Detecting nonunitary multiorbital superconductivity with Dirac points at finite energies

Determining the symmetry of the order parameter of unconventional superconductors remains a recurrent topic and nontrivial task in the field of strongly correlated electron systems. Here we show that the behavior of Dirac points away from the Fermi energy is a potential tool to unveil the orbital structure of a superconducting state. In particular, we show that gap openings in such Dirac crossings are a signature of nonunitary multiorbital superconducting order. Consequently, also spectral features at higher energy can help us to identify broken symmetries of superconducting phases and the orbital structure of nonunitary states. Our results show how angle-resolved photoemission spectroscopy measurements can be used to detect nonunitary multiorbital superconductivity.

Figure 1

(a) Sketch of the normal-state band structure, featuring a Dirac crossing away from the chemical potential. (b) Sketch of the band structure in a unitary superconducting state, showing that the Dirac point stays closed. In stark comparison, the onsite of a nonunitary superconducting state opens up the Dirac point away from the chemical potential as shown in (c). The magnitude of the gap opening at the Dirac crossing depends on the nonunitarity $\mathrm{{\rm Y}}$ and the distance to the chemical potential as shown in (d).

Figure 2

Normal-state band structure (a), (b) for a Dirac band system in the normal state, showing Dirac crossings away from the chemical potential. (a) Shows a doped honeycomb lattice without spin-orbit coupling, whereas (b) shows a half-filled honeycomb system with noncentrosymmetric spin-orbit coupling (b). (c) Shows the BdG spectra in the presence of nonunitary superconducting term for (a), and (d) the analogous spectra for the case (b). In both scenarios, it is observed that the nonunitary superconducting state drives a gap opening at the Dirac point. (e), (f) Show a comparison between the numeric and analytic results for Dirac gap opening. In particular, (e) shows the Dirac gap as a function of the distance to the chemical potential $\mathrm{\Lambda }$ (e), and (f) the Dirac gap as a function of the nonunitary superconducting term. We took ${\mathrm{\Lambda }}_1=0.7t$ and ${\mathrm{\Lambda }}_2=0$ for (a), (c), ${\mathrm{\Lambda }}_1=0.0$ and ${\mathrm{\Lambda }}_2=0.7t$ for (b), (d), ${\mathrm{\Lambda }}_2=0$ for panels (e), (f), and ${\mathrm{\Delta }}_0=0.71\mathrm{\Lambda }$ for (c), (d).

Figure 3

Electron spectral function $A(\omega ,\mathbf{k})$ for a system with a Dirac point away from the chemical potential due to doping (a), (c), (e) and due to spin-orbit coupling (b), (d), (f). (a), (b) Show the spectral function in the normal state, depicting a Dirac crossing away form the chemical potential. In the presence of unitary orbital superconductivity, a gap opens up at the chemical potential, but not at the Dirac crossings (c), (d). In contrast, in the presence of a nonunitary superconducting state, a gap opens up at the Dirac crossing (e), (f). We took ${\mathrm{\Lambda }}_1=1.5{\mathrm{\Delta }}_0$ and ${\mathrm{\Lambda }}_2=0$ in (a), (c), (e), ${\mathrm{\Lambda }}_1=0$ and ${\mathrm{\Lambda }}_2={\mathrm{\Delta }}_0$ in (b), (d), (f).

Figure 4

(a) Sketch of an interface between two superconducting domains with different nonunitary orders ${\mathrm{\Delta }}_A$ and ${\mathrm{\Delta }}_B$. (b), (c) Show the bulk spectral function of the superconductor in the absence of a superconducting domain wall, showing the gap opening in the buried Dirac point (b) and (d). In contrast, at the interface between the two superconducting domains, a chiral excitation appears inside the Dirac opening as shown in (c) and (e). We took $\mathrm{\Lambda }=1.5{\mathrm{\Delta }}_0$.