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The quantum J1−J2 Blume–Capel model

Guerrero, Alejandra I.

Published Jan 1, 2025• 0 citations
OpenAlex

Abstract

In this study, it is investigated the effect of quantum fluctuations on the J1−J2 Blume–Capel model. The cluster mean field theory in the four site approximation is used to find the minimal (free) energy solutions. The ground state and finite temperature phase diagrams for the model with competitive ferromagnetic nearest neighbors (J1) and antiferromagnetic next nearest neighbors interactions (J2) are explored. It is found that the nature of phase transitions changes when quantum fluctuations and single-ion anisotropy parameter are varied. It is discussed the existence of tricritical points.

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In this study, it is investigated the effect of quantum fluctuations on the J1−J2 Blume–Capel model. The cluster mean field theory in the four site approximation is used to find the minimal (free) energy solutions. The ground state and finite temperature phase diagrams for the model with competitive ferromagnetic nearest neighbors (J1) and antiferromagnetic next nearest neighbors interactions (J2) are explored. It is found that the nature of phase transitions changes when quantum fluctuations and single-ion anisotropy parameter are varied. It is discussed the existence of tricritical points.
Using a 1/z expansion we study the correlation of fluctuations in a spin-1 Ising model with anisotropy. Previously, the 1/z expansion has been applied only to the second-order phase transitions of ferromagnets and granular superconductors. By imposing conditions on the free energy F and the parameter x=1-〈S1z2{\mathit{S}}_{1\mathit{z}}^{2}S1z2​〉, we use the 1/z technique to examine both the first- and second-order parts of the Blume-Capel phase diagram. In agreement with many numerical simulations and experiments on $^{3}\mathrm{\ensuremath{-}}^{4}$He mixtures, the value of x at the tricritical point is not shifted from its mean-field value of 2/3 to order 1/z. On the other hand, the tricritical values of the anisotropy parameter A and of the transition temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ are altered by the correlation of fluctuations. At least to order 1/z, all the derivatives of Tc{\mathit{T}}_{\mathit{c}}Tc​ are continuous across the tricritical point.
In this work, we investigate the thermodynamic properties of the quantum Blume-Capel model with spin S=5/2 S = 5/2 S=5/2 in the presence of transverse and random crystalline fields. The system is described by a Hamiltonian that includes ferromagnetic exchange interactions between nearest neighbors, a longitudinal single-ion anisotropy, and a transverse single-ion anisotropy. Using a mean-field approach based on Bogoliubov's inequality for the Gibbs free energy, we derive the fundamental thermodynamic potential and the equation of state for the magnetization. The influence of the longitudinal and transverse anisotropy parameters on the magnetic ordering and phase transitions is analyzed in detail. We present magnetization versus temperature diagrams for various combinations of the anisotropies, exploring both positive and negative values. Our results reveal that the system exhibits standard second-order phase transitions for most parameter ranges, with no evidence of tricritical behavior. However, for certain positive values of the anisotropies, the model displays a first-order phase transition within the ordered phase, characterized by a jump from a higher-spin ordered state to a lower-spin ordered state. The critical temperatures are shown to be sensitive to the magnitude and sign of the anisotropy parameters. In particular, negative transverse anisotropies favor magnetic order, raising the critical temperature, while positive anisotropies promote disorder, lowering the critical temperature. This study provides a comprehensive analysis of the phase diagram of the S=5/2 S = 5/2 S=5/2 quantum Blume-Capel model and highlights the role of transverse fields in modifying the critical behavior.
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 exploit the quantum cluster variational method (QCVM) to study the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model for quantum Ising spins. We first describe the QCVM and discuss how it is related to other mean field approximations. The phase diagram of the model is studied at the level of the Kikuchi approximation in square lattices as a function of the ratio between g=J2/J1g={J}_{2}/{J}_{1}g=J2​/J1​, the temperature and the longitudinal and transverse external fields. Our results show that quantum fluctuations may change the order of the transition and induce a gap between the ferromagnetic and the stripe phases. Moreover, when both longitudinal and transverse fields are present, thermal fluctuations and quantum effects contribute to the appearance of a nematic phase.
We present the zero-temperature phase diagram of a Heisenberg antiferromagnet on a frustrated triangular lattice with nearest-neighbor (J1)({J}_{1})(J1​) and next-nearest-neighbor (J2)({J}_{2})(J2​) interactions, in a magnetic field. We show that the classical model has an accidental degeneracy for all J2/J1{J}_{2}/{J}_{1}J2​/J1​ and all fields, but the degeneracy is lifted by quantum fluctuations. We show that at large spin SSS, for J2/J1<1/8{J}_{2}/{J}_{1}<1/8J2​/J1​<1/8, quantum fluctuations select the same sequence of three sublattice co-planar states in a field as for J2=0{J}_{2}=0J2​=0, and for 1/8<J2/J1<11/8<{J}_{2}/{J}_{1}<11/8<J2​/J1​<1, they select the canted stripe state for all nonzero fields. The transition between the two states is first order in all fields, with the hysteresis width set by quantum fluctuations. We study the model with arbitrary SSS, including S=1/2S=1/2S=1/2, near the saturation field by exploring the fact that near saturation the density of bosons is small for all SSS. We show that for S>1S>1S>1, the transition remains first order, with a finite hysteresis width, but for S=1/2S=1/2S=1/2 and, possibly, S=1S=1S=1, there appears a new intermediate phase without a quasiclassical long-range order.