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arXiv:1012.0316 · Physical Review B

Quantum paramagnetic ground states on the honeycomb lattice and field-induced Néel order

R. Ganesh, D. N. Sheng, Young‐June Kim, Arun Paramekanti

Published Apr 18, 2011• 76 citations
arXivHTMLPDF

Abstract

Bi${}_{3}$Mn${}_{4}$O${}_{12}$(NO${}_{3}$) is a recently synthesized spin-3/2 bilayer honeycomb antiferromagnet which behaves as a spin liquid down to very low temperatures. Beyond a magnetic field of about 5 T, it develops long-range N\'eel order. Motivated by this observation, we have studied spin-$S$ Heisenberg models with next neighbor frustrating interactions as well as bilayer couplings on the honeycomb lattice. For a model with frustrating second-neighbor exchange, ${J}_{2}$, we use a Lindemann-like criterion within spin-wave theory to show that N\'eel order melts beyond a critical ${J}_{2}$. The critical ${J}_{2}$ is found to increase in the presence of a magnetic field, implying the existence of a field-induced paramagnet-N\'eel transition over a range of parameters. For the bilayer model, we use a spin-$S$ generalization of bond operator mean-field theory to show that there is a N\'eel-dimer transition for various spin values with increasing bilayer coupling and that the resulting interlayer dimer state undergoes a field-induced transition into a state with transverse N\'eel order. Motivated by a broader interest in such paramagnets, we have also studied a spin-3/2 model which interpolates between the nearest-neighbor Heisenberg model and the Affleck-Kennedy-Lieb-Tasaki (AKLT) parent Hamiltonian. Using exact diagonalization, we have found that there is a single N\'eel-AKLT quantum phase transition in this model. Computing the fidelity susceptibility and assuming a transition in the O(3) universality class, we have located the critical point of this model. In addition, we have obtained the spin gap of the AKLT parent Hamiltonian. Our numerics indicate that the AKLT state also undergoes a field-induced N\'eel ordering transition. We discuss implications of some of our results for experiments on Bi${}_{3}$Mn${}_{4}$O${}_{12}$(NO${}_{3}$) and for numerics on the honeycomb lattice Hubbard model.

