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arXiv:1603.02601

Layered frustrated antiferromagnetic Heisenberg spin model: role of inplane frustration and interlayer coupling

Haque, Md Mahfoozul, M. A. H. Ahsan, Jamal Berakdar, Dipan K. Ghosh

Published Mar 8, 2016• 0 citations
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Abstract

We present an exact diagonalization study on layered $J_{1}-J_{2}$ antiferromagnetic Heisenberg spin model to examine the role of frustration induced by inplane next-nearest neighbor coupling $J_{2}$, in presence of interlayer antiferromagnetic coupling $J_{\perp}$. A finite lattice of 24 spins in layered geometry of $(4\times 3)\oplus (4\times 3)$ is considered and the resulting Hamiltonian matrix diagonalized using Davidson iterative algorithm to obtain the ground and a few low-lying excited states. The lattice size ($24$ spins with Hilbert space dimensionality of $2704156$ in $S_{z}^{tot}=0$ subspace) has been kept relatively small because of the large number of runs required to sample $J_{1}-J_{2}-J_{\perp}$ parameter space. Quantities like spin-gap, Shannon entropy, spin-spin correlation, static spin structure-factor, magnetic specific-heat and magnetic susceptibility are calculated for various values of spin-spin coupling parameters. With increase in interlayer coupling, the system is driven to states with long range order and the interval of quantum paramagnetically disordered state, sandwiched between Néel and collinear ordered states, narrows on the scale of inplane frustration parameter.

DOI
10.48550/arxiv.1603.02601

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We study the ground-state (GS) properties of the frustrated spin-$\frac{1}{2}$ ${J}_{1}$--${J}_{2}$--${J}_{3}$ Heisenberg model on the two-dimensional honeycomb lattice with ferromagnetic nearest-neighbor (${J}_{1}=\ensuremath{-}1$) exchange and frustrating antiferromagnetic next-nearest-neighbor (${J}_{2}>0$) and next-next-nearest-neighbor (${J}_{3}>0$) exchanges, for the case ${J}_{3}={J}_{2}$. We use the coupled-cluster method implemented to high orders of approximation, complemented by the Lanczos exact diagonalization of a large finite lattice with 32 sites, in order to calculate the GS energy, magnetic order parameter, and spin-spin correlation functions. In one scenario we find a quantum phase transition point between regions characterized by ferromagnetic order and a form of antiferromagnetic (``striped'') collinear order at ${J}_{2}^{c}\ensuremath{\approx}0.1095\ifmmode\pm\else\textpm\fi{}0.0005$, which is below the corresponding hypothetical transition point at ${J}_{2}^{\mathrm{cl}}=\frac{1}{7}$ ($\ensuremath{\approx}0.143$) for the classical version of the model, in which we momentarily ignore the intervening noncollinear spiral phase in the region $\frac{1}{10}<{J}_{2}<\frac{1}{5}$. Hence we see that quantum fluctuations appear to stabilize somewhat the collinear antiferromagnetic order in preference to the ferromagnetic order in this model. We compare results for the present ferromagnetic case (with ${J}_{1}=\ensuremath{-}1$) to previous results for the corresponding antiferromagnetic case (with ${J}_{1}=+1$). The magnetic order parameter is found to behave similarly for the ferromagnetic and the antiferromagnetic models for large values of the frustration parameter ${J}_{2}$. However, there are considerable differences in the behavior of the order parameters for the two models for ${J}_{2}/|{J}_{1}|\ensuremath{\lesssim}0.6$. For example, the quasiclassical collinear magnetic long-range order for the antiferromagnetic model (with ${J}_{1}=+1$) breaks down at ${J}_{2}^{{c}_{2}}\ensuremath{\approx}0.60$, whereas the ``equivalent'' point for the ferromagnetic model (with ${J}_{1}=\ensuremath{-}1$) occurs at ${J}_{2}^{c}\ensuremath{\approx}0.11$. Unlike in the antiferromagnetic model (with ${J}_{1}=+1$), where a plaquette valence-bond crystal phase intrudes between the two corresponding quasiclassical antiferromagnetic phases (with N\'eel and striped order) for ${J}_{2}^{{c}_{1}}<{J}_{2}<{J}_{2}^{{c}_{2}}$, with ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.47$, we find no clear indications at all in the ferromagnetic model for an intermediate magnetically disordered phase between the corresponding phases exhibiting ferromagnetic and striped order. Instead the evidence for the ferromagnetic model (with ${J}_{1}=\ensuremath{-}1$) points to one of two scenarios: either there is a direct first-order transition between the two magnetically ordered phases, as mentioned above; or there exists an intervening phase between them in the very narrow range $0.10\ensuremath{\lesssim}{J}_{2}\ensuremath{\lesssim}0.12$, which is probably a remnant of the spiral phase that exists in the classical counterpart of the model over the larger range $\frac{1}{10}<{J}_{2}<\frac{1}{5}$.

