Share Icon Twitter Facebook Reddit LinkedIn Reprints and Permissions Cite Search Site Citation Dietrich Stauffer, Amnon Aharony, Sidney Redner; Introduction to Percolation Theory. Physics Today 1 April 1993; 46 (4): 64. https://doi.org/10.1063/1.2808877 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Dropdown Menu input auto suggest filter your All ContentPhysics Advanced

The mean square distance reached after $t$ random-walk steps on a percolating cluster is shown to behave as ${t}^{\frac{2}{(2+\ensuremath{\theta})}}$ (instead of $t$) for short times (for which the seen be self-similar). exponent $\ensuremath{\theta}$ related that describing dc conductivity near percolation. Averaging over all clusters yields ac and dielectric constant percolation in high-frequency limit, when polarization medium unimportant.

We discuss the temperature-concentration phase diagram of doped ${\mathrm{La}}_{2}$${\mathrm{CuO}}_{4}$. The addition holes introduces a local ferromagnetic exchange coupling between Cu spins. resulting frustration destroys 3D N\'eel state characterizing pure ${\mathrm{La}}_{2}$${\mathrm{CuO}}_{4}$, and generates new spin-glass phase. In paramagnetic phase, strongly correlated spins in planes are canted by holes, yielding an oscillating dipole-dipole attraction holes. possible relevance to...

We show that long-range ferroelectric and incommensurate magnetic order appear simultaneously in a single phase transition ${\mathrm{Ni}}_{3}{\mathrm{V}}_{2}{\mathrm{O}}_{8}$. The temperature magnetic-field dependence of the spontaneous polarization strong coupling between orders. determine symmetry using Landau theory for continuous transitions, which shows spin structure alone can break spatial inversion leading to order. This phenomenological explains our experimental observation is...

A uniform magnetic field is shown to generate random local fields in uniaxially anisotropic antiferromagnets with exchange interactions. This leads a stronger divergence of the temperature derivative static susceptibility even at zero field, and drastic change critical exponents Ising-like antiferromagnetic transition as well tricritical or bicritical points occurring finite fields.

We report a comprehensive neutron-scattering study of the evolution magnetic excitations in ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Sr}}_{\mathit{x}}$${\mathrm{CuO}}_{4}$ for 0\ensuremath{\le}x\ensuremath{\le}0.04. first present accurate measurements correlation length and sublattice magnetization carrier-free ${\mathrm{La}}_{2}$${\mathrm{CuO}}_{4}$ crystal analyze these context recent theoretical predictions. then systematically investigate influence different dopants...

Measurements of the magnetic moment antiferromagnetic ${\mathrm{La}}_{2}$Cu${\mathrm{O}}_{4}$ at high fields reveal a new phase boundary originating from previously undetected canting ${\mathrm{Cu}}^{2+}$ spins out Cu${\mathrm{O}}_{2}$ planes. This canting, together with exponential temperature dependence two-dimensional correlation length, accounts quantitatively for susceptibility peak N\'eel temperature. Enhancement conductivity in ferromagnetic demonstrates strong connection between...

Renormalization-group techniques are applied to Ising-model spins placed on the sites of several self-similar fractal lattices. The resulting critical properties shown vary with (noninteger) dimensionality $D$, but also topological factors: ramification, connectivity, lacunarity, etc. For any $D>~1$, there exist systems both ${T}_{c}=0$, and ${T}_{c}>0$; hence a lower is not defined. nonvanishing values ${T}_{c}$ exponents depend all these factors.

We prove that to all orders in perturbation expansion, the critical exponents of a phase transition $d$-dimensional ($4<d<6$) system with short-range exchange and random quenched field are same as those ($d\ensuremath{-}2$)-dimensional pure system. Heuristic arguments given discuss both this result random-field Ising model for $2<d<6$.

Mean-field theory and renormalization-group arguments are used to show that the phase transition in a system with random ordering field becomes first order at sufficiently low temperature, provided (symmetric) random-field distribution function has minimum zero field. The first-order region is separated from second-order by tricritical point. Both critical exponents $d>4$ dimensions shown be same as for pure $d\ensuremath{-}2$ dimensions. relevance spin glasses other systems discussed. new...

A nontrivial family of $d$-dimensional scale-invariant fractal lattices is described, on which statistical mechanics and conductivity problems are exactly solvable for every $d$. These fractals finitely ramified but not quasi one dimensional, hence can be used to model the important geometrical features percolating cluster's backbone. Critical exponents calculated this agree with those "real" systems at low dimensionalities.

