Competing phases of the Hubbard model on a triangular lattice: Insights from the entropy

In this Rapid Communication, we present a comprehensive study of the Hubbard model on the isotropic triangular lattice by using the recently developed ladder dual-fermion approach. This method is a nonlocal extension of the dynamical mean-field theory and is free of finite-size effect. In addition to confirming the much-discussed phase diagram at half-filling, our work provides insights into both hole- and electron-doped regimes and, in particular, the finite-temperature phase diagrams. We find the triangular system to be short-range correlated with an associated magnetic phase diagram, which is asymmetric with respect to hole and electron doping. In contrast to the unfrustrated lattice, it can adiabatically be cooled by increasing the interactions. Strikingly, at the electron-doped side, the entropy displays a maximum. This latter example, as well as other results of our work, may provide insights for a variety of correlated triangular materials, such as the water-intercalated sodium-doped cobaltates.

Figure 1

(a) $T$-$U$ phase diagram of the half-filled ITH model. The monotonic decrease of the constant-entropy curve reveals the possibility of adiabatic cooling in this frustrated system. $T^{*}$ is the corresponding temperature at which the double occupancy in Fig. 1 is minimized. The NMI phases are found at lower temperature region ($T\le 0.08$) and for $U$ slightly larger than the bandwidth ($U\in [9.4,9.55]$) (see text for more details). (b) Comparison of the total energy $E_U\left(T\right)$ calculated from the DMFT (empty symbols) and the LDFA (solid symbols) calculations. Their difference results from the nonlocal correlations. The QMC results for $U=8$ are extracted from Ref. [13]. (c) The double occupancy as a function of temperature for different interactions (From top to bottom: $U=4,6,8,9,10,12,14$, respectively.)

Figure 2

Magnetic phase diagram of the doped triangular Hubbard model as a function of the Coulomb strength at $T=0.1$. Two stable magnetic phases are found, i.e., a spiral(${120}^{\circ }$)-AF (antiferromagnetic) and a ferromagnetic (FM) phase. Short-ranged ${120}^{\circ }$-AF and FM states are found in the vicinity of long-range ${120}^{\circ }$-AF and FM phases.

Figure 3

Doping dependence of the spin susceptibilities ${\chi }_{{\Omega }_m=0}^{\mathrm{spin}}\left(\mathbf{Q}\right)$, extracted from the LDFA scheme for six different doping levels and $T=0.1$, $U=W$ with $W$ the bandwidth. $\Gamma$, $\mathbf{M}$, and $\mathbf{K}$ are the high-symmetry points of the first Brillouin zone (BZ) of the triangular lattice. The first BZ is replotted here equivalently as a square.

Figure 4

(a) The doping dependence of the entropy for four different temperatures with $U=W$. The two color-filled areas show the entropy plateau at half-filling and maximization at $\langle n\rangle \sim 1.35$. (b) Temperature dependence of the chemical potential for a range of filling levels. From bottom to top, the curves correspond to $n=0.65$ to $n=1.4$ with $0.05$ as interval.