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Physical Review E

Phase diagrams of a spin-1 Ising system with competing short- and long-range interactions

Octavio D. Rodrı́guez Salmon, J. Ricardo de Sousa, Minos A. Neto

Published Sep 15, 2015• 14 citations
Publisher

Abstract

We have studied the phase diagrams of the one-dimensional spin-1 Blume-Capel model with anisotropy constant D, in which equivalent-neighbor ferromagnetic interactions of strength -J are superimposed on nearest-neighbor antiferromagnetic interactions of strength K. A rich critical behavior is found due to the competing interactions. At zero temperature two ordered phases exist in the D/J-K/J plane, namely the ferromagnetic (F) and the antiferromagnetic one (AF). For lower values of D/J(D/J<0.25) these two ordered phases are separated by the point K_{c}=0.25J. For 0.25<D/J≤0.50, the paramagnetic phase P emerges in a region separated between the lines determined by D/J=0.5-K/J and D/J=K/J. For D/J>0.5, only phases AF and F exist and are separated by a line given by D/J=K/J. At finite temperatures, we found that the ferromagnetic region of the phase diagram in the k_{B}T/J-D/J plane is enriched by another ferromagnetic phase F^{^{'}} above a first-order line for 0.195<K/J<0.250. This first-order line, which separates phases F and F^{^{'}}, begins at a coexistence point, where phases F,F^{^{'}}, and P coexist, and ends at an ordered critical point. Similarly, we found that the phase F^{^{'}} is present in the phase diagram in the k_{B}T/J-K/J plane for 0.228<D/J<0.286.

