Metal-insulator transition and intermediate phases in the kagome lattice Hubbard model

Motivated by the recent discovery of metallic kagome lattice materials, ${\mathrm{AV}}_3{\mathrm{Sb}}_5$ (A = K, Rb, Cs), we investigate the ground state of the half-filled kagome lattice Hubbard model by employing the density-matrix renormalization group method. We identify a metal-insulator transition around $U\sim U_{c1}$ and four distinct phases as a function of $U/t$ on narrower cylinders, including a metallic phase at $U<U_{c1}$, two insulating intermediate phases (a translational symmetry breaking phase at $U_{c1}<U<U_{c2}$ and a quantum spin liquid phase at $U_{c2}<U<U_{c3}$), and the kagome antiferromagnetic phase at $U>U_{c3}$. We confirm that the translational symmetry breaking phase is robust for wider cylinders, while the quantum spin liquid phase is smoothly connected to the kagome antiferromagnetic phase with increasing the system width. Moreover, our numerical observations indicate a continuous metal-insulator transition at $U_{c1}$ whose nature is consistent with Slater's transition scenario. The magnetic phase transition between two insulating intermediate phases at $U_{c2}$ is first order. Our findings may provide insights into exotic kagome lattice materials.

Figure 1

(a) Real-space lattice geometry of a kagome cylinder. ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ are two primary vectors. $L_x$ and $L_y$ indicate the number of unit cells along ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ directions, respectively. The figure shows a cylinder with $L_y=3$ (YC6 geometry, see [24]) and $L_x=5$. (b), (c) The first and the extended Brillouin zone (BZ) of the kagome lattice. The blue cuts indicate the accessible momentum points of the finite cylinders and the blue dots highlight the accessible (quasi-)high-symmetric points for (b) $L_y=2$ and (c) $L_y=3$.

Figure 2

(a) The schematic phase diagram of the half-filled kagome lattice Hubbard model, where TBI donates translational symmetry broken insulator, QSL denotes quantum spin liquid, and KAFM denotes kagome antiferromagnetism. $U_{c1}$, $U_{c2}$, and $U_{c3}$ indicate the approximate phase transitions. $U_{c3}/t\sim 19.5$ only exists in YC4 cylinders. (b) Double occupancy $n_d$ as a function of $U/t$ near $U_{c1}/t\approx 5.4$ (i) and $U_{c2}/t\approx 7.9$ (ii) for YC4 cylinders. (c) $n_d{U}^2$ as a function of $U/t$ for YC4 cylinders. Gray shades indicate the possible phase boundaries. Empty red stars in (b ii) and (c) highlight these data obtained from DMRG calculations which converge to a metastable state with a slightly higher energy. For more details about the hysteresis loop, see the Supplemental Material [25].

Figure 3

The charge gap ${\mathrm{\Delta }}_c$ as a function of $U/t$ for the YC4 cylinder with different $L_x$. The gray shades indicate possible phase boundaries obtained in Fig. 2. The orange line in the QSL phase is obtained by a linear fitting of $L_x=24$ data to guide the eyes.

Figure 4

The contour plots of the spin structure factor ${\mathbf{S}}_{\mathbf{q}}$ (upper row) and the line cuts of ${\mathbf{S}}_{\mathbf{q}}$ (lower row) in the three insulating phases for YC4 cylinders with $L_x=24$ [(a), (b) in the TBI phase for $U/t=7$, (c), (d) in the QSL phase for $U/t=10$, and (e), (f) in the KAFM phase for $U/t=20]$. The plots in the lower row illustrates the cuts of ${\mathbf{S}}_{\mathbf{q}}$ along the lines with the same color in the corresponding contour plot.

Figure 5

The contour plots of the spin structure factor ${\mathbf{S}}_{\mathbf{q}}$ (upper row) and the line cuts of ${\mathbf{S}}_{\mathbf{q}}$(lower row) in the two insulating phases for YC6 cylinders with $L_x=12$ [(a), (b) in the TBI phase for $U/t=7$ and (c), (d) in the QSL phase for $U/t=20]$. Other legends to this figure, refer to Fig. 4.

Figure 6

The normalized absolute values of NN spin-spin correlation $\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle$ in the three insulating phases for YC4 cylinders with $L_x=24$ [(a) in the TBI phase for $U/t=7$, (b) in the QSL phase for $U/t=10$, and (c) in the KAFM phase for $U/t=20]$. We only illustrate the segment including the central eight unit cells along ${\mathbf{a}}_1$ direction to reduce the boundary effects.

Figure 7

The normalized absolute values of NN spin-spin correlation $\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle$ in the two insulating phases for YC6 cylinders with $L_x=12$ [(a) in the TBI phase for $U/t=7$ and (b) in the QSL phase for $U/t=20]$. We only illustrate the segment including the central eight unit cells along ${\mathbf{a}}_1$ direction to reduce the boundary effects.