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arXiv:1403.6283 · Journal of the Physical Society of Japan

Critical Temperature of Site-Diluted Spin-1/2 Systems with Long-Range Ferromagnetic Interactions

Karol Szałowski, T. Balcerzak

Published Mar 24, 2014• 4 citations

Abstract

In the paper the Pair Approximation (PA) method for studies of the site-diluted spin-1/2 systems of arbitrary dimensionality with the long-range ferromagnetic interactions is adopted. The method allows to take into account arbitrary anisotropy of the interactions in the spin space, so it is not limited to purely Ising couplings. Within this approach, the Gibbs free energy is obtained, which allows to derive all the further interesting thermodynamic properties. In particular, we obtain an equation for the critical temperature of the second-order phase transitions for the model in question. In the study we focus our attention on the systems with ferromagnetic interactions decaying with the distance according to the power law $J(r)\propto r^{-n}$. We discuss the dependence of the critical temperature on the concentration of magnetic component and the index $n$ for selected one-, two- and three-dimensional lattices.We confirm the absence of the critical concentration for a diluted magnet with infinite interaction range. In the regime of the low concentrations of magnetic component, we find a non-linear increase of the critical temperature with the concentration in the form of $T_{c}\propto p^{n/d}$, depending on the system dimensionality $d$ and the index $n$, whereas $n > d$.

DOI: 10.7566/jpsj.83.044002

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In the paper the Pair Approximation (PA) method for studies of the site-diluted spin-1/2 systems of arbitrary dimensionality with the long-range ferromagnetic interactions is adopted. The method allows to take into account arbitrary anisotropy of the interactions in the spin space, so it is not limited to purely Ising couplings. Within this approach, the Gibbs free energy is obtained, which allows to derive all the further interesting thermodynamic properties. In particular, we obtain an equation for the critical temperature of the second-order phase transitions for the model in question. In the study we focus our attention on the systems with ferromagnetic interactions decaying with the distance according to the power law

J(r)\propto r^{-n}

. We discuss the dependence of the critical temperature on the concentration of magnetic component and the index

n

for selected one-, two- and three-dimensional lattices.We confirm the absence of the critical concentration for a diluted magnet with infinite interaction range. In the regime of the low concentrations of magnetic component, we find a non-linear increase of the critical temperature with the concentration in the form of

T_{c}\propto p^{n/d}

, depending on the system dimensionality

d

and the index

n

, whereas

n > d

.

Ferromagnetism in one dimension is a observation which has been reported in a recent work [Gambardella et al., Nature (London) 416, 301 (2002)], and it is thought that anisotropy barriers are responsible in that relevant effect. In the present work, transitions between two different magnetic ordering phases are obtained as a result of an alternative approach. The critical temperature has been estimated by the Binder method. Ferromagnetic long-range interactions have been included in a special Hamiltonian through a power law that decays at large interparticle distance

r

as ${r}^{\ensuremath{-}\ensuremath{\alpha}}$ for $\ensuremath{\alpha}\ensuremath{\geqslant}0$. If the range of interactions decreases $(\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\infty})$, the trend of the critical temperature disappears, but if the range of interactions increases $(\ensuremath{\alpha}\ensuremath{\rightarrow}0)$, the trend of the critical temperature approaches to the mean field approximation. The crossover between these two limiting situations is discussed.

The spin- 1 / 2 XY model with quenched impurity sites is studied on three cubic lattices. Series of degree seven in inverse temperature at fixed magnetic-ion concentration and series of the same degree in magnetic-ion concentration at fixed temperature are generated for both the zero-field free energy and the zero-field fluctuation in the long-range order. The high-temperature series are too short for satisfactory analysis. The low-concentration series for the fluctuation however yield estimates for the critical exponent gamma , which appear to depend on temperature. The phase transition curve is estimated for all three lattices.

The thermodynamic behavior of systems with long range interactions is anomalous, because there are some problems about defining the thermodynamic limit. A way to solve the problem is to use scaled thermodynamic quantities. In this work, we use a nonextensive scaling into Hamiltonian and characterize transitions between two different magnetic ordering phases. The critical temperature is estimated by Binder method. Ferromagnetic long range interactions are included in a special Hamiltonian through a power law that decays at large interparticle distance r as r -α for α ≥ 0. In addition, we improve the known nonextensive scaling and obtain the critical temperature for several values of α.

We study the effects of dilution to the critical properties of site-diluted Ising model in two dimensions using Monte Carlo simulations. Quenched disorder from the dilution is incorporated into the Ising model via random empty sites on the square lattice of Ising spins. Thermodynamic quantities such as the magnetization

M

per spin, energy

E

per spin, magnetic susceptibility

χ

per spin, and specific heat

C

per spin are then calculated after the system has equilibrated. At small dilution concentrations

d<0.1

, we find that the value of the critical exponent

β

does not deviate from its pure Ising value. At higher dilution concentrations

d>0.1

, however, we find

β

to strongly depend on the value of

d

. We are able to locate a critical temperature

T_c

and a critical dilution concentration

d_c

where the phase transition occurs. We find

T_c

to depend linearly on

d

. In the phase diagrams of

M

,

E

,

χ

, and

C

, we find that the phase transition line eventually disappears at high dilutions. Our results suggest that there is a transition from Strong Universality at low dilution to Weak Universality at high dilution. Lastly, we find a wide and nonsingular increase in the magnetic susceptibility

χ

at the low temperature and high dilution region.

We report extensive Monte Carlo simulations of disordered Ising systems in the ferromagnetic region with concentrations of magnetic sites between p=1.0 and 0.5. The magnetization, the susceptibility, and the caloric properties have been studied in the critical region. The critical exponents \ensuremath{\beta} and \ensuremath{\gamma} as well as the universal amplitude A of the magnetization and the ratio

{\mathit{C}}_{+}

/${\mathit{C}}_{\mathrm{\ensuremath{-}}}$ of the susceptibility amplitudes have been determined with high precision. In addition to the cusplike specific heat, we have also measured the magnetization-energy correlation function \ensuremath{\Gamma}, which is a divergent thermal quantity. The corresponding critical exponent \ensuremath{\zeta}, which is related to the other exponents by \ensuremath{\zeta}=1-\ensuremath{\beta}=(\ensuremath{\alpha}+\ensuremath{\gamma})/2, has been determined. We have found that all quantities show power-law behavior within the temperature range of our simulation. All critical exponents change continuously with dilution. Even in the range of weak dilution (p\ensuremath{\ge}0.8), the effective critical exponents are concentration dependent and are clearly different from their pure system values. In the strongly diluted regime the critical exponents gradually reach new asymptotic behavior at p=0.5--0.6 with values of \ensuremath{\beta}=0.335\ifmmode\pm\else\textpm\fi{}0.01 and \ensuremath{\gamma}=1.49\ifmmode\pm\else\textpm\fi{}0.02. The exponent \ensuremath{\alpha} of the specific heat becomes -0.17\ifmmode\pm\else\textpm\fi{}0.04, which corresponds to a cusplike singularity. We conclude that disorder profoundly changes the critical behavior for weakly as well as strongly disordered spin systems.