We study the long-distance behavior of the $O(N)$ model in the presence of random fields and random anisotropies correlated as $\ensuremath{\sim}1∕{x}^{d\ensuremath{-}\ensuremath{\sigma}}$ for large separation $x$ using the functional renormalization group. We compute the fixed points and analyze their regions of stability within a double $\ensuremath{\epsilon}=d\ensuremath{-}4$ and $\ensuremath{\sigma}$ expansion. We find that the long-range disorder correlator remains analytic but generates short-range disorder whose correlator develops the usual cusp. This allows us to obtain the phase diagrams in $(d,\ensuremath{\sigma},N)$ parameter space and compute the critical exponents to first order in $\ensuremath{\epsilon}$ and $\ensuremath{\sigma}$. We show that the standard renormalization group methods with a finite number of couplings used in previous studies of systems with long-range correlated random fields fail to capture all critical properties. We argue that our results may be relevant to the behavior of $^{3}\mathrm{He}\text{\ensuremath{-}}\mathrm{A}$ in aerogel.We study the long-distance behavior of the $O(N)$ model in the presence of random fields and random anisotropies correlated as $\ensuremath{\sim}1∕{x}^{d\ensuremath{-}\ensuremath{\sigma}}$ for large separation $x$ using the functional renormalization group. We compute the fixed points and analyze their regions of stability within a double $\ensuremath{\epsilon}=d\ensuremath{-}4$ and $\ensuremath{\sigma}$ expansion. We find that the long-range disorder correlator remains analytic but generates short-range disorder whose correlator develops the usual cusp. This allows us to obtain the phase diagrams in $(d,\ensuremath{\sigma},N)$ parameter space and compute the critical exponents to first order in $\ensuremath{\epsilon}$ and $\ensuremath{\sigma}$. We show that the standard renormalization group methods with a finite number of couplings used in previous studies of systems with long-range correlated random fields fail to capture all critical properties. We argue that our results may be relevant to the behavior of $^{3}\mathrm{He}\text{\ensuremath{-}}\mathrm{A}$ in aerogel.

arXiv:cond-mat/0701256Physical Review BMay 21, 200729 citations

Crossover between a short-range and a long-range Ising model

Taro Nakada, Per Arne Rikvold, Takashi Mori +2 more

Recently, it has been found that an effective long-range interaction is realized among local bistable variables (spins) in systems where the elastic interaction causes ordering of the spins. In such systems, we generally expect both long-range and short-range interactions to exist. In the short-range Ising model, the correlation length diverges at the critical point. In contrast, in the long-range interacting model the spin configuration is always uniform and the correlation length is zero. As long as a system has nonzero long-range interactions, it shows criticality in the mean-field universality class, and the spin configuration is uniform beyond a certain scale. Here we study the crossover from the pure short-range interacting model to the long-range interacting model. We investigate the infinite-range model (Husimi-Temperley model) as a prototype of this competition, and we study how the critical temperature changes as a function of the strength of the long-range interaction. This model can also be interpreted as an approximation for the Ising model on a small-world network. We derive a formula for the critical temperature as a function of the strength of the long-range interaction. We also propose a finite-size scaling form for the spin correlation length at the critical point, which is finite as long as the long-range interaction is included, though it diverges in the limit of the pure short-range model. These properties are confirmed by extensive Monte Carlo simulations.Recently, it has been found that an effective long-range interaction is realized among local bistable variables (spins) in systems where the elastic interaction causes ordering of the spins. In such systems, we generally expect both long-range and short-range interactions to exist. In the short-range Ising model, the correlation length diverges at the critical point. In contrast, in the long-range interacting model the spin configuration is always uniform and the correlation length is zero. As long as a system has nonzero long-range interactions, it shows criticality in the mean-field universality class, and the spin configuration is uniform beyond a certain scale. Here we study the crossover from the pure short-range interacting model to the long-range interacting model. We investigate the infinite-range model (Husimi-Temperley model) as a prototype of this competition, and we study how the critical temperature changes as a function of the strength of the long-range interaction. This model can also be interpreted as an approximation for the Ising model on a small-world network. We derive a formula for the critical temperature as a function of the strength of the long-range interaction. We also propose a finite-size scaling form for the spin correlation length at the critical point, which is finite as long as the long-range interaction is included, though it diverges in the limit of the pure short-range model. These properties are confirmed by extensive Monte Carlo simulations.

arXiv:1104.1834Physical Review BAug 10, 201134 citations

Experimental investigation of the correlation function near the critical point

Mikhail A. Anisimov, A. M. Evtyushenkov, Yu. F. Kiyachenko +1 more

Sep 1, 19741 citation

Critical properties of S>12 Ising chains with long-range interactions

M. Barati, Abdolali Ramazani

The phase diagram and critical exponent of the correlation length for an Ising chain with a long-range interaction in the form of ${1/r}^{1+\ensuremath{\sigma}}$ for $S=1/2,1,3/2,2$ is calculated by the finite-range scaling technique. The spin dependence of the critical temperature and critical exponent $\ensuremath{\nu}$ in the classical and nonclassical regions is also studied. It is observed that, with a good approximation, the critical exponent of the correlation length is independent of the magnitude of spin in both the classical and nonclassical regions.The phase diagram and critical exponent of the correlation length for an Ising chain with a long-range interaction in the form of ${1/r}^{1+\ensuremath{\sigma}}$ for $S=1/2,1,3/2,2$ is calculated by the finite-range scaling technique. The spin dependence of the critical temperature and critical exponent $\ensuremath{\nu}$ in the classical and nonclassical regions is also studied. It is observed that, with a good approximation, the critical exponent of the correlation length is independent of the magnitude of spin in both the classical and nonclassical regions.

Physical review. B, Condensed matterNov 1, 200010 citations

We study the long-distance behavior of the

O(N)

model in the presence of random fields and random anisotropies correlated as $\ensuremath{\sim}1∕{x}^{d\ensuremath{-}\ensuremath{\sigma}}$ for large separation

x

using the functional renormalization group. We compute the fixed points and analyze their regions of stability within a double $\ensuremath{\epsilon}=d\ensuremath{-}4$ and $\ensuremath{\sigma}$ expansion. We find that the long-range disorder correlator remains analytic but generates short-range disorder whose correlator develops the usual cusp. This allows us to obtain the phase diagrams in $(d,\ensuremath{\sigma},N)$ parameter space and compute the critical exponents to first order in $\ensuremath{\epsilon}$ and $\ensuremath{\sigma}$. We show that the standard renormalization group methods with a finite number of couplings used in previous studies of systems with long-range correlated random fields fail to capture all critical properties. We argue that our results may be relevant to the behavior of $^{3}\mathrm{He}\text{\ensuremath{-}}\mathrm{A}$ in aerogel.

The phase diagram and critical exponent of the correlation length for an Ising chain with a long-range interaction in the form of ${1/r}^{1+\ensuremath{\sigma}}$ for

S=1/2,1,3/2,2

is calculated by the finite-range scaling technique. The spin dependence of the critical temperature and critical exponent $\ensuremath{\nu}$ in the classical and nonclassical regions is also studied. It is observed that, with a good approximation, the critical exponent of the correlation length is independent of the magnitude of spin in both the classical and nonclassical regions.