I'll use this question as an occasion to advertise to mathematicians a interesting piece of terminology commonly used in condensed matter physics: <b>\``degenerate ground state manifold''</b>.
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''.
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Let me analyze the individual terms of the phrase:

<b><i>State:</i></b> Here, a \``state'' is an eigenvector of the Hamiltonian.

<b><i>ground:</i></b> A state is a \``ground state'' if its corresponding eigenvalue (=energy) is the lowest.

<b><i>degenerate:</i></b> 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.

<b><i>Manifold:</i></b> 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).