The inhomogeneous nature of critical phenomena in semi-infinite systems is studied by calculating the full crossover for both the two-point correlation function and the surface susceptibility ${\ensuremath{\chi}}_{11}$(z) as a function of the distance z from the surface at z=0 for a semi-infinite system near the special transition. We introduce z-dependent counterterms to derive the renormalization-group (RG) equation describing this full crossover. The solution to the RG equation in the one-loop approximation is obtained explicitly and in the long-distance regime is shown to approach bulk behavior exponentially fast. In contrast to the usual approach, our RG equation also contains all information about the short-distance (near-surface) behavior of the theory and its crossover to the bulk. Applications of our method to the descriptions of crossovers for other problems of critical phenomena with several competing lengths are also discussed.The inhomogeneous nature of critical phenomena in semi-infinite systems is studied by calculating the full crossover for both the two-point correlation function and the surface susceptibility ${\ensuremath{\chi}}_{11}$(z) as a function of the distance z from the surface at z=0 for a semi-infinite system near the special transition. We introduce z-dependent counterterms to derive the renormalization-group (RG) equation describing this full crossover. The solution to the RG equation in the one-loop approximation is obtained explicitly and in the long-distance regime is shown to approach bulk behavior exponentially fast. In contrast to the usual approach, our RG equation also contains all information about the short-distance (near-surface) behavior of the theory and its crossover to the bulk. Applications of our method to the descriptions of crossovers for other problems of critical phenomena with several competing lengths are also discussed.

Physical review. B, Condensed matterSep 1, 19873 citations

Critical Temperatures of Finite Samples at Finite Observation Times

Olena Vedmedenko, N. Mikuszeit, T. Stapelfeldt +4 more

Bulletin of the American Physical SocietyMar 15, 20100 citations

High-temperature series for the bond-diluted Ising model in 3, 4, and 5 dimensions

Meik Hellmund, Wolfhard Janke E

In order to study the influence of quenched disorder on second-order phase transitions, high-temperature series expansions of the susceptibility and the free energy are obtained for the quenched bond-diluted Ising model in $d=3--5$ dimensions. They are analyzed using different extrapolation methods tailored to the expected singularity behaviors. In $d=4$ and 5 dimensions we confirm that the critical behavior is governed by the pure fixed point up to dilutions near the geometric bond percolation threshold. The existence and form of logarithmic corrections for the pure Ising model in $d=4$ are confirmed, and our results for the critical behavior of the diluted system are in agreement with the type of singularity predicted by renormalization group considerations. In three dimensions we find large crossover effects between the pure Ising, percolation, and random fixed points. We estimate the critical exponent of the susceptibility to be $\ensuremath{\gamma}=1.305(5)$ at the random fixed point.In order to study the influence of quenched disorder on second-order phase transitions, high-temperature series expansions of the susceptibility and the free energy are obtained for the quenched bond-diluted Ising model in $d=3--5$ dimensions. They are analyzed using different extrapolation methods tailored to the expected singularity behaviors. In $d=4$ and 5 dimensions we confirm that the critical behavior is governed by the pure fixed point up to dilutions near the geometric bond percolation threshold. The existence and form of logarithmic corrections for the pure Ising model in $d=4$ are confirmed, and our results for the critical behavior of the diluted system are in agreement with the type of singularity predicted by renormalization group considerations. In three dimensions we find large crossover effects between the pure Ising, percolation, and random fixed points. We estimate the critical exponent of the susceptibility to be $\ensuremath{\gamma}=1.305(5)$ at the random fixed point.

arXiv:cond-mat/0606320Physical Review BOct 12, 200613 citations

Criticality in Equilibrium Systems

IMPERIAL COLLEGE PRESS eBooksNov 1, 20100 citations

Diffusion Constants near the Critical Point for Time-Dependent Ising Models. I

Kyozi Kawasaki

The diffusion constant of spins in ferromagnets or of molecules in binary mixtures near the critical point is discussed employing a time-dependent Ising model in which spin interactions are replaced by certain temperature-dependent transition probabilities of spin exchange. The spin diffusion constant is calculated with the single approximation of replacing a reduced spin distribution function by its value for local equilibrium with a given inhomogeneous spin density. The behavior of the diffusion constant near the critical point is dominated by a factor ${\ensuremath{\chi}}^{\ensuremath{-}1}$, where $\ensuremath{\chi}$ is the magnetic susceptibility. This problem is also studied with the use of the Bethe lattice. The effects of surrounding spins on the transition probability for spin exchange are found to be essential for obtaining the critical slowing-down near the critical point. In view of this, Koci\ifmmode \acute{n}\else \'{n}\fi{}ski's calculation of the spin diffusion constant is critically discussed.The diffusion constant of spins in ferromagnets or of molecules in binary mixtures near the critical point is discussed employing a time-dependent Ising model in which spin interactions are replaced by certain temperature-dependent transition probabilities of spin exchange. The spin diffusion constant is calculated with the single approximation of replacing a reduced spin distribution function by its value for local equilibrium with a given inhomogeneous spin density. The behavior of the diffusion constant near the critical point is dominated by a factor ${\ensuremath{\chi}}^{\ensuremath{-}1}$, where $\ensuremath{\chi}$ is the magnetic susceptibility. This problem is also studied with the use of the Bethe lattice. The effects of surrounding spins on the transition probability for spin exchange are found to be essential for obtaining the critical slowing-down near the critical point. In view of this, Koci\ifmmode \acute{n}\else \'{n}\fi{}ski's calculation of the spin diffusion constant is critically discussed.

Physical ReviewMay 6, 1966616 citations

d=3--5

dimensions. They are analyzed using different extrapolation methods tailored to the expected singularity behaviors. In

d=4

and 5 dimensions we confirm that the critical behavior is governed by the pure fixed point up to dilutions near the geometric bond percolation threshold. The existence and form of logarithmic corrections for the pure Ising model in

d=4

are confirmed, and our results for the critical behavior of the diluted system are in agreement with the type of singularity predicted by renormalization group considerations. In three dimensions we find large crossover effects between the pure Ising, percolation, and random fixed points. We estimate the critical exponent of the susceptibility to be $\ensuremath{\gamma}=1.305(5)$ at the random fixed point.