Gapless symmetry-protected topological phase of fermions in one dimension

We consider a one-dimensional, time-reversal-invariant system with attractive interactions and spin-orbit coupling. Such a system is gapless due to the strong quantum fluctuations of the superconducting order parameter. However, we show that a sharply defined topological phase with protected, exponentially localized edge states exists. If one of the spin components is conserved, the protection of the edge modes can be understood as a consequence of the presence of a spin gap. In the more general case, the localization of the edge states arises from a gap to single-particle excitations in the bulk. We consider specific microscopic models and demonstrate both analytically and numerically (using density matrix renormalization group calculations) that they can support the topologically nontrivial phase.

Figure 1

Low-energy configurations of ${\varphi }_{\sigma }$ in a finite-size gapless topological superconductor. Configurations plotted in red dashed lines correspond to total spin of $\pm \frac{1}{2}$ in the system, i.e., an odd number of particles, while configurations plotted in blue solid lines correspond to zero total spin, i.e., an even number of particles. To calculate the local tunneling density of states we calculate the matrix element of a single particle creation operator ${\mathrm{\Psi }}_{\uparrow }^{†}\left(x\right)$ between the states denoted by $|0\rangle$ and $|1\rangle$.

Figure 2

The ground-state energy of a system with open boundary conditions as the number of particles is varied for (a) a system in the trivial phase, and (b) a system in the topological phase. In both cases the contribution of the chemical potential is subtracted, $\mathrm{\Delta }$ indicates the single-particle gap. In the topological phase a low-energy state appears for an odd number of particles due to the presence of the low-energy edge state. Note that a single-particle gap is still present in the bulk of the system. The data is obtained using DMRG study of the model given by Eq. (16) for a system of length $L=100$ sites with model parameters $t=1,U=-1$ and spin-orbit coupling $v=1$ in panel (a) and $v=0$ in panel (b). The chemical potential that is subtracted in both cases is found from a linear fit of $E_N$ to $N$. The red solid lines are fits to parabolas. The finite curvature of the parabolas is due to the charging energy and depends on the size of the system.

Figure 3

Generic spectrum for a system without inversion symmetry. $R_1,R_2(L_1,L_2)$ denote the right (left) movers in each band. Under time-reversal symmetry $R_1\to L_2$ and $R_2\to -L_1$.

Figure 4

DMRG results for the model described by Eq. (16) in absence of spin-orbit coupling, $v=0$. A system of length $L=100$ sites with model parameters $t=1,U=-1$ is considered. (a) Expectation value of the $\widehat{z}$ component of the spin along the chain in the ground states of the system for even and odd numbers of particles. The blue and green solid curves correspond to the two degenerate ground states of a system with $N=20$ particles and zero net spin. Accumulation of spin at the edge of the system is observed, with the integrated spin in the left (right) half of the system being $\pm \frac{1}{4}$. The red dashed curve corresponds to the ground state of a system with $N=21$ particles and total spin $S=+\frac{1}{2}$. Due to the nonzero spin localized at the edge, and power-law decaying spin density wave correlations expected in a phase with ${\varphi }_{\sigma }$ pinned to $\frac{\pi }{2}$ [see Eq. (5)], a spin density wave pattern is formed in the bulk. (b) Matrix elements for the transition between each of the two ground states with $N=20$ particles and the ground state with $N=21$ particles by adding a spin-up particle at a position $i$ along the lattice. As can be seen, the matrix elements are nonzero only at either end of the system depending on the initial state. This is in agreement with the existence of low-energy single-particle states at the edges of the system.

Figure 5

Pair binding energy [see Eq. (7)] as function of system size for the model Hamiltonian given in Eq. (16) with parameters $t=1,U=-1$, spin-orbit couplings $v=0$ and $v=1$, and fixed density $n=\frac{2N}{L}=0.2$. The red solid lines are fits to parabolic curves. For $v=0$ the system is in the topological phase and the pair binding energy tends to zero as $1/L\to 0$. For $v=1$ the system is in the trivial phase and $E_B$ tends to a finite constant equal to the single-particle gap in the system.

Figure 6

Phase diagram obtained for the model described by Eq. (16) with model parameters $t=1,U=-1$ and total particle density $n=\frac{N_{\uparrow }+N_{\downarrow }}{L}=0.2$. As the spin-orbit coupling strength is increased, a phase transition from the topological to the trivial phase is observed. For $v<v_c\approx 0.22$ the gap (defined as the energy difference between the first-excited state and the ground state) in a system with an even number of particles decreases exponentially with the system size, as expected for the topological phase. Data points marked by a blue cross correspond to the inverse correlation length obtained from the finite-size scaling of the gap $\mathrm{\Delta }E=\frac{1}{L}{e}^{-L/\xi }$. For $v>v_c$ we plot the the pair binding energy $E_B$ [see Eq. (7)] for different system sizes. In the limit $L\to \infty$ the pair binding energy tends to a nonzero value, increasing with $v$, indicating an opening of a trivial single-particle gap in the system.

Figure 7

(a) Realization of the topological phase in a semiconducting quantum wire with spin-orbit coupling and repulsive short-range interactions, $U>0$, coupled to a one-dimensional superconductor with intrinsic short-range attractive interactions $V<0$. Spin-orbit coupling is assumed to have similar magnitude and opposite sign on the two wires, and the chemical potential is taken to be such that the inner modes in both wires are at $k=0$ to simplify the analysis. (b) Energy dispersions in the two wires and the relevant and marginal scattering processes. Two independent pair tunnelings of the inner and the outer modes are denoted by $g_1$ and $g_2$, respectively. The latter is suppressed due to the mismatch in the energy dispersions for each of the spin flavors, as a result of the spin-orbit coupling. Back-scattering process due to the repulsive interactions in the semiconducting wire is denoted by $g$.

Figure 8

Phase diagram for the composite semiconducting-superconducting one-dimensional system, as function of the bare dimensionless pair tunneling couplings $y_{1,2}(t=0)$. Initial conditions corresponding to spin-isotropic interactions in the quantum wire are used, $\frac{1}{2}{K}_{\sigma }(t=0)=1+y(t=0)$, where the strength of the repulsive interactions in the wire is taken to be $y(t=0)=0.1$. We assume the noninteracting values for the charge sectors in each wire $K_{{\rho }_{1,2}}(t=0)=2$. The scale $t^{*}$ at which the integration of the RG equations (21) is stopped and the field ${\varphi }_1$ is replaced by its mean value was defined such that $y_1\left(t^{*}\right)=0.5$. As argued in the text, increasing the spin-orbit-coupling strength in the wires suppresses $y_2$ with respect to $y_1$. As can be seen from the phase diagram this drives the system into the topological phase. The topological phase is obtained also for $y_2\gg y_1$ because all the analysis is symmetric in $y_{1,2}$.