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physica status solidi (b)

Phase diagram of the three‐dimensional anisotropic Heisenberg anti‐ferromagnetic model

J. C. S. Rocha, B. V. Costa

Published Sep 9, 2009• 0 citations
Publisher

Abstract

Abstract In this work we use numerical Monte Carlo techniques to study the phase diagram of the three‐dimensional classical Heisenberg anti‐ferromagnetic (AF) model with single‐ion anisotropy model with an uniform magnetic field applied along the easy axis. Early experimental as well as theoretical calculations have shown that the phase diagram of the model is quite rich. We simulated the model on a simple cubic lattices using the Wang–Landau sampling and Metropolis algorithm. Finite size scaling theory is used to obtain the phase diagram. At low field and low temperature the system exhibits AF order separated from a spin‐flop (SF) phase by a first‐order line. As temperature grows a paramagnetic phase is reached by crossing a second‐order line both from the AF phase and from the SF phase. The critical exponents are obtained for each transition line. Contrary to early works we find that the line separating the SF phase from the paramagnetic phase is in the Ising class of universality.

DOI
10.1002/pssb.200844461

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Abstract In this work we use numerical Monte Carlo techniques to study the phase diagram of the three‐dimensional classical Heisenberg anti‐ferromagnetic (AF) model with single‐ion anisotropy model with an uniform magnetic field applied along the easy axis. Early experimental as well as theoretical calculations have shown that the phase diagram of the model is quite rich. We simulated the model on a simple cubic lattices using the Wang–Landau sampling and Metropolis algorithm. Finite size scaling theory is used to obtain the phase diagram. At low field and low temperature the system exhibits AF order separated from a spin‐flop (SF) phase by a first‐order line. As temperature grows a paramagnetic phase is reached by crossing a second‐order line both from the AF phase and from the SF phase. The critical exponents are obtained for each transition line. Contrary to early works we find that the line separating the SF phase from the paramagnetic phase is in the Ising class of universality.
We study the phase diagram of the three-dimensional classical ferromagnetic Heisenberg model with an easy-plane crystalline anisotropy and an easy-axis exchange anisotropy through Monte Carlo simulations. We employ the Metropolis algorithm together with single-histogram techniques in order to characterize the transitions in each region of the phase diagram. Our results reveal, besides the disordered phase, the existence of Ising-like and XY-like ordered phases which are separated by a first-order transition line.
We study the ±J three-dimensional (3D) Ising model with a spatially uniaxial anisotropic bond randomness on the simple cubic lattice. The ±J random exchange is applied on the xy planes, whereas, in the z direction, only a ferromagnetic exchange is used. After sketching the phase diagram and comparing it with the corresponding isotropic case, the system is studied at the ferromagnetic-paramagnetic transition line using parallel tempering and a convenient concentration of antiferromagnetic bonds (p(z)=0;p(xy)=0.176). The numerical data clearly point out a second-order ferromagnetic-paramagnetic phase transition belonging in the same universality class with the 3D random Ising model. The smooth finite-size behavior of the effective exponents, describing the peaks of the logarithmic derivatives of the order parameter, provides an accurate estimate of the critical exponent 1/ν=1.463(3), and a collapse analysis of magnetization data gives an estimate of β/ν=0.516(7). These results are in agreement with previous papers and, in particular, with those of the isotropic ±J three-dimensional Ising model at the ferromagnetic-paramagnetic transition line, indicating the irrelevance of the introduced anisotropy.
Abstract A great triumph of statistical physics in the latter part of the 20th century was the understanding of critical behavior and universality at second-order phase transitions. In contrast, first-order transitions were believed to have no common features. However, we argue that the classic, first-order “spin-flop” transition (between the antiferromagnetic and the rotationally degenerate, canted state) in an anisotropic antiferromagnet in a magnetic field exhibits a new kind of universality. We present a finite-size scaling theory for a first-order phase transition where a continuous symmetry is broken using an approximation of Gaussian probability distributions with a phenomenological degeneracy factor “q” included, where “q” characterizes the relative degeneracy of the ordered phases. Predictions are compared with high resolution Monte Carlo simulations of the three-dimensional, XXZ Heisenberg antiferromagnet in a field to study the finite-size behavior for L×L×L simple cubic lattices. The field dependence of all moments of the order parameters exhibit universal intersections at the spin-flop transition. Our Monte Carlo data agree with theoretical predictions for asymptotic large L behavior. Our theory yields q = π, and we present numerical evidence that is compatible with this prediction. The agreement between the theory and simulation implies a heretofore unknown universality.
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.
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 have studied the anisotropic three-dimensional nearest-neighbor Ising model with competitive interactions in an uniform longitudinal magnetic field HHH. The model consists of ferromagnetic interaction Jx(Jz)J_{x}(J_{z})Jx​(Jz​) in the x(z)x(z)x(z) direction and antiferromagnetic interaction JyJ_{y}Jy​ in the yyy direction. We have compared our calculations within a effective-field theory in clusters with four spins (EFT-4) in the simple cubic (sc) lattice with traditional Monte Carlo (MC) simulations. The phase diagrams in the h−kBT/Jxh-k_{B}T/J_{x}h−kB​T/Jx​ plane (h=H/Jxh=H/J_{x}h=H/Jx​) were obtained for the particular case λ1=Jy/Jx(λ2=Jz/Jx)=1λ_{1}=J_{y}/J_{x} (λ_{2}=J_{z}/J_{x})=1λ1​=Jy​/Jx​(λ2​=Jz​/Jx​)=1 (anisotropic sc). Our results indicate second-order frontiers for all values of HHH for the particular case λ2=0λ_{2}=0λ2​=0 (square lattice), while in case λ1=λ2=1λ_{1}=λ_{2}=1λ1​=λ2​=1, we observe first- and second-order phase transitions in the low and high temperature limits, respectively, with presence of a tricritical point. Using EFT-4, a reentrant behavior at low temperature was observed in contrast with results of MC.