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

TL;DR

This paper derives exact eigenstates of a truncated Hamiltonian for weakly interacting bosons, revealing differences from traditional Bogoliubov and Gross-Pitaevskii theories, especially regarding dissipation mechanisms.

Contribution

It provides the exact solutions for a class of eigenstates of the truncated Hamiltonian, extending understanding beyond approximations like BCA and GPT.

Findings

Exact eigenstates of the truncated Hamiltonian are derived in the thermodynamic limit.

Differences between exact solutions and BCA/GPT persist even at large system sizes.

The emission of pseudobosons may differ from traditional Landau dissipation predictions.

Abstract

In a gas of $N$ weakly interacting bosons \cite{Bogo1, Bogo2}, a truncated canonic Hamiltonian $\widetilde{h}_c$ follows from dropping all the interaction terms between free bosons with momentum $\hbar\mathbf{k}\ne\mathbf{0}$. Bogoliubov Canonic Approximation (BCA) is a further manipulation, replacing the number \emph{operator} $\widetilde{N}_{in}$ of free particles in $\mathbf{k}=\mathbf{0}$, with the total number $N$ of bosons. BCA transforms $\widetilde{h}_c$ into a different Hamiltonian $H_{BCA}=\sum_{\mathbf{k}\ne\mathbf{0}}\epsilon(k)B^\dagger_\mathbf{k}B_\mathbf{k}+const$, where $B^\dagger_\mathbf{k}$ 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 $|\:S,\:\mathbf{k}\:\rangle_{c}$, 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 $\eta\hbar\mathbf{k}$ ($\eta=\:1,\:2,\: \dots$). 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.