The Bogoliubov Method in the Theory of the Crystal State

W. Younes, Daniel Marc Gogny, J. F. Berger

V. V. Voronov, R. V. Jolos, N. N. Arsenyev +1 more

It is shown that an idea of the superfluidity of nuclear matter and the u -v transformation of Bogoliubov, time-dependent Hartree-Fock-Bogoliubov method have made a strong impact on the development of the modern nuclear theory. Some applications of the Bogoliubov methods to describe properties of the low-lying collective nuclear states and the giant resonances are demonstrated.

Edward V. Gevorkyan

A J Vidal, G. E. Astrakharchik, Leandra Vranješ Markić +1 more

The ground-state properties of one-dimensional electron-spin-polarized hydrogen 1 H, deuterium 2 H, and tritium 3 H are obtained by means of quantum Monte Carlo methods. The equations of state of the three isotopes are calculated for a wide range of linear densities. The pair correlation function and the static structure factor are obtained and interpreted within the framework of the Luttinger liquid theory. We report the density dependence of the Luttinger parameter and use it to identify different physical regimes: Bogoliubov Bose gas, super-Tonks–Girardeau gas, and quasi-crystal regimes for bosons; repulsive, attractive Fermi gas, and quasi-crystal regimes for fermions. We find that the tritium isotope is the one with the richest behavior. Our results show unambiguously the relevant role of the isotope mass in the properties of this quantum system.

A. M. Ettouhami

As is well-known, in the conventional formulation of Bogoliubov's theory of an interacting Bose gas, the Hamiltonian is written as a decoupled sum of contributions from different momenta of the form . Then, each of the single-mode Hamiltonians is diagonalized separately, and the resulting ground state wavefunction of the total Hamiltonian is written as a simple product of the ground state wavefunctions of each of the single-mode Hamiltonians . While this way of diagonalizing the total Hamiltonian may seem to be valid from the perspective of the standard, number non-conserving Bogoliubov's method, where the k = 0 state is removed from the Hilbert space and hence the individual Hilbert spaces where the Hamiltonians {} are diagonalized are disjoint from one another, we argue that from a number-conserving perspective this diagonalization method may not be adequate since the true Hilbert spaces where the Hamiltonians {} should be diagonalized all have the k = 0 state in common, and hence the ground state wavefunction of the total Hamiltonian may not be written as a simple product of the ground state wavefunctions of the 's. In this paper, we give a thorough review of Bogoliubov's method, and discuss a variational and number-conserving formulation of this theory in which the k = 0 state is restored to the Hilbert space of the interacting gas, and where, instead of diagonalizing the Hamiltonians separately, we diagonalize the total Hamiltonian . When this is done, we find that the ground state energy is lowered below the Bogoliubov result, and the depletion of bosons is significantly reduced with respect to the one obtained in the number non-conserving treatment. We also find that the spectrum of the usual αk excitations of Bogoliubov's method changes from a gapless one, as predicted by the standard, number non-conserving formulation of this theory, to one which exhibits a finite gap in the k → 0 limit. We discuss the presence of a gap in the spectrum of the αk's in light of Goldstone's theorem, and show that there is no contradiction with the latter.

George P. Flessas