Ground-state properties and excitation spectra of non-Galilean-invariant interacting Bose systems

Physical Review B - Condensed Matter and Materials Physics

Volumn 64, Issue 10, 2001, Pages

  Jackeli, G ^a   Ranninger, J ^a  

^a CNRS   (France)

Author keywords

[No Author keywords available]

Indexed keywords

ARTICLE; ELEMENTARY PARTICLE; ENERGY; MATHEMATICAL ANALYSIS; MOLECULAR INTERACTION; SOUND; VELOCITY;

EID: 18544394077     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.64.104513     Document Type: Article

References (11)

1. Bose-Einstein Condensation, edited by A. Griffin, D. W. Snoke, and S. Stringari (Cambridge University Press, Cambridge, 1995).
2. N N. Bogoliubov, J. Phys. (Moscow)5, 23 (1947).
3. S T. Beliaev, Sov. Phys. JETP7, 299 (1958).
4. V N. Popov, Functional Integrals in Quantum Field theory and Statistical Physics (Reidel, Dortrecht, 1983), Chap. 6.
5. For a review see, H. Shi and A. Griffin, Phys. Rep.304, 1 (1998).
6. N M. Hugenholtz and D. Pines, Phys. Rev.116, 489 (1959).
7. J. Gavoret and P. Nozières, Ann. Phys. (N.Y.)28, 349 (1964).
8. G. Jackeli and G. Ranninger, Phys. Rev. B63, 184512 (2001).
9. See Eqs. (4.7) and (4.9) in Ref. 7. We point out that in deriving these equations no Galilean invariance has been invoked in Ref. 7, and hence they still hold in the present case.
10. First we point out, that the diagram given in Fig. 11(a) that has been already evaluated Eq. (3) is the basic element of the diagrams (formula presented) (formula presented), and (formula presented) shown in Fig. 11(d). Based on this observation one finds (formula presented) and (formula presented). One farther verifies, that diagram (formula presented), which can be obtained from (formula presented) by reversing an arrow of one of the boson propagators, is twice bigger. This leads to (formula presented) since in the former one (formula presented) both two poles (in lower and upper half of the complex plane) of the phonon propagator contribute. The same holds for (formula presented) and (formula presented)—and it is straightforward to show (formula presented). Putting all these contributions together and noting that the diagrams (formula presented) (formula presented) (formula presented), and (formula presented) enters with the symmetric factor 2, one obtains (formula presented). Substituting in the last expression those corresponding values of the diagrams, discussed above, one recovers Eq. (9).
11. The second order corrections to the chemical potential (formula presented) have been discussed in details in Ref. 8. The diagrammatic representation of the contribution to (formula presented) due to the boson phonon coupling is shown in Fig. 7 of that paper. The contributions due to the boson-boson interaction is obtained by replacing the phonon propagator by the boson-boson interaction line in these diagrams. The leading order in density terms, can then be obtained by replacing the dressed boson propagators in these diagrams by corresponding perturbation series and keeping only the leading, linear in density terms. In this way the same diagrams as those shown in Fig. 11(d) of the present paper are generated and one recovers the analytical formula for (formula presented) presented in the text.