Physical review. B, Solid state

Renormalization of Critical Exponents in a Compressible Ising Model

K. A. Penson

Published Sep 1, 1973• 8 citations

Abstract

The critical behavior of a compressible Ising system of the Baker-Essam type is examined for positive pressures. In addition to satisfying the assumptions of Fisher's theory of renormalization of critical exponents, the model, allows one to calculate the temperature region around ${T}_{c}$ where the renormalized exponents should show up. The region is $\ensuremath{\Delta}{T}_{\ifmmode\times\else\texttimes\fi{}}\ensuremath{\sim}{(P{J}_{1}^{2})}^{\frac{1}{\ensuremath{\alpha}}}$ for small pressures, where ${J}_{1}$ is the spin-lattice coupling and $\ensuremath{\alpha}$ is the specific-heat exponent of the corresponding rigid Ising system.

DOI: 10.1103/physrevb.8.2119

Related papers

Effects of Lattice Compressibility on Critical Behavior

George Allen Baker Jr., J W Essam

We present a model of a spin system on a compressible lattice whose solution we are able to express in terms of that for the Ising model on a rigid lattice. We find that the critical indices are renormalized when observed both at constant pressure and at constant volume. In particular the infinite Ising-model specific-heat peak is rounded into dynamic instability.

Physical Review LettersMar 2, 1970137 citations

Renormalized Critical Behavior or First-Order Phase Transitions?

Leon Gunther, David J. Bergman, Y. Imry

Physical Review LettersAug 30, 197158 citations

Fisher renormalization in systems with finite specific heat

V. Dohm

The current theory of critical phenomena in constrained systems is re-examined. It is found that renormalization of critical exponents is possible even if alpha <0 in the unconstrained system. The renormalized behaviour occurs at a tricritical point which is of the same type as found recently in magnetothermomechanics. A soluble model illustrates this behaviour.

Journal of Physics C Solid State PhysicsApr 21, 197416 citations

Visibility of Critical-Exponent Renormalization

Michael Ellis Fisher, Paul E. Scesney

Physical Review ASep 1, 1970115 citations

A position space renormalisation group approach to the compressible Ising model

Mustansir Barma, Deepak Kumar

Journal of Physics C Solid State PhysicsApr 14, 19792 citations

Critical behaviour of compressible Ising models at marginal dimensionalities

M. Vallade, Janine Lajzerowicz

Journal de physiqueJan 1, 197912 citations

Physical review. B, Solid state

Renormalization of Critical Exponents in a Compressible Ising Model

K. A. Penson

Published Sep 1, 1973• 8 citations

Publisher

Abstract

{T}_{c}

where the renormalized exponents should show up. The region is

\ensuremath{\Delta}{T}_{\ifmmode\times\else\texttimes\fi{}}\ensuremath{\sim}{(P{J}_{1}^{2})}^{\frac{1}{\ensuremath{\alpha}}}

for small pressures, where

{J}_{1}

is the spin-lattice coupling and

\ensuremath{\alpha}

is the specific-heat exponent of the corresponding rigid Ising system.

DOI: 10.1103/physrevb.8.2119

Related papers

Effects of Lattice Compressibility on Critical Behavior

George Allen Baker Jr., J W Essam

Physical Review LettersMar 2, 1970137 citations

Renormalized Critical Behavior or First-Order Phase Transitions?

Leon Gunther, David J. Bergman, Y. Imry

The exactly solvable Baker-Essam model for a compressible Ising lattice is re-examined. We find that, depending on the type of constraint imposed upon the system, the second-order phase transition either gets renormalized or is changed into a first-order transition. We resolve the apparent disagreement of this result with Fisher's renormalization theory by showing that, in fact, our result is typical of the correct state of affairs to be expected on the basis of the general theory of critical points in systems with hidden variables.The exactly solvable Baker-Essam model for a compressible Ising lattice is re-examined. We find that, depending on the type of constraint imposed upon the system, the second-order phase transition either gets renormalized or is changed into a first-order transition. We resolve the apparent disagreement of this result with Fisher's renormalization theory by showing that, in fact, our result is typical of the correct state of affairs to be expected on the basis of the general theory of critical points in systems with hidden variables.

