  Spectral functions on Euclidean Jordan algebras arise frequently in convex
models. Despite the success of primal-dual conic interior point solvers, there
has been little work on enabling direct support for spectral cones, i.e. proper
nonsymmetric cones defined from epigraphs and perspectives of spectral
functions. We propose simple logarithmically homogeneous barriers for spectral
cones and we derive efficient, numerically stable procedures for evaluating
barrier oracles such as inverse Hessian operators. For two useful classes of
spectral cones - the root-determinant cones and the matrix monotone derivative
cones - we show that the barriers are self-concordant, with nearly optimal
parameters. We implement these cones and oracles in our open source solver
Hypatia, and we write simple, natural formulations for four applied problems.
Our computational benchmarks demonstrate that Hypatia often solves the natural
formulations more efficiently than advanced solvers such as MOSEK 9 solve
equivalent extended formulations written using only the cones these solvers
support.
