Suppose we have all the possible combinations for each spin type, ↑ and ↓, i.e. B ↑Ip and B ↓Ip respectively. We would like to combine these two, to produce a complete list of determinants. In order to do this, it must be realized these systems independent, given any configuration, e.g. for determinant ↑ 7 B ↑7p , we would like all the configurations for the ↓-type i.e. B ↓Jp . Therefore, the approach is to concatenate (this operation is given by the two-vertical-lines) the following outer-product: