Andreev reflection and the semiclassical Bogoliubov-de Gennes Hamiltonian: resonant states

Vortex Lattices and the Bogoliubov-De Gennes Equations

We consider the Bogoliubov-de Gennes equations giving an equivalent formulation of the BCS theory of superconductivity. We are interested in static solutions with the magnetic field present. We carefully formulate the equations in the basis independent form, discuss their general features and isolate key physical classes of solutions (normal and vortex lattice states) which are the candidates for the ground state. We prove existence of the normal and vortex lattice states and stability of the normal states for large temperature or magnetic fields and their instability for small temperature and…

Bogoliubov-de Gennes方程中的配对

Pairing in the Bogoliubov-de Gennes equations

结果表明，Bogoliubov-de Gennes方程对电子的状态是正常状态的线性组合。因此，需要BCS样归约过程来选择正确的配对。对于齐次系统，我们指出由Bogoliubov-de Gennes方程导出的自洽方程的核需要受到BCS配对条件的约束。另一方面，在存在普通杂质的情况下，Bogoliubov-de Gennes方程应补充Anderson的配对条件，以通过相应的幺正变换获得正确的真空态。这导致声子介导的相互作用的局部化校正。

The one dimensional semi-classical Bogoliubov-de Gennes Hamiltonian with PT symmetry: generalized Bohr-Sommerfeld quantization rules

We present a method for computing first order asymptotics of semiclassical spectra for 1-D Bogoliubov-de Gennes (BdG) Hamiltonian from Supraconductivity, which models the electron/hole scattering through two SNS junctions. This involves: 1) reducing the system to Weber equation near the branching point at the junctions, 2) constructing local sections of the fibre bundle of microlocal solutions, 3) normalizing these solutions for the "flux norm" associated to the microlocal Wronskians, 4) finding the relative monodromy matrices in the gauge group that leaves invariant the flux norm, 5) from thi…

Electron-hole coherent states for the Bogoliubov-de Gennes equation

We construct a new set of generalized coherent states, the electron-hole coherent states, for a (quasi-)spin particle on the infinite line. The definition is inspired by applications to the Bogoliubov-de Gennes equations where the quasi-spin refers to electron- and hole-like components of electronic excitations in a superconductor. Electron-hole coherent states generally entangle the space and the quasi-spin degrees of freedom. We show that the electron-hole coherent states allow obtaining a resolution of unity and form minimum uncertainty states for position and velocity where the velocity op…

The Dirac-Frenkel Principle for Reduced Density Matrices, and the Bogoliubov-de-Gennes Equations

The derivation of effective evolution equations is central to the study of non-stationary quantum many-body sytems, and widely used in contexts such as superconductivity, nuclear physics, Bose-Einstein condensation and quantum chemistry. We reformulate the Dirac-Frenkel approximation principle in terms of reduced density matrices, and apply it to fermionic and bosonic many-body systems. We obtain the Bogoliubov-de-Gennes and Hartree-Fock-Bogoliubov equations, respectively. While we do not prove quantitative error estimates, our formulation does show that the approximation is optimal within the…

The Bogoliubov/de Gennes system, the AKNS hierarchy, and nonlinear quantum mechanical supersymmetry

We show that the Ginzburg-Landau expansion of the grand potential for the Bogoliubov-de Gennes Hamiltonian is determined by the integrable nonlinear equations of the AKNS hierarchy, and that this provides the natural mathematical framework for a hidden nonlinear quantum mechanical supersymmetry underlying the dynamics.