Stripe order in the doped Hubbard model on the honeycomb lattice

We study the ground state properties of the doped Hubbard model with strong interactions on the honeycomb lattice by the density matrix renormalization group (DMRG) method. At half-filling, due to the absence of minus sign problem, it is now well established by large-scale quantum Monte Carlo calculations that a Dirac semimetal to antiferromagnetic Mott insulator transition occurs with the increase of the interaction strength $U$ for the Hubbard model on the honeycomb lattice. However, an understanding of the fate of the antiferromagnetic Mott insulator when holes are doped into the system is still lacking. In this work, by calculating the local spin and charge density for width-4 cylinders with DMRG, we discover a half-filled stripe order in the doped Hubbard model on the honeycomb lattice. We also perform complementary large-scale mean-field calculations with renormalized interaction strength. We observe half-filled stripe order and find stripe states with filling close to one half are nearly degenerate in energy.

Figure 1

Left: sketch of the honeycomb lattice. Blue and red dots represent sites for the two sublattices, respectively. We choose the two primitive vectors as $v_1$ and $v_2$. The dotted rhombus contains a $L_1\times L_2=4\times 4$ super-cell. Right: in DMRG calculation, we arrange the super-cell into a square lattice which is then mapped into a one-dimensional chain in the ordinary manner. The $4\times 4$ cell in the left panel is transformed into a square lattice with size $4\times 8$.

Figure 2

DMRG results of the hole and spin density for a $4\times 8$ cylinder with $U=8$ and 4 holes. Top: color map of the hole (left) and spin density (right). Middle (Bottom): plots of hole (staggered spin) density along the $L_2$ direction. Both results with finite kept state $m$ and the extrapolated to zero truncation error results are shown. The details of the extrapolation can be found in the Supplemental Material [43].

Figure 3

Similar as Fig. 2. The system is a $4\times 16$ cylinder with $U=8$ and 8 holes.

Figure 4

Color map of mean-field results of hole density on a $16\times 16$ system with different dopings and $U=3$. Half-filled stripe order is developed at $N_{\mathrm{hole}}=32$, which corresponds to $1/16$ doping.

Figure 5

Mean-field results for 8 $\times$ 48 system with $U$ = 3. Top: color map of hole (left) and spin density (right) for $N_{\mathrm{hole}}=8,24$, and 48. Middle (Bottom): plots of hole (staggered spin) density along the $L_2$ direction with $N_{\mathrm{hole}}=24$.