Predominant magnetic states in the Hubbard model on anisotropic triangular lattices

By using an optimization variational Monte Carlo method, we study the half-filled-band Hubbard model on anisotropic triangular lattices, as a continuation of the preceding study [T. Watanabe, H. Yokoyama, Y. Tanaka, and J. Inoue, J. Phys. Soc. Jpn. 75, 074707 (2006)]. We introduce two new trial states: (i) A coexisting state of ${\Psi }_Q^{\text{co}}$-antiferromagnetic (AF) and a $d$-wave singlet gaps, in which we allow for a band renormalization effect, and (ii) a state with an AF order of $120°$ spin structure. In both states, a first-order metal-to-insulator transition occurs at smaller $U/t$ than that of the pure $d$-wave state. In insulating regimes, magnetic orders always exist; an ordinary $\left(\pi ,\pi \right)$-AF order survives up to $t^{\prime }/t\sim 0.9$ $\left(U/t=12\right)$, and a $120°$-AF order becomes dominant for $t^{\prime }/t$. The regimes of the robust superconductor and of the nonmagnetic insulator the preceding study proposed give way to these magnetic domains.

Figure 1

(Color online) Schematic representation of spin structure in two AF orders studied in this paper for the anisotropic triangular lattice: (a) an ordinary $\left(\pi ,\pi \right)$-AF order and (b) an AF order with $120°$ spin structure.

Figure 2

(Color online) Schematic explanation of Hartree–Fock approximation for an AF order with $120°$ spin structure. In (a), it is shown how we divide the anisotropic triangular lattice into six sublattices $\left(A\text{−}F\right)$ with different directions of a spin quantization axis, which are illustrated in (b): the axis of B (C,D,E,F,A) sublattice is obtained by turning that of A (B,C,D,E,F) sublattice by 60 degrees. For these sublattices, we suppose that the gap parameter is staggered, namely ${\Delta }_{120}$, $-{\Delta }_{120}$, ${\Delta }_{120},\cdots$, leading to the formation of a $120°$-AF order in Fig. 1b.

Figure 3

(Color online) Total energies of the coexisting state ${\Psi }_Q^{\text{co}}$ $\left[d+\text{AF}\right]$, the $120°$-AF state ${\Psi }_Q^{120}$ [120 AF], and the $d$-wave state ${\Psi }_Q^d$ $\left[d\right]$ are compared as a function of the correlation strength for (a) $t^{\prime }/t=0.8$ and (b) 1.0. The critical values of Mott transitions $U_c/t$ are indicated by arrows for respective states. Although the data for $L=10$ and 12 are plotted, the system-size dependence is almost negligible in this scale. For $t^{\prime }/t=1$, the system of $L=10$ is not used because the closed-shell condition is not satisfied.

Figure 4

(Color online) Comparison of total energies in the insulating regime $\left(U/t=12\right)$ as a function of $t^{\prime }/t$ among various states: a coexisting state ${\Psi }_Q^{\text{co}}$ $\left[d+\text{AF}\right]$, a $120°$-AF state ${\Psi }_Q^{120}$ [120 AF], and three singlet states: a $d$ wave ${\Psi }_Q^d$ $\left[d\right]$, an $\text{ext}.s+d_{xy}$ wave ${\Psi }_Q^{s+d^{\prime }}$ $\left[\text{ext}.s+d_{xy}\right]$, and a $d_{x^2\text{−}{y}^2}+i{d}_{xy}$ wave ${\Psi }_Q^{d+id}$ $\left[d+id\right]$. The latter two states will be discussed in Sec. 3D. The arrows indicate the boundary values between $t_c^{\prime }/t$ and $t_{c2}^{\prime }/t$, satisfying $E^{120}=E^{\text{co}}$ and $E^{120}=E^{s+d^{\prime }}$, respectively.

Figure 5

(Color online) Comparison of the condensation energy $E_c/t$ among ${\Psi }_Q^{\text{co}}$ $\left(d+\text{AF}\right)$, ${\Psi }_Q^{120}$ (120 AF), and ${\Psi }_Q^d$ $\left(d\right)$ for (a) $t^{\prime }/t=0.6$, (b) 0.8, and (c) 1.0. The arrow on the horizontal axis in each panel indicates the critical point of the metal-insulator transition arising at the smallest $U_{rmc}$ $\left(\equiv U_{rmc}^{\text{min}}\right)$ among those states.

