I'll use this question as an occasion to advertise to mathematicians a interesting piece of terminology commonly used in condensed matter physics: ``degenerate ground state manifold''. In mathematical terms, this translates to ``eigenspace for the lowest eigenvalue of the Hamiltonian, whose dimension is $\ge2$, and that does not have a preferred basis of eigenvectors''.

State: Here, a ``state'' is an eigenvector of the Hamiltonian.

ground: A state is a ``ground state'' if its corresponding eigenvalue (=energy) is the lowest.

degenerate: Generically, the eigenspaces corresponding to the various eigenvalues of the Hamiltonian will be one-dimensional . When that doesn't happen, an energy level is called degenerate. The word ``degeneracy'' is then used to refer to its dimension.

Manifold: Here, physicists use the term ``manifold'' here because the lowest energy eigenspace does not have a natural basis (or has more than one natural basis that one could write down). This is somewhat similar to the use of the term ``manifold'' by mathematicians: a manifold is a space on which one does not have preferred choices of coordinates (or one has multiple choices of coordinate systems).