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Register a new userBoundary-diagnosing topological invariants beyond symmetry indicators: A case study of two-fold rotational symmetric superconductors
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Topological crystalline superconductors are known to have possible higher-order topology, which results in Majorana modes on $d-2$ or lower dimensional boundaries. Given the rich possibilities of boundary signatures, it is desirable to have topological invariants that can predict the type of Majorana modes from band structures. Although symmetry indicators, a type of invariants that depends only on the band data at high-symmetry points, have been proposed for certain crystalline superconductors, there exist symmetry classes in which symmetry indicators fail to distinguish superconductors with different Majorana boundaries. Here, we systematically obtain topological invariants for an example of this kind, the two-dimensional time-reversal symmetric superconductors with two-fold rotational symmetry $C_2$. First, we show that the non-trivial topology is independent of band data on the high-symmetry points by conducting a momentum-space classification study. Then from the resulting K groups, we derive calculable expressions for four $mathbb{Z}_2$ invariants defined on the high-symmetry lines or general points in the Brillouin zone. Finally, together with a real-space classification study, we establish the bulk-boundary correspondence and show that the four $mathbb{Z}_2$ invariants can predict Majorana boundary types from band structures. Our proposed invariants can fuel practical material searches for $C_2$-symmetric topological superconductors featuring Majorana edge and corner modes.
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Topological crystalline superconductors have attracted rapidly rising attention due to the possibility of higher-order phases, which support Majorana modes on boundaries in $d-2$ or lower dimensions. However, although the classification and bulk topological invariants in such systems have been well studied, it is generally difficult to faithfully predict the boundary Majoranas from the band-structure information due to the lack of well-established bulk-boundary correspondence. Here we propose a protocol for deriving symmetry indicators that depend on a minimal set of necessary symmetry data of the bulk bands and can diagnose boundary features. Specifically, to obtain indicators manifesting clear bulk-boundary correspondence, we combine the topological crystal classification scheme in the real space and a twisted equivariant K group analysis in the momentum space. The key step is to disentangle the generally mixed strong and weak indicators through a systematic basis-matching procedure between our real-space and momentum-space approaches. We demonstrate our protocol using an example of two-dimensional time-reversal odd-parity superconductors, where the inversion symmetry is known to protect a higher-order phase with corner Majoranas. Symmetry indicators derived from our protocol can be readily applied to ab initio database and could fuel material predictions for strong and weak topological crystalline superconductors with various boundary features.
Fully gapped two-dimensional superconductors coupled to dynamical electromagnetism are known to exhibit topological order. In this work, we develop a unified low-energy description for spin-singlet paired states by deriving topological Chern-Simons field theories for $s$-wave, $d+id$, and chiral higher even-wave superconductors. These theories capture the quantum statistics and fusion rules of Bogoliubov quasiparticles and vortices and incorporate global continuous symmetries - specifically, spin rotation and conservation of magnetic flux - present in all singlet superconductors. For all such systems, we compute the Hall response for these symmetries and investigate the physics at the edge. In particular, the weakly-coupled phase of a chiral $d+id$ chiral state has a spin Hall coefficient $ u_s=2$ and a vanishing Hall response for the magnetic flux symmetry. We argue that the latter is a generic result for two-dimensional superconductors with gapped photons, thereby demonstrating the absence of a spontaneous magnetic field in the ground state of chiral superconductors. It is also shown that the Chern-Simons theories of chiral spin-singlet superconductors derived here fall into Kitaevs 16-fold classification of topological superconductors.
We show that a Weyl superconductor can absorb light via a novel surface-to-bulk mechanism, which we dub the topological anomalous skin effect. This occurs even in the absence of disorder for a single-band superconductor, and is facilitated by the topological splitting of the Hilbert space into bulk and chiral surface Majorana states. In the clean limit, the effect manifests as a characteristic absorption peak due to surface-bulk transitions. We also consider the effects of bulk disorder, using the Keldysh response theory. For weak disorder, the bulk response is reminiscent of the Mattis-Bardeen result for $s$-wave superconductors, with strongly suppressed spectral weight below twice the pairing energy, despite the presence of gapless Weyl points. For stronger disorder, the bulk response becomes more Drude-like and the $p$-wave features disappear. We show that the surface-bulk signal survives when combined with the bulk in the presence of weak disorder. The topological anomalous skin effect can therefore serve as a fingerprint for Weyl superconductivity. We also compute the Meissner response in the slab geometry, incorporating the effect of the surface states.
We propose a general theoretical framework for both constructing and diagnosing symmetry-protected higher-order topological superconductors using Kitaev building blocks, a higher-dimensional generalization of Kitaevs one-dimensional Majorana model. For a given crystalline symmetry, the Kitaev building blocks serve as a complete basis to construct all possible Kitaev superconductors that satisfy the symmetry requirements. Based on this Kitaev construction, we identify a simple but powerful bulk Majorana counting rule that can unambiguously diagnose the existence of higher-order topology for all Kitaev superconductors. For a systematic construction, we propose two inequivalent stacking strategies using the Kitaev building blocks and provide minimal tight-binding models to explicitly demonstrate each stacking approach. Notably, some of our Kitaev superconductors host higher-order topology that cannot be captured by the existing symmetry indicators in the literature. Nevertheless, our Majorana counting rule does enable a correct diagnosis for these beyond-indicator models. We conjecture that all Wannierizable superconductors should yield a decomposition in terms of our Kitaev building blocks, up to adiabatic deformations. Based on this conjecture, we propose a universal diagnosis of higher-order topology that possibly works for all Wannierizable superconductors. We also present a realistic example of higher-order topological superconductors with fragile Wannier obstruction to verify our conjectured universal diagnosis. Our work paves the way for a complete topological theory for superconductors.
Searching for topological insulators/superconductors is one central subject in recent condensed matter physics. As a theoretical aspect, various classification methods of symmetry-protected topological phases have been developed, where the topology of a gapped Hamiltonian is investigated from the viewpoint of its onsite/crystal symmetry. On the other hand, topological physics also appears in semimetals, whose gapless points can be characterized by topological invariants. Stimulated by the backgrounds, we shed light on the topology of nodal superconductors. In this paper, we review our modern topological classification theory of superconducting gap nodes in terms of symmetry. The classification method elucidates nontrivial gap structures arising from nonsymmorphic symmetry or angular momentum, which cannot be predicted by a conventional theory.
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