Are indexes need for parts of primary key?
A compound index can also support queries that compare columns that are at the beginning of the indexed values (but aren't all). So an index on (field1, field2) (which is implicitly created for a primary key) would also support queries that compare only field1.
But it cannot support queries that only compare columns of its end (but aren't all). So (field1, field2) won't support queries that only compare field2.
So yes, you need to create an extra index only on (field2) to have a chance it supports queries only comparing field2.