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We study the localization transitions which arise in both one and two dimensions when quantum mechanical particles described by a random Schrödinger equation are subjected to a constant imaginary vector potential. A path-integral formulation relates the transition to flux lines depinned from columnar defects by a transverse magnetic field in superconductors. The theory predicts that the transverse Meissner effect is accompanied by stretched exponential relaxation of the field into the bulk and a diverging penetration depth at the transition.PACS numbers: 05.30. Jp, 72.15.Rn, 74.60.Ge Although forbidden in conventional quantum mechanics, exponentiated non-Hermitian quantum Hamiltonians do appear in the transfer matrices of classical statistical mechanics problems. A nonequilibrium process can be described as the time-evolution of a non-Hermitian system [1]. Another example is the XXZ spin chain mapped onto the asymmetric six-vertex model [2].In this Letter, we investigate a non-Hermitian quantum Hamiltonian with randomness. The study is motivated by a mapping of flux lines in a d+1-dimensional superconductor to the world lines of d-dimensional bosons. Columnar defects in the superconductor, which were introduced experimentally in order to pin the flux lines [3], give rise to random potential in the boson system [4]. Although the field component H z parallel to the columns acts like a chemical potential for the bosons, the component perpendicular to the columns results in a constant imaginary vector potential [5].We study localization in this simple example of nonHermitian quantum mechanics, and thereby show how a flux line is depinned from columnar defects by an increasing perpendicular magnetic field H ⊥ . It is generally believed that all eigenstates are localized in conventional one-and two-dimensional non-interacting quantum systems with randomness. On the other hand, it is almost obvious that a flux line is depinned from defects by a strong perpendicular field component. This indicates that the present non-Hermitian system has extended states in a large H ⊥ region, and that there must be a delocalization transition at a certain strength of H ⊥ .The non-Hermitian Hamiltonian treated hereafter has the form H ≡ (p + ih) 2 /(2m) + V (x), where p ≡ (h/i)∇, and V (x) is a random potential. The nonHermitian field h originates in the transverse magnetic field as h = φ 0 H ⊥ /(4π), where φ 0 is the flux quantum [4,5]. Figure 1 shows a vortex whose "world line" is described by this Hamiltonian with periodic boundary conditions in one dimension. The mass m is equivalent to the tilt modulus of the flux line. The Planck parameter h corresponds to the temperature of the superconductor, while the inverse temperature of the quantum system corresponds to the thickness L τ of the superconductor. Interactions between many particles (or flux lines) can be treated approximately by forbidding multiple occupancy of localized state in a tight binding model (see below) [4]. Interactions in the delocalized regime will be discussed...