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We present an exact diagonalization study on layered J1−J2J_{1}-J_{2}J1​−J2​ antiferromagnetic Heisenberg spin model to examine the role of frustration induced by inplane next-nearest neighbor coupling J2J_{2}J2​, in presence of interlayer antiferromagnetic coupling J⊥J_{\perp}J⊥​. A finite lattice of 24 spins in layered geometry of (4×3)⊕(4×3)(4\times 3)\oplus (4\times 3)(4×3)⊕(4×3) is considered and the resulting Hamiltonian matrix diagonalized using Davidson iterative algorithm to obtain the ground and a few low-lying excited states. The lattice size (242424 spins with Hilbert space dimensionality of 270415627041562704156 in Sztot=0S_{z}^{tot}=0Sztot​=0 subspace) has been kept relatively small because of the large number of runs required to sample J1−J2−J⊥J_{1}-J_{2}-J_{\perp}J1​−J2​−J⊥​ parameter space. Quantities like spin-gap, Shannon entropy, spin-spin correlation, static spin structure-factor, magnetic specific-heat and magnetic susceptibility are calculated for various values of spin-spin coupling parameters. With increase in interlayer coupling, the system is driven to states with long range order and the interval of quantum paramagnetically disordered state, sandwiched between Néel and collinear ordered states, narrows on the scale of inplane frustration parameter.
We use the coupled cluster method to study the zero-temperature quantum phase diagram of the spin-$\frac{1}{2}{J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{1}^{\ensuremath{\perp}}$ model on the honeycomb bilayer lattice. In each layer, we include both nearest-neighbor and frustrating next-nearest-neighbor antiferromagnetic exchange couplings, of strength J1>0{J}_{1}>0J1​>0 and ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, respectively. The two layers are coupled by an interlayer nearest-neighbor exchange, with coupling constant ${J}_{1}^{\ensuremath{\perp}}\ensuremath{\equiv}\ensuremath{\delta}{J}_{1}>0$. We calculate directly in the infinite-lattice limit both the ground-state energy per spin and the N\'eel magnetic order parameter, as well as the triplet spin gap. By implementing the method to very high orders of approximation we obtain an accurate estimate for the full boundary of the N\'eel phase in the $\ensuremath{\kappa}\ensuremath{\delta}$ plane. For each value $\ensuremath{\delta}<{\ensuremath{\delta}}_{c}^{>}(0)\ensuremath{\approx}1.70(5)$, we find an upper critical value ${\ensuremath{\kappa}}_{c}(\ensuremath{\delta})$, such that N\'eel order is present for $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{c}(\ensuremath{\delta})$. Conversely, for each value $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{c}(0)\ensuremath{\approx}0.19(1)$, we find an upper critical value ${\ensuremath{\delta}}_{c}^{>}(\ensuremath{\kappa})$, such that N\'eel order persists for $0<\ensuremath{\delta}<{\ensuremath{\delta}}_{c}^{>}(\ensuremath{\kappa})$. Most interestingly, for values of $\ensuremath{\kappa}$ in the range ${\ensuremath{\kappa}}_{c}(0)<\ensuremath{\kappa}<{\ensuremath{\kappa}}^{>}\ensuremath{\approx}0.215(2)$, we find a reentrant behavior such that N\'eel order exists only in the range ${\ensuremath{\delta}}_{c}^{<}(\ensuremath{\kappa})<\ensuremath{\delta}<{\ensuremath{\delta}}_{c}^{>}(\ensuremath{\kappa})$, with ${\ensuremath{\delta}}_{c}^{<}(\ensuremath{\kappa})>0$. These latter upper and lower critical values coalesce when $\ensuremath{\kappa}={\ensuremath{\kappa}}^{>}$, such that ${\ensuremath{\delta}}_{c}^{<}({\ensuremath{\kappa}}^{>})={\ensuremath{\delta}}_{c}^{>}({\ensuremath{\kappa}}^{>})\ensuremath{\approx}0.25(5)$.
Quantum phase diagram of a frustrated spin-12\frac{1}{2}21​ J1J_{1}J1​--J2J_{2}J2​--J1⊥J_{1}^{\perp}J1⊥​ Heisenberg model on an AAAAAA-stacked bilayer honeycomb lattice