Physical Review LettersAug 30, 197158 citations

Fisher renormalization in systems with finite specific heat

V. Dohm

Journal of Physics C Solid State PhysicsApr 21, 197416 citations

Visibility of Critical-Exponent Renormalization

Michael Ellis Fisher, Paul E. Scesney

The extent to which the renormalization of critical-point behavior should be visible experimentally is investigated on the basis of detailed numerical calculations for a three-dimensional soluble model (a mobile-electron Ising ferromagnet). If a dilution parameter $x$ is defined such that the change in critical temperature from the "pure" or unrenormalized system is $|{T}_{c}(x)\ensuremath{-}{T}_{c}^{0}|=\frac{x{T}_{c}}{f}$, where $f=0.6\ensuremath{-}0.9$, then we conclude that the effective exponents ${\ensuremath{\beta}}_{\mathrm{fit}}(x)$ and ${\ensuremath{\gamma}}_{\mathrm{fit}}(x)$ which will be observed experimentally, vary roughly as ${\ensuremath{\beta}}_{\mathrm{fit}}\ensuremath{\approx}\ensuremath{\beta}+x\ensuremath{\Delta}{\ensuremath{\beta}}_{X}$ and ${\ensuremath{\gamma}}_{\mathrm{fit}}\ensuremath{\simeq}\ensuremath{\gamma}+\frac{1}{5}x\ifmmode\times\else\texttimes\fi{}(1+2{x}^{2})\ensuremath{\Delta}{\ensuremath{\gamma}}_{X}$. Here $\ensuremath{\beta}$ and $\ensuremath{\gamma}$ are the ideal exponents for the order parameter and total fluctuation or susceptibility of the pure system, while $\ensuremath{\Delta}{\ensuremath{\beta}}_{X}={\ensuremath{\beta}}_{X}\ensuremath{-}\ensuremath{\beta}$ and $\ensuremath{\Delta}{\ensuremath{\gamma}}_{X}={\ensuremath{\gamma}}_{X}\ensuremath{-}\ensuremath{\gamma}$, in which ${\ensuremath{\beta}}_{X}=\frac{\ensuremath{\beta}}{(1\ensuremath{-}{\ensuremath{\alpha}}^{\ensuremath{'}})}$ and ${\ensuremath{\gamma}}_{X}=\frac{\ensuremath{\gamma}}{(1\ensuremath{-}\ensuremath{\alpha})}$ are the fully renormalized exponents, while $\ensuremath{\alpha}$ and ${\ensuremath{\alpha}}^{\ensuremath{'}}$ (assumed positive) describe the divergence of the specific heats of the pure system. [Theoretically the true limiting asymptotic behavior at the transition is described by $\ensuremath{\beta}(x)={\ensuremath{\beta}}_{X}$ and $\ensuremath{\gamma}(x)={\ensuremath{\gamma}}_{X}$ for all $x>0$ .] The renormalized specific heats are found to be sensitive to $x$ but their true renormalized behavior is not evident until $x\ensuremath{\gtrsim}0.3$. Various techniques of data analysis such as logarithmic, semilogarithmic, Heller-Benedek, and Kouvel-Fisher plots have been tested.The extent to which the renormalization of critical-point behavior should be visible experimentally is investigated on the basis of detailed numerical calculations for a three-dimensional soluble model (a mobile-electron Ising ferromagnet). If a dilution parameter $x$ is defined such that the change in critical temperature from the "pure" or unrenormalized system is $|{T}_{c}(x)\ensuremath{-}{T}_{c}^{0}|=\frac{x{T}_{c}}{f}$, where $f=0.6\ensuremath{-}0.9$, then we conclude that the effective exponents ${\ensuremath{\beta}}_{\mathrm{fit}}(x)$ and ${\ensuremath{\gamma}}_{\mathrm{fit}}(x)$ which will be observed experimentally, vary roughly as ${\ensuremath{\beta}}_{\mathrm{fit}}\ensuremath{\approx}\ensuremath{\beta}+x\ensuremath{\Delta}{\ensuremath{\beta}}_{X}$ and ${\ensuremath{\gamma}}_{\mathrm{fit}}\ensuremath{\simeq}\ensuremath{\gamma}+\frac{1}{5}x\ifmmode\times\else\texttimes\fi{}(1+2{x}^{2})\ensuremath{\Delta}{\ensuremath{\gamma}}_{X}$. Here $\ensuremath{\beta}$ and $\ensuremath{\gamma}$ are the ideal exponents for the order parameter and total fluctuation or susceptibility of the pure system, while $\ensuremath{\Delta}{\ensuremath{\beta}}_{X}={\ensuremath{\beta}}_{X}\ensuremath{-}\ensuremath{\beta}$ and $\ensuremath{\Delta}{\ensuremath{\gamma}}_{X}={\ensuremath{\gamma}}_{X}\ensuremath{-}\ensuremath{\gamma}$, in which ${\ensuremath{\beta}}_{X}=\frac{\ensuremath{\beta}}{(1\ensuremath{-}{\ensuremath{\alpha}}^{\ensuremath{'}})}$ and ${\ensuremath{\gamma}}_{X}=\frac{\ensuremath{\gamma}}{(1\ensuremath{-}\ensuremath{\alpha})}$ are the fully renormalized exponents, while $\ensuremath{\alpha}$ and ${\ensuremath{\alpha}}^{\ensuremath{'}}$ (assumed positive) describe the divergence of the specific heats of the pure system. [Theoretically the true limiting asymptotic behavior at the transition is described by $\ensuremath{\beta}(x)={\ensuremath{\beta}}_{X}$ and $\ensuremath{\gamma}(x)={\ensuremath{\gamma}}_{X}$ for all $x>0$.] The renormalized specific heats are found to be sensitive to $x$ but their true renormalized behavior is not evident until $x\ensuremath{\gtrsim}0.3$. Various techniques of data analysis such as logarithmic, semilogarithmic, Heller-Benedek, and Kouvel-Fisher plots have been tested.

