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Physical Review B

Field-induced quantum critical point in planar Heisenberg ferromagnets with long-range interactions: Two-time Green’s function framework

L. S. Campana, L. De Cesare, Ugo Esposito, Maria Teresa Mercaldo, Ileana Rabuffo

Published Jul 13, 2010• 20 citations
Publisher

Abstract

The two-time Green's function method is used to study the critical properties and crossover phenomena near the field-induced quantum critical point (QCP) of a $d$-dimensional spin-$S$ planar Heisenberg ferromagnet with long-range interactions decaying as ${r}^{\ensuremath{-}\ensuremath{\alpha}}$ (with $\ensuremath{\alpha}>d$) with the distance $r$ between spins. We adopt the Callen scheme for spin $S$ and the Tyablikov decoupling procedure which is expected to provide suitable results at low temperatures. Different quantum critical regimes are found in the $(\ensuremath{\alpha},d)$ plane and the global structure of the phase diagram is determined showing the typical V-shaped region close to the QCP. Depending on the values of $\ensuremath{\alpha}$, we find that also for dimensionalities $d\ensuremath{\leqslant}2$ a finite-temperature critical line, ending in the QCP, exists with asymptotic behaviors and crossovers which can be employed as a useful guide for experimental studies. Moreover, these crossovers are shown to be suitably described in terms of $(\ensuremath{\alpha},d)$-dependent scaling functions and effective critical exponents.

DOI
10.1103/physrevb.82.024409

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The two-time Green's function method is used to study the critical properties and crossover phenomena near the field-induced quantum critical point (QCP) of a ddd-dimensional spin-SSS planar Heisenberg ferromagnet with long-range interactions decaying as r\ensuremath−\ensuremathα{r}^{\ensuremath{-}\ensuremath{\alpha}}r\ensuremath−\ensuremathα (with \ensuremathα>d\ensuremath{\alpha}>d\ensuremathα>d) with the distance rrr between spins. We adopt the Callen scheme for spin SSS and the Tyablikov decoupling procedure which is expected to provide suitable results at low temperatures. Different quantum critical regimes are found in the (\ensuremathα,d)(\ensuremath{\alpha},d)(\ensuremathα,d) plane and the global structure of the phase diagram is determined showing the typical V-shaped region close to the QCP. Depending on the values of \ensuremathα\ensuremath{\alpha}\ensuremathα, we find that also for dimensionalities d\ensuremath⩽2d\ensuremath{\leqslant}2d\ensuremath⩽2 a finite-temperature critical line, ending in the QCP, exists with asymptotic behaviors and crossovers which can be employed as a useful guide for experimental studies. Moreover, these crossovers are shown to be suitably described in terms of (\ensuremathα,d)(\ensuremath{\alpha},d)(\ensuremathα,d)-dependent scaling functions and effective critical exponents.
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.
The low-temperature critical properties and crossovers of a spin-½ planar ferromagnet in a longitudinal magnetic field are explored 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 an averaging over dynamic degrees of freedom and the classical Wilson renormalization group transformation. Within this framework we get the phase boundary, ending in a quantum critical point, and general expressions for the correlation length and susceptibility as functions of the temperature and the applied magnetic field within the disordered phase. In particular, two crossovers occur decreasing the temperature with the magnetic field fixed at its quantum critical point value, which might be actually observable in complex magnetic compounds, as suggested by recent experiments.
We analyze the effects induced by single-ion anisotropy on quantum criticality in a d -dimensional spin-3/2 planar ferromagnet. To tackle this problem we employ the two-time Green's function method, using the Tyablikov decoupling for exchange interactions and the Anderson-Callen decoupling for single-ion anisotropy. In our analysis the role of non-thermal control parameter which drives the quantum phase transition is played by a longitudinal external magnetic field. We find that the single-ion anisotropy has substantial effects on the structure of the phase diagram close to the quantum critical point.
Quantum Heisenberg ferromagnets with long-range interactions in low dimensions are inves- tigated by means of the Green's frunction method. The model Hamiltonian is given by scrH=-(J0{\mathit{J}}_{0}J0​/2)${\mathcal{J}}_{\mathit{i}\mathrm{\ensuremath{\ne}}\mathit{j}}$${\mathit{r}}_{\mathit{i}\mathit{j}}^{\mathrm{\ensuremath{-}}\mathit{p}}$Si{\mathbf{S}}_{\mathit{i}}Si​\ensuremath{\cdot}Sj{\mathbf{S}}_{\mathit{j}}Sj​ -HJj{\mathcal{J}}_{\mathit{j}}Jj​Sjz{\mathit{S}}_{\mathit{j}}^{\mathit{z}}Sjz​. It is shown that there exists a finite-temperature phase transition in the region dp2d for the d-dimensional case and that no transitions at any finite temperature exist for p\ensuremath{\ge}2d; the critical temperature is also estimated. We study the magnetic properties of this model. We calculate the critical exponents' dependence on p; these exponents also satisfy a scaling relation. Some of the results were also found using the modified spin-wave theory and are in remarkable agreement with each other.
The low-temperature critical properties and crossovers of a spin-12\frac{1}{2}21​ 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<42<d<42<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.
The thermodynamic properties of a classical spin-SSS Heisenberg ferromagnetic chain with long-range interactions in the presence of an external magnetic field are investigated by means of the spectral density method within the framework of the two-time Green's functions in classical statistical mechanics. For exchange interactions decaying as ${r}^{\ensuremath{-}\ensuremath{\alpha}},$ it is shown that long-range order exists at finite temperature for $1<\ensuremath{\alpha}<2$ and that no transition to a ferromagnetic phase at any finite temperature occurs for $\ensuremath{\alpha}>~2,$ consistently with a recent extension of the Mermin-Wagner theorem to spin models with long-range interactions. Besides, by means of a suitable modification of the lowest-order moment equations for the spectral density, the critical temperature and the main critical exponents for $1<\ensuremath{\alpha}<2$ and the low-temperature behavior of the paramagnetic susceptibility for $\ensuremath{\alpha}>~2$ are also obtained. Our results appear in a reasonable agreement with the available exact ones and Monte Carlo calculations.