Find gauged symmetries
When the Hamiltonian includes gauge fields, it happens that qsymm misses some symmetries.
For example, when the superconducting order parameter has a constant complex value, in
qsymm this is represented as two terms with real coefficients
Re Δ and
Im Δ. Treating both of these as free parameters,
qsymm.symmetries finds no time-reversal symmetry. However, any constant
Δ is gauge-equivalent to a purely real
Δ, which has TR invariance. The reason TR is not found in the general case is that the unitary part of TR depends on the gauge choice (the phase of
qsymm only looks for symmetries with constant unitary parts.
Looking for symmetries with constant unitary part works fine usually, as physical symmetries should be independent of the specific parameters in the system. For gauge-fields, however, there is an inherent gauge freedom, which is not physical, and the symmetries may depend on the gauge choice. For now it is the user's responsibility to carry out a gauge-fixing before looking for symmetries, but this is not always obvious.
For example a Hamiltonian family might include non-physical gauge-like symmetries, i.e. when parts of the family are related by a family of unitary transformations. This typically happens when changing the phase of a complex valued parameter results purely in a unitary transformation of the Hamiltonian. Like in the above example with
Δ, it may happen that the family lacks a symmetry with constant unitary part for arbitrary phase of the parameter, but has this symmetry for any fixed value of the phase. Such a situation might result in mysterious degeneracies, seemingly not associated with any symmetry.
Would be nice to automate detection of such gauge-like symmetries, gauge-fixing, or allow some form of gauge-dependence in the symmetries that the symmetry finder looks for.