The critical behavior of random-field systems is characterized by exponentially long relaxation times. They may differ by orders of magnitude and slow modes have to be considered quenched with respect to the faster ones. This requires a drastic modification of the renormalization process in which the degrees of freedom are integrated out. A new replica-space renormalization procedure, to carry out the coarse-graining of the time intervals, is presented. We relate by recursion-relations the effective reduced dimensions of consecutive time scales. Their stable fixed-points yield the apparent dimensionalities for the longest and shortest time scales, which are insensitive to the exact behavior on intermediate scales. For Ising systems we find that d = 2 is the lower critical dimension in both regimes. The thermal exponents for d = 3, in the short-time observable regime, are related to those of the pure system in d = 2. This is consistent with the observations in field-cooled random antiferromagnets of the correlation length exponent nu approx. = 1 (by neutron scattering) and of the symmetric logarithmic divergence in the specific-heat (by linear birefringence). 17 refs., 3 figs.The critical behavior of random-field systems is characterized by exponentially long relaxation times. They may differ by orders of magnitude and slow modes have to be considered quenched with respect to the faster ones. This requires a drastic modification of the renormalization process in which the degrees of freedom are integrated out. A new replica-space renormalization procedure, to carry out the coarse-graining of the time intervals, is presented. We relate by recursion-relations the effective reduced dimensions of consecutive time scales. Their stable fixed-points yield the apparent dimensionalities for the longest and shortest time scales, which are insensitive to the exact behavior on intermediate scales. For Ising systems we find that d = 2 is the lower critical dimension in both regimes. The thermal exponents for d = 3, in the short-time observable regime, are related to those of the pure system in d = 2. This is consistent with the observations in field-cooled random antiferromagnets of the correlation length exponent nu approx. = 1 (by neutron scattering) and of the symmetric logarithmic divergence in the specific-heat (by linear birefringence). 17 refs., 3 figs.

Jan 1, 19850 citations

Dynamics of the Ising Model near the Critical Point

Masuo Suzuki, 英雄猪狩

Jul 20, 19680 citations

Nonperturbative Critical Behavior of Random-Field Systems

Yonathan Shapir

The thermal critical properties of random-field systems in the observable, linear, local-response regime are considered. A consistent nonperturbative approach yields an effective reduced dimension $\overline{d}=\frac{[d+\frac{1}{\ensuremath{\nu}(\overline{d})}]}{2}$ for the thermal exponents if $\ensuremath{\alpha}(\overline{d})<~0$. The consequences for Ising systems are particularly reliable at the upper and lower critical dimensions ( ${d}_{\mathrm{u}.\mathrm{c}.\mathrm{d}.}=6$ and ${d}_{\mathrm{l}.\mathrm{c}.\mathrm{d}.}=2$ , respectively) as well as at $d=3$ where $\ensuremath{\alpha}=0$ (logarithmic divergence). The results are in agreement with measurements on random antiferromagnets of the specific heat (by linear birefringence) and the correlation length (by neutron scattering).The thermal critical properties of random-field systems in the observable, linear, local-response regime are considered. A consistent nonperturbative approach yields an effective reduced dimension $\overline{d}=\frac{[d+\frac{1}{\ensuremath{\nu}(\overline{d})}]}{2}$ for the thermal exponents if $\ensuremath{\alpha}(\overline{d})<~0$. The consequences for Ising systems are particularly reliable at the upper and lower critical dimensions (${d}_{\mathrm{u}.\mathrm{c}.\mathrm{d}.}=6$ and ${d}_{\mathrm{l}.\mathrm{c}.\mathrm{d}.}=2$, respectively) as well as at $d=3$ where $\ensuremath{\alpha}=0$ (logarithmic divergence). The results are in agreement with measurements on random antiferromagnets of the specific heat (by linear birefringence) and the correlation length (by neutron scattering).

Physical Review LettersJan 14, 198544 citations

Self-similar replication approach to random-fields systems

Yonathan Shapir

Very long relaxation times are present in random-fields systems near their critical points. As the correlation length grows, already above the transition it is necessary to differentiate between the short-time (local) regime and the long-time (global) one. A self-similar replication procedure is proposed in order to separate iteratively between slow and fast modes. Recursion relations among effective exponents on successive time scales are obtained and apparent reduced dimensions, in both regimes, are deduced. For the short time behaviour of Ising systems we the authors find a dimensional reduction by one, both in d=3 and in d=2 (the lower critical dimension). The results are in agreement with the measurements of the specific heat and the correlation length performed on random antiferromagnets by linear birefringence and neutron scattering respectively.Very long relaxation times are present in random-fields systems near their critical points. As the correlation length grows, already above the transition it is necessary to differentiate between the short-time (local) regime and the long-time (global) one. A self-similar replication procedure is proposed in order to separate iteratively between slow and fast modes. Recursion relations among effective exponents on successive time scales are obtained and apparent reduced dimensions, in both regimes, are deduced. For the short time behaviour of Ising systems we the authors find a dimensional reduction by one, both in d=3 and in d=2 (the lower critical dimension). The results are in agreement with the measurements of the specific heat and the correlation length performed on random antiferromagnets by linear birefringence and neutron scattering respectively.

Journal of Physics C Solid State PhysicsOct 30, 198414 citations

{d}_{\mathrm{u}.\mathrm{c}.\mathrm{d}.}=6

and

{d}_{\mathrm{l}.\mathrm{c}.\mathrm{d}.}=2

, respectively) as well as at

d=3

where $\ensuremath{\alpha}=0$ (logarithmic divergence). The results are in agreement with measurements on random antiferromagnets of the specific heat (by linear birefringence) and the correlation length (by neutron scattering).