Physical review. B, Condensed matter

Ordering under random fields: Renormalization-group arguments

A. Nihat Berker

Published May 1, 1984• 49 citations

Abstract

The possibility of long-range order under quenched random magnetic fields can be deduced from strong-coupling rescaling behavior. The lower critical dimensions of ${d}_{l}=2$ and 4 are indicated, respectively, for Ising and $n$-component spin models with $n>1$.

DOI: 10.1103/physrevb.29.5243

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Physical review. B, Condensed matter

Ordering under random fields: Renormalization-group arguments

A. Nihat Berker

Published May 1, 1984• 49 citations

Publisher

Abstract

The possibility of long-range order under quenched random magnetic fields can be deduced from strong-coupling rescaling behavior. The lower critical dimensions of

{d}_{l}=2

and 4 are indicated, respectively, for Ising and

n

-component spin models with

n>1

DOI: 10.1103/physrevb.29.5243

Related papers

Dimensional reduction with correlated random fields. A superspace renormalization-group calculation

Mehran Kardar A, Bruce McClain, Cyrus C. Taylor

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Quasi-long-range order in the random anisotropy Heisenberg model: Functional renormalization group in 4−ε dimensions

Dima E. Feldman

The large-distance behaviors of the random field and random anisotropy $O(N)$ models are studied with the functional renormalization group in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions. The random anisotropy Heisenberg $(N=3)$ model is found to have a phase with an infinite correlation length at low temperatures and weak disorder. The correlation function of the magnetization obeys a power law $〈\mathbf{m}({\mathbf{r}}_{1})\mathbf{m}({\mathbf{r}}_{2})〉\ensuremath{\sim}|{\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{2}{|}^{\ensuremath{-}0.62\ensuremath{\epsilon}}.$ The magnetic susceptibility diverges at low fields as $\ensuremath{\chi}\ensuremath{\sim}{H}^{\ensuremath{-}1+0.15\ensuremath{\epsilon}}.$ In the random field $O(N)$ model the correlation length is found to be finite at the arbitrarily weak disorder for any $N>3.$ The random field case is studied with a simple method, based on a rigorous inequality. This approach allows one to avoid the integration of the functional renormalization-group equations.The large-distance behaviors of the random field and random anisotropy $O(N)$ models are studied with the functional renormalization group in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions. The random anisotropy Heisenberg $(N=3)$ model is found to have a phase with an infinite correlation length at low temperatures and weak disorder. The correlation function of the magnetization obeys a power law $〈\mathbf{m}({\mathbf{r}}_{1})\mathbf{m}({\mathbf{r}}_{2})〉\ensuremath{\sim}|{\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{2}{|}^{\ensuremath{-}0.62\ensuremath{\epsilon}}.$ The magnetic susceptibility diverges at low fields as $\ensuremath{\chi}\ensuremath{\sim}{H}^{\ensuremath{-}1+0.15\ensuremath{\epsilon}}.$ In the random field $O(N)$ model the correlation length is found to be finite at the arbitrarily weak disorder for any $N>3.$ The random field case is studied with a simple method, based on a rigorous inequality. This approach allows one to avoid the integration of the functional renormalization-group equations.

arXiv:cond-mat/9907122Physical review. B, Condensed matterJan 1, 200056 citations

The possibility of long-range order under quenched random magnetic fields can be deduced from strong-coupling rescaling behavior. The lower critical dimensions of

{d}_{l}=2

and 4 are indicated, respectively, for Ising and

n

-component spin models with

n>1

{\mathit{T}}_{\mathit{c}}

-T)/

{\mathit{T}}_{\mathit{c}}

is varied from 0 to 0.2947\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}16}$. However, a rigorous argument is given showing that the magnetization M cannot be truly discontinuous at phase transitions controlled by a strong-coupling random-field fixed distribution, in renormalization-group studies that employ the Migdal-Kadanoff local recursion relations.

We consider the random transverse-field Ising model in

d=3

dimensions with long-range ferromagnetic interactions which decay as a power $\ensuremath{\alpha}>d$ with the distance. Using a variant of the strong-disorder renormalization group method we study numerically the phase-transition point from the paramagnetic side. We find that the fixed point controlling the transition is of the strong-disorder type, and based on experience with other similar systems, we expect the results to be qualitatively correct, but probably not asymptotically exact. The distribution of the (sample dependent) pseudocritical points is found to scale with

1/lnL, L

being the linear size of the sample. Similarly, the critical magnetization scales with ${(lnL)}^{\ensuremath{\chi}}/{L}^{d}$ and the excitation energy behaves as ${L}^{\ensuremath{-}\ensuremath{\alpha}}$. Using extreme-value statistics we argue that extrapolating from the ferromagnetic side the magnetization approaches a finite limiting value and thus the transition is of mixed order.

We consider the Ginzburg-Landau

\mathrm{MN}

model that describes

MN

-vector cubic models with

O(M)

-symmetric couplings. We compute the renormalization-group functions to six-loop order in

d=3.

We focus on the limit $\stackrel{\ensuremath{\rightarrow}}{N}0$ which describes the critical behavior of an M-vector model in the presence of weak quenched disorder. We perform a detailed analysis of the perturbative series for the random Ising model

(M=1).

We obtain for the critical exponents: $\ensuremath{\gamma}=1.330(17),\ensuremath{\nu}=0.678(10),\ensuremath{\eta}=0.030(3),\ensuremath{\alpha}=\ensuremath{-}0.034(30),\ensuremath{\beta}=0.349(5),\ensuremath{\omega}=0.25(10).$ For

M>~2

we show that the

O(M)

fixed point is stable, in agreement with general nonperturbative arguments, and that no random fixed point exists.

The large-distance behaviors of the random field and random anisotropy

O(N)

models are studied with the functional renormalization group in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions. The random anisotropy Heisenberg

(N=3)

model is found to have a phase with an infinite correlation length at low temperatures and weak disorder. The correlation function of the magnetization obeys a power law $〈\mathbf{m}({\mathbf{r}}_{1})\mathbf{m}({\mathbf{r}}_{2})〉\ensuremath{\sim}|{\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{2}{|}^{\ensuremath{-}0.62\ensuremath{\epsilon}}.$ The magnetic susceptibility diverges at low fields as $\ensuremath{\chi}\ensuremath{\sim}{H}^{\ensuremath{-}1+0.15\ensuremath{\epsilon}}.$ In the random field

O(N)

model the correlation length is found to be finite at the arbitrarily weak disorder for any

N>3.

The random field case is studied with a simple method, based on a rigorous inequality. This approach allows one to avoid the integration of the functional renormalization-group equations.