Majorana vortex phases in time-reversal invariant higher-order topological insulators and topologically trivial insulators

TL;DR

This work shows that Majorana vortex end modes can emerge in time-reversal-invariant higher-order topological insulators and even in topologically trivial insulators when two copies of a TI are coupled with TRS-preserving mass terms. The key mechanism is that the vortex induces an effective 1D D-class problem with a $\mathbb{Z}_2$ classification, yielding MVEMs whenever the chemical potential lies between two distinct vortex-transition points $\mu_c^{(1)}$ and $\mu_c^{(2)}$, and this persists across intrinsic THOTIs, extrinsic THOTIs, and Bi-based models, as well as in fully gapped trivial systems. The results extend to magnetic materials and show MVEMs can persist even without gapless surface states, broadening candidate material platforms for Majorana physics. Overall, the paper demonstrates that MVEMs are a robust feature of systems with double band inversions and TRS, not limited to conventional topological phases, with potential implications for topological quantum computation in a wider class of materials.

Abstract

Majorana vortex phases have been extensively studied in topological materials with conventional superconducting pairing. Inspired by recent experimental progress in realizing time-reversal invariant higher-order topological insulators (THOTIs) and inducing superconducting proximity effects, we investigate Majorana vortex phases in these systems. We construct THOTIs as two copies of a topological insulator (TI) with time-reversal symmetry-preserving mass terms that anisotropically gap the surface states. We find that these mass terms have a negligible impact on the vortex phase transitions of double TIs when treated as perturbations, and no additional topological phase transitions are induced. Consequently, $\mathbb{Z}_2$-protected Majorana vortex end modes (MVEMs) emerge when the chemical potential lies between the critical chemical potentials $μ_c^{(1)}$ and $μ_c^{(2)}$ of the two TI vortex phase transitions. We demonstrate this behavior across multiple THOTI models, including rotational symmetry-protected THOTI, inversion symmetry-protected THOTI, rotational and inversion symmetries-protected THOTI bismuth, and extrinsic THOTI. Remarkably, MVEMs persist even when all surfaces are gapped with the same sign, rendering the system topologically trivial in both first- and second-order classifications. Our findings establish that MVEMs can be realized in time-reversal invariant systems with fully gapped surfaces, encompassing both topologically nontrivial and trivial insulators, thus significantly broadening the solid state material platforms for hosting Majorana vortex phases.

Figures (5)

• Figure 1: Schematical illustration for (a) double TIs residing in the subspace $\tau_z=1$ and $\tau_z=-1$, respectively, (b) rotational symmetry-protected THOTI, (c) extrinsic THOTI with trivial bulk topology, (d) inversion symmetry-protected THOTI. Red and blue arrows indicate hinge states with opposite flow directions. (e)--(h) The energy spectrum of the BdG Hamiltonian $\mathcal{H}_{\text{BdG}}(x, y, k_z = 0)$ as a function of chemical potential $\mu$ associated with the systems depicted in (a)--(d), respectively. In (e), (g), and (h), the color represents the expectation value $\langle \tau_z \rangle$. In (f), the color encodes the eigenvalue of the $\mathcal{C}_{4z}$ symmetry. In our calculations, we take $\eta_0=0, \eta_1=0.4$ in (f), $\eta_0=0.1,\eta_1=0.4$ in (g), and $V=0.1$ in (h). Common model parameters are $\lambda = 0.2$, $\Delta = 0.4$, and lattice size $L_x = L_y = 30$.
• Figure 2: (a) Schematical illustration of MVEMs for the case with the $\mathcal{C}_{4z}$ symmetry. The number of MVEMs is two for $\mu<\mu_c^{(1)}$, one for $\mu_c^{(1)}<\mu<\mu_c^{(2)}$, and none for $\mu_c^{(2)}<\mu$. (b) Schematical illustration of MVEMs for the case without the $\mathcal{C}_{4z}$ symmetry and one MVEMs appear for $\mu_c^{(1)}<\mu<\mu_c^{(2)}$.
• Figure 3: (a) Schematical illustration of HHMs in THOTI Bi. $L$ denotes the length of side of the (111) surface. (b) Vortex energy spectra of $\mathcal{H}_{\text{BdG}}(x,y,k_z=\pi)$, as a function of $\mu$, for superconducting bismuth. In (b), $L=24$ and $\Delta=0.4$.
• Figure 4: (a) and(e) Schematical illustration of trivial insulators for double TIs and TI, respectively, with uniform Dirac mass $m$ for surfaces states. (b) and (f) Vortex energy spectra of $\mathcal{H}_{\text{BdG}}(x,y,k_z=0)$ associated with the systems depicted in (a) and (e), respectively. (c) and (g) Energies close to zero for the superconducting trivial systems in (a) and (e), respectively, under the open boundary conditions. (d) and (h) The spatial wave function distribution of the two MVEMs, represented red dots in (c) and (g), respectively. In (b)-(d), $\eta_0=0.1$, $\eta_1=0$, $\lambda=0.2$, and $\Delta=0.4$. In (f)-(h), $m=0.5$ and $\Delta=0.4$.
• Figure 5: (a) and (b) Vortex energy spectra of the Hamiltonian $\mathcal{H}_{\text{BdG}}(x,y,k_z=0)$ associated with the normal-state Hamiltonian $H_{\text{HOTI}}^{(1)}$. (c) and (d) Vortex energy spectra of the Hamiltonian $\mathcal{H}_{\text{BdG}}(x,y,k_z=0)$ associated with the normal-state Hamiltonian $H_{\text{HOTI}}^{(2)}$. The same model parameters are used as those in Fig. \ref{['fig1']}.