Interaction Distance: Measuring Many-Body Freedom via Quantum Correlation Structure*

The entanglement spectrum [Li, Haldane (2008)], obtained by reducing a pure state,
has been shown a valuable quantity in identifying universal features of a many-body quantum state.
Ground states of free-fermion Hamiltonians exhibit particular patterns in their entanglement spectra.

For any state, we define the `interaction distance' as the minimal trace distance of a generic entanglement spectrum
to any free-fermion entanglement spectrum.
Surprisingly, it is possible that ground states of interacting systems show vanishing interaction distance.

We present results for the 1D Ising model with transverse and longitudinal fields, where we find that the
interaction distance diagnoses the phase diagram by application on the ground state [Nat. Commun. 8, 14926 (2017)].
Furthermore, we propose parafermion chains [Alicea, Fenldey (2015)] as examples of intrinsically interacting systems in terms of fermions.
This is identified by the interaction distance being finite, and in particular cases, almost maximal [arXiv:1705.09983].
Finally, we propose to bypass the optimisation problem of computing the interaction distance by the use of machine learning methods.