Thermal depinning and transverse-field tilting transitions in a planar vortex array pinned by a columnar defect

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We study the thermal and transverse-magnetic-field response of a vortex line array confined to a plane with a single columnar pinning defect. By integrating out ``bulk'' degrees of freedom away from the columnar defect we reduce this two-dimensional problem to a one-dimensional one, localized on the defect and exhibiting a long-range elasticity along the defect. We show that as a function of temperature, for a magnetic field aligned with the defect this system exhibits a one-dimensional analog of a roughening transition, with a low-temperature ``smooth'' phase corresponding to a vortex array pinned by the defect, and a high-temperature ``rough'' phase in which at long scales thermal fluctuations effectively average away pinning by the defect. We also find that in the low-temperature pinned phase, the vortex lattice tilt response to a transverse magnetic field proceeds via a soliton proliferation ``transition,'' governed by an integrable sine-Hilbert equation and analogous to the well-known commensurate-incommensurate transition in sine-Gordon systems. The distinguishing feature here is the long-range nature of the one-dimensional elasticity, leading to a logarithmic soliton energy and interaction. We predict the transverse-field--temperature phase diagram and discuss extension of our results to a bulk vortex array in the presence of a dilute concentration of columnar defects.