Nonperturbative linked-cluster expansions for the trimerized ground state of the spin-one kagome Heisenberg model

We use nonperturbative linked-cluster expansions to determine the ground-state energy per site of the spin-one Heisenberg model on the kagome lattice. To this end, a parameter is introduced allowing us to interpolate between a fully trimerized state and the isotropic model. The ground-state energy per site of the full graph decomposition up to graphs of six triangles (18 spins) displays a complex behavior as a function of this parameter close to the isotropic model which we attribute to divergencies of partial series in the graph expansion of quasi-1D unfrustrated chain graphs. More concretely, these divergencies can be traced back to a quantum critical point of the one-dimensional unfrustrated chain of coupled triangles. Interestingly, the reorganization of the nonperturbative linked-cluster expansion in terms of clusters with enhanced symmetry yields a ground-state energy per site of the isotropic two-dimensional model that is in quantitative agreement with other numerical approaches in favor of a spontaneous trimerization of the system. Our findings are of general importance for any nonperturbative linked-cluster expansion on geometrically frustrated systems.

Figure 1

(a) A schematic illustration of the spin-one Heisenberg model on the kagome lattice. Spins are located on the vertices of the lattice highlighted as circles. The Heisenberg exchange on up- (down-)triangles is $J$ ($\lambda J$) and illustrated as black (gray/cyan) lines. The black boxes mark the center of up-triangles. (b) The effective triangular lattice of up-triangles is shown. Note that up- and down-pointing triangles (light/green and dark/red areas) in the effective lattice are topologically distinct, since they correspond to different plaquettes (hexagons and down-triangles) in the original kagome lattice.

Figure 2

Illustration of all graphs used in the full graph expansion up to $N_{\mathrm{tr}}=3$ triangles. The left panel corresponds to the original kagome lattice, while the right panel shows the same graphs in the effective lattice where up-triangles (shown in black) are replaced by supersites (black boxes).

Figure 3

Illustration of all graphs used in the reorganized expansion except the one consisting just of a single supersite. The idea is to use the two inequivalent triangles of the effective triangular lattice as the building blocks of a NLCE calculation.

Figure 4

Illustration of the one-dimensional spin-one triangle chain. Spins are located on the vertices of the lattice highlighted as black circles. The Heisenberg exchange on up-triangles is $J$ (black lines), while the unfrustrated Heisenberg exchange is $\overline{J}$ between triangles forming dimers (shown as gray/cyan lines). In the limit $J\to 0$ one can derive an effective spin-one Heisenberg chain in second-order perturbation theory in $J/\overline{J}$ for the spin ones building the tips of the black triangles (dark/green circles). The effective Heisenberg exchange $J_{\mathrm{eff}}$ is antiferromagnetic and illustrated as dotted lines.

Figure 5

Spin gap $\mathrm{\Delta }$ as a function of $\lambda$ for the one-dimensional triangle chain with periodic boundary conditions $J=1$ and number of triangles $N_{\mathrm{tr}}$. The dashed vertical line marks the location of the quantum critical point in the thermodynamic limit. Inset left: Minimal spin gap ${\mathrm{\Delta }}_c$ as a function of $1/N_{\mathrm{tr}}$. The dashed line represents a linear fit through the data points signaling a quantum phase transition in the thermodynamic limit. Inset middle: Ratio ${\lambda }_c$ corresponding to the minimal value of the spin gap ${\mathrm{\Delta }}_c$ as a function of $1/N_{\mathrm{tr}}$. The dashed line represents a linear fit through the data points signaling the location of the quantum critical point in the thermodynamic limit. Inset right: Second derivative ${\partial }_{\lambda }^2{\epsilon }_0$ as a function of $\lambda$.

Figure 6

Ground-state energy per site ${\epsilon }_0^{\left(N_{\mathrm{tr}}\right)}$ of the one-dimensional triangle chain as a function of $\lambda$ calculated by NLCE using a full graph expansion for different $N_{\mathrm{tr}}$ and setting $J=1$. Upper inset: Reduced contributions ${\epsilon }_{0,\mathrm{red}}^{\left(N_{\mathrm{tr}}\right)}$ as a function of $\lambda$ for different $N_{\mathrm{tr}}$. Lower inset: Reduced contributions $|{\epsilon }_{0,\mathrm{red}}^{\left(N_{\mathrm{tr}}\right)}|$ as a function of $1/N_{\mathrm{tr}}$ for fixed values of $\lambda$ in a double logarithmic plot. Dashed black line represents an algebraic fit $a{N}_{\mathrm{tr}}^{-b}$ for the displayed data with $\lambda =0.8$.

Figure 7

Comparison of the ground-state energy per site ${\epsilon }_0$ as a function of $\lambda$ for $J=1$ calculated by NLCE using a full graph expansion for different $N_{\mathrm{tr}}$ (shown as symbols) and by iPEPS (shown as solid black line) taken from Ref. [11]. The gray/cyan symbols correspond to the Wynn extrapolation $e_3^{\left(2\right)}$ and $e_1^{\left(4\right)}$, where all contributions from $N_{\mathrm{tr}}=2$ to $N_{\mathrm{tr}}=6$ are taken into account.

Figure 8

Total contribution $-{\nu }^{N_{\mathrm{tr}}}{\epsilon }_{0,\mathrm{red}}^{\left(N_{\mathrm{tr}}\right)}$ of unfrustrated chain graphs as a function of $N_{\mathrm{tr}}$ for $\lambda =1$, where ${\nu }^{N_{\mathrm{tr}}}$ is the embedding factor and ${\epsilon }_{0,\mathrm{red}}^{\left(N_{\mathrm{tr}}\right)}$ the reduced contribution to the energy per site. Lower inset: Embedding factor ${\nu }^{N_{\mathrm{tr}}}$ of unfrustrated chain graphs as a function of $N_{\mathrm{tr}}$ for $\lambda =1$.

Figure 9

Comparison of the ground-state energy per site ${\epsilon }_0$ for $J=1$ as a function of $\lambda$ calculated by NLCE using reorganized graphs of different size (shown as symbols) and by iPEPS (shown as solid black line) taken from Ref. [11]. Gray/cyan symbols correspond to different Wynn extrapolations. The star and the plus symbol belong to $e_1^{\left(4\right)}$ and $e_3^{\left(2\right)}$, while gray/cyan circles represent $e_{1,\mathrm{even}}^{\left(2\right)}$ where all (even order) contributions are taken into account. Inset: Zoom of the same data close to $\lambda =1$.