Physical Review ASep 1, 1970115 citations

A position space renormalisation group approach to the compressible Ising model

Mustansir Barma, Deepak Kumar

A position space renormalisation group procedure is developed for the compressible Ising model. The procedure is essentially a generalisation of the cell variable approach of Niemeijer and Van Leeuwen (1974) in that vibrational cell variables are introduced in addition to spin cell variables. The procedure is applied to the two-dimensional triangular lattice, the shearless square lattice. In both cases the rigid Ising fixed point is stable to spin-lattice coupling as long as the latter is small. Beyond a critical strength of the spin-lattice coupling, the system develops instability leading to, presumably, a first-order transition. For the shearless square lattice the conclusion is at variance with the exact result of Baker and Essam (1971), so the approximation is regarded as unreliable for larger values of the spin-lattice coupling (1971).A position space renormalisation group procedure is developed for the compressible Ising model. The procedure is essentially a generalisation of the cell variable approach of Niemeijer and Van Leeuwen (1974) in that vibrational cell variables are introduced in addition to spin cell variables. The procedure is applied to the two-dimensional triangular lattice, the shearless square lattice. In both cases the rigid Ising fixed point is stable to spin-lattice coupling as long as the latter is small. Beyond a critical strength of the spin-lattice coupling, the system develops instability leading to, presumably, a first-order transition. For the shearless square lattice the conclusion is at variance with the exact result of Baker and Essam (1971), so the approximation is regarded as unreliable for larger values of the spin-lattice coupling (1971).

