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Progress of Theoretical Physics

A Method of Approximate Second Quantization in the Theory of Superconductivity

Zygmunt M. Galasiewicz

Published Feb 1, 1960• 5 citations
Publisher

Abstract

The Hamiltonian of a dynamical system of Fermi-particles is transformed by means of the general unitary transformation proposed by N. N. Bogolubov. From this Hamiltonian we obtain the approximate second quantization (a. s. q.) Hamiltonian, introducing the Bose-amplitudes with two indices. This Hamiltonian is diagonalized and the collective oscillations considered, especially for the pairs of particles with parallel spins. For the forced collective oscillations the paramagnetic term in the Hamiltonian is also taken into account. Thie term leads to the additional “spin” current, connected with the elementary excitations with spin-moment ±1. If the transfer momentum tends to zero, this current vanishes as the terms omitted in obtaining the Meissner-Ochsenfeld effect.

DOI
10.1143/ptp.23.197

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The Hamiltonian of a dynamical system of Fermi-particles is transformed by means of the general unitary transformation proposed by N. N. Bogolubov. From this Hamiltonian we obtain the approximate second quantization (a. s. q.) Hamiltonian, introducing the Bose-amplitudes with two indices. This Hamiltonian is diagonalized and the collective oscillations considered, especially for the pairs of particles with parallel spins. For the forced collective oscillations the paramagnetic term in the Hamiltonian is also taken into account. Thie term leads to the additional “spin” current, connected with the elementary excitations with spin-moment ±1. If the transfer momentum tends to zero, this current vanishes as the terms omitted in obtaining the Meissner-Ochsenfeld effect.
Abstract The method of canonical transformations proposed by one of the authors ten years ago in connection with a microscopic theory of superfluidity for Bose systems, is generalized here to Fermi systems, and forms the basis of a method for investigating the problem of superconductivity. Starting from Fröhlich's Hamiltonian, the energy of the superconducting ground state and the one‐Fermion and collective excitations corresponding to this state are obtained. It turns out that the final formulae for the ground state and one‐Fermion excitations recently obtained by Bardeen, Cooper and Schrieffer are correct in the first approximation. The physical picture appears to be closer to the one proposed by Schafroth, Butler and Blatt. The effect on superconductivity of the Coulomb interaction between the electrons is analyzed in detail. A criterion for the superfluidity of a Fermi system with a four‐line vertex Hamiltonian is established.
A method of constructing ’’atomic second quantization’’ representations is described, based on the introduction of redundant modes (’’ideal atom variables’’) which are then given physical content by carrying out a suitable unitary transformation, the ’’generalized Tani transformation.’’ In such a representation bound atoms or molecules are described by elementary Bose or Fermi operators, the field operators for nuclei and electrons referring only to unbound particles. The Hamiltonian thus obtained contains not only nucleus–nucleus, electron–electron, and nucleus–electron Coulomb interactions, but also atom–atom, atom–nucleus, and atom–electron Coulomb and exchange interactions, including breakup and recombination terms. All possible scattering and reaction channels are exhibited simultaneously in this transformed Hamiltonian. The method is applicable to any species or mixture of species of composite particles, but for simplicity the derivation is restricted here to the case of atomic hydrogen.
Taking inspiration from the state-of-the art knowledge of the Bose-Hubbard (BH) model and recent methodological developments in its fermionic counterpart, this work deals with the study of the collective dynamics of a lattice Bose gas beyond the mean-field picture through a quantum description of its elementary excitations. The Hamiltonian quantization, performed via a Bogoliubov quadratization of the BH action within the Gutzwiller approach, allows to expand the effective action of the theory up to second order in the fluctuations around the mean-field solution, as well as to prefigure the possibility of identifying the main decay vertices of the collective modes and other effects that are not evident at the second-order level. This quantum description extends the standard Bogoliubov approach to the study of superfluid Bose systems for comprising higher excitation branches, including the Higgs mode in the superfluid phase, and identifies their physical meaning together with appropriate observables which could be taken into consideration for their experimental characterization. The ultimate aim of the quantization procedure is the determination of fundamental quantities as the depletion of the condensate and the effective superfluid fraction, which are not accessible by a mean-field description or not completely characterized in the regime of strong interactions.
The effects of possible superfluidity on the cranking moment of a large many-fermion system moving under periodic boundary conditions are investigated within the framework of the theory of superconductivity recently formulated by Bogolyubov. The Hamiltonian is initially subjected to Bogolyubov's general unitary transformation. The collective excitations of the fermions are then considered in the usual pair approximation; the appropriate cranking terms are linear in the boson pair operators. On performing a unitary transformation which transforms away these linear terms, one obtains an expression for the moment of inertia of the system which includes both the effects of possible superfluidity and collective excitation. This expression, by virtue of its being stationary with respect to arbitrary variations in the amplitude associated with the latter unitary transformation, is then utilized as a variational principle for the moment of inertia. For the normal state, the result previously obtained by the author, that the moment of inertia has the rigid value, is rederived in more compact form. For the superfluid state, one finds that collective excitations effect a marked increase in the superfluid moment at intermediate coupling strengths although the resulting moment is still quite small compared to the rigid value. In contrast to the normal state case, where particle-hole pairs play a major role, this increase is almost entirely due to excitations consisting of particle-pairs or hole-pairs. The precise magnitude of the apparent resonance in the moment produced by the ddd-wave part of the cranking interaction is dependent to some extent on the features of the particle-particle potential which leads to the superfluid state. Variational expressions for the moment are exhibited for both Yukawa and delta-function shell potentials. These results are identical in charged and neutral Fermi systems. A calculation of the cranking moment at finite temperatures is presented in an Appendix along with an interpretation of it in terms of Bardeen's two-fluid model of superconductivity.
Summary of the lectures: 1. Creation and annihilation operators for the system of indistinguishable particles 1.1 The permutation group and the states of a system of indistinguishable particles 1.2 Dimension of the Hilbert space of a system of indistinguishable particles 1.3 Definition and properties of the creation and annihilation operators 1.4 TheFockspace 1.5 The representations of state vectors and operators 1.5.1 N-particlewave-functions 1.5.2 The second-quantization representation of operators 1.5.3 Examples 1.6 Evolution of operators in the Heisenberg picture 1.7 Statistical operators of indistinguishable particles 1.7.1 The averages of the s-particle operators 1.7.2 The general structure of the one-particle statistical operator 2 Quadratic Hamiltonian and the diagonalization 2.1 Diagionalization of the Hamiltonian quadratic in the fermion operators 2.2 Diagionalization of the Hamiltonian quadratic in the boson operators 2.2.1 Diagonalization of the quadratic bosonic Hamiltonian possessing a zero mode 2.3 Long range order, condensation and the Bogoliubov spectrum of weakly interacting Bose gas 2.3.1 The excitation spectrum of weakly interacting Bose gas inabox 2.3.2 The excitation spectrum of an interacting Bose gas in an externalpotential 2.4 The Jordan-Wigner transformation: fermionization of interacting 1D spin chains