Logarithmic corrections in the two-dimensional Ising model in a random surface field

Pleimling, Michel; Bagaméry, Farkas Á.; Turban, L.; Iglói, Ferenc

doi:10.1088/0305-4470/37/37/003

Cited by 7 publications

(9 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then one calculates β 1 = 0.499 (2). It is in good agreement with the surface exponent β s = 1/2 for the free surface [56]. Now, the exponent β 0 /ν is estimated to be 0.374.…”

Section: A Magnetizationsupporting

confidence: 67%

“…For a large s, careful analysis shows that the power-law behavior is not perfect [55]. In the equilibrium state, one may show that the surface exponent β s of the disordered surface remains 1/2, but with a logarithmic correction to scaling [56]. Therefore, we fit the timedependent magnetization at x = 0.5 with a logarithmic correction to scaling, i.e., M(t) = c 1 t −α 1 /(1 + c 2 ln(t)) 1/2 , and derive α 1 = 0.231, consistent with β 1 /νz = 0.231(1) for the free surface.…”

Section: A Magnetizationmentioning

confidence: 99%

See 1 more Smart Citation

Non-equilibrium critical dynamics with domain interface

Zhou

Zheng

2007

Europhys. Lett.

View full text Add to dashboard Cite

With Monte Carlo simulations, we investigate the relaxation dynamics with a domain wall for magnetic systems at the critical temperature. The dynamic scaling behavior is carefully analyzed, and a dynamic roughening process is observed. For comparison, similar analysis is applied to the relaxation dynamics with a free or disordered surface.In the past years, much effort of physicists has been devoted to the understanding of nonequilibrium dynamic processes. Phase ordering dynamics, spin glass dynamics, structural glass dynamics and interface growth etc are important examples. Since the pioneer work by Janssen et al [1], the universal dynamic scaling form in critical dynamics has been explored up to the macroscopic short-time regime [1-10], when the system is still far from equilibrium.Although the spatial correlation length is still short in the beginning of the time evolution, the short-time dynamic scaling form is induced by the divergent correlating time around a continuous phase transition. Based on the short-time dynamic scaling form, new methods for the determination of both dynamic and static critical exponents as well as the critical temperature have been developed [4,5,[7][8][9][10][11]. Since the measurements are carried out in the short-time regime, one does not suffer from critical slowing down.In understanding the dynamic scaling form far from equilibrium, we should keep in mind that it holds after a time scale t mic , which is sufficiently long in the microscopic sense, but still short in the macroscopic sense. More importantly, the macroscopic initial condition should be taken into account in the dynamic scaling form [1,9,12]. For the dynamic relaxation starting from an ordered state, i.e., a state with an initial magnetization m 0 = 1, for example, the magnetization decays by a power law [8,9,12]. If m 0 is smaller but close to 1, there emerge corrections to scaling. For the dynamic relaxation starting from a random state, i.e., a state without spatial correlations and with a small m 0 , however, the magnetization does not decay, and rather shows an initial increase in the macroscopic short-time regime. An independent critical exponent x 0 must be introduced to describe the scaling dimension of the initial magnetization [1,6,9,10]. If m 0 = 0, the magnetization naturally remains zero during the dynamic evolution, but x 0 is still needed to describe the auto-correlation function etc. This critical exponent also explains the power-law decay of the remanent magnetization in spin glasses [2,13,14]. On the other hand, the short-time dynamic scaling form is universal, in the sense that it does not depend on the microscopic details of the dynamic system, such as the lattice types, interactions, and updating schemes etc. Up to now, the dynamic relaxation with the ordered and random initial states has been systematically investigated.Recent progress in the non-equilibrium critical dynamics and its applications includes, for

show abstract

“…Then one calculates β 1 = 0.499 (2). It is in good agreement with the surface exponent β s = 1/2 for the free surface [56]. Now, the exponent β 0 /ν is estimated to be 0.374.…”

Section: A Magnetizationsupporting

confidence: 67%

Section: A Magnetizationmentioning

confidence: 99%

Non-equilibrium critical dynamics with domain interface

Zhou

Zheng

2007

Europhys. Lett.

View full text Add to dashboard Cite

show abstract

“…In the two-dimensional semi-infinite Ising model the perturbation caused by the random surface field is marginal so that the criterion does not yield a definite answer. This case was studied subsequently in [242,243,244,245] where it was shown that the surface critical behaviour of the 2d Ising model is described by Ising critical exponents with logarithmic corrections to scaling. For the special transition point, the Harris like criterion of [227] indicates that random surface fields are relevant in dimensions d ≤ 4.…”

Section: Semi-infinite Systems With Surface Imperfectionsmentioning

confidence: 99%

Critical phenomena at perfect and non-perfect surfaces

Pleimling¹

2004

J. Phys. A: Math. Gen.

Self Cite

100

135

View full text Add to dashboard Cite

Abstract. In the past perfect surfaces have been shown to yield a local critical behaviour that differs from the bulk critical behaviour. On the other hand surface defects, whether they are of natural origin or created artificially, are known to modify local quantities. It is therefore important to clarify whether these defects are relevant or irrelevant for the surface critical behaviour.The purpose of this review is two-fold. In the first part we summarise some of the important results on surface criticality at perfect surfaces. Special attention is thereby paid to new developments as for example the study of surface critical behaviour in systems with competing interactions or of surface critical dynamics. In the second part the effect of surface defects (presence of edges, steps, quenched randomness, lines of adatoms, regular geometric patterns) on local critical behaviour in semi-infinite systems and in thin films is discussed in detail. Whereas most of the defects commonly encountered are shown to be irrelevant, some notable exceptions are highlighted. It is shown furthermore that under certain circumstances non-universal local critical behaviour may be observed at surfaces.

show abstract

“…This type of star-like geometry has already been investigated for different type of problems: for the classical and quantum Ising models [26][27][28][29][30][31][32], wetting [33], self-avoiding random walks, percolation [32] etc. For the Ising model, the limit M → 0 corresponds to the problem of random boundary field [26,27,34]. The multiple junction could be a simplified model for describing the behavior of the contact process on complex networks composed of long, one-dimensional segments and rarely located junction points, in the vicinity of junctions.…”

Section: Introductionmentioning

confidence: 99%

Mixed-order phase transition of the contact process near multiple junctions

Juhász

Iglói

2017

Phys. Rev. E

View full text Add to dashboard Cite

We have studied the phase transition of the contact process near a multiple junction of M semi-infinite chains by Monte Carlo simulations. As opposed to the continuous transitions of the translationally invariant (M=2) and semi-infinite (M=1) system, the local order parameter is found to be discontinuous for M>2. Furthermore, the temporal correlation length diverges algebraically as the critical point is approached, but with different exponents on the two sides of the transition. In the active phase, the estimate is compatible with the bulk value, while in the inactive phase it exceeds the bulk value and increases with M. The unusual local critical behavior is explained by a scaling theory with an irrelevant variable, which becomes dangerous in the inactive phase. Quenched spatial disorder is found to make the transition continuous in agreement with earlier renormalization group results.

show abstract

Logarithmic corrections in the two-dimensional Ising model in a random surface field

Cited by 7 publications

References 30 publications

Non-equilibrium critical dynamics with domain interface

Non-equilibrium critical dynamics with domain interface

Critical phenomena at perfect and non-perfect surfaces

Mixed-order phase transition of the contact process near multiple junctions

Contact Info

Product

Resources

About