Classical phase diagram of the stuffed honeycomb lattice

We investigate the classical phase diagram of the stuffed honeycomb Heisenberg lattice, which consists of a honeycomb lattice with a superimposed triangular lattice formed by sites at the center of each hexagon. This lattice encompasses and interpolates between the honeycomb, triangular, and dice lattices, preserving the hexagonal symmetry while expanding the phase space for potential spin liquids. We use a combination of iterative minimization, classical Monte Carlo, and analytical techniques to determine the complete ground state phase diagram. It is quite rich, with a variety of noncoplanar and noncollinear phases not found in the previously studied limits. In particular, our analysis reveals the triangular lattice critical point to be a multicritical point with two new phases vanishing via second order transitions at the critical point. We analyze these phases within linear spin wave theory and discuss consequences for the $S=1/2$ spin liquid.

Figure 1

Stuffed honeycomb lattice model. This model contains three sublattices, A (red) and B (blue) on the honeycomb sites, and C (green) on the triangular sites. $J_1$ (solid, black) is the nearest-neighbor coupling between honeycomb sites, while $J^{\prime }$ (dashed, purple) is the coupling between honeycomb and C sites. All three sublattices have next-nearest-neighbor $J_2$'s that we take to be identical (dotted, blue). The lattice vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ as well as the basis vectors ${\delta }_B$ and ${\delta }_C$ are shown.

Figure 2

Classical ground state phase diagram as a function of $J_2/J_1$ versus $J^{\prime }/J_1$. This phase diagram interpolates between the honeycomb limit (far left) to the triangular limit (middle) and beyond. Thick dotted lines separating the phases indicate a first order phase transition while solid lines imply a continuous transition. Details of each phase are given in the following sections: Néel*, spiral*, triple conical (TC), and “triangle of triangles” (TT) are described in Sec. 5; the double conical phases, DC I and DC II are described in Sec. 7; the spiral, collinear, interpolating, and ferrimagnetic phases are described in Sec. 6; the noncollinear phases I and II are discussed in Sec. 8.

Figure 3

Classical ground state phase diagram as a function of $J_2/J^{\prime }$ and $J_1/J^{\prime }$; this phase diagram interpolates between the triangular (far left) and dice limits (far right). Dotted lines indicate a first order phase transition, while solid lines imply a continuous transition. Details of each phase are given in the following sections: the double conical phase DC I is described in Sec. 7, the spiral, collinear, interpolating, and ferrimagnetic phases are described in Sec. 6, and noncollinear phase I is discussed in Sec. 8.

Figure 4

Néel* phase. The A (red) and B (blue) spins are Néel ordered while the decoupled C (green) spins form ${120}^{\circ }$ order. This particular arrangement is one of the six favored by thermal and quantum fluctuations.

Figure 5

Spiral* phase: (a) There is an incommensurate coplanar spiral ordering on the A and B sublattices with $Q_{sp}$ being a function of $J_2$ only. The decoupled C spins have a ${120}^{\circ }$ order. (b) $Q_x\left(Q_y\right)$ plotted as a function of $J_2$. For $J_2/J_1\le 1/6$, $Q_{sp}$ limits to a constant value, and the Néel* phase is a special case of the spiral phase. Note that all parameters of this phase are independent of $J^{\prime }$.

Figure 6

Spin configurations in the triple conical phase plotted using a common origin plot, with A, B, C spins shown in red, blue, green, respectively. Three representative configurations are plotted for $J_2/J_1=.3$, with increasing $J^{\prime }/J_1$. While the configuration in (a) is very close to the spiral* phase, the AB spirals wave out of the plane with increasing $J^{\prime }$, and the C spins distort into cones around the original ${120}^{\circ }$ axes. (a) $J^{\prime }/J_1=0.05$, (b) $J^{\prime }/J_1=0.075$, (c) $J^{\prime }/J_1=0.1$.

Figure 7

The triangle of triangles phase exists in a wedge near the honeycomb axis. The unit cell is shown in (a) with A, B, C spins in red, blue, and green, respectively. The triangle of triangles feature is particularly emphasized by the solid blue and red triangles on the three distinct types of ferromagnetic hexagons. (b) While the AB spins lie in a plane, the C spins are oriented out of the plane; their orientations are shown via a common origin plot. These plots are for $J_2/J_1=0.275$ and $J^{\prime }/J_1=0.15$.