DOI
10.1103/physrevb.83.144414

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Bi3{}_{3}3​Mn4{}_{4}4​O12{}_{12}12​(NO3{}_{3}3​) is a recently synthesized spin-3/2 bilayer honeycomb antiferromagnet which behaves as a spin liquid down to very low temperatures. Beyond a magnetic field of about 5 T, it develops long-range N'eel order. Motivated by this observation, we have studied spin-SSS Heisenberg models with next neighbor frustrating interactions as well as bilayer couplings on the honeycomb lattice. For a model with frustrating second-neighbor exchange, J2{J}_{2}J2​, we use a Lindemann-like criterion within spin-wave theory to show that N'eel order melts beyond a critical J2{J}_{2}J2​. The critical J2{J}_{2}J2​ is found to increase in the presence of a magnetic field, implying the existence of a field-induced paramagnet-N'eel transition over a range of parameters. For the bilayer model, we use a spin-SSS generalization of bond operator mean-field theory to show that there is a N'eel-dimer transition for various spin values with increasing bilayer coupling and that the resulting interlayer dimer state undergoes a field-induced transition into a state with transverse N'eel order. Motivated by a broader interest in such paramagnets, we have also studied a spin-3/2 model which interpolates between the nearest-neighbor Heisenberg model and the Affleck-Kennedy-Lieb-Tasaki (AKLT) parent Hamiltonian. Using exact diagonalization, we have found that there is a single N'eel-AKLT quantum phase transition in this model. Computing the fidelity susceptibility and assuming a transition in the O(3) universality class, we have located the critical point of this model. In addition, we have obtained the spin gap of the AKLT parent Hamiltonian. Our numerics indicate that the AKLT state also undergoes a field-induced N'eel ordering transition. We discuss implications of some of our results for experiments on Bi3{}_{3}3​Mn4{}_{4}4​O12{}_{12}12​(NO3{}_{3}3​) and for numerics on the honeycomb lattice Hubbard model.
We use a combination of analytical and numerical techniques to study the phase diagram of the frustrated Heisenberg model on the bilayer honeycomb lattice. Using the Schwinger-boson description of the spin operators followed by a mean-field decoupling, the magnetic phase diagram is studied as a function of the frustration coupling J2{J}_{2}J2​ and the interlayer coupling ${J}_{\ensuremath{\perp}}$. The presence of both magnetically ordered and disordered phases is investigated by means of the evaluation of ground-state energy, spin gap, local magnetization, and spin-spin correlations. We observe a phase with a spin gap and short-range N\'eel correlations that survives for nonzero next-nearest-neighbor interaction and interlayer coupling. Furthermore, we detect signatures of a reentrant behavior in the melting of the N\'eel phase and symmetry restoring when the system undergoes a transition from an on-layer nematic valence-bond crystal phase to an interlayer valence-bond crystal phase. We complement our work with exact diagonalization on small clusters and dimer-series expansion calculations, together with a linear spin-wave approach to study the phase diagram as a function of the spin SSS, the frustration, and the interlayer couplings.
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 ${\ensuremath{\delta}}_{c}^{<}({\ensuremath{\kappa}}^{>})={\ensuremath{\delta}}_{c}^{>}({\ensuremath{\kappa}}^{>})\ensuremath{\approx}0.25(5)$.
We study the zero-temperature quantum phase diagram of a spin-1 Heisenberg antiferromagnet on the honeycomb lattice with both nearest-neighbor exchange coupling J1>0{J}_{1}>0J1​>0 and frustrating next-nearest-neighbor coupling ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, using the coupled cluster method implemented to high orders of approximation, and based on model states with different forms of classical magnetic order. For each we calculate directly in the bulk thermodynamic limit both ground-state low-energy parameters (including the energy per spin, magnetic order parameter, spin stiffness coefficient, and zero-field uniform transverse magnetic susceptibility) and their generalized susceptibilities to various forms of valence-bond crystalline (VBC) order, as well as the energy gap to the lowest-lying spin-triplet excitation. In the range $0<\ensuremath{\kappa}<1$ we find evidence for four distinct phases. Two of these are quasiclassical phases with antiferromagnetic long-range order, one with two-sublattice N\'eel order for $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{1}}=0.250(5)$, and another with four-sublattice N\'eel-II order for $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{2}}=0.340(5)$. Two different paramagnetic phases are found to exist in the intermediate region. Over the range ${\ensuremath{\kappa}}_{{c}_{1}}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{c}^{i}=0.305(5)$ we find a gapless phase with no discernible magnetic order, which is a strong candidate for being a quantum spin liquid, while over the range ${\ensuremath{\kappa}}_{c}^{i}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{2}}$ we find a gapped phase, which is most likely a lattice nematic with staggered dimer VBC order that breaks the lattice rotational symmetry.
