Degenerate topological line surface phonons in quasi-1D double helix crystal SnIP

Degenerate points/lines in the band structures of crystals have become a staple of the growing number of topological materials. The bulk-boundary correspondence provides a relation between bulk topology and surface states. While line degeneracies of bulk excitations have been extensively characterised, line degeneracies of surface states are not well understood. We show that SnIP, a quasi-one-dimensional van der Waals material with a double helix crystal structure, exhibits topological nodal rings/lines in both the bulk phonon modes and their corresponding surface states. Using a combination of first-principles calculations, symmetry-based indicator theories and Zak phase analysis, we find that two neighbouring bulk nodal rings form doubly degenerate lines in their drumhead-like surface states, which are protected by the combination of time-reversal symmetry $${{{\mathcal{T}}}}$$ T and glide mirror symmetry $${\bar{M}}_{y}$$ M ¯ y . Our results indicate that surface degeneracies can be generically protected by symmetries such as $${{{\mathcal{T}}}}{\bar{M}}_{y}$$ T M ¯ y , and phonons provide an ideal platform to explore such degeneracies.

The topological counterparts of non-Hermitian SSH models

Inspired by the relevance between the asymmetric coupling amplitude and the imaginary gauge field, we construct the counterpart of the non-Hermitian SSH model. The idea is the nonzero imaginary magnetic flux vanishing when the boundary condition changes from periodic to open. The zero imaginary magnetic flux of the counterpart leads to the eliminating of the non-Hermitian skin effect and the non-Hermitian Aharonov-Bohm effect which ensures the recovery of the conventional bulk-boundary correspondence from the non-Bloch bulk-boundary correspondence. We explain how some the non-Hermitian models can be transformed to the non-Hermitian SSH models and how the non-reciprocal hopping in the non-Hermitian SSH models can be transformed from one term to the other terms by the similarity transformations. We elaborate why the effective imaginary magnetic flux disappears due to the interplay of the non-reciprocal hoppings in the partner of the non-Hermitian SSH model. As the results, we obtain the topological invariants of the non-Hermitian SSH model in analytical form defined in conventional Brillouin zone. The non-Hermitian SSH model in domain configuration on a chain is discussed with this method. The technique gives an alternative way to study the topological properties of non-Hermitian systems.

Tunable zero modes and quantum interferences in flat-band topological insulators

We investigate the interplay between Aharonov-Bohm (AB) caging and topological protection in a family of quasi-one-dimensional topological insulators, which we term CSSH ladders. Hybrids of the Creutz ladder and the SSH chain, they present a regime with completely flat bands, and a rich topological phase diagram, with several kinds of protected zero modes. These are reminiscent of the Creutz ladder edge states in some cases, and of the SSH chain edge states in others. Furthermore, their high degree of tunability, and the fact that they remain topologically protected even in small systems in the rungless case, due to AB caging, make them suitable for quantum information purposes. One of the ladders can belong to the BDI, AIII and D symmetry classes depending on its parameters, the latter being unusual in a non-superconducting model. Two of the models can also harbor topological end modes which do not follow the usual bulk-boundary correspondence, and are instead related to a Chern number. Finally, we propose some experimental setups to implement the CSSH ladders with current technology, focusing on the photonic lattice case.

Topological properties of non-Hermitian Creutz Ladders

In this work we study topological properties of the one-dimensional Creutz ladder model with different non-Hermitian asymmetric hoppings and on-site imaginary potentials, and obtain phase diagrams regarding the presence and absence of an energy gap and in-gap edge modes. The non-Hermitian skin effect (NHSE), which is known to break the bulk-boundary correspondence (BBC), emerges in the system only when the non-Hermiticity induces certain unbalanced non-reciprocity along the ladder. The topological properties of the model are found to be more sophisticated than that of its Hermitian counterpart, whether with or without the NHSE. In one scenario without the NHSE, the topological winding is found to exist in a two-dimensional plane embedded in a four-dimensional space of the complex Hamiltonian vector. The NHSE itself also possesses some unusual behaviors in this system, including a high spectral winding without the presence of long-range hoppings, and a competition between two types of the NHSE, with the same and opposite inverse localization lengths for the two bands, respectively. Furthermore, it is found that the NHSE in this model does not always break the conventional BBC, which is also associated with whether the band gap closes at exceptional points under the periodic boundary condition.

Non-Hermitian Bulk-Boundary Correspondence and Singular Behaviors of Generalized Brillouin Zone

The bulk boundary correspondence, which connects the topological invariant, the continuum band and energies under different boundary conditions, is the core concept in the non-Bloch band theory, in which the generalized Brillouin zone (GBZ), appearing as a closed loop generally, is a fundamental tool to rebuild it. In this work, it can be shown that the recovery of the open boundary energy spectrum by the continuum band remains unchanged even if the GBZ itself shrinks into a point. Contrastively, if the bizarreness of the GBZ occurs, the winding number will become illness. Namely, we find that the bulk boundary correspondence can still be established whereas the GBZ has singularities from the perspective of the energy, but not from the topological invariant. Meanwhile, regardless of the fact that the GBZ comes out with the closed loop, the bulk boundary correspondence can not be well characterized yet because of the ill-definition of the topological number. Here, the results obtained may be useful for improving the existing non-Bloch band theory.

Symmetry-protected solitons and bulk-boundary correspondence in generalized Jackiw–Rebbi models

We investigate the roles of symmetry and bulk-boundary correspondence in characterizing topological edge states in generalized Jackiw–Rebbi (JR) models. We show that time-reversal (T), charge-conjugation (C), parity (P), and discrete internal field rotation ($$Z_n$$ Z n ) symmetries protect and characterize the various types of edge states such as chiral and nonchiral solitons via bulk-boundary correspondence in the presence of the multiple vacua. As two representative models, we consider the JR model composed of a single fermion field having a complex mass and the generalized JR model with two massless but interacting fermion fields. The JR model shows nonchiral solitons with the $$Z_2$$ Z 2 rotation symmetry, whereas it shows chiral solitons with the broken $$Z_2$$ Z 2 rotation symmetry. In the generalized JR model, only nonchiral solitons can emerge with only $$Z_2$$ Z 2 rotation symmetry, whereas both chiral and nonchiral solitons can exist with enhanced $$Z_4$$ Z 4 rotation symmetry. Moreover, we find that the nonchiral solitons have C, P symmetries while the chiral solitons do not, which can be explained by the symmetry-invariant lines connecting degenerate vacua. Finally, we find the symmetry correspondence between multiply-degenerate global vacua and solitons such that T, C, P symmetries of a soliton inherit from global minima that are connected by the soliton, which provides a novel tool for the characterization of topological solitons.

Novel topological end modes and breaking of bulk boundary correspondence in skin-effect absent superconductive chain

Topological nontrivial systems feature isolated gapless edge modes, and play a key role in advancing our understanding of quantum matter. A most profound way to characterize edge modes above is through bulk topological invariants, which is known as bulk boundary correspondence. Recent studies on non-Hermitian physics have pronounced the broken bulk-boundary correspondence with the presence of skin effect. Here, we propose a new type of fermionic topological edge modes η, satisfying η+= iη,η2=-i. Remarkably, we demonstrate that for both two cases: superconductive chain with purely η modes and quantum chain with η, Majorana modes γ on different ends, fermion parity can be well defined. Interestingly, for the latter case, broken bulk boundary correspondence is observed despite the absence of skin effects . The phenomenon above is unique to open quantum systems. For the junction with both η,γ modes, the current will not remain sinusoid form but decay exponentially. The exchange of η modes obeys the rules of non-abelian statistics, and can find its applications in topological quantum computing.