Exact canonic eigenstates of the truncated Bogoliubov Hamiltonian in an interacting bosons gas

Abstract

In a gas of $N$ weakly interacting bosons \cite{Bogo1, Bogo2}, a truncated canonic Hamiltonian ${\tilde{h}}_c$ follows from dropping all the interaction terms between free bosons with momentum $\hslash \mathbf{k}\ne 0$. Bogoliubov Canonic Approximation (BCA) is a further manipulation, replacing the number \emph{operator} ${\tilde{N}}_{in}$ of free particles in $\mathbf{k}=0$, with the total number $N$ of bosons. BCA transforms ${\tilde{h}}_c$ into a different Hamiltonian $H_{BCA}={\sum }_{\mathbf{k}\ne 0}\epsilon (k)B_{\mathbf{k}}^{†}{B}_{\mathbf{k}}+const$, where $B_{\mathbf{k}}^{†}$ and $B_{\mathbf{k}}$ create/annihilate non interacting pseudoparticles. The problem of the \emph{exact} eigenstates of the truncated Hamiltonian is completely solved in the thermodynamic limit (TL) for a special class of eigensolutions $|S,\mathbf{k}{⟩}_c$, denoted as \textquoteleft s-pseudobosons\textquoteright, with energies ${\mathcal{E}}_S(k)$ and \emph{zero} total momentum. Some preliminary results are given for the exact eigenstates (denoted as \textquoteleft $\eta$-pseudobosons\textquoteright), carrying a total momentum $\eta \hslash \mathbf{k}$ ($\eta =1,2,\dots$). A comparison is done with $H_{BCA}$ and with the Gross-Pitaevskii theory (GPT), showing that some differences between exact and BCA/GPT results persist even in the TL. Finally, it is argued that the emission of $\eta$-pseudobosons, which is responsible for the dissipation $\acute{a}$ \emph{la} Landau \cite{L}, could be significantly different from the usual picture, based on BCA pseudobosons.

Cited by in corpus (2)

• Bogoliubov theory of interacting bosons: new insights from an old problem
• Exact diagonalization of the truncated Bogoliubov Hamiltonian