Topological properties of the long-range Kitaev chain with Aubry-André-Harper modulation

We present a detailed study of the topological properties of the Kitaev chain with long-range pairing terms and in the presence of an Aubry-André-Harper on-site potential. Specifically, we consider algebraically decaying superconducting pairing amplitudes; the exponent of this decay is found to determine a critical pairing strength, below which the chain remains topologically trivial. Above the critical pairing, topological edge modes are observed in the central gap. For sufficiently fast decay of the pairing, these modes are identified as Majorana zero modes. However, if the pairing term decays slowly, the modes become massive Dirac modes. Interestingly, these massive modes still exhibit a true level crossing at zero energy, which points towards an intimate relation to Majorana physics. We also observe a clear lack of bulk-boundary correspondence in the long-range system, where bulk topological invariants remain constant, while dramatic changes appear in the behavior at the edge of the system. In addition to the central gap around zero energy, the Aubry-André-Harper potential also leads to other energy gaps at nonzero energy. As for the analogous short-range model, the edge modes in these gaps can be characterized through a 2D Chern invariant. However, in contrast to the short-range model, this topological invariant does not correspond to the number of edge mode crossings anymore. This provides another example of the weakening of the bulk-boundary correspondence occurring in this model. Finally, we discuss possible realizations of the model with ultracold atoms and condensed matter systems.

Figure 1

Energy spectrum for the system with OBC (green) and APBC (blue) for $q=233$, $L=1$, $\mathrm{\Delta }=0.5t$, and $\varphi =0$. We consider $f\left(i\right)=1$ in (a) and (c) and AAH chemical potential in (b) and (d). (a) and (b) depict the energy spectrum for the short-range system ($\alpha =5.0$) and thus, exhibit MMs. (c) and (d) depict the energy spectrum for the long-range system ($\alpha =0.5$), which harbours MDMs in the central gap.

Figure 2

Real-space winding number for $L=\{1,3,5\}$ and infinite system winding number. In both cases, we considered $q=233$, $\mathrm{\Delta }=0.5t$, and $\varphi =0$. We consider $\alpha =5.0$ and $f\left(i\right)=1$ in (a), $\alpha =5.0$ and AAH chemical potential in (b), $\alpha =0.5$ and $f\left(i\right)=1$ in (c) and $\alpha =0.5$ and AAH chemical potential in (d). We see that, when increasing $L$, the real-space winding number converges to the one for the infinite system. The convergence is slower for long-range systems (c) and (d). Note that the computation of winding number is not valid for the values of $\mu$ for which there is no gap at zero energy (shaded regions).

Figure 3

Energy spectrum (blue) and winding number (red) for $f\left(i\right)=1$, $\alpha =5.0$, and $\mu /t=0.5$ (a), AAH chemical potential, $\alpha =5.0$ and $\mu /t=1.2$ (b), $f\left(i\right)=1$, $\alpha =0.5$, and $\mu /t=0.5$ (c) and AAH chemical potential, $\alpha =0.5$ and $\mu /t=1.2$ (d). In (a) and (c), the system exhibits edge modes for any ${\mathrm{\Delta }}_C>0$. For short-range pairing, the edge states are zero-energy modes and the winding number is equal to one, see (a). For long-range pairing, the edge states become massive, and the winding number becomes a half-integer, see (c). In contrast, in presence of an AAH potential, there is a region $\mathrm{\Delta }<{\mathrm{\Delta }}_C$ for which the central gap is closed, for both short-range pairing (b) and long-range pairing (d). In the latter case, the combined effect of AAH onsite potential and long-range pairing leads to a succession of topologically differentiated regions separated by gap closings, see (d). As for the case of constant potential, the winding numbers in presence of AAH potential are integer in the short-ranged system, and half-integer in the long-ranged system. All quantities are computed using the real-space Hamiltonian for a system with $q=233$ and $L=1$.

Figure 4

Real-space winding number for the system with $f\left(i\right)=1$ and $\mu /t\in \{0.5,1,1.5,2\}$ [(a)–(d)]. We observe two regions, corresponding to $\alpha <1$ (long-range) and $\alpha >1$ (short-range). At $\mu /t=1$, (b) there is a closing of the gap, for which the winding number is not well-defined. When we go from $\mu /t<1$ (a) to $\mu /t>1$ [(c) and (d)] we see a topological phase transition, in which the winding number for short-range goes from 1 to 0 and for long-range from 0.5 to $-0.5$. Here, ${\mathrm{\Delta }}_C=0$. The real-space winding number is computed for $q=233$ and $L=1$. We consider that the central gap is closed when $\mathrm{\Delta }E<0.035t$ (black regions).

