The 64th regiment on foot stayed only from 1813 to 1815 in Halifax before returning to Europe.
How old were the children? If they are more that 2 years old, they were probably born elsewhere.
The story of the 64th regiment in well detailed on line "wiki". Also it seems that the records are available at the Public Records Office in Kew (London), record class WO17.
Hope it helps.
Ginette