We have performed Monte Carlo studies of the three-dimensional $XY$ model with random uniaxial anisotropy, which is a model for randomly pinned spin-density waves. We study $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ simple cubic lattices using $L$ values in the range 16--64, and with random anisotropy strengths of $D/2J=1$ , 2, 3, 6, and $\ensuremath{\infty}$. There is a well-defined finite-temperature critical point, ${T}_{c}$ , for each of these values of $D/2J$ . We present results for the angle-averaged magnetic structure factor $S(k)$ at ${T}_{c}$ for $L=64$ . We also use finite-size scaling analysis to study scaling functions for the critical behavior of the specific heat, the magnetization, and the longitudinal magnetic susceptibility. Good data collapse of the scaling functions over a wide range of $T$ is seen for $D/2J=6$ and $\ensuremath{\infty}$. For our finite values of $D/2J$ the scaled magnetization function increases with $L$ below ${T}_{c}$ and appears to approach an $L$ -independent limit for large $L$ . This suggests that the system is ferromagnetic below ${T}_{c}$ .We have performed Monte Carlo studies of the three-dimensional $XY$ model with random uniaxial anisotropy, which is a model for randomly pinned spin-density waves. We study $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ simple cubic lattices using $L$ values in the range 16--64, and with random anisotropy strengths of $D/2J=1$, 2, 3, 6, and $\ensuremath{\infty}$. There is a well-defined finite-temperature critical point, ${T}_{c}$, for each of these values of $D/2J$. We present results for the angle-averaged magnetic structure factor $S(k)$ at ${T}_{c}$ for $L=64$. We also use finite-size scaling analysis to study scaling functions for the critical behavior of the specific heat, the magnetization, and the longitudinal magnetic susceptibility. Good data collapse of the scaling functions over a wide range of $T$ is seen for $D/2J=6$ and $\ensuremath{\infty}$. For our finite values of $D/2J$ the scaled magnetization function increases with $L$ below ${T}_{c}$ and appears to approach an $L$-independent limit for large $L$. This suggests that the system is ferromagnetic below ${T}_{c}$.

arXiv:0902.4049Physical Review BJun 24, 20098 citations

Critical Behavior in Random Magnetic Systems

Alfred Nash

Jan 1, 19910 citations

Finite-size scaling on random magnetic structures

Fabio D. A. Aarao Reis

We study the ferromagnetic Ising model on very long strips of random widths using transfer matrix techniques. The width distribution is Gaussian with mean L and rms deviation \ensuremath{\Delta}L, which may be constant for any L (type I) or proportional to L (type II). We calculate the initial susceptibility for strips with 4\ensuremath{\leqslant}L\ensuremath{\leqslant}12, obtaining estimates of the ratio of exponents (\ensuremath{\gamma}/\ensuremath{\nu} ${)}_{\mathrm{L}}$ and the pseudocritical temperature ${\mathrm{T}}^{\mathrm{*}}$ (L,L-1). In strips of type I, those estimates satisfy finite-size scaling, with finite-size corrections increasing with \ensuremath{\Delta}L, and in strips of type II it is satisfied only for very large lengths L. We discuss the influence of nonuniform thicknesses on the magnetism of low-dimensional systems, comparing the finite-size corrections of this problem to those which fit the experimental data of magnetic thin films.We study the ferromagnetic Ising model on very long strips of random widths using transfer matrix techniques. The width distribution is Gaussian with mean L and rms deviation \ensuremath{\Delta}L, which may be constant for any L (type I) or proportional to L (type II). We calculate the initial susceptibility for strips with 4\ensuremath{\leqslant}L\ensuremath{\leqslant}12, obtaining estimates of the ratio of exponents (\ensuremath{\gamma}/\ensuremath{\nu}${)}_{\mathrm{L}}$ and the pseudocritical temperature ${\mathrm{T}}^{\mathrm{*}}$(L,L-1). In strips of type I, those estimates satisfy finite-size scaling, with finite-size corrections increasing with \ensuremath{\Delta}L, and in strips of type II it is satisfied only for very large lengths L. We discuss the influence of nonuniform thicknesses on the magnetism of low-dimensional systems, comparing the finite-size corrections of this problem to those which fit the experimental data of magnetic thin films.

Physical review. B, Condensed matterMay 1, 19976 citations

Finite-Size Scaling at Criticality

WORLD SCIENTIFIC eBooksAug 1, 20000 citations

We have performed Monte Carlo studies of the three-dimensional

XY

model with random uniaxial anisotropy, which is a model for randomly pinned spin-density waves. We study $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ simple cubic lattices using

L

values in the range 16--64, and with random anisotropy strengths of

D/2J=1

, 2, 3, 6, and $\ensuremath{\infty}$. There is a well-defined finite-temperature critical point,

{T}_{c}

, for each of these values of

D/2J

. We present results for the angle-averaged magnetic structure factor

S(k)

{T}_{c}

for

L=64

. We also use finite-size scaling analysis to study scaling functions for the critical behavior of the specific heat, the magnetization, and the longitudinal magnetic susceptibility. Good data collapse of the scaling functions over a wide range of

T

is seen for

D/2J=6

and $\ensuremath{\infty}$. For our finite values of

D/2J

the scaled magnetization function increases with

L

below

{T}_{c}

and appears to approach an

L

-independent limit for large

L

. This suggests that the system is ferromagnetic below

{T}_{c}

{)}_{\mathrm{L}}

and the pseudocritical temperature

{\mathrm{T}}^{\mathrm{*}}

(L,L-1). In strips of type I, those estimates satisfy finite-size scaling, with finite-size corrections increasing with \ensuremath{\Delta}L, and in strips of type II it is satisfied only for very large lengths L. We discuss the influence of nonuniform thicknesses on the magnetism of low-dimensional systems, comparing the finite-size corrections of this problem to those which fit the experimental data of magnetic thin films.