Figure 8

The interpolating phase occupies the entire $J_2=0$ axis out to $J_2/J_1=2$ and is described by a single spin orientation on each sublattice. (a) depicts the angle between the sublattice spins, while (b)–(d) highlight the three key limits for (b) $J^{\prime }=0$ (Néel order on the honeycomb), (c) $J^{\prime }=J_1({120}^{\circ }$ order), and (d) $J^{\prime }>2J_1$ (the ferrimagnetic order of the dice lattice). (a),(b) Néel, (c) ${120}^{\circ }$, (d) Ferri.

Figure 9

Magnetization of the interpolating phase as a function of $J^{\prime }/J_1$. We plot the magnitude of the net moment per unit cell in units of $S$. The moment is maximal for $J^{\prime }=0$, and again for $J^{\prime }=2J_1$, only vanishing on the triangular lattice for $J^{\prime }=J_1$.

Figure 10

The collinear phase consists of ferromagnetic stripes along one triangular axis (here, the $\widehat{x}$ axis) and antiferromagnetic stripes along the other two axes. This particular arrangement is selected via order by disorder. More generally, it is a four sublattice phase labeled by the sites $ABCD$ in the figure, with ${\vec{S}}_A+{\vec{S}}_B+{\vec{S}}_C+{\vec{S}}_D=0$ the only ground state condition.

Figure 11

In the spiral phase, all three sublattices spiral with a generically incommensurate wave vector ${\mathbf{Q}}_{sp}$ that is a function of both $J_2/J^{\prime }$ and $J_1/J^{\prime }$. (a) shows the real space ordering for $J_2/J^{\prime }=0.3$ and $J_1/J^{\prime }=0.1$. (b) shows $(Q_x,Q_y)$ as a function of $J_2$ for $J_1=0.1$; the plateau in $Q_x$ begins at the transition to the collinear phase, while $Q_y$ becomes nonzero when the spiral phase is reentered.

Figure 12

Double conical I for $J^{\prime }<J_1$. (a) This phase consists of double conical spirals on each of the A, B, and C sublattices, with conical angles ${\theta }_A,{\theta }_B=\pi -{\theta }_A$, and ${\theta }_C\ll {\theta }_A$. The AB and C spins are shown in common origin plots on the left and right, respectively. The ordering wave vectors are generally incommensurate in the plane, ${\mathbf{Q}}_1$, although here we plot the special commensurate case where ${\mathbf{Q}}_1=\xi {\mathbf{Q}}_{\mathrm{coll}}$ with $\xi =2/5$. The corresponding $J_2/J_1$ and $J^{\prime }/J_1$ values are 0.198 and 0.29575, respectively. There is a small angle, $\eta$ between the A,B spins, as well as a small angle ${\eta }_C$ between the A and C spins (not shown). Note: as ${\theta }_C$ is very small, we plot a conical angle of $10{\theta }_C$ for clarity. (b) DC I parameters as functions of $J_2/J_1$ and $J^{\prime }/J_1$. The variation of the conical angles ${\theta }_{AB}$ (yellow) and ${\theta }_C$ (blue) is shown from the boundary of DC I shared with the noncollinear II phase to the critical $J_2$ beyond which it becomes collinear.

Figure 13

The dice lattice version of DC I. (a) The AB and C spins are shown in common origin plots on the left and right, respectively. Each sublattice forms a double conical spiral with conical angles ${\theta }_A,{\theta }_B=\pi -{\theta }_A$, and ${\theta }_C$ respectively. Here, ${\mathbf{Q}}_1$ takes the special commensurate value, $\xi {\mathbf{Q}}_{\mathrm{coll}}$ with $\xi =2/5$, which occurs for $J_1/J^{\prime }=.4$ and $J_2/J^{\prime }=.24$. (b) The variation of the double conical I parameters as a function of $J_2$ for two values of $J_1/J^{\prime }$ values is shown. The conical angles, ${\theta }_{AB}$ (yellow) and ${\theta }_C$ (blue), are plotted from the boundary of DC I and noncollinear II out to the critical $J_2$ beyond which it becomes collinear.

Figure 14

Double conical II: The A and B sublattices are each in a collinear configuration, while the C sublattice forms a double conical structure, with the conical axis oriented along the A/B spins. (a) shows the real space lattice configuration. (b) shows the double conical structure of the C spins in a common-origin plot, where ${\mathbf{Q}}_1={\mathbf{Q}}_{\mathrm{tri}}$ and ${\mathbf{Q}}_2={\mathbf{Q}}_{\mathrm{coll}}$. The conical angle ${\theta }_C$ is a function of both $J_2$ and $J^{\prime }$.

