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Advances in Condensed Matter Physics

Thermodynamics of the Classical Planar Ferromagnet Close to the Zero-Temperature Critical Point: A Many-Body Approach

L. S. Campana, A. Cavallo, L. De Cesare, Ugo Esposito, Adele Naddeo

Published Jan 1, 2012• 0 citations
Publisher

Abstract

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.

DOI
10.1155/2012/619513

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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.
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 ${r}^{\ensuremath{-}\ensuremath{\alpha}}$ (with $\ensuremath{\alpha}>d$) 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 $(\ensuremath{\alpha},d)$ 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.
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.
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.
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.
We face the problem of detecting and featuring footprints of quantum criticality in the finite-temperature behavior of quantum many-body systems. Our strategy is that of comparing the phase diagram of a system displaying a T=0T=0T=0 quantum phase transition with that of its classical limit, in order to single out the genuinely quantum effects. To this aim, we consider the one-dimensional Ising model in a transverse field: while the quantum S=1∕2S=1∕2S=1∕2 Ising chain is exactly solvable and extensively studied, results for the classical limit $(S\ensuremath{\rightarrow}\ensuremath{\infty})$ of such model are lacking, and we supply them here. They are obtained numerically, via the transfer-matrix method, and their asymptotic low-temperature behavior is also derived analytically by self-consistent spin-wave theory. We draw the classical phase diagram according to the same procedure followed in the quantum analysis, and the two phase diagrams are found unexpectedly similar: Three regimes are detected also in the classical case, each characterized by a functional dependence of the correlation length on temperature and field analogous to that of the quantum model. What discriminates the classical from the quantum case are the different values of the exponents entering such dependencies, a consequence of the different nature of zero-temperature quantum fluctuations with respect to the thermal ones.
The thermodynamics of low-dimensional spin-12\frac{1}{2}21​ Heisenberg ferromagnets (HFMs) in an external magnetic field is investigated within a second-order two-time Green function formalism in the wide temperature and field ranges. A crucial point of the proposed scheme is a proper account of the analytical properties for the approximate transverse commutator Green function obtained as a result of the decoupling procedure. A good quantitative description of the correlation functions, magnetization, susceptibility, and heat capacity of the HFMs on a chain and square and triangular lattices is found for both infinite- and finite-sized systems. The dependences of the thermodynamic functions of two-dimensional HFMs on the cluster size are studied. The obtained results agree well with the corresponding data found by Bethe ansatz, exact diagonalization, high temperature series expansions, and quantum Monte Carlo simulations.