We calculate the spectrum of the amplitude mode, the analog of the Higgs mode in high-energy physics, for the $d$-density wave state proposed to describe the pseudogap phase of the high-$T_c$ cuprates. Even though the state breaks translational symmetry by a lattice spacing and is described by a particle-hole singlet order parameter at the wave vector $q=Q=(\pi ,\pi )$, remarkably, we find that the amplitude mode spectrum can have peaks at both $q=(0,0)$ and $q=Q=(\pi ,\pi )$; we shall set lattice spacing to unity. In general, the spectrum is nonuniversal and, depending on the microscopic parameters, can have one or two peaks in the Brillouin zone, signifying the existence of two kinds of magnetic excitations. Our theory sheds important light on how multiple inelastic neutron peaks at different wave vectors can, in principle, arise even with an order parameter that condenses at $Q=(\pi ,\pi )$.

• Received 16 September 2012

DOI:https://doi.org/10.1103/PhysRevB.87.224503