Quantum Fluctuation Of The Cyclic States Of Spin-2Condensates In A Weak Magnetic Field

In the mean-field level, some ground-state configurations of a quantum many-bodysystem are degenerate. But when the disorderly quantum fluctuations are concluded, thecorrection of the mean-field energy will lift the degeneracy and give the real ground state,which is orderly. This "order-by-disorder" phenomenon has been widely studied in thecontext of quantum magnetism, but has not been observed in experiments until now.Recently, Fei Zhou' group in Canada and Demler' group at Harvard have shown that thisphenomenon may be observed in the nematic phase of spin-2condensates, provided that theexternal magnetic field is weak enough. However, in experiments, the fluctuation of themagnetic field can not be suppressed. We in the present paper provide a model that the"order-by-disorder" phenomenon may be observed in finite magnetic field. for a non-zeroWe consider the cyclic states of a dilute spin-2atomic gas in the presence of only thelinear Zeeman term, which are degenerate. But the symmetry of these states are diferent,meaning that they should not be degenerate. This is confirmed by the calculation of thetwo-order quantum fluctuation, where the corrections of the mean-field energies are indeeddiferent. Then the real ground state beyond the mean-field approximation is obtained.It is shown that for the condensates with repulsive interaction, when the total spin arepositive, the ground state is the state C_+for the antiferromagnetic condensates, and theground state C_-for the ferromagnetic condensates. The state C can never be the groundstate in the presence of the non-zero total spin.