The paper by Hatano and Nelson explores localization transitions in non-Hermitian quantum mechanics, specifically in one and two dimensions, when quantum mechanical particles are subjected to a constant imaginary vector potential. The authors use a path-integral formulation to relate the transition to flux lines depinned from columnar defects by a transverse magnetic field in superconductors. They predict 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. The study is motivated by the mapping of flux lines in a \(d+1\)-dimensional superconductor to the world lines of \(d\)-dimensional bosons. Columnar defects in the superconductor introduce random potential in the boson system, and the perpendicular component of the magnetic field acts as a constant imaginary vector potential. The authors investigate how a flux line is depinned from these defects by an increasing perpendicular magnetic field, indicating the presence of extended states in the large perpendicular field region. The non-Hermitian Hamiltonian is given by \(\mathcal{H} \equiv (\boldsymbol{p} + i\boldsymbol{h})^2/(2m) + V(\boldsymbol{x})\), where \(\boldsymbol{p}\) is the momentum operator and \(V(\boldsymbol{x})\) is a random potential. The imaginary gauge transformation applies to the eigenfunctions and eigenvalues, leading to the appearance of complex eigenvalues at the delocalization point. The authors solve the one-dimensional system with a single attractive impurity and show that the ground state is localized only for \(h < \hbar \kappa_{\mathrm{gs}}\), while all other states are extended. In the two-dimensional case, they find bands of localized energies bounded by a mobility edge, with extended and localized states mixed near the band center. The paper also discusses the probability distribution of a flux line near a free surface and the relaxation process of flux lines from the surface to occupied states. The authors derive a stretched exponential form for the displacement of the flux line near a free surface, characterized by a diverging surface localization length. Overall, the study provides insights into the behavior of non-Hermitian quantum systems and the localization transitions in the presence of random potentials and magnetic fields.The paper by Hatano and Nelson explores localization transitions in non-Hermitian quantum mechanics, specifically in one and two dimensions, when quantum mechanical particles are subjected to a constant imaginary vector potential. The authors use a path-integral formulation to relate the transition to flux lines depinned from columnar defects by a transverse magnetic field in superconductors. They predict 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. The study is motivated by the mapping of flux lines in a \(d+1\)-dimensional superconductor to the world lines of \(d\)-dimensional bosons. Columnar defects in the superconductor introduce random potential in the boson system, and the perpendicular component of the magnetic field acts as a constant imaginary vector potential. The authors investigate how a flux line is depinned from these defects by an increasing perpendicular magnetic field, indicating the presence of extended states in the large perpendicular field region. The non-Hermitian Hamiltonian is given by \(\mathcal{H} \equiv (\boldsymbol{p} + i\boldsymbol{h})^2/(2m) + V(\boldsymbol{x})\), where \(\boldsymbol{p}\) is the momentum operator and \(V(\boldsymbol{x})\) is a random potential. The imaginary gauge transformation applies to the eigenfunctions and eigenvalues, leading to the appearance of complex eigenvalues at the delocalization point. The authors solve the one-dimensional system with a single attractive impurity and show that the ground state is localized only for \(h < \hbar \kappa_{\mathrm{gs}}\), while all other states are extended. In the two-dimensional case, they find bands of localized energies bounded by a mobility edge, with extended and localized states mixed near the band center. The paper also discusses the probability distribution of a flux line near a free surface and the relaxation process of flux lines from the surface to occupied states. The authors derive a stretched exponential form for the displacement of the flux line near a free surface, characterized by a diverging surface localization length. Overall, the study provides insights into the behavior of non-Hermitian quantum systems and the localization transitions in the presence of random potentials and magnetic fields.