Topological two-band electron-hole superconductors with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math>-wave symmetry: Absence of Dirac quasiparticle annihilation in magic-angle twisted trilayer graphene

We discuss a two-band model for two-dimensional superconductors with electron and hole bands separated by an energy gap and singlet $d$-wave pairing in each band. This type of model exhibits a V-shaped to U-shaped transition in the density of the states of the superconductor, and was phenomenologically used as a possible interpretation of recent tunneling experiments in magic-angle twisted trilayer graphene (MATTG) [H. Kim et al., Nature (London) 606, 494 (2022)]. Performing a microscopic investigation, we find that such a qualitative difference in behavior occurs when the electron and hole chemical potentials change, leading to topological quantum phase transitions (TQPTs) between gapless and gapped $d$-wave superconducting states, due to the annihilation of chiral Dirac fermions at the phase boundaries. This transition requires the vanishing of the coherence peaks in the density of states at zero energy when the phase boundary is crossed, but this is not seen in the experimental data of Kim et al. We also show that direct thermodynamic signatures of these topological quantum phase transitions arise in the theoretical compressibility, which exhibits logarithmic singularities at the transition points. Measurements of the compressibility may illuminate the interpretation of the experimental data of Kim et al. and provide additional information about topological quantum phase transitions in the superconducting state of MATTG. Based on our analysis, we are led to conclude that the V-shaped to U-shaped transition observed is not related to the annihilation of Dirac fermion quasiparticles and its associated TQPTs, but is possibly connected to a change in symmetry of the order parameter from a nodal to a non-nodal superconducting phase.