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Physical review. B, Condensed matter

Renormalization-group treatment of the long-ranged one-dimensional Ising model with random fields

P. O. Weir, N. Read, J. Michael Kosterlitz

Published Oct 1, 1987• 6 citations
Publisher

Abstract

We construct a renormalization-group theory for a one-dimensional Ising model with longranged interactions and random fields with short-ranged correlations. We find recursion relations near the zero-temperature, finite randomness fixed point, valid for $\ensuremath{\sigma}$ near \textonehalf{}, where the interaction $\frac{J}{(r)}=\frac{J}{{r}^{1+\ensuremath{\sigma}}}$. We also show that the probability distribution for the random fields becomes strongly non-Gaussian under the scaling procedure.

DOI
10.1103/physrevb.36.5760

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We construct a renormalization-group theory for a one-dimensional Ising model with longranged interactions and random fields with short-ranged correlations. We find recursion relations near the zero-temperature, finite randomness fixed point, valid for \ensuremathσ\ensuremath{\sigma}\ensuremathσ near \textonehalf{}, where the interaction J(r)=Jr1+\ensuremathσ\frac{J}{(r)}=\frac{J}{{r}^{1+\ensuremath{\sigma}}}(r)J​=r1+\ensuremathσJ​. We also show that the probability distribution for the random fields becomes strongly non-Gaussian under the scaling procedure.
The one-dimensional Ising model in an external magnetic field with uniform long-range interactions and random short-range interactions satisfying bimodal annealed distributions is studied. This generalizes the random model discussed by Paladin et al. (J. Phys. I France 4, 1994, p. 1597). Exact results are obtained for the thermodynamic functions at arbitrary temperatures, and special attention is given to the induced and spontaneous magnetization. At low temperatures the system can exist in a “ferrimagnetic” phase with magnetization 0 < σ < 1, in addition to the usual paramagnetic, ferromagnetic and antiferromagnetic phases. For a fixed distribution of the random variables the system presents up to three tricritical points for different intensities of the long-range interactions. Field-temperature diagrams can present up to four critical points. 1
A highly detailed renormalization-group study is applied to the equilibrium statistical mechanics of the random-field Ising model in three dimensions, with analysis of the renormalization-group flows of over 8 500 000 quantities that determine the quenched probability distribution. We determine the strong-coupling fixed distribution that controls the finite-temperature phase boundary. This boundary is second order but has a magnetization behavior that is phenomenally close to a discontinuity, with the exponent \ensuremath{\beta}=0.0200\ifmmode\pm\else\textpm\fi{}0.0005. Magnetization curves are obtained, demonstrating that, for example, the calculated M changes from 0 to 0.407, almost half saturation, as (Tc{\mathit{T}}_{\mathit{c}}Tc​-T)/Tc{\mathit{T}}_{\mathit{c}}Tc​ 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 study the critical behaviour of the 1D random field Ising model (RFIM) with long-range interactions (∝ r −( d + σ ) ) by the nonperturbative functional renormalization group. We find two distinct regimes of critical behaviour as a function of σ , separated by a critical value σ c . What distinguishes these two regimes is the presence or not of a cusp-like nonanalyticity in the functional dependence of the renormalized cumulants of the random field at the fixed point. This change of behaviour can be associated with the characteristics of the large-scale avalanches present in the system at zero temperature. We propose some ways to check these predictions through lattice simulations. We also discuss the differences with the RFIM on the Dyson hierarchical lattice.
The one-dimensional Ising model with ferromagnetic interactions which decay as 1/${\mathit{r}}^{\mathrm{\ensuremath{\alpha}}}$ is considered. Using a real-space renormalization group scheme (RG) we calculate the critical temperature and the correlation-length critical exponent as a function of \ensuremath{\alpha}. General asymptotic properties are obtained for arbitrary values of the rescaling length b of the RG transformation. Several rigorous results are recovered exactly in the limit b\ensuremath{\rightarrow}\ensuremath{\infty}. We obtain a b=\ensuremath{\infty} extrapolation of the critical temperature for arbitrary values of \ensuremath{\alpha}>1, which we conjecture aproximates with high precision the exact one. In particular, we obtain the value Tc{\mathit{T}}_{\mathit{c}}Tc​/J=${\mathrm{\ensuremath{\pi}}}^{2}$/12 for the 1/r2{\mathit{r}}^{2}r2 model.