Figure 15

The sublattice configuration of the noncollinear phases. Each noncollinear phase has a 12 sublattice structure, wherein each of the A, B, and C sublattices has four different spin configurations, arranged as shown in the figure, where the AB and C sublattices are labeled with numbers and greek letters, respectively.

Figure 16

Noncollinear phase I consists of spins on two cones: one for the AB spins (up, with conical angle ${\theta }_A)$ and one for the C spins (inverted, with ${\theta }_C)$. The spin components perpendicular to the conical axis form opposing pairs, (1,3) and (2,4); $(\alpha ,\gamma )$ and $(\beta ,\delta )$.

Figure 17

(a) Variation of conical angles for the noncollinear phases along a path $J^{\prime }=\alpha ,J_2=\frac{3}{20}(\alpha -1)\theta (\alpha -1)+\frac{1}{8}$ (shown in inset). The C spins are planar for $\alpha <1$, with ${\theta }_C=\pi /2$; ${\theta }_C$ then decreases with increasing $\alpha$ as the spins tilt down into an increasingly narrowing cone. The AB spins are similarly planar exactly at $\alpha =1$, with decreasing conical angles away from the triangular axis in both phases. (b) Ferromagnetic moment in the noncollinear phases. The inset at the left corner shows the path with the same parametrization as before, along which the variation of the moment is plotted. The path and the $x$ axis are also color matched to further elucidate this fact. (a) Conical angles, (b) magnetic moment.

Figure 18

Noncollinear II is similar to noncollinear I, but here, the A and B cones are inverted with respect to one another and the C spins lie in the plane. This phase has the same classical free angle $\eta$ between pairs of spins.

Figure 19

(a) For $\eta \ne 0$, noncollinear I has a striped pattern of chirality, as shown by the four values of chirality $\left(\pm {\kappa }_{1,2}\right)$ on nearest-neighbor triangles, indicated with four colors. Each hexagon has a distinct sign of chirality, which is arranged in a collinear pattern. (b) Noncollinear II similarly has a striped pattern of chirality, with ${\kappa }_1$ corresponding to ${\theta }_C=\pi /2$ in Eq. (26) while the zero chirality comes from the antiparallel orientation of spins $A_1$ and $B_3$ and also $A_2$ and $B_4$ as shown in Fig. 18. The flipping of the signs of ${\kappa }_1$ is also due to the opposite orientation of the A and B cones. (a) Chirality for noncollinear I, (b) chirality for noncollinear II.

Figure 20

Spin wave dispersion for noncollinear phase I for $\eta =0$, which is selected via order by disorder. (a) shows both the original hexagonal Brillouin zone with the three sublattices and the new rectangular Brillouin zone for this six site unit cell. As this six site unit cell derives from the collinear phase, there are linear modes at each of the M points of the original Brillouin zone: one at the reciprocal lattice vectors of the new Brillouin zone $({\mathbf{Q}}_{\mathrm{coll}}$, indicated by red circles) and two at the corners (indicated by red squares). (b) shows the spin wave dispersion. In total, there are three Goldstone modes, as expected for noncollinear orders. There is also a quadratic mode at $\mathrm{\Gamma }$ associated with the accidental classical degeneracy in $\eta$ that will be gapped out by $1/S$ corrections [67]. (a) New and old Brillouin zones, (b) spin wave dispersion.

Figure 21

Critical spin for the noncollinear phases, along the parametric path defined in Fig. 17 and shown in the inset. There is a substantial region around the triangular lattice critical point where $S_c>1/2$, and we naively expect a quantum disordered state in this region, as is found on the triangular lattice line.

Figure 22

$\langle |{\mathcal{N}}_{ab}{|}^2\rangle$ as a function of temperature for a lattice of $10\times 10$ unit cells at $J_2/J_1=0.28,J^{\prime }/J_1=1.6$. The nematic phase transition is seen at $T_N=J_1/4$, with some broadening due to the finite system size. As $T\to 0$, the $\langle |{\mathcal{N}}_{ab}{|}^2\rangle$ approaches the line $4{sin}^4{\theta }_A=2.747$, indicating that the coplanar $\eta =0$ classical angle is selected.