Renormalization-group Recursion Relations Research Articles

Physical realizations ofn≥4-component vector models. III. Phase transitions in Cr, Eu, MnS2, Ho, Dy, and Tb

We use the symmetry arguments of Laudau and Lifshitz, described in the preceding papers, to derive the Landau-Ginzburg-Wilson Hamiltonians corresponding to several antiferromagnetic systems. The phase transitions associated with these models are then studied using the exact renormalization-group technique in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions. We find that the fcc type-III antiferromagnets $\stackrel{\ensuremath{\rightarrow}}{\mathrm{m}}\ensuremath{\perp}[100],\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}=(\frac{1}{2},0,1)$ and the spiral magnets Eu and Cr are described by two $n=12$ models. For both models, the renormalization-group recursion relations have no stable fixed points. This might explain the existence of first-order transitions in Cr and Eu. We also find that the antiferromagnet Mn${\mathrm{S}}_{2}$ is described by the $n=6$ model derived by Mukamel and Krinsky to represent Tb${\mathrm{D}}_{2}$, Nd, and ${\mathrm{K}}_{2}$Ir${\mathrm{Cl}}_{6}$. Since this model has one stable fixed point, it is predicted that the four compounds belong to the same universality class. Similarly, the spiral magnets Tb, Dy, and Ho correspond to the $n=4$ model which was used to describe Nb${\mathrm{O}}_{2}$, Tb${\mathrm{Au}}_{2}$, and Dy${\mathrm{C}}_{2}$, and it is predicted that they all have the same critical behavior. Existing experimental data are discussed and several experiments are suggested.

Equations of state and renormalization-group recursion relations

Renormalization-group recursion relations are used to construct equations of state to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ for continuous spin models of critical behavior. The recursion relations map Hamiltonians out of the critical regime into regions of small correlation length, where Landau theory with fluctuation corrections is employed. This technique is used to discuss critical behavior along a $\ensuremath{\lambda}$ line of critical points, together with the effect of imposing an ordering field. An application of the method to tricritical phenomena is described briefly.

Coexistence-curve singularities in isotropic ferromagnets

The critical behavior of a classical isotropic ferromagnet ($n\ensuremath{\ge}2$) is studied to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$. Using renormalization-group recursion relations, we obtain expressions for the equation of state and longitudinal susceptibility which describe singular behavior both at the critical point and on the coexistence curve. It is found that, although the diverging susceptibility on the coexistence curve should not be expected in real magnetic crystals, it can produce a large effect on the initial susceptibility measured below ${T}_{c}$ in hexagonal layered crystals such as Cr${\mathrm{Cl}}_{3}$. Specifically, we find that the susceptibility should diverge as $T\ensuremath{\rightarrow}{T}_{c}$ from below as ${\ensuremath{\chi}}_{L}\ensuremath{\sim}{|t|}^{\ensuremath{-}{\ensuremath{\gamma}}^{\ensuremath{'}}}$, with ${\ensuremath{\gamma}}^{\ensuremath{'}}\ensuremath{\simeq}\ensuremath{\gamma}+0.79\ensuremath{\simeq}2.12$.

Equations of state for bicritical points. I. Calculations in the disordered phase

Equations of state are calculated for anisotropic spin Hamiltonians suitable for describing the spin-flop transition in materials such as GdAl${\mathrm{O}}_{3}$ and Mn${\mathrm{F}}_{2}$. Crossover scaling functions for the specific heat, nonordering susceptibility, and parallel and transverse susceptibilities are derived to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ directly from renormalization-group recursion relations.

Equations of State for Tricritical Points

Renormalization-group recursion relations are used to calculate to order $\ensuremath{\epsilon}=4\ensuremath{-}d$ the free-energy, ordering-susceptibility, and magnetization-crossover scaling functions for tricritical points in isotropic $n$-vector models. The equations of state describe critical and tricritical behavior in both ordered and disordered phases.

Critical behavior of compressible magnets

A model of a compressible $n$-component magnet which includes shearing forces is solved by the renormalization-group recursion relations to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$. Four fixed points are found and their relevance for critical behavior is discussed. The critical exponents of Heisenberg magnets have their rigid-lattice values. The leading correction to scaling has the exponent $\ensuremath{\alpha}{\ensuremath{\nu}}^{\ensuremath{-}1}$ with respect to inverse length. In Ising-like systems the transition is of the first order, but may appear as a second order.

On the calculation of critical exponents by renormalisation group techniques

The application of renormalisation group recursion relations to the calculation of critical exponents in d dimensions to second order in epsilon =4-d is discussed. It is shown that in the limit of a large scale factor b, the fixed point value, u* of the four-spin coupling constant is correctly given to second order by a renormalisation group equation in which all irrelevant variables, mass renormalisation graphs, and mass terms in the propagators are ignored. Related approximations in the calculation of the critical exponents are justified. The unexpected equality to order epsilon 2 of u* and u0c, the Feynman graph coupling constant, is discussed.

Exact Renormalization Group Exhibiting a Tricritical Fixed Point for a Spin-1 Ising Model in One Dimension

Exact renormalization-group recursion relations are studied for a spin-1 Ising model in one dimension. In this model, critical lines are marked by a double degeneracy of the largest eigenvalue of the transfer matrix, and the tricritical point by a triple degeneracy. This is directly related to the tricritical fixed point being more unstable than the critical fixed points.

Critical Behavior of Magnets with Lattice Coupling

The exact spin Hamiltonian, induced by linear exchange coupling to a harmonic lattice with fixed periodic boundary conditions, is considered in the framework of renormalization-group recursion relations. Neglecting irrelevant variables, the Hamiltonian amounts to a replacement of the four-spin amplitude ${u}_{0}$ by $a(T\ensuremath{-}{T}_{1})$, with ${T}_{1}$ proportional to the lattice compressibility. Hence the system exhibits a critical point with unrenormalized exponent values when ${T}_{c}g{T}_{t}$, but presumably a first-order transition for ${T}_{c}l{T}_{t}$, where ${T}_{t}\ensuremath{\simeq}{T}_{1}$. The point ${T}_{c}={T}_{t}$ is expected to be a classical tricritical point. Experiments on N${\mathrm{H}}_{4}$Cl are considered briefly.