Majorana multipole response: General theory and application to wallpaper groups

Whereas identification of Cooper-pair symmetry is the first and crucial step in the investigation of unconventional superconductors, only a few have been established so far because of their own difficulties. To solve this problem, we develop a theory for identification of pairing symmetry using knowledge of topological superconductivity. Establishing the multipole theory of emergent Majorana fermions in time-reversal-invariant topological superconductors, we discover a one-to-one correspondence between the electromagnetic response of Majorana fermions and Cooper-pair symmetry. The emergent Majorana fermions host magnetic structures that share the same irreducible representation with Cooper pairs under crystalline symmetry. We furthermore reveal that Majorana fermions in high-spin or nonsymmorphic superconductors may exhibit magnetic octupole responses, which give a direct evidence of these exotic superconducting states. Electric responses of multiple Majorana Kramers pairs are also clarified. Our theory provides the fundamentals for identification of unconventional Cooper pairings through surface-spin-sensitive measurements as well as that for manipulation of Majorana fermions by external electromagnetic fields.

Figure 1

(a) The band structure of Eq. (86) with parameters $(m_0,m_1,t_{xy},t_z,\alpha ,\beta ,\gamma )=(2.2,5,-1.3,-2.5,6,0.5,1)$. (b) The (001) surface state in the fully gapped superconducting states. Here, we choose the chemical potential and the amplitude of the gap funcion as $\mu =0$ and ${\mathrm{\Delta }}_0=1$ and the ${\mathrm{B}}_1$ and ${\mathrm{B}}_2$ gap functions coexist such that ${\eta }_1={\eta }_2={\eta }_1^{\prime }={\eta }_2^{\prime }=0.5$. (c) Applying the Zeeman magnetic field $h_{\mathrm{Z}}=g\mathit{B}·\mathit{s}$, the energy gap of the MKP is illustrated as a function of $\mathit{B}$: (c1) ${\mathrm{\Delta }}_{B_1}$ only, (c2) ${\mathrm{\Delta }}_{B_2}$ only, and (c3) the mixture of ${\mathrm{\Delta }}_{B_1}$ and ${\mathrm{\Delta }}_{B_2}$.

Figure 2

(a) The band structure of Eq. (91) with parameters $(m_0,t_1,t_2,t_3,\alpha ,\beta ,\gamma )=(-1,0.1,2.5,0.25,-1,0.3,2)$. (b) The (01) surface state in the superconducting state with $\mu =1, {\mathrm{\Delta }}_0=1, {\eta }_1=0.5$, and ${\eta }_2=0.1$. (c) The energy gap of the MKP as a function of $\mathit{B}$ under the Zeeman magnetic field $h_{\mathrm{Z}}=g\mathit{B}·\mathit{s}$: The view from (c1) [001], (c2) [100], and (c3) [111] directions.