Fig. 1: Wheel geometry, ladder geometry and the single-particle dispersion.
From: Symmetry-protected Bose-Einstein condensation of interacting hardcore bosons
The main plot illustrates the single-particle dispersion relation (middle, red curve) of the wheel geometry (a), emerging from projecting down the dispersion from the ladder geometry (b) (upper orange and lower blue curve). Note the appearance of two single-particle states at k0 = 0 (red crosses). This is because the Hilbert space of the wheel is obtained by projecting out all modes on the inner ring, except for the zero momentum states \({\big\vert {N}_{\odot ,k = 0}\big\rangle }_{\odot }\). Momentum conservation then couples this central mode to the particular mode on the outer ring respecting the k0-modulated ring-to-center hopping, which generates an extensively scaling level splitting (red circle and crosses).
