Chern Dartboard Superconductors

TL;DR

The work analyzes how particle-hole symmetry and mirror-related sub-Brillouin zone topology constrain or enable reduced Chern numbers in Chern dartboard superconductors (CDSCs). By proximitizing CDIs with $n=0,2$, or shifted/FFLO-enabled $n=1$, the authors show that even-$n$ CDIs can sustain nontrivial $sBZ$ topology in CDSCs, including minimal spinless phases with $ N_r=oxed{ ext{±}1}$ and, in some cases, nonzero total Chern numbers; odd-$n$ CDIs are generally constrained unless symmetry is effectively altered via momentum shifts or FFLO pairing. A key finding is that certain $n=2$ CDSCs preserve a well-defined, quantized crystalline response tied to the Berry curvature quadrupole $Q_{xy}$, inherited from the CDI, while $n=1$ CDSCs under standard PH do not exhibit such a quantized electromagnetic response. The results broaden the landscape of $sBZ$ topology, link it to robust bulk responses, and propose concrete experimental probes via edge spectra and strain-related responses. Overall, the paper extends the classification of topological phases in superconducting hybrids and highlights routes to realize and detect sub-BZ topological phenomena in solid-state systems.

Abstract

We investigate the interplay of particle-hole symmetry and sub-Brillouin zone (sBZ) topology by coupling a so-called Chern dartboard insulator (CDI) to a superconductor (SC) via the proximity effect. We dub the hybrid system, and equivalent intrinsically superconducting phases, a \emph{Chern dartboard superconductor} (CDSC). We show that a CDSC can have nontrivial sBZ topology if it arises from a CDI that has an even number of mirror symmetries $n$. On the other hand, particle-hole symmetry constrains a CDSC that arises from an odd-$n$ CDI to have trivial sBZ topology. However, we can circumvent this constraint for $n=1$ by inducing an FFLO-type pairing or shifting the CDI in momentum space, converting the mirror symmetry to a momentum-space nonsymmorphic mirror symmetry. With a superconducting pairing that preserves the (nonsymmorphic) mirror symmetries, even-$n$ CDIs and the shifted $n=1$ CDI can realize the minimal spinless phase that has a trivial total Chern number and nontrivial reduced Chern numbers. With a pairing that breaks the mirror symmetries, the hybrid system can realize phases that have nontrivial total and reduced Chern numbers, expanding the classification of phases that have sub-Brillouin zone (sBZ) topology. We also predict that some types of $n=2$ CDSCs inherit the quantized crystalline response of the $n=2$ CDI, providing experimentalists with a well-defined way to probe the CDSC. Our work motivates further exploration of sBZ topology, bulk topology, and quantized response.

Figures (15)

• Figure 1: Sub-Brillouin zones (sBZs) and reduced Chern numbers for an $n=1$ and $n=2$ CDI. Dashed lines indicate the mirror high-symmetry lines (HSLs) demarcating the sBZs. An $n=1$ CDI has sBZs which are not fully bounded by HSLs.
• Figure 2: The $n=2$ CDI has doubly degenerate counter-propagating gapless edge modes. The energy spectra from 0 to $\pi$ and $-\pi$ to 0 are identical, as required by the mirror symmetries.
• Figure 3: Phase diagram of the CDSC system when a mirror symmetry-preserving $p_x$-wave pairing described by $\Delta({\bf k})=\Delta\sin k_x$ is proximity-induced in the $n=2$ CDI. Pairs $(\mathcal{N}_r^a,\mathcal{N}_r^b)$ label the reduced Chern numbers for each phase. The CDI and NI phases are well-defined only along $\Delta=0$. The starred points represent parameter values correlated to the edge state spectra shown in Fig. \ref{['fig:kx_edge_spec']}a,b.
• Figure 4: Energy spectra in a cylinder geometry for the $n=2$ CDSC generated by proximitizing a CDI with $p_x$-wave pairing. We have shown only the spectra for $0<k_x<\pi$; the spectra for $-\pi<k_x<0$ are the same due to the mirror symmetry relating the sBZs. (a) and (b) are the spectra at the starred points in the $\mathcal{N}_r=\pm2,\pm1$ phases in Fig. \ref{['fig:kx_phase']}, respectively. The solid lines in (c) and (d) show the real-space probability distribution of the edge states in (a) and (b), respectively, using $L_y=80$. In (c), comparing the solid ($\Delta=0.4$) and circle-marked ($\Delta=0.6$) lines demonstrates that the edge states start to delocalize into the bulk as $\Delta$ increases.
• Figure 5: Schematic evolution of the edge states of the $n=2$ CDSC having mirror-preserving pairing. Each edge state line with an arrow represents a chiral Majorana mode in the edge sBZ $0<k_x<\pi$ (an identical picture occurs simultaneously for the sBZ $-\pi<k_x<0$). Pairs $(\mathcal{N}_r^a,\mathcal{N}_r^b)$ label the reduced Chern numbers, and each phase has vanishing total Chern number. (a) The CDI has a pair of complex fermion counter-propagating modes on each edge, which we represent as two pairs of counter-propagating chiral Majorana modes. Hence, the CDSC in the $\Delta=0$ limit two pairs of counter-propagating chiral Majorana modes on an edge. (b) The pairs become distinct at nonzero $\Delta.$ As $\Delta$ increases, one pair on each edge delocalizes into the bulk and annihilates with the pair from the other edge. (c) The resulting state has a single counter-propagating pair of chiral Majorana modes. (d) Further changes in the parameters induce a phase transition into a trivial SC state that has no Majorana edge modes.