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arXiv:1912.09310 · Journal of Physics Condensed Matter

Dynamical properties of Néel and valence-bond phases in the J 1 – J 2 model on the honeycomb lattice

Francesco Ferrari, Federico Becca

Published Mar 12, 2020• 23 citations
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Abstract

Abstract By using a variational Monte Carlo technique based upon Gutzwiller-projected fermionic states, we investigate the dynamical structure factor of the antiferromagnetic S = 1/2 Heisenberg model on the honeycomb lattice, in presence of first-neighbor ( J 1 ) and second-neighbor ( J 2 ) couplings, for J 2 < 0.5 J 1 . The ground state of the system shows long-range antiferromagnetic order for J 2 / J 1 ≲ 0.23 (Néel phase), plaquette valence-bond order for 0.23 ≲ J 2 / J 1 ≲ 0.36, and columnar dimer order for J 2 / J 1 ≳ 0.36. Within the Néel phase, a well-defined magnon mode is observed, whose dispersion is in relatively good agreement with linear spin-wave approximation for J 2 = 0. When a nonzero second-neighbor super-exchange is included, a roton-like mode develops around the K point (i.e., the corner of the Brillouin zone). This mode softens when J 2 / J 1 is increased and becomes gapless at the transition point, J 2 / J 1 ≈ 0.23. Here, a broad continuum of states is clearly visible in the dynamical spectrum, suggesting that nearly-deconfined spinon excitations could exist, at least at relatively high energies. For larger values of J 2 / J 1 , valence-bond order is detected and the spectrum of the system becomes clearly gapped, with a triplon mode at low energies. This is particularly evident for the spectrum of the dimer valence-bond phase, in which the triplon mode is rather well separated from the continuum of excitations that appears at higher energies.

