Abstract 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 a resolution of unity to be obtained and form minimum uncertainty states for position and velocity where the velocity operator is defined using the Bogoliubov–de Gennes Hamiltonian. The usefulness and the limitations of electron–hole coherent states and the phase space representations built from them are discussed in terms of basic applications to the Bogoliubov–de Gennes equation such as Andreev reflection.
Abstract 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 a resolution of unity to be obtained and form minimum uncertainty states for position and velocity where the velocity operator is defined using the Bogoliubov–de Gennes Hamiltonian. The usefulness and the limitations of electron–hole coherent states and the phase space representations built from them are discussed in terms of basic applications to the Bogoliubov–de Gennes equation such as Andreev reflection.
We investigate generalizations of coherent states as a means of representing the dynamics of excitations of the superconducting ground state. We also analyse the propagation of generalized coherent state wave packets under the Bogoliubov-de Gennes Hamiltonian. The excitations of the superconducting ground state are superpositions of electron and hole quasi-particles described by the Bogoliubov-de Gennes equations, that can only exist at energies outside the band gap. A natural generalization relevant to the excitations of the superconducting ground state is the tensor product of canonical and spin coherent states. This state will quickly become de-localized on phase space under evolution by the Bogoliubov-de Gennes Hamiltonian due to the opposite velocities of the quasi-spin components. We therefore define the electron-hole coherent states which remain localised on phase space over longer times. We show that the electron-hole coherent states though entangled retain many defining features of coherent states. We analyse the propagation of both product and electron hole coherent states in a superconductor with a spatially homogeneous superconducting band gap. The dispersion relation indicates that wavepackets defined on the band gap have a zero group velocity, but we will show that interference effects can create states on the band gap that propagate at the Fermi velocity. We also consider the two semiclassical, short wavelength regimes, hbar->0$ and the large Fermi energy limit mu->infinity. In general these limits produce behaviour analogous to the canonical coherent states except for isolated cases. Finally we analyse the dynamics of the Andreev Reflection of a Gaussian wavepacket incident on a discontinuous normal-superconducting interface. We show that restricting the energy bandwidth of the incident state inside the superconducting band gap precludes the wavepacket from fully entering the superconducting region. We again consider the two semiclassical regimes.
A new quantum Bogoliubov-de Gennes (BdeG) formalism is developed to study the self-consistent motion of holes and spin excitations in a quantum antiferromagnet within the generalized t-J model. On the one hand, the effects of local distortion of spin configurations and the renormalization of the hole motion due to virtual excitations of the distorted spin background are treated on an equal footing to obtain the hole wave function and its spectrum, as well as the effective mass for a propagating hole. On the other hand, the change of the spin excitation spectrum and the spin correlations due to the presence of dynamical holes are studied within the same adiabatic approximation. The stability of the hole states with respect to such changes justifies the self-consistency of the proposed formalism.
Starting with a microscopic Hamiltonian, we derive the Schr\"odinger equation of a superconductor with a vortex. The analysis describes the many-body state of a superconductor as a delocalized superposition of Bogoliubov-de Gennes mean-field vortex states. Taking a suitable one-band extended Hubbard model as the microscopic Hamiltonian of electrons in high-Tc compounds, solutions to the vortex Schr\"odinger equation are computed. We compute the vortex mass. Tuning the Hamiltonian parameters, we obtain solutions for the case of an s-wave order parameter and for the d-wave case and consider consequences of order parameter symmetry for vortex directional motion.