Figure 5

Real-space winding number for the system with AAH onsite potential and $\mu /t\in \{0.5,1,1.5,2\}$ [(a)–(d)]. When $\mu /t<1$ (a), we only observe two regions, corresponding to $\alpha <1$ (long-range) and $\alpha >1$ (short-range). Here, the system behaves like in the homogeneous case in Fig. 4, and thus ${\mathrm{\Delta }}_C=0$. When $\mu /t\ge 1$, the critical superconducting pairing ${\mathrm{\Delta }}_C\ne 0$ both for the short-range and the long-range cases. For the short-range case, there is simply a region where the gap is closed for $\mathrm{\Delta }<{\mathrm{\Delta }}_C$ and a region with winding number 1 for $\mathrm{\Delta }>{\mathrm{\Delta }}_C$. For the long range case, we see three different regions: first the gap is closed, then it opens with winding number $-0.5$, then it closes again and it re-opens with winding number 0.5. The real-space winding number is obtained for a system with size $q=233$ and $L=1$. We consider that the central gap is closed when $\mathrm{\Delta }E<0.025t$ (black regions).

Figure 6

Energy spectrum (a), quantum fidelity $F$ (b) and ground state fermion parity $P$ (c) for a system with size $q=233$ and $L=1$. Here we considered $\mathrm{\Delta }=0.5t$, $\alpha =0.5$, and $\varphi =\pi$. The quantum fidelity $F$, which is computed following the MDM indicated in green, takes the value 1 everywhere except for the closings of the gap at $|\mu /t|=1.2$ and the crossing at $\mu /t=-0.85$. The ground state fermion parity switches at the gap closing at $\mu /t=-1.2$ and at the crossing at $\mu /t=-0.85$.

Figure 7

Energy splitting $\mathrm{\Delta }E$ of edge modes (a), quantum fidelity (b), ground state fermion parity (c) and real-space winding number (d) for different values of $\mu$ and $\alpha$. The energy splitting $\mathrm{\Delta }E$, the quantum fidelity $F$ and the ground state fermion parity switches allow us to detect the crossings, while the winding number $\nu$ does not detect it. All the computations consider a system with $q=233,L=1$, and $\varphi =\pi$. The winding number is computed using the real-space approach. Gapless regions (defined by a threshold $\mathrm{\Delta }E<0.02t$), where the winding number becomes ill-defined, are colored black in (c). Note that in (b) and (c) the region that corresponds to $\alpha >1$ is expected to host MMs, but even in this regime, there is a finite-size splitting of the two modes and a true energy crossing at zero energy occurs.

Figure 8

Finite-size scaling of the energy splitting $\mathrm{\Delta }E$ of the edge modes vs. $1/L$, where $L$ is the number of supercells, each containing $q=233$ sites. We choose $\mu /t=-1$, $\mathrm{\Delta }=0.5t$, $\varphi =\pi$, so that we are on the left side of the crossing in Fig. 6. We plot the data for $\alpha =0.8$ (a) and $\alpha =1.1$ (b), and compare both cases with exponential fits (dashed lines) and polynomial fits (dotted lines). For both fits, we have assumed that $\mathrm{\Delta }E\to 0$ for $L\to \infty$ (see text). Discrepancy with both fits at $\alpha =0.8$ suggest that the splitting $\mathrm{\Delta }E$ remains finite in this case. (c) Scaling of the infinite system winding number with the discretization $D$ defining the computational grid of ${\delta }_k=\frac{2\pi }{D}$.

Figure 9

Energy spectrum as a function of $\varphi \in [0,2\pi ]$, in the vicinity of the energy gap around $E=0.5t$. (a) and (c) correspond to the short-range system ($\alpha =5$). In (a) we see the energy spectrum for the full period of the dephasing, while in (c) we zoom into one of the avoided crossings. Here, we also compute the quantum fidelity $F$ (dashed red line). (b) and (d) correspond to the long-range scenario ($\alpha =0.5$). Again, (b) shows the energy spectrum for the full period of the dephasing and (d) zooms into the crossing and also considers the quantum fidelity $F$ (dashed red line). To compute the quantum fidelity, we follow the edge state highlighted in green. We considered a system with OBC and $q=233$, $L=1$, $\mu /t=1$, and $\mathrm{\Delta }=0.3t$.

Figure 10

(a) Energy of the AAH edge states vs. pairing strength $\mathrm{\Delta }$ for fixed $\alpha =0.5$. (b) Energy of the AAH edge states vs. decay exponent $\alpha$ for fixed $\mathrm{\Delta }=0.3t$. In both plots, (a) and (b), we take the value of the phase at the crossing $\varphi =5.07$. In (a), the edge states split as we increase $\mathrm{\Delta }$, while in (b) the edge states come together as we increase $\alpha$. The plots are obtained for a system with $q=233$, $L=1$, and $\mu /t=1.2$.

Figure 11

(a) Energy gap between the edge states at $\varphi =5.07$, and (b) minimum of the quantum fidelity $F$ in the region $\varphi \in [5,5.1]$. The plots are obtained for a system size of $q=233$, $L=1$, and $\mu /t=0.5$. They both indicate an avoided crossing of the AAH modes for the region where $\alpha <1$.

Figure 12

Energy spectrum with OBC and real-space winding number for a system with $\mu /t=0.5$, $\alpha =0.5$, and $\varphi =0$. We see that the real-space winding number changes sign at $\mathrm{\Delta }=0$, while the edge states exist at both sides of the gap closing. The plot is obtained for a system with size of $q=233$ and $L=1$.