DOI
10.1103/physreve.92.032120

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We have studied the phase diagrams of the one-dimensional spin-1 Blume-Capel model with anisotropy constant D, in which equivalent-neighbor ferromagnetic interactions of strength -J are superimposed on nearest-neighbor antiferromagnetic interactions of strength K. A rich critical behavior is found due to the competing interactions. At zero temperature two ordered phases exist in the D/J-K/J plane, namely the ferromagnetic (F) and the antiferromagnetic one (AF). For lower values of D/J(D/J<0.25) these two ordered phases are separated by the point K_{c}=0.25J. For 0.25<D/J≤0.50, the paramagnetic phase P emerges in a region separated between the lines determined by D/J=0.5-K/J and D/J=K/J. For D/J>0.5, only phases AF and F exist and are separated by a line given by D/J=K/J. At finite temperatures, we found that the ferromagnetic region of the phase diagram in the k_{B}T/J-D/J plane is enriched by another ferromagnetic phase F^{^{'}} above a first-order line for 0.195<K/J<0.250. This first-order line, which separates phases F and F^{^{'}}, begins at a coexistence point, where phases F,F^{^{'}}, and P coexist, and ends at an ordered critical point. Similarly, we found that the phase F^{^{'}} is present in the phase diagram in the k_{B}T/J-K/J plane for 0.228<D/J<0.286.
Abstract We study the effect of competing interactions on ensemble inequivalence. We consider a one-dimensional Ising model with ferromagnetic mean-field interactions and short-range nearest-neighbor (NN) and next-NN couplings which can be either ferromagnetic or antiferromagnetic. Despite the relative simplicity of the model, our calculations in the microcanonical ensemble reveal a rich phase diagram. The comparison with the corresponding phase diagram in the canonical ensemble shows the presence of phase transition points and lines which are different in the two ensembles. As an example, in a region of the phase diagram where the canonical ensemble shows a critical point and a critical end point, the microcanonical ensemble has an additional critical point and also a triple point. The regions of ensemble inequivalence typically occur at lower temperatures and at larger absolute values of the competing couplings. The presence of two free parameters in the model allows us to obtain a fourth-order critical point, which can be fully characterized by deriving its Landau normal form.
The d=1 Ising ferromagnet and spin glass with long-range power-law interactions J r^-a is studied for all interaction range exponents a by a renormalization-group transformation that simultaneously projects local ferromagnetism and antiferromagnetism. In the ferromagnetic case, J>0, a finite-temperature ferromagnetic phase occurs for interaction range 0.742, the phase transition temperature discontinuously drops to zero and for a>2 there is no ordered phase above zero temperature, also as predicted by rigorous results. At the other end, on approaching a=0.74 from above, namely increasing the range of the interaction, the phase transition temperature diverges to infinity, meaning that, at all non-infinite temperatures, the system is ferromagnetically ordered. Thus, the equivalent-neighbor interactions regime is entered before (a > 0) the neighbors become equivalent, namely before the interactions become equal for all separations. The critical exponents alpha,beta, gamma,delta,eta,nu are calculated, from a large recursion matrix, varying as function of a. For antiferromagnetic J<0, all triplets of spins at all ranges have competing interactions and this highly frustrated system does not have an ordered phase. In the spin-glass system, where all couplings for all separations are randomly ferromagnetic or antiferromagnetic (with probability p), a finite-temperatures spin-glass phase is obtained in the absence of antiferromagnetic phase. In the spin-glass phase, the signature chaotic behavior under scale change occurs in a richer version than previously: In the long-range interaction of this system, the interactions at every separation become chaotic, yielding a piecewise chaotic interaction function.
The phase diagram of the anisotropic quantum spin-1/2 Heisenberg antiferromagnet in the presence of competitive nearest (J1)({J}_{1})(J1​)- and next-nearest $({J}_{2}=\ensuremath{\alpha}{J}_{1})$-neighbor interactions on a simple cubic lattice is studied within the framework of the differential operator technique. The Hamiltonian is solved by using an effective-field theory in a two-spin cluster. We propose a function for the free energy, which is similar to the Landau expansion, in order to find the ground state (T=0)(T=0)(T=0) and the finite-TTT phase diagram. At zero temperature, we find a first-order quantum phase transition in the point ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta})$ between the antiferromagnetic (AF) and superantiferromagnetic (SAF) phases. This is dependent on the anisotropy parameter $\ensuremath{\Delta}$ ($\ensuremath{\Delta}=0$ and 1 correspond to the isotropic Heisenberg and Ising models, respectively). The quantum critical values of ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}\ensuremath{\ne}1)={\ensuremath{\alpha}}_{c}^{q}(\ensuremath{\Delta})$ are smaller than those of the classical model ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}=1)={\ensuremath{\alpha}}_{c}^{c}=1/4$ (for example, for $\ensuremath{\Delta}=0$, we find ${\ensuremath{\alpha}}_{c}^{q}(0)=0.21$). The former monotonically increases with the anisotropy parameter $\ensuremath{\Delta}$. The phase diagram in $T\text{\ensuremath{-}}\ensuremath{\alpha}$ plane presents a second-order transition at ${T}_{2}(\ensuremath{\alpha})$ between the AF and paramagnetic (P) phases. The phase transition between the SAF and P phases is of first-order ${T}_{1}(\ensuremath{\alpha})$ for all values of $\ensuremath{\Delta}∊[0,1]$. The phase transition lines ${T}_{1}(\ensuremath{\alpha})$ and ${T}_{2}(\ensuremath{\alpha})$ merge in a critical end point (CEP) that is dependent on $\ensuremath{\Delta}$ [i.e., ${T}_{\text{CEP}}(\ensuremath{\Delta})$ and ${\ensuremath{\alpha}}_{\text{CEP}}(\ensuremath{\Delta})$ are the temperature and frustration parameters at the CEP]. The ordered phases (AF and SAF) are separated by first-order transitions and exhibit a nontrivial re-entrant behavior at low temperatures.
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.
We perform Monte-Carlo simulations of the anti-ferromagnetic (AF) spin-1 Blume- Capel (BC) model and the AF Ising model on triangular lattice. We estimate the exact critical magnetic fields for both models at zero temperature using the Wang-Landau sampling method. We also show the phase diagrams and the critical lines for the models using the joint density functions. We find that the shapes of critical lines for the models are identical, but the phase transitions across the critical lines are different.
Phase diagram and critical phenomena of a frustrated Blume–Capel model with alternating single-ion anisotropy ±D, nearest- and next-nearest-neighbour exchange interactions with equal magnitude J are studied by the Monte Carlo simulations using Wang–Landau sampling and nonequilibrium relaxation method. The collinear ordered phase due to next-nearest-neighbour interaction and partial ordered phase due to alternating anisotropy are confirmed together with a paramagnetic phase. We obtained the numerical results which strongly suggests that a single phase transition at D=0 splits into two Ising lines as soon as D is switched on. With the increase of D, the critical point of the easy-axis sublattice stays on a usual Ising-like critical line while that of the easy-plane sublattice passes through the tricritical point and the transition becomes of the first order as in the usual Blume–Capel model.