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arXiv:cond-mat/0506033 · Physical Review B

Ferromagnetism in one dimension: Critical temperature

Sergio Curilef, Luis Alberto del Pino, P. A. Orellana

Published Dec 8, 2005• 16 citations
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Abstract

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.

DOI
10.1103/physrevb.72.224410

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•Jan 1, 2015•0 citations
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 rrr as r\ensuremath−\ensuremathα{r}^{\ensuremath{-}\ensuremath{\alpha}}r\ensuremath−\ensuremathα for \ensuremathα\ensuremath⩾0\ensuremath{\alpha}\ensuremath{\geqslant}0\ensuremathα\ensuremath⩾0. If the range of interactions decreases (\ensuremathα\ensuremath→\ensuremath∞)(\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\infty})(\ensuremathα\ensuremath→\ensuremath∞), the trend of the critical temperature disappears, but if the range of interactions increases (\ensuremathα\ensuremath→0)(\ensuremath{\alpha}\ensuremath{\rightarrow}0)(\ensuremathα\ensuremath→0), the trend of the critical temperature approaches to the mean field approximation. The crossover between these two limiting situations is discussed.
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.
The one-dimensional Ising model in a random field and with power-law decaying ferromagnetic bonds is studied at zero temperature. Comparing the scaling of the energy contributions of the ferromagnetic domain wall flip and of the random field \`a la Imry-Ma, a threshold value for the power $\ensuremath{\rho}$ of the long-range interaction can be determined, beyond which no critical behavior occurs. The critical threshold value is ${\ensuremath{\rho}}_{c}=3/2$, at a difference with the zero field model in which ${\ensuremath{\rho}}_{c}=2$. This prediction is analyzed by numerical computation of the ground states below, at, and above this threshold value. The analogy between the critical behavior of long-range one-dimensional systems and in short-range DDD-dimensional systems is investigated. At the critical threshold value of $\ensuremath{\rho}$, corresponding to the lower critical dimension, numerical evidence is found for a zero temperature transition at a finite critical field. Possible finite size crossover effects are discussed in this case. Some implications for the critical behavior of spin glasses in a field are conjectured.
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)∝r−nJ(r)\propto r^{-n}J(r)∝r−n. We discuss the dependence of the critical temperature on the concentration of magnetic component and the index nnn 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 Tc∝pn/dT_{c}\propto p^{n/d}Tc​∝pn/d, depending on the system dimensionality ddd and the index nnn, whereas n>dn > dn>d.
The behavior of the spherical model of a ferromagnet with an interaction energy between the magnetic spins which varies with distance as $\frac{1}{{r}^{d+\ensuremath{\sigma}}}$ (where ddd is the dimensionality of the lattice and $\ensuremath{\sigma}>0$) is analyzed. It is shown that the model exhibits a ferromagnetic transition in one and two dimensions, providing $0<\ensuremath{\sigma}<d$. (The usual spherical model with nearest-neighbor interactions does not have a transition in one and two dimensions.) The critical-point behavior is investigated. It is found that the singularities in the specific heat and susceptibility are dependent on $\ensuremath{\sigma}$ and ddd, but the behavior of the magnetization is independent of $\ensuremath{\sigma}$ and ddd. In three dimensions the susceptibility diverges as ${(T\ensuremath{-}{T}_{c})}^{\ensuremath{-}\ensuremath{\gamma}}$, where $\ensuremath{\gamma}=1$ for $0<\ensuremath{\sigma}<\frac{3}{2}$, $\ensuremath{\gamma}=\frac{\ensuremath{\sigma}}{(3\ensuremath{-}\ensuremath{\sigma})}$ for $\frac{3}{2}<\ensuremath{\sigma}<2$ and $\ensuremath{\gamma}=2$ for $\ensuremath{\sigma}>2$. The asymptotic form of the spin-spin correlation function $\ensuremath{\Gamma}(\mathrm{r})$ is studied in the neighborhood of the critical temperature Tc{T}_{c}Tc​. At T=TcT={T}_{c}T=Tc​, $\ensuremath{\Gamma}(\mathrm{r})$ decays for large rrr as $\frac{1}{{r}^{d\ensuremath{-}\ensuremath{\sigma}}}$. Several two-dimensional models with long-range interactions falling off as 1r2\frac{1}{{r}^{2}}r21​ in certain directions only are also investigated.
We study the nature of the phase transition of the qqq-state Potts model with long-range ferromagnetic interactions decaying as $1/{r}^{d+\ensuremath{\sigma}}$, in dimension d=1d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1d=1, using a histogram Monte Carlo (MC) technique. The model can exhibit a first-order transition or a second-order phase transition with nonstandard critical exponents. The critical value of qqq above which a first-order transition occurs decreases with decreasing $\ensuremath{\sigma}$, from qc=8{q}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}8qc​=8 for $\ensuremath{\sigma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ to qc=2{q}_{c}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2qc​=2 for $\ensuremath{\sigma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.3$. Detailed results for various $\ensuremath{\sigma}$ will be shown and discussed. Mean-field calculation confirms the tendency of our MC results.
This report will study the theory of renormalization group used on a ferromagnet to calculate magnetic phase transitions. As the ferromagnet's temperature changes, so does interatomic interaction, causing either a lingering magnetization at low temperatures, or a random state at high temperatures. At the phase transition, where the magnetization first appears/disappears, depending from which side the critical temperature is approached, the ferromagnet will behave in a certain way that causes some quantities to receive power law behaviours, so called critical exponents. The theory of renormalization group is used to find the critical point and these power units on two different two-dimensional lattices, the square lattice and the triangular lattice. The first step is to study how atoms couple to each other in one-dimension. It is possible to group atoms together into blocks. These blocks will interact with each other through new couplings. It is possible to calculate the critical point and critical exponents by comparing the block couplings with the former atomic couplings and then find the exponents. The theory of renormalization does seem to work in general, as the result will be able to explain experimentally observed relation between exponents.