Journal of Physics C Solid State PhysicsApr 14, 19792 citations

Critical behaviour of compressible Ising models at marginal dimensionalities

M. Vallade, Janine Lajzerowicz

Renormalization group methods are applied to study the critical behaviour of a compressible n-component Ising model with short range interactions at d = 4 and a one component Ising model with dipolar interactions at d = 3. The recursion equations are exactly solved in the case of an elastic system of spherical symmetry (d = 4) or cylindrical symmetry (d = 3); new types of logarithmic corrections, corresponding to a Fisher renormalization at marginal dimensions, are found. It is shown that the system exhibits a first-order transition for constant pressure external conditions or when anisotropy is taken into account. The relevance of the calculations to the critical behaviour of uniaxial ferroelectrics is discussed.Renormalization group methods are applied to study the critical behaviour of a compressible n-component Ising model with short range interactions at d = 4 and a one component Ising model with dipolar interactions at d = 3. The recursion equations are exactly solved in the case of an elastic system of spherical symmetry (d = 4) or cylindrical symmetry (d = 3); new types of logarithmic corrections, corresponding to a Fisher renormalization at marginal dimensions, are found. It is shown that the system exhibits a first-order transition for constant pressure external conditions or when anisotropy is taken into account. The relevance of the calculations to the critical behaviour of uniaxial ferroelectrics is discussed.

Journal de physiqueJan 1, 197912 citations

{T}_{c}

where the renormalized exponents should show up. The region is

\ensuremath{\Delta}{T}_{\ifmmode\times\else\texttimes\fi{}}\ensuremath{\sim}{(P{J}_{1}^{2})}^{\frac{1}{\ensuremath{\alpha}}}

for small pressures, where

{J}_{1}

is the spin-lattice coupling and

\ensuremath{\alpha}

is the specific-heat exponent of the corresponding rigid Ising system.

x

is defined such that the change in critical temperature from the "pure" or unrenormalized system is $|{T}_{c}(x)\ensuremath{-}{T}_{c}^{0}|=\frac{x{T}_{c}}{f}$, where $f=0.6\ensuremath{-}0.9$, then we conclude that the effective exponents ${\ensuremath{\beta}}_{\mathrm{fit}}(x)$ and ${\ensuremath{\gamma}}_{\mathrm{fit}}(x)$ which will be observed experimentally, vary roughly as ${\ensuremath{\beta}}_{\mathrm{fit}}\ensuremath{\approx}\ensuremath{\beta}+x\ensuremath{\Delta}{\ensuremath{\beta}}_{X}$ and ${\ensuremath{\gamma}}_{\mathrm{fit}}\ensuremath{\simeq}\ensuremath{\gamma}+\frac{1}{5}x\ifmmode\times\else\texttimes\fi{}(1+2{x}^{2})\ensuremath{\Delta}{\ensuremath{\gamma}}_{X}$. Here $\ensuremath{\beta}$ and $\ensuremath{\gamma}$ are the ideal exponents for the order parameter and total fluctuation or susceptibility of the pure system, while $\ensuremath{\Delta}{\ensuremath{\beta}}_{X}={\ensuremath{\beta}}_{X}\ensuremath{-}\ensuremath{\beta}$ and $\ensuremath{\Delta}{\ensuremath{\gamma}}_{X}={\ensuremath{\gamma}}_{X}\ensuremath{-}\ensuremath{\gamma}$, in which ${\ensuremath{\beta}}_{X}=\frac{\ensuremath{\beta}}{(1\ensuremath{-}{\ensuremath{\alpha}}^{\ensuremath{'}})}$ and ${\ensuremath{\gamma}}_{X}=\frac{\ensuremath{\gamma}}{(1\ensuremath{-}\ensuremath{\alpha})}$ are the fully renormalized exponents, while $\ensuremath{\alpha}$ and ${\ensuremath{\alpha}}^{\ensuremath{'}}$ (assumed positive) describe the divergence of the specific heats of the pure system. [Theoretically the true limiting asymptotic behavior at the transition is described by $\ensuremath{\beta}(x)={\ensuremath{\beta}}_{X}$ and $\ensuremath{\gamma}(x)={\ensuremath{\gamma}}_{X}$ for all

x>0

.] The renormalized specific heats are found to be sensitive to

x

but their true renormalized behavior is not evident until $x\ensuremath{\gtrsim}0.3$. Various techniques of data analysis such as logarithmic, semilogarithmic, Heller-Benedek, and Kouvel-Fisher plots have been tested.