Quantum Phase Diagram and Spontaneously Emergent Topological Chiral Superconductivity in Doped Triangular-Lattice Mott Insulators

The topological superconducting state is a highly sought-after quantum state hosting topological order and Majorana excitations. In this Letter, we explore the mechanism to realize the topological superconductivity (TSC) in the doped Mott insulators with time-reversal symmetry (TRS). Through large-scale density matrix renormalization group study of an extended triangular-lattice $t\text{−}J$ model on the six- and eight-leg cylinders, we identify a $d+id$-wave chiral TSC with spontaneous TRS breaking, which is characterized by a Chern number $C=2$ and quasi-long-range superconducting order. We map out the quantum phase diagram with by tuning the next-nearest-neighbor (NNN) electron hopping and spin interaction. In the weaker NNN-coupling regime, we identify a pseudogaplike phase with a charge stripe order coexisting with fluctuating superconductivity, which can be tuned into $d$-wave superconductivity by increasing the doping level and system width. The TSC emerges in the intermediate-coupling regime, which has a transition to a $d$-wave superconducting phase with larger NNN couplings. The emergence of the TSC is driven by geometrical frustrations and hole dynamics which suppress spin correlation and charge order, leading to a topological quantum phase transition.

Figure 1

Global quantum phase diagram. (a) Schematic figure of the triangular $t\text{−}J$ model with the NN and NNN hoppings $t_1$, $t_2$ and spin interactions $J_1$, $J_2$. ${\theta }_F$ is the magnetic flux threading in the cylinder. ${\mathrm{\Delta }}_{a,b,c}$ define the pairing order parameters of the NN bonds along the $e_{a,b,c}$ directions. (b) The relative phases between ${\mathrm{\Delta }}_{\alpha }=|{\mathrm{\Delta }}_{\alpha }|e^{i{\theta }_{\alpha }}$ ($\alpha =a$, $b$, $c$), defined as ${\theta }_{\alpha \beta }={\theta }_{\alpha }-{\theta }_{\beta }$. (c) The quantum phase diagram obtained on the $L_y=6$ cylinder with doping level $\delta =1/12$. We identify a pseudogaplike (PGL) phase with $\mathrm{CDW}/\mathrm{SDWF}$, a $d+id$-wave TSC phase, and a $d$-wave SC phase. The dotted dashed line denotes $J_2/J_1=(t_2/t_1{)}^2$. The symbols mark the studied parameters, and the cyan triangle marks the studied parameter in Ref. [55]. (d)–(f) The charge density profile in the three phases. $n(x)$ is the charge density per site in each column $x$, obtained on the $40\times 6$ cylinder with $M=12000$.

Figure 2

Identifying the TSC phase and phase transitions along $(t_2/t_1{)}^2=J_2/J_1$. (a) Spin pumping simulation by adiabatically inserting flux ${\theta }_F$ for $J_2/J_1=0.05$. $m$ is the $U(1)$ bond dimension. By inserting a flux quantum, we obtain the Chern number $C=\mathrm{\Delta }{Q}_s\approx 2$ with an error smaller than $\pm 0.03$. The inset shows the flux dependence of ground-state energy per site $E_0$. (b) Coupling dependence of the obtained Chern number with $m=8000$. (c) Spin chiral order $⟨\chi ⟩=⟨{\widehat{\mathit{S}}}_i·({\widehat{\mathit{S}}}_j\times {\widehat{\mathit{S}}}_k)⟩$ of the triangles in each column versus the column position $x$ for $J_2/J_1=0.05$. $M$ is the $SU(2)$ bond dimension. (d) Double-logarithmic plot of the pairing correlation $|P_{bb}(r)|$ obtained with $M=12000$.

Figure 3

Spin structure factor $S(\mathbf{k})$ and electron density in momentum space $n(\mathbf{k})$ in the three phases. The results are obtained using the middle $N_m=24\times 6$ sites of a long cylinder, which are calculated with $M=12000$ and well converged. The dashed hexagon denotes the Brillouin zone. (a) and (d) belong to the $\mathrm{CDW}+\mathrm{SDWF}$ phase, (b) and (e) belong to the TSC phase, (c) and (f) belong to the $d$-wave SC phase.

Figure 4

Correlation functions and SC orders in the $\mathrm{CDW}+\mathrm{SDWF}$ phase with extrapolated $M\to \infty$ data for (a)–(d). (a) Logarithmic-linear plots of the spin correlations on the $40\times 6$ and $24\times 9$ cylinders, with the correlation length ${\xi }_S=9.2(2)$ (6.9(2)) for $L_y=6$ (9). The number in the bracket gives the standard deviation from linear fitting. (b) Comparing correlations which are rescaled with doping ratio for a direct comparison. The fittings give ${\xi }_S=9.2(2)$ and ${\xi }_G=3.7(6)$. (c) Double-logarithmic plot of pairing correlation $|P_{bb}(r)|$. The extrapolated results with $r\le 10$ can be fitted algebraically with $K_{\mathrm{SC}}^{\prime }=1.05(8)$. (d) Comparing correlations where the fittings give ${\xi }_S=5.22(8)$ and ${\xi }_G=2.7(4)$. We choose the reference site at $x_0=L_x/4$ for demonstrating correlations. (e) Different SC orders ${\mathrm{\Delta }}_{\alpha }$ versus each column $x$ for a system in a grand canonical ensemble with $M=8000$ and the averaged electron density $n(x)$. The coupling parameters are the same as (d). A varying chemical potential ${\mu }_i=\mu (x)={\mu }_0+x/L_x[a+b(x/L_x)]$ is used to adjust the range of $n(x)$.

Figure 5

Correlation functions for $(t_2/t_1{)}^2=J_2/J_1=0.05$ in the TSC phase using the extrapolated data. (a) Double-logarithmic plot of the pairing correlations $|P_{bb}(r)|$ obtained by keeping different $SU(2)$ bond dimensions. The extrapolated correlations decay algebraically with $K_{\mathrm{SC}}=1.03(6)$. (b) $d+id$-wave pairing symmetry identified by the phase differences of pairing correlations on the $L_y=6$ (8) cylinder using bond dimensions $M=15000$ (20 000). (c) The ratios of the magnitudes of the pairing correlations at different bonds. The dashed dotted line indicates the averaged ratio of 1.2(1) for the TSC phase. The dotted line indicates the averaged ratio of 0.45(5) for the $d$-wave SC phase. We choose $r\le L_x/2$ to calculate the averages to minimize the boundary effect. (d) Comparing the correlations which are rescaled with the doping ratio. The fittings give ${\xi }_S=2.2(1)$ and ${\xi }_G=3.3(2)$. We choose $x_0=L_x/4$ and fit the data to the distance $r=L_x/2$ to avoid a boundary effect.