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Metric spaces are characterized by distances between pairs of elements.Systems that are physically similar are expected to present smaller distances(between their densities, wave functions and potentials) than systems thatpresent different physical behaviors. For this reason metric spaces are goodcandidates for probing quantum phase transitions, since they could identifyregimes of distinct phases. Here we apply metric space analysis to explore thetransitions between the several phases in spin imbalanced systems. Inparticular we investigate the so-called FFLO (Fulde-Ferrel-Larkin-Ovchinnikov)phase, which is an intriguing phenomenon in which superconductivity andmagnetism coexist in the same material. This is expected to appear for examplein attractive fermionic systems with spin-imbalanced populations, due to theinternal polarization produced by the imbalance. The transition between FFLOphase (superconducting phase) and the normal phase (non-superconducting) andtheir boundaries have been subject of discussion in recent years. We considerthe Hubbard model in the attractive regime for which Density MatrixRenormalization Group calculations allow us to obtain the exact densityfunction of the system. We then analyze the exact density distances as afunction of the polarization. We find that our distances display signatures ofthe distinct quantum phases in spin-imbalanced fermionic systems: with respectto a central reference polarization, systems without FFLO present a verysymmetric behavior, while systems with phase transitions are asymmetric.

Metric-space approach for distinguishing quantum phase transitions in spin-imbalanced systems