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Physical Review B

Low-temperature critical properties and crossovers of a spin- 12 planar ferromagnet in 4−ε dimensions

Maria Teresa Mercaldo, Alvaro Caramico D’Auria, L. De Cesare, Ileana Rabuffo

Published May 20, 2008• 14 citations
Publisher

Abstract

The low-temperature critical properties and crossovers of a spin-$\frac{1}{2}$ planar ferromagnet in a longitudinal magnetic field are explored in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions in terms of an anisotropic bosonic action, suitable to describe the spin model in the low-temperature regime. This is performed adopting a procedure, which combines a preliminary averaging over dynamic degrees of freedom and a next static Wilson renormalization group transformation. We work to one-loop level to obtain results which are exact to first order in $\ensuremath{\epsilon}$ and believed to be qualitatively correct for $2<d<4$. Within this static framework, we get the phase boundary, ending in a quantum critical point, and the general expressions for the correlation length and susceptibility as functions of the temperature and the applied magnetic field within the disordered phase. With these basic ingredients, the critical properties and a rich crossover scenario in the influence domain of the quantum critical point are derived in a simple and transparent way. Remarkably, one can speculate that the crossovers, found to occur decreasing the temperature with the magnetic field fixed at its quantum critical point value, might be actually observable in complex magnetic compounds, as suggested by recent experiments.

DOI
10.1103/physrevb.77.184424

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We explore the low-temperature thermodynamic properties and crossovers of a d -dimensional classical planar Heisenberg ferromagnet in a longitudinal magnetic field close to its field-induced zero-temperature critical point by employing the two-time Green’s function formalism in classical statistical mechanics. By means of a classical Callen-like method for the magnetization and the Tyablikov-like decoupling procedure, we obtain, for any d , a low-temperature critical scenario which is quite similar to the one found for the quantum counterpart. Remarkably, for d>2 the discrimination between the two cases is found to be related to the different values of the shift exponent which governs the behavior of the critical line in the vicinity of the zero-temperature critical point. The observation of different values of the shift-exponent and of the related critical exponents along thermodynamic paths within the typical V-shaped region in the phase diagram may be interpreted as a signature of emerging quantum critical fluctuations.
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The exact renormalization-group approach of Wilson is used to study the critical behavior for T>TcT>{T}_{c}T>Tc​, H=0H=0H=0, and small $\ensuremath{\epsilon}>0$, of an isotropic ferromagnetic system in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions, with exchange and dipolar interactions between ddd-component spins. Normal isotropic Heisenberg behavior with $\frac{1}{\ensuremath{\gamma}}\ensuremath{\approx}\frac{1}{2\ensuremath{\nu}}\ensuremath{\approx}1\ensuremath{-}\frac{\ensuremath{\epsilon}}{4}$ (to first order in $\ensuremath{\epsilon}$) is retained for $t=(\frac{T}{{T}_{c}})\ensuremath{-}1\ensuremath{\ll}\frac{G}{J{a}^{d}}$, where $G=\frac{{(g{\ensuremath{\mu}}_{B})}^{2}}{2}$ measures the strength of the dipole-dipole interactions, JJJ is the short-range exchange parameter, and aaa is the lattice spacing. When ${t}^{\ensuremath{\varphi}}\ensuremath{\approx}\frac{G}{J{a}^{d}}$, where $\ensuremath{\varphi}\ensuremath{\approx}1+\frac{\ensuremath{\epsilon}}{4}$, crossover occurs to a characteristic dipolar behavior described by a new fixed point of the recursion relations. For $t\ensuremath{\ll}\frac{G}{J{a}^{d}}$ one thus finds $\frac{1}{\ensuremath{\gamma}}\ensuremath{\approx}\frac{1}{2\ensuremath{\nu}}\ensuremath{\approx}1\ensuremath{-}\frac{9\ensuremath{\epsilon}}{34}$ {and, for spins of ndn dnd components, $\frac{1}{\ensuremath{\gamma}}\ensuremath{\approx}\frac{1}{2\ensuremath{\nu}}\ensuremath{\approx}1\ensuremath{-}[\frac{(6n+3)}{2(6n+11)}]\ensuremath{\epsilon}$, which agrees with spherical-model results when $n\ensuremath{\rightarrow}\ensuremath{\infty}$}. In the dipolar regime the spin-correlation function $〈{s}_{q}^{\ensuremath{\alpha}}{s}_{\ensuremath{-}q}^{\ensuremath{\beta}}〉$ has a factor [${\ensuremath{\delta}}_{\ensuremath{\alpha}\ensuremath{\beta}}\ensuremath{-}(\frac{{q}^{\ensuremath{\alpha}}{q}^{\ensuremath{\beta}}}{{q}^{2}})$], which suppresses longitudinal spin fluctuations; the susceptibilities ${\ensuremath{\chi}}^{\ensuremath{\alpha}\ensuremath{\alpha}}$ display the expected demagnetization effects. It is found that dipolar anistropies derving from the lattice structure produce weak instabilities which should be hard to detect although their effects are not fully elucidated. Extensions of the results to nonzero magnetic fields, and to anistropic exchange interactions are indicated; the experimental situation is mentioned briefly.
A nonconventional renormalization-group (RG) treatment close to and below four dimensions is used to explore, in a unified and systematic way, the low-temperature properties of a wide class of systems in the influence domain of their quantum critical point. The approach consists in a preliminary averaging over quantum degrees of freedom and a successive employment of the Wilsonian RG transformation to treat the resulting effective classical Ginzburg-Landau free energy functional. This allows us to perform a detailed study of criticality of the quantum systems under study. The emergent physics agrees, in many aspects, with the known quantum critical scenario. However, a richer structure of the phase diagram appears with additional crossovers which are not captured by the traditional RG studies. In addition, in spite of the intrinsically static nature of our theory, predictions about the dynamical critical exponent, which parametrizes the link between statics and dynamics close to a continuous phase transition, are consistently derived from our static results.
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