Ground-state phase diagram of the triangular lattice Hubbard model by the density-matrix renormalization group method

Two-dimensional density-matrix renormalization group method is employed to examine the ground-state phase diagram of the Hubbard model on the triangular lattice at half-filling. The calculation reveals two discontinuities in the double occupancy with increasing the repulsive Hubbard interaction $U$ at $U_{\mathrm{c}1}\sim 7.8t$ and $U_{\mathrm{c}2}\sim 9.9t$ ($t$ being the hopping integral), indicating that there are three phases separated by first-order transitions. The absence of any singularity in physical quantities for $0\le U<U_{\mathrm{c}1}$ implies a metallic phase in this regime. For $U>U_{\mathrm{c}2}$, the local spin density induced by an applied pinning magnetic field exhibits a three sublattice feature, which is compatible with the ${120}^{\circ }$ Néel-ordered state realized in the limit of $U\to \infty$. For $U_{\mathrm{c}1}<U<U_{\mathrm{c}2}$, a response to the applied pinning magnetic field is comparable to that in the metallic phase with a relatively large spin correlation length, but showing neither valence bond nor chiral magnetic order, which therefore resembles gapless spin liquid. However, the spin structure factor for the intermediate phase exhibits the maximum at the $\mathrm{K}$ and ${\mathrm{K}}^{\prime }$ points in the momentum space, which is not compatible to spin liquid with a large spinon Fermi surface. The calculation also finds that the pairing correlation function monotonically decreases with increasing $U$ and thus the superconductivity is unlikely in the intermediate phase.

Figure 1

The (a) 36-, (b) 32-, and (c) 48-site clusters. Open (periodic) boundary conditions are imposed in the $x$ direction ($y$ direction). The indexing of bonds ($l=1,2,3,\cdots ,21$) as well as elementary triangles (${▵}_i=0,1,2,\cdots ,9$) with their chiral directions (arrows) are indicated in (a).

Figure 2

Ground-state energy per site ${\varepsilon }_0$ as a function of the discarded weight ${\delta }_m$ for (a) $U=6t$, (b) $U=8.5t$, and (c) $U=11t$. The cluster used here is the 36-site cluster shown in Fig. 1. The number $m$ of eigenstates of the reduced density-matrix kept in the DMRG calculations is indicated beside each data point. A red straight line shows a linear fit to the three data points with $m=10000$, $12000$, and $14000$.

Figure 3

(a) Ground-state energy per site ${\epsilon }_0$, (b) double occupancy $n_d$, and (c) $n_d{U}^2$ for three different clusters. Insets in (b) and (c) show the results for small values of $U$ calculated in the 36-site cluster. The ground-state phase diagram is shown schematically in the top panel, where QSL denotes quantum spin liquid. The phase boundaries are indicated by gray shades.

Figure 4

Local spin density $S_i^z$ for (a) $U=6t$, (b) $U=8.5t$, and (c) $U=11t$ when a pinning magnetic field $h=0.005t$ is applied along the $z$ direction at a single site located at the edge of the 36-site cluster (indicated by a green open circle). For comparison, the results for the spin-1/2 antiferromagnetic Heisenberg model on the triangular lattice with the nearest-neighbor exchange interaction $J$ is also shown in (d). Here, the same 36-site cluster is used as in (a)–(c) and the applied pinning magnetic field is $h=0.005J$. The dashed line in (c) and (d) indicates a domain wall separating the cluster into two pieces, each of which exhibits a three sublattice pattern, expected for the ${120}^{\circ }$ Néel order. Note that the $z$ component of total spin is kept zero, i.e., ${\sum }_{i=1}^N{S}_i^z=0$, in the calculations.

Figure 5

Same as Figs. 4–4 but in the logarithmic scale. (a) $U=6t$, (b) $U=8.5t$, and $U=11t$.

Figure 6

Spin correlation function $S_{i,j}/\sqrt{\left|S_{i,j}\right|}$ between a reference site $j$ at the center of the cluster (indicated by a red circle) and other sites $i$ for (a) $U=6t$, (b) $U=8.5t$, and (c) $U=11t$. Here $max|S_{i,j}{|}^{1/2}$ represents the maximum value of $|S_{i,j}{|}^{1/2}$ for each $U$. Yellow-green shaded circle indicates the correlation length $\xi$. A three sublattice pattern expected for the ${120}^{\circ }$ Néel order is indicated in (c) by “A”, “B”, and “C”, where $S_{i,j}>0$ ($S_{i,j}<0$) if sites $i$ and $j$ belong to the same (different) sublattice. The cluster used here is the 36-site cluster shown in Fig. 1.

Figure 7

Intensity plot of spin structure factor $S\left(\mathbf{q}\right)$ for (a) $U=6t$, (b) $U=8.5t$, and (c) $U=11t$ calculated in the 36-site cluster and for (d) $U=3t$ obtained by the random phase approximation. Notice that the intensity is doubled in (d) for clarity. The Brillouin zone boundaries are indicated by yellow-green lines.

Figure 8

Nearest-neighbor spin correlation $S_{〈i,j〉}$ for (a) $U=6t$, (b) $U=8.5t$, and (c) $U=11t$. Here $max|S_{〈i,j〉}|$ represents the maximum value of $|S_{〈i,j〉}|$ for each $U$. (d) $S\left(l\right)=S_{〈i,j〉}$ along $x$ direction, where the bond index $l$ connecting neighboring sites $i$ and $j$ is shown in Fig. 1. The 36-site cluster is used for all figures.

Figure 9

Nearest-neighbor hopping amplitude $B_{〈i,j〉}$ for (a) $U=6t$, (b) $U=8.5t$, and (c) $U=11t$. Here $max\left[B_{〈i,j〉}\right]$ is the maximum value of $B_{〈i,j〉}$ for each $U$. (d) $B\left(l\right)=B_{〈i,j〉}$ along the $x$ direction, where the bond index $l$ connecting neighboring sites $i$ and $j$ is denoted in Fig. 1. Notice that the increase of kinetic energy, proportional to $-t{\sum }_{l}B\left(l\right)$, is nicely observed with increasing $U/t$. The 36-site cluster is used for all figures.

Figure 10

(a) Sign and (b) amplitude of chiral correlation function $C\left(l\right)$ for $U=8.5t$ and $11t$. The cluster used here is the 36-site cluster shown in Fig. 1.

Figure 11

(a) Singlet ($\nu =\mathrm{s}$) and (b) triplet ($\nu =\mathrm{t}$) pairing correlation function $P_{\nu }\left(r\right)=P_{\nu }(i,j,k,l)$ for $U=6t$, $8.5t$, and $11t$ calculated in the 36-site cluster. A pair of integer numbers $(l_1,l_2)$ beside each data point for $U=6t$ denotes a pair of bond indexes $l_1$ and $l_2$ [for the definition, see Fig. 1], representing nearest-neighbor sites $(i,j)$ and $(k,l)$, respectively, for which the pairing operators ${\mathrm{\Delta }}_{\nu }(i,j)$ and ${\mathrm{\Delta }}_{\nu }^{†}(k,l)$ are chosen in Eq. (20). $r$ is the distance between the centers of bonds $l_1$ and $l_2$.

Figure 12

Entanglement gaps $\mathrm{\Delta }{\xi }_C$ and $\mathrm{\Delta }{\xi }_S$ for the charge and spin sectors, respectively. A shaded region indicates phase II determined from the discontinuities of the double occupancy in the 48-site cluster.