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We discuss a two-band model for two-dimensional superconductors with electronand hole bands separated by an energy gap and singlet $d$-wave pairing in eachband. This model exhibits a V-shaped to U-shaped transition in the density ofthe states of the superconductor, which is similar to that found in recenttunneling experiments in magic-angle twisted trilayer graphene [H. Kim et al.,Nature 606, 494 (2022)]. This qualitative difference in behavior occurs whenthe electron and hole chemical potentials change, leading to topologicalquantum phase transitions between gapless and gapful $d$-wave superconductingstates, due to the annihilation of chiral Dirac fermions at the criticalpoints. Lastly, we show that direct thermodynamic evidence of these topologicalquantum phase transitions can be found in measurements of the compressibility,which exhibits logarithmic singularities at the transition points.

Topological two-band electron-hole superconductors with $d$-wave symmetry: a possible application to magic-angle twisted trilayer graphene