Phase coherence in two-dimensional type-II superconductors.

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We show that in two-dimensional type-II superconductors, the vortex liquid-solid transition is not accompanied by a simultaneous divergence of the phase correlation length. Even in the ``vortex-solid'' phase, despite a finite shear modulus and quasi-long-range density correlations, the phase coherence remains short ranged, with a correlation length of order one magnetic length. We also find that, in the solid phase, there is no ``Bose condensation,'' in the sense that any particular Abrikosov lattice state has zero statistical weight in the thermodynamic limit. The vortex-solid phase in two dimensions is therefore best described as a Cooper-pair charge-density wave with no superconducting phase order.