The zero-temperature quantum phase diagram of the spin-12\frac{1}{2}21​ J1J_{1}J1​--J2J_{2}J2​--J1⊥J_{1}^{\perp}J1⊥​ model on an AAAAAA-stacked bilayer honeycomb lattice is investigated using the coupled cluster method (CCM). The model comprises two monolayers in each of which the spins, residing on honeycomb-lattice sites, interact via both nearest-neighbor (NN) and frustrating next-nearest-neighbor isotropic antiferromagnetic (AFM) Heisenberg exchange iteractions, with respective strengths J1>0J_{1} > 0J1​>0 and J2≡κJ1>0J_{2} \equiv \kappa J_{1}>0J2​≡κJ1​>0. The two layers are coupled via a comparable Heisenberg exchange interaction between NN interlayer pairs, with a strength J1⊥≡δJ1J_{1}^{\perp} \equiv \delta J_{1}J1⊥​≡δJ1​. The complete phase boundaries of two quasiclassical collinear AFM phases, namely the Neel and Neel-II phases, are calculated in the κδ\kappa \deltaκδ half-plane with κ>0\kappa > 0κ>0. Whereas on each monolayer in the Neel state all NN pairs of spins are antiparallel, in the Neel-II state NN pairs of spins on zigzag chains along one of the three equivalent honeycomb-lattice directions are antiparallel, while NN interchain spins are parallel. We calculate directly in the thermodynamic (infinite-lattice) limit both the magnetic order parameter MMM and the excitation energy Δ\DeltaΔ from the sTz=0s^{z}_{T}=0sTz​=0 ground state to the lowest-lying ∣sTz∣=1|s^{z}_{T}|=1∣sTz​∣=1 excited state (where sTzs^{z}_{T}sTz​ is the total zzz component of spin for the system as a whole, and where the collinear ordering lies along the zzz direction) for both quasiclassical states used (separately) as the CCM model state, on top of which the multispin quantum correlations are then calculated to high orders (n≤10n \leq 10n≤10) in a systematic series of approximations involving nnn-spin clusters. The sole approximation made is then to extrapolate the sequences of nnnth-order results for MMM and Δ\DeltaΔ to the exact limit, n→∞n \to \inftyn→∞.
The thermodynamics of coupled frustrated ferromagnetic chains is studied within a spin-rotation-invariant Green's-function approach. We consider an isotropic Heisenberg spin-half system with a ferromagnetic in-chain coupling J1<0{J}_{1}<0J1​<0 between nearest neighbors and a frustrating antiferromagnetic next-nearest-neighbor in-chain coupling J2>0{J}_{2}>0J2​>0. We focus on the moderate strength of frustration J2<∣J1∣/4{J}_{2}<|{J}_{1}|/4J2​<∣J1​∣/4 such that the in-chain spin-spin correlations are predominantly ferromagnetic. We consider two interchain couplings (ICs) ${J}_{\ensuremath{\perp},y}$ and ${J}_{\ensuremath{\perp},z}$, corresponding to the two axes perpendicular to the chain, where ferromagnetic as well as antiferromagnetic ICs are taken into account. We discuss the influence of frustration on the ground-state properties for antiferromagnetic ICs, where the ground state is of a quantum nature. The major part of our study is devoted to the finite-temperature properties. We calculate the critical temperature Tc{T}_{c}Tc​ as a function of the competing exchange couplings ${J}_{2},{J}_{\ensuremath{\perp},y},{J}_{\ensuremath{\perp},z}$. We find that for fixed ICs, Tc{T}_{c}Tc​ decreases monotonically with increasing frustration J2{J}_{2}J2​, where as ${J}_{2}\ensuremath{\rightarrow}|{J}_{1}|/4$ the Tc(J2){T}_{c}({J}_{2})Tc​(J2​) curve drops down rapidly. To characterize the magnetic ordering below and above Tc{T}_{c}Tc​, we calculate the spin-spin correlation functions $\ensuremath{\langle}{\mathbf{S}}_{0}{\mathbf{S}}_{\mathbf{R}}\ensuremath{\rangle}$, the magnetic order parameter MMM, the uniform static susceptibility ${\ensuremath{\chi}}_{0}$, as well as the correlation length $\ensuremath{\xi}$. Moreover, we discuss the specific heat CV{C}_{V}CV​ and the temperature dependence of the excitation spectrum ${\ensuremath{\omega}}_{\mathbf{q}}$. As ${J}_{2}\ensuremath{\rightarrow}|{J}_{1}|/4$, some unusual frustration-induced features were found, such as an increase of the in-chain spin stiffness (in the case of ferromagnetic ICs) or of the in-chain spin-wave velocity (in the case of antiferromagnetic ICs) with growing temperature.
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