The exact renormalization-group approach of Wilson is used to study the critical behavior for $T>{T}_{c}$, $H=0$, and small $\ensuremath{\epsilon}>0$, an isotropic ferromagnetic system in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions, with exchange dipolar interactions between $d$-component spins. Normal Heisenberg $\frac{1}{\ensuremath{\gamma}}\ensuremath{\approx}\frac{1}{2\ensuremath{\nu}}\ensuremath{\approx}1\ensuremath{-}\frac{\ensuremath{\epsilon}}{4}$ (to first order...

The hypothesis of two-scale-factor universality, originally proposed by Stauffer, Ferer, and Wortis, is shown to follow from the renormalization-group approach, for systems close their critical point. Values universal ratios involving correlation length specific-heat amplitudes are obtained $\ensuremath{\epsilon}$ expansion, Ising, $X\ensuremath{-}Y$, Heisenberg models. In latter two cases function has a power-law behavior at large distances below ${T}_{c}$, (transverse) defined in terms...

Moriya's expression for the ringle-bond anisotropic superexchange interaction is shown to possess an overlooked hidden symmetry, isomorphic symmetry of isotropic case. For unfrustrated case, this results in a degeneracy macroscopic state, implying no unique value Dzyaloshinsky weak ferromagnetic moment. A emerges from only when more than single bond considered and as result frustration. This implies that symmetric part anisotropy tensor must vary among bonds. The are particularly relevant...

It is claimed that the abstract analytic continuation of hypercubic lattices to noninteger dimensionalities can be implemented explicitly by certain fractal low lacunarity. These are special examples Sierpinski carpets. Their being lacunarity means they arbitrarily close translationally invariant. The claim substantiated for Ising model in $D=1+\ensuremath{\epsilon}$ dimensions, and resistor network models with $1<D<2$.

For pt.II see ibid. vol.17, p.435 (1984). In the first two papers of this series authors considered self-similar fractal lattices with a finite order ramification R. present paper they study physical models defined on family fractals R= infinity . to characterise geometry these systems, need connectivity Q and lacunarity L, in addition dimensionality D. It is found that discrete-symmetry spin undergo phase transition at Tc>0. An approximate renormalisation group scheme constructed used find...

The critical behavior in zero field above ${T}_{c}$ of ferromagnets or ferroelectrics with a Hamiltonian cubic symmetry is studied, to order ${\ensuremath{\epsilon}}^{2}$, by exact renormalization-group techniques $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions $n$-component spins. For $\ensuremath{\epsilon}=1$, $n\ensuremath{\ge}3$, crossover from isotropic (Heisenberg) characteristic occurs, the new value...

The exact renormalization-group approach is used to study the critical behavior for $T&gt;{T}_{c}$, $H=0$ of a uniaxial ferromagnetic (or ferroelectric) system in $d$ dimensions, with exchange and dipolar interactions between (single-component) spins. Normal Ising-like retained $t=\frac{T}{{T}_{c}}\ensuremath{-}1\ensuremath{\gg}\stackrel{^}{g}=\frac{{(g{\ensuremath{\mu}}_{B})}^{2}}{J{a}^{d}}$, where $J$ parameter, $g{\ensuremath{\mu}}_{B}$ magnetic moment per spin, $a$ lattice spacing....

The phase diagrams of systems described by a Hamiltonian containing an anisotropic quadratic term the form $\frac{1}{2}g\ensuremath{\Sigma}{\ensuremath{\alpha}=1}^{n}{c}_{\ensuremath{\alpha}}\ensuremath{\int}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}}^{}{S}_{\ensuremath{\alpha}}^{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})$, and cubic...

For pt.I see ibid., vol.16, p.1267 (1983). The authors construct and investigate a family of fractals which are generalisations the Sierpinski gaskets (SGs) to all Euclidean dimensionalities. These fractal lattices have finite order ramification, can be considered 'marginal' between one-dimensional higher-dimensional geometries. Physical models defined on them exactly solvable. argue that short-range spin SG show no finite-temperature phase transitions. As examples, they solve few study...

The distributions $P(X)$ of singular thermodynamic quantities, on an ensemble $d$-dimensional quenched random samples linear size $L$ near a critical point, are analyzed using the renormalization group. For much larger than correlation length $\ensuremath{\xi}$, we recover strong self-averaging (SA): approaches Gaussian with relative squared width ${R}_{X}\ensuremath{\sim}(L/\ensuremath{\xi}{)}^{\ensuremath{-}d}$. $L\ensuremath{\ll}\ensuremath{\xi}$ show weak SA ( ${R}_{X}$ decays small...

The hypothesis of universality implies that there are four universal ratios among the six usually defined thermodynamic critical amplitudes. Theoretical information from series and $\ensuremath{\epsilon}$ expansions is presented on values these for short-ranged Ising, Heisenberg, spherical models, dipolar systems. A number real materials discussed (Xe, ${\mathrm{CO}}_{2}$, Ni, EuO, ${\mathrm{LiTbF}}_{4}$), present state our understanding systems found to be rather crude.