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arXiv:2011.11786 · Physical review. B./Physical review. B

Quantum cluster variational method and phase diagram of the quantum ferromagnetic J1−J2 model

Eduardo Domínguez, Carlos E. Lopetegui, Roberto Mulet

Published Jul 9, 2021• 10 citations
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Abstract

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 $g={J}_{2}/{J}_{1}$, 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.

DOI
10.1103/physrevb.104.014205

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We exploit the quantum cluster variational method (QCVM) to study the J1\ensuremath-J2{J}_{1}\text{\ensuremath{-}}{J}_{2}J1​\ensuremath-J2​ 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.
The J(1)-J(2) Ising model in the square lattice in the presence of an external field is studied by two approaches: the cluster variation method (CVM) and Monte Carlo simulations. The use of the CVM in the square approximation leads to the presence of a new equilibrium phase, not previously reported for this model: an Ising-nematic phase, which shows orientational order but not positional order, between the known stripes and disordered phases. Suitable order parameters are defined, and the phase diagram of the model is obtained. Monte Carlo simulations are in qualitative agreement with the CVM results, giving support to the presence of the new Ising-nematic phase. Phase diagrams in the temperature-external field plane are obtained for selected values of the parameter κ=J(2)/|J(1)| which measures the relative strength of the competing interactions. From the CVM in the square approximation we obtain a line of second order transitions between the disordered and nematic phases, while the nematic-stripes phase transitions are found to be of first order. The Monte Carlo results suggest a line of second order nematic-disordered phase transitions in agreement with the CVM results. Regarding the stripes-nematic transitions, the present Monte Carlo results are not precise enough to reach definite conclusions about the nature of the transitions.
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 the coupled-cluster method for high orders of approximation and Lanczos exact diagonalization we study the ground-state phase diagram of a quantum spin-12\frac{1}{2}21​ J1{J}_{1}J1​-J2{J}_{2}J2​ model on the square lattice with plaquette structure. We consider antiferromagnetic (J1>0{J}_{1}>0J1​>0) as well as ferromagnetic (J1<0{J}_{1}<0J1​<0) nearest-neighbor interactions together with frustrating antiferromagnetic next-nearest-neighbor interaction J2>0{J}_{2}>0J2​>0. The strength of interplaquette interaction $\ensuremath{\lambda}$ varies between $\ensuremath{\lambda}=1$ (which corresponds to the uniform J1{J}_{1}J1​-J2{J}_{2}J2​ model) and $\ensuremath{\lambda}=0$ (which corresponds to isolated frustrated 4-spin plaquettes). While on the classical level ($s\ensuremath{\rightarrow}\ensuremath{\infty}$) both versions of models (i.e., with ferro- and antiferromagnetic J1{J}_{1}J1​) exhibit the same ground-state behavior, the ground-state phase diagram differs basically for the quantum case s=1/2s=1/2s=1/2. For the antiferromagnetic case (J1>0{J}_{1}>0J1​>0) N\'eel antiferromagnetic long-range order at small J2/J1{J}_{2}/{J}_{1}J2​/J1​ and $\ensuremath{\lambda}\ensuremath{\gtrsim}0.47$ as well as collinear striped antiferromagnetic long-range order at large J2/J1{J}_{2}/{J}_{1}J2​/J1​ and $\ensuremath{\lambda}\ensuremath{\gtrsim}0.30$ appear which correspond to their classical counterparts. Both semiclassical magnetic phases are separated by a nonmagnetic quantum paramagnetic phase. The parameter region, where this nonmagnetic phase exists, increases with decreasing $\ensuremath{\lambda}$. For the ferromagnetic case (J1<0{J}_{1}<0J1​<0) we have the trivial ferromagnetic ground state at small J2/∣J1∣{J}_{2}/|{J}_{1}|J2​/∣J1​∣. By increasing J2{J}_{2}J2​ this classical phase gives way for a semiclassical plaquette phase, where the plaquette block spins of length s=2s=2s=2 are antiferromagnetically long-range ordered. Further increasing of J2{J}_{2}J2​ then yields collinear striped antiferromagnetic long-range order for $\ensuremath{\lambda}\ensuremath{\gtrsim}0.38$, but a nonmagnetic quantum paramagnetic phase $\ensuremath{\lambda}\ensuremath{\lesssim}0.38$.
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