Matrix Product Toolkit | Tools / MpWignerEckartOld

This program projects non-abelian quantum numbers onto the `axis of quantization', thereby breaking the symmetry down into an abelian subgroup.

Matrix Product Toolkit version HEAD-0.7.4.0 (subversion tree rev 802)
Compiled on Dec 10 2007 at 11:49:00
usage: mp-wigner-eckart <input-psi> <output-psi> <symmetry-list> <projection>

The new symmetry-list must map all non-abelian quantum numbers onto the abelian subgroup, and appear in the same order as the original symmetry list. A current limitation is that all non-abelian symmetries must be projected; if there is more than one {$SU(2)$} symmetry, for example, it is not currently possible to project one but not the other.

The projection is the quantum number in the new symmetry list.

Examples

For the Hubbard model with {$SO(4) \equiv SU(2) \otimes SU(2)$} symmetry, we have the symmetry list Q:SU(2),S:SU(2). The projected symmetry list will therefore contain two {$U(1)$} symmetries. We could call them anything we want, but for consistency it is best to call them Qz and Sz. Thus the new symmetry list will be Qz:U(1),Sz:U(1).

The projection quantum numbers can be chosen arbitrarily, provided that {$-Q \leq Q^z \leq Q$} and {$-S \leq S^z \leq S$}, and {$Q+Q^z$} and {$S+S^z$} are both integer (ie, if {$Q$} or {$S$} are (half-)integer, then {$Q^z$} and {$S^z$} are also (half-)integer).

For example, for a state psi-in with quantum numbers {$4,2$}, we can project onto any state from {$-4,-2$} up to {$4,2$}. The command would be

mp-wigner-eckart psi-in psi-out "Qz:U(1),Sz:U(1)" -4,2

Note that the z-component of the pseudospin is related to the particle number by {$Q^z = \frac{N-1}{2}$}. Thus, {$Q^z=0$} corresponds to half-filling, and if {$Q > 0$}, the {$Q^z = 0$} state is a half-filled excited state. The hole-doped and particle-doped groundstates correspond to {$Q^z = -Q$} and {$Q^z = Q$} respectively. To convert from {$Q^z$} to the particle number, a shift, scale, and rename (and possibly a reordering of the basis) is required.

MpWignerEckartOld

Examples

See Also