Anisotropic frustrated Heisenberg model on the honeycomb lattice

We investigate the ground-state phase diagram of an anisotropic Heisenberg model on the honeycomb lattice with competing interactions. We use quantum Monte Carlo simulations, as well as linear spin-wave and Ising series expansions, to determine the phase boundaries of the ordered magnetic phases. We find a region without any classical order in the vicinity of a highly frustrated point. Higher-order correlation functions in this region give no signal for long-range valence-bond order. The low-energy spectrum is derived via exact diagonalization to check for topological order on small-size periodic lattices.

Figure 1

(a) Classical ground states: Néel and collinear order on the honeycomb lattice. (b) Schematic phase diagram for the frustrated anisotropic Heisenberg model on the honeycomb lattice. The axes represent the frustration $V=V_2/V_1=V_3/V_1$ and the relative amplitude of the quantum fluctuations $t=t_r/V_r$. Red circles refer to actual transition points calculated with QMC simulations (see below).

Figure 2

Comparison of energies calculated from the different methods introduced in Sec. 3, at small quantum fluctuations $t=-0.10$ (top) and $t=-0.05$ (bottom). For the SE two different calculations up to fourth order (this work) and up to eighth order (Ref. 11) are shown. In the direct vicinity of the critical point $V=1/2$ both expansions become rather unreliable due to an increasing number of divergences. The LSW results underestimate the influence of quantum fluctuations, as also observed for larger $\left|t\right|$.

Figure 3

Ground-state phase diagram for the anisotropic Heisenberg model on the honeycomb lattice by means of QMC simulations (red circles and interpolation by red dashed lines). The Ising ground states survive separated by a direct first-order transition for small fluctuations $\left|t\right|$. Only for values of $\left|t\right|>0.175\left(25\right)$ can ferromagnetic order in the $x-y$ plane be detected. The solid blue line represents the first-order transition line between the antiferromagnetic states [Eq. (16)], determined from fourth-order SE. Magenta crosses represent phase boundaries determined by the condition of vanishing order parameters, provided as eighth-order SE (Ref. 11) (see the text for more details). Inset: The LSWs (green dash-dotted lines) yield phase transition lines due to stability arguments and a comparison of energies between antiferromagnetic states. The overall scenario is very similar.

Figure 4

Evolution of QMC results for energies and magnetic order parameters shown for decreasing temperatures in two different points of the phase diagram, exhibiting (a) Néel order (NO) and (b) ferromagnetic order (FO), respectively.

Figure 5

Energy and order parameters (FO, ferromagnetic in-plane order; NO, Néel order; CO, collinear order) shown for (a) a horizontal cut through the phase diagram (Fig. 3) at $t=-0.05$ (classical order is absent for $0.48\lesssim V\lesssim 0.54)$ and (b) a vertical cut at $V=0.45$ (no finite-order parameter for $0.075\lesssim -t\lesssim 0.15)$.

Figure 6

QMC results for correlation functions on an $N=288$ lattice. Each bond (and site) represents the strength of the correlation of a dimer (or spin) at the corresponding distance to the top left dimer (or spin) in a blue (dimers, top) or gray (spins, bottom) scale, respectively.

Figure 7

Direct comparison of dimer correlations—in an enlarged illustration of the top left hexagon of Figs. 6b and 6c—shows a small relative enhancement of correlations on two bonds for the disordered phase ($V=0.45$ and original lattice $L=12$). The upper bond is again the reference bond for all dimer correlations.

Figure 8

Spectrum of energy gaps $\Delta E_{\mathbf{k}}=E_{\mathbf{k}}-E_0$ for the $S_{\text{total}}^z=0$ subspace for $N=34$ at $V=0.45$ and $t=-0.1$. The smallest gap at $\mathbf{k}=0$ is stable in a finite-size analysis.

Figure 9

Scaling of the ground-state energy and two gaps with the system size $N$ from ED at $V=0.45$ and $t=-0.1$. The antiferromagnetic gap (AF, middle) inside the $S_{\text{total}}^z=0$ subspace stays finite; however, the ferromagnetic (FE, bottom) gap between the ground-state energy and the lowest eigenvalue of the $S_{\text{total}}^z=1$ subspace scales to zero.