We use the coupled cluster method (CCM) to study a frustrated spin-12J1−J2−J1⊥ Heisenberg antiferromagnet on a bilayer honeycomb lattice with AA stacking. Both nearest-neighbor (NN) and frustrating next-nearest-neighbor antiferromagnetic (AFM) exchange interactions are present in each layer, with respective exchange coupling constants J1 > 0 and J2 ≡ κJ1 > 0. The two layers are coupled with NN AFM exchanges with coupling strength J1⊥≡δJ1>0. We calculate to high orders of approximation within the CCM the zero-field transverse magnetic susceptibility χ in the Néel phase. We thus obtain an accurate estimate of the full boundary of the Néel phase in the κδ plane for the zero-temperature quantum phase diagram. We demonstrate explicitly that the phase boundary derived from χ is fully consistent with that obtained from the vanishing of the Néel magnetic order parameter. We thus conclude that at all points along the Néel phase boundary quasiclassical magnetic order gives way to a nonclassical paramagnetic phase with a nonzero energy gap. The Néel phase boundary exhibits a marked reentrant behavior, which we discuss in detail.
We study a frustrated spin-$\frac{1}{2}\phantom{\rule{0.16em}{0ex}}{J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}\text{\ensuremath{-}}{J}_{1}^{\ensuremath{\perp}}$ Heisenberg antiferromagnet on an AAAAAA-stacked bilayer honeycomb lattice. In each layer we consider nearest-neighbor (NN), next-nearest-neighbor, and next-next-nearest-neighbor antiferromagnetic (AFM) exchange couplings J1,J2{J}_{1},\phantom{\rule{0.16em}{0ex}}{J}_{2}J1​,J2​, and J3{J}_{3}J3​, respectively. The two layers are coupled with an AFM NN exchange coupling ${J}_{1}^{\ensuremath{\perp}}\ensuremath{\equiv}\ensuremath{\delta}{J}_{1}$. The model is studied for arbitrary values of $\ensuremath{\delta}$ along the line ${J}_{3}={J}_{2}\ensuremath{\equiv}\ensuremath{\alpha}{J}_{1}$ that includes the most highly frustrated point at $\ensuremath{\alpha}=\frac{1}{2}$, where the classical ground state is macroscopically degenerate. The coupled cluster method is used at high orders of approximation to calculate the magnetic order parameter and the triplet spin gap. We are thereby able to give an accurate description of the quantum phase diagram of the model in the $\ensuremath{\alpha}\ensuremath{\delta}$ plane in the window $0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}1,\phantom{\rule{0.16em}{0ex}}0\ensuremath{\le}\ensuremath{\delta}\ensuremath{\le}1$. This includes two AFM phases with N\'eel and striped order, and an intermediate gapped paramagnetic phase that exhibits various forms of valence-bond crystalline order. We obtain accurate estimations of the two phase boundaries, $\ensuremath{\delta}={\ensuremath{\delta}}_{{c}_{i}}(\ensuremath{\alpha})$, or equivalently, $\ensuremath{\alpha}={\ensuremath{\alpha}}_{{c}_{i}}(\ensuremath{\delta})$, with i=1i=1i=1 (N\'eel) and 2 (striped). The two boundaries exhibit an ``avoided crossing'' behavior with both curves being re-entrant. Thus, in this $\ensuremath{\alpha}\ensuremath{\delta}$ window, N\'eel order exists only for values of $\ensuremath{\delta}$ in the range ${\ensuremath{\delta}}_{{c}_{1}}^{<}(\ensuremath{\alpha})<\ensuremath{\delta}<{\ensuremath{\delta}}_{{c}_{1}}^{>}(\ensuremath{\alpha})$, with ${\ensuremath{\delta}}_{{c}_{1}}^{<}(\ensuremath{\alpha})=0$ for $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{{c}_{1}}(0)\ensuremath{\approx}0.46(2)$ and ${\ensuremath{\delta}}_{{c}_{1}}^{<}(\ensuremath{\alpha})>0$ for ${\ensuremath{\alpha}}_{{c}_{1}}(0)<\ensuremath{\alpha}<{\ensuremath{\alpha}}_{1}^{>}\ensuremath{\approx}0.49(1)$, and striped order similarly exists only for values of $\ensuremath{\delta}$ in the range ${\ensuremath{\delta}}_{{c}_{2}}^{<}(\ensuremath{\alpha})<\ensuremath{\delta}<{\ensuremath{\delta}}_{{c}_{2}}^{>}(\ensuremath{\alpha})$, with ${\ensuremath{\delta}}_{{c}_{2}}^{<}(\ensuremath{\alpha})=0$ for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{{c}_{2}}(0)\ensuremath{\approx}0.600(5)$ and ${\ensuremath{\delta}}_{{c}_{2}}^{<}(\ensuremath{\alpha})>0$ for ${\ensuremath{\alpha}}_{{c}_{2}}(0)>\ensuremath{\alpha}>{\ensuremath{\alpha}}_{2}^{<}\ensuremath{\approx}0.56(1)$.
We present a comprehensive computational study of the phase diagram of the frustrated S=1/2S=1/2S=1/2 Heisenberg antiferromagnet on the honeycomb lattice, with second-nearest (J2)({J}_{2})(J2​) and third-neighbor (J3)({J}_{3})(J3​) couplings. Using a combination of exact diagonalizations (EDs) of the original spin model, of the Hamiltonian projected into the nearest-neighbor short-range valence-bond basis, and of an effective quantum dimer model, as well as a self-consistent cluster mean-field theory, we determine the boundaries of several magnetically ordered phases in the region ${J}_{2},{J}_{3}\ensuremath{\in}[0,1]$, and find a sizable magnetically disordered region in between. We characterize part of this magnetically disordered phase as a plaquette valence-bond crystal phase. At larger J2{J}_{2}J2​, we locate a sizable region in which staggered valence-bond crystal correlations are found to be important, either due to genuine valence-bond crystal (VBC) ordering or as a consequence of magnetically ordered phases, which break lattice rotational symmetry. Furthermore, we find that a particular parameter-free Gutzwiller projected tight-binding wave function has remarkably accurate energies compared to finite-size extrapolated ED energies along the transition line from conventional N\'eel to plaquette VBC phases, a fact that points to possibly interesting critical behavior---such as a deconfined critical point---across this transition. We also comment on the relevance of this spin model to model the spin liquid region found in the half filled Hubbard model on the honeycomb lattice.