A ferromagnetic system with frozen-in defects is investigated within a modified spherical model: The three-dimensional lattice is formed out of homogeneous layers which are occupied at random either by $A$ or by $B$ spins. We assume three exchange constants ${J}_{A}$ , ${J}_{B}$ , ${J}_{\mathrm{AB}}$ which correspond to the coupling between $A\ensuremath{-}A$, $B\ensuremath{-}B$, and $A\ensuremath{-}B$ spins, respectively. Moreover, two types of layer configurations are considered. (i) The first one allows for the occurrence of arbitrarily large clusters of connected layers which contain the same spins only; (ii) in the second type, homogeneous clusters larger than a certain size are excluded. In case (ii) the randomness of the layer distribution prohibits long-range order in the following sense: If we add one layer to a finite lattice of thickness $d$ , then its influence on the free energy decreases exponentially with $d$ ; this theorem is proved to be true for all temperatures and any particular layer configuration. We then specialize to ${J}_{\mathrm{AB}}=0$ . In case (i) we find a finite critical temperature ${T}_{c}$ which is indicated by a molecular-field-like jump in the specific heat. This singularity turns out to belong to the class of Griffiths singularities. The surprising fact that we have here a "visible" Griffiths singularity is attributed to fluctuations in the spin length. In case (ii) no critical behavior is observed.A ferromagnetic system with frozen-in defects is investigated within a modified spherical model: The three-dimensional lattice is formed out of homogeneous layers which are occupied at random either by $A$ or by $B$ spins. We assume three exchange constants ${J}_{A}$, ${J}_{B}$, ${J}_{\mathrm{AB}}$ which correspond to the coupling between $A\ensuremath{-}A$, $B\ensuremath{-}B$, and $A\ensuremath{-}B$ spins, respectively. Moreover, two types of layer configurations are considered. (i) The first one allows for the occurrence of arbitrarily large clusters of connected layers which contain the same spins only; (ii) in the second type, homogeneous clusters larger than a certain size are excluded. In case (ii) the randomness of the layer distribution prohibits long-range order in the following sense: If we add one layer to a finite lattice of thickness $d$, then its influence on the free energy decreases exponentially with $d$; this theorem is proved to be true for all temperatures and any particular layer configuration. We then specialize to ${J}_{\mathrm{AB}}=0$. In case (i) we find a finite critical temperature ${T}_{c}$ which is indicated by a molecular-field-like jump in the specific heat. This singularity turns out to belong to the class of Griffiths singularities. The surprising fact that we have here a "visible" Griffiths singularity is attributed to fluctuations in the spin length. In case (ii) no critical behavior is observed.

Physical review. B, Solid stateMar 1, 19768 citations

On closed-form approximations for the free energy of 3-dimensional ising model

Andrzej Kossakowski

Open Systems & Information DynamicsApr 1, 19925 citations

A

or by

B

spins. We assume three exchange constants

{J}_{A}

{J}_{B}

{J}_{\mathrm{AB}}

which correspond to the coupling between $A\ensuremath{-}A$, $B\ensuremath{-}B$, and $A\ensuremath{-}B$ spins, respectively. Moreover, two types of layer configurations are considered. (i) The first one allows for the occurrence of arbitrarily large clusters of connected layers which contain the same spins only; (ii) in the second type, homogeneous clusters larger than a certain size are excluded. In case (ii) the randomness of the layer distribution prohibits long-range order in the following sense: If we add one layer to a finite lattice of thickness

d

, then its influence on the free energy decreases exponentially with

d

; this theorem is proved to be true for all temperatures and any particular layer configuration. We then specialize to

{J}_{\mathrm{AB}}=0

. In case (i) we find a finite critical temperature

{T}_{c}

which is indicated by a molecular-field-like jump in the specific heat. This singularity turns out to belong to the class of Griffiths singularities. The surprising fact that we have here a "visible" Griffiths singularity is attributed to fluctuations in the spin length. In case (ii) no critical behavior is observed.