Figure 6

(Color online) Density of doubly-occupied site (doublon) as a function of $U/t$ for the lowest-energy states: the coexisting state ${\Psi }_Q^{\text{co}}$ for $t^{\prime }/t=0.4–0.8$ and the $120°$-AF state ${\Psi }_Q^{120}$ for $t^{\prime }/t=1.0$.

Figure 7

(Color online) The momentum distribution function of the lowest-energy states is shown for various values of $U/t$ along the path $\left(0,0\right)-\left(\pi ,0\right)-\left(\pi ,-\pi \right)-\left(0,0\right)$ in the Brillouin zone (a) for $t^{\prime }/t=0.8$ (coexisting state ${\Psi }_Q^{\text{co}}$, $U_c/t\sim 6.65$) and (b) for $t^{\prime }/t=1.0$ ($120°$-AF state ${\Psi }_Q^{120}$, $U_c/t\sim 7.65$). The open (solid) symbols denote the data for $U<U_c$ $\left(U>U_c\right)$.

Figure 8

(Color online) The charge structure factor $N\left(\mathbf{q}\right)$ for the same states with those in Fig. 7 is plotted along the same path: (a) Coexisting state ${\Psi }_Q^{\text{co}}$ for $t^{\prime }/t=0.8$ and (b) $120°$-AF state ${\Psi }_Q^{120}$ for $t^{\prime }/t=1.0$. The open (solid) symbols denote the data for $U<U_c$ $\left(U>U_c\right)$.

Figure 9

(Color online) Optimized values of variational parameters in correlation factor $\mathcal{P}$ for several $t^{\prime }/t$ as function of $U/t$; (a) $g$ [onsite (Gutzwiller) correlation parameter], (b) $\mu$ [doublon-holon binding parameter in the direction of $t$], and (c) ${\mu }^{\prime }$ [the same of $t^{\prime }$]. For $t^{\prime }/t=0.4–0.8$, the parameters are optimized in the coexisting state ${\Psi }_Q^{\text{co}}$, and for $t^{\prime }/t=1.0$ in the $120°$-AF state ${\Psi }_Q^{120}$. The symbols are common to all panels.

Figure 10

(Color online) (a) Optimized gap parameter ${\Delta }_{\text{AF}}/t$ and (b) order parameter $m_s$ of a $\left(\pi ,\pi \right)$-AF order for the coexisting state ${\Psi }_Q^{\text{co}}$ $\left(t^{\prime }/t=0.4–8\right)$. (c) Optimized gap parameter ${\Delta }_{120}/t$ and (d) order parameter $m_s^{120}$ of a $120°$-AF order for the $120°$-AF state ${\Psi }_Q^{120}$ $\left(t^{\prime }/t=1.0\right)$. For the full polarization, $m_s$ and $m_s^{120}$ become 1.

Figure 11

(Color online) (a) Optimized values of $d$-wave gap parameter in ${\Psi }_Q^{\text{co}}$ for $t^{\prime }/t=0.4–0.8$ as a function of $U/t$. Averaged nearest-neighbor $d$-wave pairing correlation function in ${\Psi }_Q^{\text{co}}$ for (b) $t^{\prime }/t=0.4$ and (c) $t^{\prime }/t=0.8$. Note that we average $P_d\left(\mathbf{r}\right)$ only for large values of $\left|\mathbf{r}\right|$ (see text). For $U/t=0$, we use analytic values. The error bars in (b) and (c) include the standard deviations both of VMC calculations and by averaging with respect to r.

Figure 12

(Color online) Optimized values of the remaining variational parameters in ${\Psi }_Q^{\text{co}}$ for $t^{\prime }/t=0.4–0.8$ as a function of $U/t$; (a) $\zeta /t$ [chemical potential] and (b) $\widetilde{t^{\prime }}/t$ [band renormalization factor]. The symbols are common in both panels.

Figure 13

(Color online) Ground-state phase diagram in the $t^{\prime }\text{−}U$ plane constructed from the present VMC results of the coexisting wave function ${\Psi }_Q^{\text{co}}$ and the $120°$-AF state ${\Psi }_Q^{120}$. At the boundaries of the metallic and insulating phases, first-order magnetic transitions take place.

Figure 14

Magnitude of pairing potentials $\left|{\Delta }_{\mathbf{k}}/{\Delta }_{\text{max}}\right|$ considered in BCS state: (a) $d_{x^2\text{−}{y}^2}$, (b) $\text{ext}.s+d_{xy}$, and (c) $d_{x^2\text{−}{y}^2}+i{d}_{xy}$. $\left|{\Delta }_{\text{max}}\right|$ denotes the maximum of $\left|{\Delta }_{\mathbf{k}}\right|$ for each pairing gap.