Abstract Chapter 13 addresses Bose condensation in superfluids (and superconductors), which involves the field operator ψ having a c-number component (<ψ(x,t)>≠0), challenging number conservation. The nonlinear Gross-Pitaevskii equation is derived for this condensate wave function <ψ>=ψ−ψ˜, facilitating identification of the coherence length and the core region of vortex motion. The noncondensate Green’s function G˜1(1,1′)=−i<(ψ˜(1)ψ˜+(1′))+> and the nonvanishing anomalous correlation function F˜∗(2,1′)=−i<(ψ˜+(2)ψ˜+(1′))+> describe the dynamics and elementary excitations of the non-condensate states and are discussed in conjunction with Landau’s criterion for viscosity. Associated concepts of off-diagonal long-range order and the interpretation of <ψ> as a superfluid order parameter are also introduced. Anderson’s Bose-condensed state, as a phase-coherent wave packet superposition of number states, resolves issues of number conservation. Superconductivity involves bound Cooper pairs of electrons capable of Bose condensation and superfluid behavior. Correspondingly, the two-particle Green’s function has a term involving a product of anomalous bound-Cooper-pair condensate wave functions of the type F(1,2)=−i<(ψ(1)ψ(2))+>≠0, such that G2(1,2;1′,2′)=F(1,2)F+(1′,2′)+G˜2(1,2;1′,2′). Here, G˜2 describes the dynamics/excitations of the non-superfluid-condensate states, while nonvanishing F,F+ represent a phase-coherent wave packet superposition of Cooper-pair number states and off-diagonal long range order. Employing this form of G2 in the G1-equation couples the condensed state with the non-condensate excitations. Taken jointly with the dynamical equation for F(1,2), this leads to the Gorkov equations, encompassing the Bardeen–Cooper–Schrieffer (BCS) energy gap, critical temperature, and Bogoliubov-de Gennes eigenfunction Bogoliubons. Superconductor thermodynamics and critical magnetic field are discussed. For a weak magnetic field, the Gorkov-equations lead to Ginzburg–Landau theory and a nonlinear Schrödinger-like equation for the pair wave function and the associated supercurrent, along with identification of the Cooper pair density. Furthermore, Chapter 13 addresses the apparent lack of gauge invariance of London theory with an elegant variational analysis involving re-gauging the potentials, yielding a manifestly gauge invariant generalization of the London equation. Consistency with the equation of continuity implies the existence of Anderson’s acoustic normal mode, which is supplanted by the plasmon for Coulomb interaction. Type II superconductors and the penetration (and interaction) of quantized magnetic flux lines are also discussed. Finally, Chapter 13 addresses Josephson tunneling between superconductors.
A general free bosonic system with a pairing term is described by a bosonic Bogoliubov-de Gennes (BdG) Hamiltonian. The representation is given by a pseudo-Hermitian matrix, which is crucially different from the Hermitian representation of a fermionic BdG Hamiltonian. In fermionic BdG systems, a topological invariant of the whole particle (hole) bands can be nontrivial, which characterizes the Andreev bound states (ABS), including Majorana fermions. In bosonic cases, on the other hand, the corresponding topological invariant is thought to be trivial, owing to the stability condition of the bosonic ground state. In this paper, we consider a two-dimensional model that realizes a bosonic analogy of the ABS at the boundaries. The boundary states of this model are located outside the bulk bands and are characterized by a nontrivial Berry phase (or polarization) of the hole band. Furthermore, we investigate the zero-energy flat-band limit in which the Bloch Hamiltonian is defective, where the particle and hole states are identical to each other. In this limit, the Berry phase is Z2 quantized thanks to an emergent parity-time symmetry induced by the defective nature. This is an example of a topological invariant that uses the defective nature as a projection structure. Thus, boundary states in our model are essentially different from Hermitian topological modes and their variants. We also discuss the entanglement entropy of the system with bosonic Andreev bound states, motivated by the relationship between our model and the continuous variable surface codes.
Spin coherent states play a crucial role in defining QESM (quasi-exactly solvable models) establishing a strict correspondence between energy spectra of spin systems and low-lying quantum states for a particle moving in a potential field of a certain form. Spin coherent states are also used for finding the Wigner-Kirkwood expansion and quantum corrections to energy quantization rules. The closed equation which governs dynamics of a quantum system is obtained in the spin coherent representation directly for observable quantities.