over a wide range of µ , consistent with dimensional anal- ysis; in particular there is no sign of singular behaviour at µ = µ$_{Q}$ , although there is a systematic rise for µ > ∼ µ$_{D}$ . Fig. 4 plots quantities giving information on the nature and symmetries of the ground state. In the limit j → 0, the diquark condensate 〈 qq 〉 is an order parameter for the spontaneous breaking of U(1)$_{B}$ leading to baryon num- ber superfluidity. Although the data of Fig. 4 are taken with j = 0, implying some care must be taken with the extrapolation j → 0 at small µ [6], we are confident that this symmetry is broken for all µ > µ$_{o}$ . The approximate flatness of the curve for µ$_{Q}$ < ∼ µ < ∼ µ$_{D}$ is then evidence for a scaling 〈 qq 〉 ∝ µ 2 similar to Eqn.(7). We take this as an indication that in this region the system consists of degenerate quark matter with a Fermi surface disrupted by a BCS instability. FIG. 4: (Color online) Superfluid order parameter 〈 qq 〉 /µ 2 and Polyakov line versus µ . The Polyakov line is an order parameter for decon- finement in the limit of infinitely massive quarks – away from this limit it continues to yield information on the free energy of an isolated color source. Fig. 4 shows that QC 2 D remains confined for µ < µ$_{D}$ , but that there ap- pears to be a transition to a deconfined state for chem- ical potentials in excess of this value. In physical units µ$_{D}$ ≈ 850MeV, corresponding to quark density n$_{q}$ ≈ 16 – 32fm − $^{3}$, some 35 – 70 times nuclear density. To summarise, the simulations suggest that QC 2 D has three distinct transitions (or at least crossovers). The first, at µ = µ$_{o}$ , is a firmly established second order phase transition (in the limit j → 0) from vacuum to a BEC superfluid, and is described accurately for the most part by χ PT (the quark energy density ε$_{q}$ looks to be an important exception). Since the pion is not especially light with our choice of lattice parameters, implying only a moderate separation of Goldstone and hadronic mass scales, the µ -window within which the BEC is favoured is not particularly wide. The second transition at µ = µ$_{Q}$ looks like a BEC/BCS An interesting issue is whether QC 2 D is special in that the N$_{c}$ -quark bound states required by color confinement are also favoured by the more general renormalisation group argument that 2-body interactions are the only The apparent sensitivity of the value of µ$_{D}$ to small changes in temperature is consistent with the very weak curvature of the phase boundary between confined and deconfined phases postulated in that work, and observed in a recent study of QC 2 D matter using the PNJL model [10]. Between µ$_{Q}$ and µ$_{D}$ the system resembles the quarky- onic matter recently postulated on the basis of large- N$_{c}$ arguments [2]; namely a state of degenerate matter which is also confined, so that excitations above the ground state remain color singlet. Because we have used Wil- son fermions (with no manifest chiral symmetry) we are unfortunately unable at this point to test whether chi- ral symmetry is restored, another important aspect of the quarkyonic hypothesis; we note however that even in a conventional scenario χ PT predicts 〈 ¯ ψψ 〉 ∝ µ − 2 for µ ≥ µ$_{o}$ [5], which is likely to be difficult to distinguish from true chiral symmetry restoration in this region, par- ticularly if the transition is a crossover. An immediate concern is the validity of the deconfin- ing transition in the continuum limit. Fig. 4 shows that with a = 0 . 23fm µ$_{D}$ ≈ 600MeV, and is practically indis- tinguishable from µ$_{Q}$ ; for this reason only a deconfined quark matter phase was identified in Ref. [6] (Cf. Fig. 1). In both cases, however, the quark density in lattice units n$_{q}$a 3 = 0 . 17 (coarse) or 0.20 (fine), well short of the value 2 N$_{c}$N$_{f}$ signifying that lattice saturation artifacts have set in – indeed for a = 0 . 186fm trends in all observables look smooth out to µa = 1 . 0 corresponding to µ = 1 . 06GeV. It is therefore plausible that the observed difference in µ$_{D}$ is physical, and due to the differing temperatures of the two lattices used. The third transition at µ = µ$_{D}$ is signalled by a change in the scaling of the thermodynamic observables, notably ε$_{g}$ ( µ ) and ( T$_{µµ}$ )$_{q}$, a change in the sign of ( T$_{µµ}$ )$_{g}$, and a non-zero Polyakov loop. For µ > µ$_{D}$ the system consists of deconfined quark matter. crossover to form a ground state where the scalings of the observables n$_{q}$ ( µ ), p ( µ ), ε$_{q}$ ( µ ) and 〈 qq ( µ ) 〉 all suggest it is formed of degenerate quark matter with a well-defined Fermi sphere, albeit one whose surface is disrupted by a BCS condensate. We note that effective treatments based on both χ PT and NJL models predict n$_{q}$/n$_{SB}$ to be monotonic decreasing in this regime, and are unable to fit the lattice data [9]. The distinct nature of this region is also supported by the diverging behaviours of ( T$_{µµ}$ )$_{g}$ and ( T$_{µµ}$ )$_{q}$ for µ > ∼ µ$_{Q}$ seen in Fig. 2, although the reason for the peculiar behaviour of ( T$_{µµ}$ )$_{q}$ is not well understood at present. The transition at µ ≈ µ$_{Q}$ is most likely a smooth crossover, but the exact nature of this putative transition requires further study, in particular a careful extrapolation to the limit of zero diquark source. 4