Re-examining Bogoliubov's theory of an interacting Bose gas TeX Export

@misc{citeulike:1177168, abstract = {We point out a few questionable features of Bogoliubov's theory of an interacting Bose gas. In this theory, one does not diagonalize the part \$\hat{H}\_2\$ of the Hamiltonian that is quadratic in the boson creation and annihilation operators, but instead one diagonalizes the ``grand canonical Hamiltonian" \$\tilde{H}\_2 = \hat{H}\_2 - \mu \hat{N}\_1\$, where \$\mu\$ is the chemical potential and the operator \$\hat{N}\_1\$ counts the number of bosons outside the condensate. We show that such a procedure leads to an energy cost of exciting a single boson from the condensate to the single-particle state of momentum \${\bf k}\$ which diverges like \$1/k\$ in the long wavelength limit. We also show that the above divergence does not occur if one diagonalizes \$\hat{H}\_2\$ directly, which results in a gapped excitation spectrum, both for bosons and for quasiparticles. Gapped excitation spectra for bosons were derived previously, by a number of authors, and were deemed unphysical on the grounds that such spectra violate Goldstone's theorem. Here, we argue that such an interpretation stems from an improper identification of the collective Goldstone modes of the system, which have to do with fluctuations of the phase of the condensate wavefunction. More importantly, we explicitly show that the so-called "quasiparticle" modes of \$\tilde{H}\_2\$ represent single-particle excitations, {\em not} collective sound waves. Finally, we show that the average interaction energy between condensed and depleted bosons is negative, even for strictly repulsive interactions, which we argue seriously undermines the validity of the present formulation of Bogoliubov's theory.}, archivePrefix = {arXiv}, author = {Ettouhami, A. M.}, citeulike-article-id = {1177168}, citeulike-linkout-0 = {http://arxiv.org/abs/cond-mat/0703498,%20noraise}, citeulike-linkout-1 = {http://arxiv.org/pdf/cond-mat/0703498,%20noraise}, day = {19}, eprint = {cond-mat/0703498,\%20noraise}, keywords = {bogoliubov, bose\_gas}, month = {Mar}, posted-at = {2007-03-20 10:42:52}, priority = {2}, title = {Re-examining Bogoliubov's theory of an interacting Bose gas}, url = {http://arxiv.org/abs/cond-mat/0703498,%20noraise}, year = {2007} }

TY - ELEC ID - citeulike:1177168 L3 - citeulike-article-id:1177168 N2 - We point out a few questionable features of Bogoliubov's theory of an interacting Bose gas. In this theory, one does not diagonalize the part $\hat{H}_2$ of the Hamiltonian that is quadratic in the boson creation and annihilation operators, but instead one diagonalizes the ``grand canonical Hamiltonian" $\tilde{H}_2 = \hat{H}_2 - \mu \hat{N}_1$, where $\mu$ is the chemical potential and the operator $\hat{N}_1$ counts the number of bosons outside the condensate. We show that such a procedure leads to an energy cost of exciting a single boson from the condensate to the single-particle state of momentum ${\bf k}$ which diverges like $1/k$ in the long wavelength limit. We also show that the above divergence does not occur if one diagonalizes $\hat{H}_2$ directly, which results in a gapped excitation spectrum, both for bosons and for quasiparticles. Gapped excitation spectra for bosons were derived previously, by a number of authors, and were deemed unphysical on the grounds that such spectra violate Goldstone's theorem. Here, we argue that such an interpretation stems from an improper identification of the collective Goldstone modes of the system, which have to do with fluctuations of the phase of the condensate wavefunction. More importantly, we explicitly show that the so-called "quasiparticle" modes of $\tilde{H}_2$ represent single-particle excitations, {\em not} collective sound waves. Finally, we show that the average interaction energy between condensed and depleted bosons is negative, even for strictly repulsive interactions, which we argue seriously undermines the validity of the present formulation of Bogoliubov's theory. TI - Re-examining Bogoliubov's theory of an interacting Bose gas KW - bogoliubov KW - bose_gas AU - Ettouhami, AM PY - 2007/Mar/19/ UR - http://arxiv.org/abs/cond-mat/0703498,%20noraise ER -