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We consider flux lines in the presence of a dense square array of columnar defects, with insulating core and radius of the order of the coherence length. The properties of the phase transition from the flux line regime to the localized superconducting state are determined here. We show that the localized superconducting state is a new vortex state because of the counterclockwise and clockwise supercurrent flows found around each defect. We propose here a general operator to count the vorticity of any cavity inside a superconductor and give a method to obtain the reversible magnetization of superconductors with multiply connected geometries. ͓S0163-1829͑99͒11541-8͔Recent improvements in the resolution of electron beam lithography have allowed the fabrication of ordered arrays of columnar defects ͑CD's͒ with radius R not much larger than the coherence length . 1-3 Flux lines ͑FL's͒ in presence of a regular lattice of CD's show collective properties such as matching and multiple quantum trapping. Matching occurs every time the artificial CD and FL lattices are commensurate. According to some theories, 4 CD can trap many FL's ͑Ref. 5͒ up to the saturation limit of R/2(T), where T is the temperature.A regular array of CD's turns the superconductor into a multiply connected geometry. This system presents the interesting property of surface superconductivity around the CD edges, which is crucially dependent on the properties of the medium exterior to the superconductor, as found long ago by Saint-James and de Gennes. 6 This medium, played here by the CD core, must be insulating such that the supercurrent flow is constrained to the superconductor's interior. 7 For a nonperforated sample above the upper critical field H c2 , superconductivity disappears in the bulk. However, in the case of a perforated sample, such as the case of a regular array of CD's, superconductivity remains above H c2 around the CD edges, in the form of a thin layer of thickness (T), and this lasts until the field H c3 is reached. Bezryadin, Buzdin, and Pannetier 8 ͑BBP͒ determined several properties of this localized superconducting state ͑LSS͒, such as the ratio H c3 /H c2 whose large R limit recovers the single planar interface problem, first considered by Saint-James and de Gennes. BBP also found that there are many possible LSS's, each characterized by P, the number of FL's trapped inside the CD. They determined that the transition between these LSS's yields discontinuities in the magnetization ͑first-order transi-tion͒. Since they assume the LSS around each CD from the beginning, their conclusions only apply for fields above the transition. The interaction between the FL lattice and a single CD was also investigated below the transition where the local symmetry group caused by the deformation was determined. 9 In this paper we obtain the magnetic properties of the transition to the LSS, and propose a Ginzburg-Landau free-energy expansion that describes the interaction between FL's and a dense regular lattice of CD's parallel to them with radiu...

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