One-dimensional topology and topolectrics of nonsymmorphic Kramers degenerate systems

Abstract

We describe nonsymmorphic four-band tight-binding models in one dimension with Kramers degeneracy, and propose topolectric-circuit realizations of their topological phases. We begin with a representative model in the nonsymmorphic AII class with symmorphic time-reversal symmetry and nonsymmorphic charge-conjugation and chiral symmetries, resulting in a $\mathbb{Z}_2$ invariant. We also provide a Bogoliubov-de Gennes model in the nonsymmorphic D class with symmorphic charge-conjugation symmetry and nonsymmorphic time-reversal and chiral symmetries, which supports a $\mathbb{Z}_4$ classification. To compute these invariants, we extend an open-path winding-number formulation previously used for non-Kramers-degenerate nonsymmorphic $\mathbb{Z}_2$ phases to Kramers-degenerate four-band systems. We propose topolectric circuit implementations of nonsymmorphic systems, beginning with a comparison between the symmorphic Su-Schrieffer-Heeger and nonsymmorphic charge-density-wave models. We then extend this methodology to topolectric realizations of the AII and $\mathbb{Z}_4$ models, finding that the impedance response reproduces the predicted phase boundaries and associated zero energy modes. Finally, we analyze disorder in the $\mathbb{Z}_4$ model and find that, although disorder breaks the nonsymmorphic symmetries, certain domain-wall zero modes in the minimal nearest-neighbor model remain pinned at zero energy because the disorder has no first-order coupling to the soliton subspace; longer-range terms satisfying relevant symmetries lift this emergent property.