From the triangular to the kagome lattice: Following the footprints of the ordered state

We study the spin-$1∕2$ Heisenberg model in a lattice that interpolates between the triangular and the kagome lattices. The exchange interaction along the bonds of the kagome lattice is $J$, and the one along the bonds connecting kagome and nonkagome sites is $J^{\prime }$, so that $J^{\prime }=J$ corresponds to the triangular limit and $J^{\prime }=0$ to the kagome one. We use variational and exact diagonalization techniques. We analyze the behavior of the order parameter for the antiferromagnetic phase of the triangular lattice, the spin gap, and the structure of the spin excitations as functions of $J^{\prime }∕J$. Our results indicate that the antiferromagnetic order is not affected by the reduction of $J^{\prime }$ down to $J^{\prime }∕J\simeq 0.2$. Below this value, antiferromagnetic correlations grow weaker, a description of the ground state in terms of a Néel phase renormalized by quantum fluctuations becomes inadequate, and the finite-size spectra develop features that are not compatible with antiferromagnetic ordering. However, this phase does not appear to be connected to the kagome phase as well, as the low-energy spectra do not evolve with continuity for $J^{\prime }\to 0$ to the kagome limit. In particular, for any nonzero value of $J^{\prime }$, the latter interaction sets the energy scale for the low-lying spin excitations, and a gapless triplet spectrum, destabilizing the kagome phase, is expected.

Figure 1

The depleted triangular lattice. Filled and empty circles are the kagome and nonkagome sites; solid and dashes lines indicate $J$ and $J^{\prime }$ bonds of the Hamiltonian (1). The letters A,B,C label the three different spin directions oriented $120°$ apart of the $\sqrt{3}\times \sqrt{3}$ classical Néel state.

Figure 2

Results on the $N=36$ cluster. Upper panel: Relative error on the ground-state energy, for the spin-wave (empty triangles) and FN (full triangles) wave functions. Stars refer to the accuracy of the upper bound on the energy given by the lowest eigenvalue of the FN Hamiltonian, $E_0^{\text{FN}}$. Lower panel: average sign of both the spin-wave and FN wave function (circles) and their overlap with the exact ground state (same symbols as above). Inset: antiferromagnetic order parameter. The circles are the exact ground-state values, while empty and full triangles correspond to the results obtained with spin wave and FN wave functions, respectively. Lines are guides for the eye.

Figure 3

Size scaling of the antiferromagnetic order parameter for the spin-wave (empty symbols and dashed lines), and the FN (full symbols and dotted line) wave functions: $J^{\prime }∕J=0.2$ (triangles), $J^{\prime }∕J=1.0$ (squares).

Figure 4

Size scaling of the ground-state energy per spin for $J^{\prime }∕J=0.2$: Spin-wave wave function (empty triangles), FN (full triangles).

Figure 5

Energy spectra as functions of $S(S+1)$ for a cluster with $N=12$ spins, periodic boundary conditions, and $J^{\prime }∕J=0.8$, 0.6, 0.4, 0.2 (left to right and top to bottom). The crosses (circles) indicate energy levels belonging to symmetry representations compatible (incompatible) with the $\sqrt{3}\times \sqrt{3}$ magnetic order.

Figure 6

Energy spectra as functions of $S_z^2$ for a cluster with $N=16$ spins and twisted boundary conditions. Details are the same as in Fig. 5.

Figure 7

Lowest energy levels within the different $S_z^2$ subspaces for $N=28$ and twisted boundary conditions. Details are the same as in Fig. 5.

Figure 8

Detail of the low-lying energy levels close to the kagome limit for a cluster with $N=12$ spins and periodic boundary conditions. The lower panel shows the behavior of the low energy levels at the kagome limit in a cluster with $N=9$ sites. Details are the same as in Fig. 5.

Figure 9

Detail of the low-lying energy levels close to the kagome limit for a cluster with $N=28$ spins and twisted boundary conditions. Details are the same as in Fig. 5.

Figure 10

Gap to the lowest excitation with $\Delta S_z=1$ for $N=12$ (circles), $N=28$ (triangles), and (in the ${\Gamma }_1$ representation) $N=36$ (squares). The open circle and square at $J^{\prime }=0$ indicate the value of the spin gap for the $N=9$ and $N=27$ kagome cluster (corresponding to the $N=12$ and $N=36$ depleted triangular clusters). The cross at $J^{\prime }=0$ corresponds to a kagome cluster with $N=36$. Inset: The ground-state energy as function of $J^{\prime }$ (open diamonds). The filled diamond indicates the ground-state energy of the kagome cluster with $N=27$ spins.