Bosonic excitations in random media

We consider classical normal modes and noninteracting bosonic excitations in disordered systems. We emphasize generic aspects of such problems and parallels with disordered, noninteracting systems of fermions, and discuss in particular the relevance for bosonic excitations of symmetry classes known in the fermionic context. We also stress important differences between bosonic and fermionic problems. One of these follows from the fact that ground-state stability of a system requires all bosonic excitation energy levels to be positive, while stability in systems of noninteracting fermions is ensured by the exclusion principle, whatever the single-particle energies. As a consequence, simple models of uncorrelated disorder are less useful for bosonic systems than for fermionic ones, and it is generally important to study the excitation spectrum in conjunction with the problem of constructing a disorder-dependent ground state: we show how a mapping to an operator with chiral symmetry provides a useful tool for doing this. A second difference involves the distinction for bosonic systems between excitations which are Goldstone modes and those which are not. In the case of Goldstone modes we review established results illustrating the fact that disorder decouples from excitations in the low-frequency limit, above a critical dimension $d_c,$ which in different circumstances takes the values $d_c=2\mathrm{}$ and $d_c=0.\mathrm{}$ For bosonic excitations which are not Goldstone modes, we argue that an excitation density varying with frequency as $\rho \left(\omega \right)\propto {\omega }^4$ is a universal feature in systems with ground states that depend on the disorder realization. We illustrate our conclusions with extensive analytical and some numerical calculations for a variety of models in one dimension.