Topological superconductivity by doping symmetry-protected topological states

We propose an exotic scenario in which topological superconductivity can emerge by doping strongly interacting fermionic systems whose spin degrees of freedom form a bosonic symmetry protected topological (SPT) state. Specifically, we study a one-dimensional (1D) example where the spin degrees of freedom form a spin-1 Haldane phase. Before doping, the charge and spin degrees of freedom are both gapped. Upon doping, the charge channel becomes gapless and is described by a $c=1$ compactified bosonic conformal field theory (CFT), while the spin channel remains gapped and still form a bosonic SPT state. Interestingly, an instability toward $p$-wave topological superconductivity is induced, coexisting with the symmetry protected spin edge modes that are inherited from the Haldane phase. This scenario is confirmed by density-matrix renormalization group simulation of a concrete lattice model, where we find that topological superconductivity is robust against interactions. We further show that by stacking doped Haldane phases an exotic 2D anisotropic superconductor can be realized, where the boundaries transverse to the chain direction are either gapless or spontaneously symmetry broken due to the Lieb-Schultz-Mattis (LSM) anomaly.

Figure 1

Bosonization phase diagram with light doping for the Hamiltonian with interaction (3). $c$ represents the central charge. The charge channel is always gapless with $c=1$. The dominant instability of the red regime is superconductivity which is ${\mathrm{\Delta }}_{\mathrm{SC}}={\sum }_{\alpha }{\psi }_{\alpha }\left(x\right){\psi }_{\alpha }(x+a)$ while that of the green regime is density wave instability which is $n\left(x\right)={\sum }_{\alpha }{n}_{\alpha }\left(x\right)$. The Luttinger liquid is gapless in both spin and charge channels.

Figure 2

Ground state phase diagram of the model in Eq. (3) at doping level $\delta =5\%$ as a function of $U$ and $J$.

Figure 3

Superconducting $\mathrm{\Phi }\left(r\right)$ and density-density $D\left(r\right)$ correlation functions for the model in Eq. (3) at doping level $\delta =5\%$ with $U=4$ and $J=1.0$ in a double-logarithmic plot. The solid lines are fits to $\mathrm{\Phi }\left(r\right)\sim r^{-K_{\mathrm{SC}}}$ (black) and $D\left(r\right)\sim r^{-K_c}$ (red), respectively. The inset is the spin-spin correlation function $F\left(r\right)$ in semilogarithmic scales for the system, and the solid line is the fit to $F\left(r\right)\sim e^{-r/{\xi }_s}$.

Figure 4

The 100 lowest levels of the entanglement spectrum $-ln\left({\lambda }_i\right)$ for system of length $L=160$ at $\delta =5\%$ and $U=4$ as a function of $J$, where ${\lambda }_i$ are the eigenvalues of the reduced density matrix. The two largest eigenvalues surrounded by the dashed blue oval are degenerate.

Figure 5

The extracted central charge $c$ for the model in Eq. (2) (in the main text) at doping level $\delta =5\%$ and $U=4$ as a function of $J$. Inset: von Neumann entanglement entropy $S$ at $U=4$ and $J=0.3$, where $x^{\prime }=\frac{L}{\pi }sin\left(\frac{x\pi }{L}\right)$ and the linear line is the fit to $S=\frac{c}{6}ln\left(x^{\prime }\right)+\tilde{c}$.

Figure 6

The density profile $n_i$ in the superconducting (black) and phase separation (red) regions. Here, $U=4$.