted two-dimensional Ising models at and below the critical temperature Tc of the undiluted Ising model, but above the critical temperature of the diluted system. Using damage spreading (or rather damage "healing"), we are able to measure down to much lower final magnetizations (10–9) and to much larger times than previous authors. Nevertheless, we do not yet find the predicted asymptotic behavior in the Griffiths phase T < Tc. But we can at least exclude a stretched exponential decay as found in previous papers, for T < Tc. Finally, we discuss the case T = Tc where a stretched exponential decay can be proven to hold, at least for p < pc. We indeed do see a stretched exponential for p = pc (and T = Tc), but we show that it cannot describe the asymptotic behavior either.
Abstract The ferromagnet to paramagnet phase transition temperature depends on the dilution concentration in the site diluted Ising ferromagnet. Though this model is a bistable system, few studies reported dynamic phase transition (DPT) in diluted Ising ferromagnet. We study dilution-dependent DPT in diluted Ising ferromagnet via Monte Carlo simulation under a time-varying external field tuning the system’s temperature on several system sizes. The nature of the transition is characterized by employing the finite-size scaling study.
A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.