with charges, that is neighbored by empty diagonals on both sides. Repetition of this pattern alone produces the ground state configuration for the case of $f=1/3$ with spatial periodicity three.

Second component pattern (type II pattern) consists of a sequence of five diagonals that are [*empty, filled, empty, filled*]{}, and [*empty* ]{} respectively, or [**(01010)**]{} in our notation. This may be termed as a double filled diagonal because two filled diagonals are positioned in parallel at second neighbor. This forms the basis of the ground state configuration for the case of $f=2/5$ with lattice periodicity five.

Lastly, the third component pattern (type III pattern) consists of a sequence of seven diagonals that are sequentially [*empty, filled, empty, partially filled, empty, filled*]{}, and [*empty* ]{} [*i.e.*]{}, [**(010p010)**]{} in our notation where [**p**]{} refers to a partially filled diagonal where only part of the diagonal sites are occupied by positive charges. This pattern is essentially a partially filled diagonal enveloped by two filled diagonals on both sides at second neighbor diagonal position, which may be termed as a [*channel*]{} structure. This can form a basis with spatial lattice periodicity seven.

We leave the detailed description of the low temperature charge patterns for 