In Lorentz-invariant systems spontaneously broken gauge symmetry results in three types of fundamental excitations: density excitations, Higgs bosons (amplitude modes), and Goldstone bosons (phase modes). The density and phase modes are coupled by electromagnetic interactions while the amplitude modes are not. In $s$-wave superconductors, the Higgs mode, which can be observed only under special conditions, has been detected. We show that unconventional $d$-wave superconductors, such as the high-temperature cuprate superconductors, should have a rich assortment of Higgs bosons, each in a different irreducible representation of the point-group symmetry of the lattice. We also show that these modes have a characteristic singular spectral structure and discuss conditions for their observability.

• Received 16 July 2012

DOI:https://doi.org/10.1103/PhysRevB.87.054503