Tunable stripe order and weak superconductivity in the Moiré Hubbard model

The moiré Hubbard model describes correlations in certain homobilayer twisted transition metal dichalcogenides. Using exact diagonalization and density matrix renormalization group methods, we find magnetic Mott insulating and metallic phases which, upon doping, exhibit intertwined charge and spin ordering and, in some regimes, pair binding of holes. The phases are highly tunable via an interlayer potential difference. Remarkably, the hole-binding energy is found to be highly tunable, revealing an experimentally accessible regime where holes become attractive. In this attractive regime, we study the superconducting correlation function and point out the possibility of weak superconductivity.

Figure 1

Approximate ground state phase diagram at half-filling on the $N_s=12$ cluster from exact diagonalization. The colors denote different quantum numbers of the first excited state. Red (blue) colors correspond to $S_z=0$ ($S_z=1$) states at different momenta and point group symmetry representations, see Appendix. The magnetic insulating regime at large $U$ extends close to $U=0$ for $\varphi =\pi /6,\pi /2,5\pi /6$. Two ${120}^{\circ }$ Néel ordered states in the $x\text{−}y$ plane with opposite chirality are realized as well as an $x\text{−}y$ ferromagnet. The flux pattern of the moiré Hubbard model and the high symmetry momenta in the Brillouin zone are shown bottom left and right.

Figure 2

Magnetic order and gaps of the $N_s=12$ cluster from ED. The magnetic structure factor $S_m\left(\mathbf{q}\right)$ evaluated at (a) points $\mathbf{q}=K$ and (b) Brillouin zone center $\mathbf{q}=\mathrm{\Gamma }$, indicating ${120}^{\circ }$ Néel order and ferromagnetic order in the $x\text{−}y$ plane, respectively. We observe a discontinuity as small values of $U/t$ around special values of $\varphi =\pi /6,\pi /2,5\pi /6$. (c) The single-particle gap ${\mathrm{\Delta }}_p$ indicates the insulating and metallic regimes. (d) An enhanced spin gap ${\mathrm{\Delta }}_s$ is observed close to the transition from the magnetic insulator to the metallic regime, possibly indicating nonmagnetic insulating states.

Figure 3

Hole density $1-\langle n_i\rangle$ and spin correlation $\langle S_i^x{S}_0^x+S_i^y{S}_0^y\rangle$ of the ground state on the $36\times 3$ YC3 cylinder for $U/t=8$ and small hole doping $p\approx 0.074$, corresponding to $n_h=8$ holes at $\varphi =\pi /2$. The reference site of the spin correlations is marked with the black cross. We observe charge modulations with two holes per stripe. The spin correlation switches the sign at the maxima of the hole density, indicating intertwined spin and charge ordering.

Figure 4

Intertwined charge and spin ordering on the $72\times 3$ YC3 cylinder at hole doping $p=1/18$ and $U/t=8$. Blue and orange indicate the cut through the Brillouin zone shown in the insets. The peaks of spin structure factors in (a,b) are shifted by $\delta =\pi \sqrt{3}p$ from the ordering vectors $K_0$ (a) of the ${120}^{\circ }$ Néel order for $\varphi =\pi /6$ and $\mathrm{\Gamma }=(0,0)$ in (b) for $\varphi =\pi /2$. The charge structure factor $S_c\left(\mathbf{k}\right)$ in (c) is identical for $\varphi =\pi /6$ and $\varphi =\pi /2$ and is peaked at a wave vector $2\delta$, indicating stripe order. For $\varphi =\pi /3$ in (d) no peak indicating stripe order is observed. The behavior $S_{\text{c}}\left(\mathit{k}\right)\approx \alpha \left|k_x\right|$ at small $|k_x|$ indicates a metallic state.

Figure 5

Rung-averaged density $n_R$ as a function of position $x$ at small hole doping for $U/t=8.0$ in the case of (a) $\varphi =0$, (b) $\varphi =\pi /6,\pi /2$, and (c) $\varphi =\pi /3$. The number of holes is denoted by $n_h$. The $\varphi =\pi /6$ and $\varphi =\pi /2$ yield identical density profiles as guaranteed by the symmetry discussed in the main text. This density profile indicates that single holes are separated at $\varphi =0$, while pairs of holes bind together to form a charge modulation at $\varphi =\pi /6,\pi /2$. At $\varphi =\pi /3$ no clear charge density wave patterns are formed.

Figure 6

Hole-binding energy $E_{h-h}$ and electron-binding energy $E_{e-e}$ as a function of $\varphi$ evaluated on $N_s=12$ for various values of $U/t$. Negative hole-binding energies favor the formation of bound hole pairs, consistent with the density profiles shown in Fig. 5. Electron- and hole-binding energies are related by symmetry.

Figure 7

Singlet-pairing correlations ${\rho }_S(r,x)={\rho }_S(\mathit{r},\alpha |\mathit{x},\beta )$ for $U/t=8$ and $p=1/18$ on the $72\times 3$ cylinder as a function of $x=|{\mathit{r}}_i-{\mathit{r}}_j|$ for (a) $\varphi =0$ and (b) $\varphi =\pi /6$. The reference point is chosen as $\mathit{r}=(5,0)$ and we choose $\alpha =\beta =(1,0)$. We extrapolate data from finite bond dimension $D$ to infinite bond dimension by fitting a second-order polynomial to the truncated weight $\xi$ in DMRG, as shown in the insets. We fit both algebraic as well as exponential decay to the extrapolated correlations. The long-distance behavior of $\varphi =0$ is well described by an exponential decay with correlation length $\approx 4.47$, while for $\varphi =\pi /6$ both an algebraic decay with exponent $\approx 3.34$ and an exponential decay with correlation length $\approx 9.63$ can be fitted.

Figure 8

Spectrum of the singlet-pairing two-body density matrix as defined in Eq. (11) at $U/t=8$ on the $72\times 3$ cylinder for (a) $\varphi =0$ and (b) $\varphi =\pi /6$. We compare different hole dopings with $n_h=0,6,12,18$ and show the largest 50 eigenvalues only. For finite doping at $\varphi =\pi /6$ in (b) a small gap to the continuum of residual eigenvalues is observed upon doping the system. The number of dominant eigenvalues equals the number of stripes, indicating a fragmentation by the stripes of the superconducting condensate. The absence of a large gap indicates only the possibility of a weak superconductivity. For $\varphi =0$ in (a), we do not observe a gap in the eigenvalues and can therefore exclude superconducting order.

Figure 9

(a) Wigner-Seitz cell of the $N_s=12$ simulation cluster used for exact diagonalization. (b) Momentum resolution in reciprocal space of the $N_s=12$ cluster. This cluster features the highly symmetric $K$ and $M$ points.