One-dimensional topological superconductors with nonsymmorphic symmetries

TL;DR

This work develops a comprehensive framework for one-dimensional Bogoliubov–de Gennes systems with built-in charge-conjugation symmetry, incorporating both symmorphic and nonsymmorphic time-reversal symmetries. By constructing a four-band BdG model from coupled Rice–Mele chains and exploiting translational symmetry, the authors classify 1D topological phases across several symmetry classes, introducing a novel $\mathbb{Z}_4$ topological index for nonsymmorphic D, alongside a $\mathbb{Z}_2$ index for nonsymmorphic AII. They extend a generalized winding-number approach to compute indices in the $\mathbb{Z}_4$ and related nonsymmorphic settings and explore disorder and soliton effects on these phases, including robust zero-energy modes under certain conditions. The paper further demonstrates experimental relevance by proposing topolectric circuits that realize the CDW and $\mathbb{Z}_4$ models and measuring topological features via impedance, highlighting practical paths to observe nonsymmorphic topology in 1D systems.

Abstract

We present example four-band Hermitian tight-binding Bogoliubov-de-Gennes (BdG) Hamiltonians and Kramer's degenerate Hamiltonians in one dimension. Starting from a generalized Rice-Mele model, we incorporate superconducting terms to obtain a four-band BdG Hamiltonian with intrinsic charge-conjugation symmetry, and constrain it using symmorphic or nonsymmorphic time-reversal symmetries. In position space we find that each form of time-reversal symmetry, when applied to random BdG matrices, results in a unique block diagonalization of the Hamiltonian when translational symmetry is also enforced. We provide representative models in all relevant symmorphic symmetry classes, including the non-superconducting CII class. For nonsymmorphic time-reversal symmetry, we identify a $\mathbb{Z}_4$ topological index with two phases supporting Majorana zero modes and two without, and study disorder effects in the presence of topological solitons. We further generalize a winding-number method, previously applied only to $\mathbb{Z}_2$ invariants without Kramer's degeneracy, to compute indices for both the $\mathbb{Z}_4$ model and a non-superconducting AII model with nonsymmorphic chiral symmetry and Kramer's degeneracy. We propose topolectric circuit implementations of the charge-density-wave and $\mathbb{Z}_4$ models which agree with the topological calculations. Finally, we show that, in one dimension, nonsymmorphic unitary symmetries do not produce new topological classifications beyond $\mathbb{Z}$ or $\mathbb{Z}_2$ indices.

Figures (24)

• Figure 1: Four distinct topological phases of the $\mathbb{Z}_4$ model in the nonsymmorphic D class, displayed as example trajectories of the winding path $E_q(k)$, Eq. (\ref{['Eqz4']}), across the Brillouin zone $-\pi/a\leq k<\pi/a$. (a) Shows a trajectory in the phase $N_\mathrm{D}^\mathrm{NS}=2$, and (b) a trajectory in the phase $N_\mathrm{D}^\mathrm{NS}=4$. Both (a) and (b) have parameter values $v=0.5$, $\Delta_s=1$, and $\phi_s=\pi/4$, they are distinguished as for (a) $\mu=-1$ and for (b) $\mu=1$. (c) Shows a trajectory in the phase $N_\mathrm{D}^\mathrm{NS}=3$, and (d) a trajectory in the phase $N_\mathrm{D}^\mathrm{NS}=1$. Both (c) and (d) have parameter values $\mu=1.7$, $v=1.2$, and $\Delta_s=0.6$, they are distinguished as for (c) $\phi_s=3\pi/4$ and for (d) $\phi_s=\pi/4$. It is possible to adiabatically deform the parameters such that the path in (a) may resemble the path in (b) without causing a phase transition, however, a change in sign of $\mu$ as displayed here must cause a transition between the two paths. A similar statement is true for a change in the sign of $\cos(\phi_s)$ between the paths in (c) and (d).
• Figure 2: The tight binding parameters of two generalised Rice-Mele model chains in parallel, where the upper chain (black) represents an electron space, and the lower chain (blue) represents a hole space. (a) Nearest-neighbor couplings with constant chemical potential $\pm\mu$, staggered onsite energy $\pm u$, and staggered hoppings $\pm v$ and $\pm v$ between adjacent A and B orbitals. The lattice constant is $a$, and a B orbital is located at intracell distance $s$ to the right of an A orbital. (b) Next-nearest-neighbor coupling $\pm t_{AA}$ between A orbitals and $\pm t_{BB}$ between B orbitals. (c) Third-nearest-neighbor coupling $\pm t_{3}$ between an A orbital and the second B orbital to its right. Bars over parameters indicate that they are complex numbers, in general.
• Figure 3: The superconducting pairing parameters between two generalized Rice-Mele model chains in parallel, where the upper chain (black) represents an electron space, the lower chain (blue) represents a hole space, and lines between them (red) represent superconducting pairings. (a) Nearest-neighbor pairings with order parameter magnitudes $\Delta_{s}$ and $\Delta_{s}^{\prime}$ between adjacent A and B orbitals with phases $\phi_s$ and $\phi_s^\prime$. The lattice constant is $a$, and a B orbital is located at intracell distance $s$ to the right of an A orbital. (b) Next-nearest-neighbor order parameter magnitude $\Delta_{p}$ between A orbitals and $\Delta_{p}^{\prime}$ between B orbitals with phases $\phi_p$ and $\phi_p^\prime$. (c) Third-nearest-neighbor order parameter magnitude $\check{\Delta}_{s}$ between an A orbital and the second B orbital to its right with phase $\check{\phi}_s$.
• Figure 4: Distributions of the ratio of consecutive level spacings for random matrices satisfying different forms of time-reversal symmetry, $T^2$, in the presence of fixed charge conjugation symmetry. Label $P$ is the periodicity of the translational symmetry in terms of the number of electron orbitals $J$, where $P=J$ is equivalent to its absence. For (a) $T^2=0$ and $P=J$, such that there are effectively no applied symmetries, for (b) $T^2=1$ and $P=J$, for (c) $T^2=-1$ and $P=J$, and for (d) time-reversal is nonsymmorphic, $T^2=\mathrm{NS}$, which enforces a translational symmetry which, here, is chosen to have periodicity $P=2$. In all plots, black solid lines show numerical data, blue dashed lines show the prediction of Poisson statistics, blue dotted lines show the prediction of the GOE ensemble, blue dot-dashed lines show the prediction of the GUE ensemble, and blue wide-dashed lines show the prediction of the GSE ensemble. We numerically diagonalized $2J\times2J$ random matrices with $J=1000$, where every independent real variable was taken from a standard normal distribution and averaged over an ensemble of 100 matrices. Only positive energy levels were used for all plots, and twofold degeneracy was neglected (for a degenerate pair of levels, only one was included).
• Figure 5: Majorana zero modes and winding number for the symmorphic superconducting BDI model ($T^{2}=1$, $C^{2}=1$, $S^{2}=1$) described by Hamiltonian (\ref{['symBDIham']}). (a) Energy levels E in position space as a function of the chemical potential $\mu$ for 48 unit cells. (b) Probability density per orbital site for an example MZM. (c) Example bulk band structure $\mathrm{E}(k)$ with the corresponding path of $E_{q}(k)$ in (d) defining a winding number $N_{\mathrm{BDI}}=1$. Parameter values are $u=0.3$, $v=0.5$, and $\Delta_{s}=0.5$ in (a) and (b) with value $\mu=0.9$ fixed in (b), and $\mu=0.7$, $u=0.3$, $v=0.5$, and $\Delta_{s}=0.5$ in (c) and (d).