Bernard and Helmholtz are expelled from society and sent to live on an island for two, twin reasons. Primarily, they joined the Savage in fomenting a rebellion and disturbing the peace. Contributing to that act, both Bernard and Helmholtz are unable to adjust their dispositions properly to the life and manners of their society. They are fundamentally at odds with the happy conformity expected of them. 

"The New World standards of conformity discourage mental and emotional freedom. Those who rebel against the established order are rehabilitated with drugs or exiled" (eNotes).

While they are not as bold or as enraged as the Savage, both Bernard and Helmholtz stand against the values and norms of the society. 

In the discussion between Helmholtz and the World Controller, Helmholtz demonstrates his romantic preferences. When asked where he would like to be exiled, Helmholtz does not choose a happy or pleasant place, as any normal person from his society would. 

"Helmholtz rose from his pneumatic chair. 'I should like a thoroughly bad climate,' he answered. 'I believe one would write better if the climate were bad. If there were a lot of wind and storms, for example...'"

In this proclamation, Helmholtz opts for inspiration over ease, a concept foreign to his peers and anathema to the morals of his society. 

Thus, the discontent that characterizes Helmholtz and Bernard can be identified as the underlying factor that leads to their exile. On the surface, however, the two are brought in to talk with the Controller because of the role they played in helping the Savage to destroy soma meant to be distributed to the working class and, in doing so, incite a near-riot.

The society of this future is predicated on ideas of stability and figures like Helmholtz and Bernard do not contribute to stability and, in fact, undermine it.

Perhaps, the Romantic ideals of Helmholtz to experience nature and inspiration represent the ebullience of human nature, rising up despite all social forces that would inhibit its expression. And perhaps the discontent of Bernard suggests that even in a society where everyone is supposed to be a winner, some sensibilities cannot help but feel left out of the game.