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Journal of Physics Condensed Matter

Critical point of a para–ferrimagnetic phase transition of the ANNNI model in a field

J. Szṅajd

Published Oct 8, 2012• 3 citations
Publisher

Abstract

The finite field para-ferrimagnetic phase transition in the axial next-nearest-neighbor Ising (ANNNI) model is studied by using the linear perturbation real space renormalization group transformation. The method is examined in systems of ferromagnetic and antiferromagnetic Ising chains coupled by ferromagnetic interactions in the longitudinal field. As one would expect, the external field in the first case destroys the continuous phase transition and in the second shifts the critical point toward a lower temperature according to the square law. For the ferromagnetic chains coupled by the competing interchain nearest-neighbor J(1) < 0 and next-nearest-neighbor J(2) < 0 interactions, the external field changes the character of the phase transition from continuous to discontinuous. However, it has been found that for the ratio J(2)/J(1) around 0.5 there is a critical value of the field for which an isolated critical point exists. The temperature dependences of the specific heat for several values of J(2)/J(1) and the external field are presented.

DOI
10.1088/0953-8984/24/43/436006

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The finite field para-ferrimagnetic phase transition in the axial next-nearest-neighbor Ising (ANNNI) model is studied by using the linear perturbation real space renormalization group transformation. The method is examined in systems of ferromagnetic and antiferromagnetic Ising chains coupled by ferromagnetic interactions in the longitudinal field. As one would expect, the external field in the first case destroys the continuous phase transition and in the second shifts the critical point toward a lower temperature according to the square law. For the ferromagnetic chains coupled by the competing interchain nearest-neighbor J(1) < 0 and next-nearest-neighbor J(2) < 0 interactions, the external field changes the character of the phase transition from continuous to discontinuous. However, it has been found that for the ratio J(2)/J(1) around 0.5 there is a critical value of the field for which an isolated critical point exists. The temperature dependences of the specific heat for several values of J(2)/J(1) and the external field are presented.
The linear perturbation renormalization group (LPRG) is used to study the phase transition of the weakly coupled Ising chains with intrachain (J) and interchain nearest-neighbor (J_{1}) and next-nearest-neighbor (J_{2}) interactions forming the triangular and rectangular lattices in a field. The phase diagrams with the frustration point at J_{2}=-J_{1}/2 for a rectangular lattice and J_{2}=-J_{1} for a triangular lattice have been found. The LPRG calculations support the idea that the phase transition is always continuous except for the frustration point and is accompanied by a divergence of the specific heat. For the antiferromagnetic chains, the external field does not change substantially the shape of the phase diagram. The critical temperature is suppressed to zero according to the power law when approaching the frustration point with an exponent dependent on the value of the field.
The magnetic critical exponent of the two-dimensional axial next-nearest-neighbor Ising model is estimated by means of the finite-size scaling of the order parameter. For a finite system without ordering field, this magnitude is computed from the two leading eigenvectors of the transfer matrix. The ferromagnetic-paramagnetic transition temperatures estimated with this method exhibit a small lattice-size dependence. Moreover, they are in good agreement with those obtained using phenomenological renormalization, and also with analytical approximations. The correlation-length critical exponent is calculated by means of phenomenological renormalization. The results found with these numerical techniques indicate that the ferromagnetic-paramagnetic transition of the model seems to be continuous, in contradiction with a previous result obtained using the persistence-length criterion, which indicated that the transition is always first order. We found that the phase transition is first order only when the ratio of the next-nearest-neighbor and nearest-neighbor interactions equals one-half.
The linear-perturbation renormalization-group transformation is used to study s=12s=\frac{1}{2}s=21​ spin chains with antiferromagnetic intrachain interactions J<0J<0J<0 coupled by weak ferromagnetic interchain interactions ${J}^{\ensuremath{'}}>0$ in the longitudinal external magnetic field H. The field dependence of the specific heat for several models (XY,(XY,(XY, anisotropic Heisenberg, and Ising) has been found. The specific heat of the one-dimensional models ${(J}^{\ensuremath{'}}=0)$ exhibits a maximum at H=Hmax{H=H}_{\mathrm{max}}H=Hmax​ which decreases and shifts to higher fields with increasing temperature, and a minimum which disappears for the temperature higher than some characteristic temperature Tg.{T}_{g}.Tg​. For the antiferromagnetic XY\mathrm{XY}XY and Ising models the specific-heat maximum is shifted linearly, whereas for the Heisenberg model the shift cannot be fitted to a single power law. For an arbitrary small ferromagnetic interchain interaction ${(J}^{\ensuremath{'}}>0)$ the specific heat as a function of temperature has a singularity characterizing the antiferromagnetic phase transition. From the flow diagram analysis the dependence of the transition temperature on the interchain interaction and the external field has been found.
The critical behavior at the phase transition of the ferromagnetic two-dimensional anisotropic Ising model with next-nearest neighbor (NNN) couplings in the presence of the field is determined using mainly Monte Carlo (MC) method. This method is used to investigate the phase diagram of the model and to verify the existence of a divergence at null temperature which often appears in two-dimensional systems. We analyze also the influence of the report of the NNN interactions [Formula: see text] and the magnetic field [Formula: see text] on the critical temperature of the system, and we show that the critical temperature depends on the magnetic field for positive values of the interaction. Finally, we have investigated other thermodynamical qualities such as the magnetic susceptibility [Formula: see text]. It has been shown that their thermal behavior depends qualitatively and quantitatively on the strength of NNN interactions and the magnetic field.
We investigate the mixed-spin (spin-1/2 and spin-1) antiferromagnetic Ising model in two dimensions through a real-space renormalization group transformation. The model includes nearest and next-nearest neighbor interactions and is suitable to describe ferrimagnetic ordering. We present the finite-temperature phase diagram of the model. The presence of the second-neighbor bonds enlarge the ferrimagnetic phase. Our results do not indicate the existence of a tricritical point, contrary to previous effective-field calculations.
We use Monte Carlo and transfer matrix methods in combination with extrapolation schemes to determine the phase diagram of the two-dimensional superantiferromagnetic (SAF) Ising model with next-nearest-neighbor (NNN) interactions in a magnetic field. The interactions between nearest-neighbor (NN) spins are ferromagnetic along x, and antiferromagnetic along Y. We find that for sufficiently low temperatures and fields, there exists a region limited by a critical line of second-order transitions separating a SAF phase from a magnetically induced paramagnetic phase. We did not find any region with either first-order transition or with reentrant behavior. The NNN couplings produce either an expansion or a contraction of the SAF phase. Expansion occurs when the interactions are antiferromagnetic, and contraction when they are ferromagnetic. There is a critical ratio R(c)=1/2 between NNN and NN couplings, beyond which the SAF phase no longer exists.