DOI
10.1088/1361-648x/ab7f6e

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Abstract By using a variational Monte Carlo technique based upon Gutzwiller-projected fermionic states, we investigate the dynamical structure factor of the antiferromagnetic S = 1/2 Heisenberg model on the honeycomb lattice, in presence of first-neighbor ( J 1 ) and second-neighbor ( J 2 ) couplings, for J 2 < 0.5 J 1 . The ground state of the system shows long-range antiferromagnetic order for J 2 / J 1 ≲ 0.23 (Néel phase), plaquette valence-bond order for 0.23 ≲ J 2 / J 1 ≲ 0.36, and columnar dimer order for J 2 / J 1 ≳ 0.36. Within the Néel phase, a well-defined magnon mode is observed, whose dispersion is in relatively good agreement with linear spin-wave approximation for J 2 = 0. When a nonzero second-neighbor super-exchange is included, a roton-like mode develops around the K point (i.e., the corner of the Brillouin zone). This mode softens when J 2 / J 1 is increased and becomes gapless at the transition point, J 2 / J 1 ≈ 0.23. Here, a broad continuum of states is clearly visible in the dynamical spectrum, suggesting that nearly-deconfined spinon excitations could exist, at least at relatively high energies. For larger values of J 2 / J 1 , valence-bond order is detected and the spectrum of the system becomes clearly gapped, with a triplon mode at low energies. This is particularly evident for the spectrum of the dimer valence-bond phase, in which the triplon mode is rather well separated from the continuum of excitations that appears at higher energies.
We employ the spin cluster perturbation theory to investigate the dynamical properties of the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the honeycomb lattice. We obtain the excitation spectra for all possible phases in the phase diagram, including the N\'eel phase, plaquette valence-bond-solid phase, dimer valence-bond-solid phase, and stripe antiferromagnetic phase. In the N\'eel phase, besides the obvious renormalization of the magnon dispersion, we find that the spectrum exhibits a dome-shaped broad continuum around the second Brillouin zone (BZ) and the additional strong continuum close to the corner of the BZ. In the valence-bond-solid phases, the spectra are dominated by a strong broad continuum all the way down to below J1{J}_{1}J1​ coexisting with the lowest-energy triplon modes characterizing the plaquette and dimer phases. We ascribe this strong broad continuum and the additional continuum close to the BZ corner in the N\'eel phase to the contributions of fractionalized spinon excitations. In the stripe phase, a clear difference from the linear-spin-wave approximation is that the spectrum is gapped at the MMM point, while that obtained by the latter is gapless due to the strong quantum fluctuations. We point out that the features observed in the N\'eel phase are consistent with the recent neutron scattering experiments on YbCl3{\mathrm{YbCl}}_{3}YbCl3​ and YbBr3{\mathrm{YbBr}}_{3}YbBr3​.
Using variational wave functions and Monte Carlo techniques, we study the antiferromagnetic Heisenberg model with first-neighbor J1{J}_{1}J1​ and second-neighbor J2{J}_{2}J2​ antiferromagnetic couplings on the honeycomb lattice. We perform a systematic comparison of magnetically ordered and nonmagnetic states (spin liquids and valence-bond solids) to obtain the ground-state phase diagram. N\'eel order is stabilized for small values of the frustrating second-neighbor coupling. Increasing the ratio J2/J1{J}_{2}/{J}_{1}J2​/J1​, we find strong evidence for a continuous transition to a nonmagnetic phase at ${J}_{2}/{J}_{1}\ensuremath{\approx}0.23$. Close to the transition point, the Gutzwiller-projected uniform resonating valence-bond state gives an excellent approximation to the exact ground-state energy. For $0.23\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.36$, a gapless Z2{Z}_{2}Z2​ spin liquid with Dirac nodes competes with a plaquette valence-bond solid. In contrast, the gapped spin liquid considered in previous works has significantly higher variational energy. Although the plaquette valence-bond order is expected to be present as soon as the N\'eel order melts, this ordered state becomes clearly favored only for ${J}_{2}/{J}_{1}\ensuremath{\gtrsim}0.3$. Finally, for $0.36\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\le}0.5$, a valence-bond solid with columnar order takes over as the ground state, being also lower in energy than the magnetic state with collinear order. We perform a detailed finite-size scaling and standard data collapse analysis, and we discuss the possibility of a deconfined quantum critical point separating the N\'eel antiferromagnet from the plaquette valence-bond solid.
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.
We study a spin-1/2 SU(2) model on the honeycomb lattice with nearest-neighbor antiferromagnetic exchange J that favors Néel order and competing six-spin interactions Q that favor a valence-bond-solid (VBS) state in which the bond energies order at the "columnar" wave vector K=(2π/3,-2π/3). We present quantum Monte Carlo evidence for a direct continuous quantum phase transition between Néel and VBS states, with exponents and logarithmic violations of scaling consistent with those at analogous deconfined critical points on the square lattice. Although this strongly suggests a description in terms of deconfined criticality, the measured threefold anisotropy of the phase of the VBS order parameter shows unusual near-marginal behavior at the critical point.
Understanding the nature of the excitation spectrum in quantum spin liquids is of fundamental importance, in particular for the experimental detection of candidate materials.However, current theoretical and numerical techniques have limited capabilities, especially in obtaining the dynamical structure factor, which gives a crucial characterization of the ultimate nature of the quantum state and may be directly assessed by inelastic neutron scattering.In this work, we investigate the low-energy properties of the S ¼ 1=2 Heisenberg model on the triangular lattice, including both nearest-neighbor J 1 and next-nearestneighbor J 2 superexchanges, by a dynamical variational Monte Carlo approach that allows accurate results on spin models.For J 2 ¼ 0, our calculations are compatible with the existence of a well-defined magnon in the whole Brillouin zone, with gapless excitations at K points (i.e., at the corners of the Brillouin zone).The strong renormalization of the magnon branch (also including rotonlike minima around the M points, i.e., midpoints of the border zone) is described by our Gutzwiller-projected state, where Abrikosov fermions are subject to a nontrivial magnetic π flux threading half of the triangular plaquettes.When increasing the frustrating ratio J 2 =J 1 , we detect a progressive softening of the magnon branch at M, which eventually becomes gapless within the spin-liquid phase.This feature is captured by the band structure of the unprojected wave function (with two Dirac points for each spin component).In addition, we observe an intense signal at low energies around the K points, which cannot be understood within the unprojected picture and emerges only when the Gutzwiller projection is considered, suggesting the relevance of gauge fields for the low-energy physics of spin liquids.
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.