You have to understand, most of these people are not ready to be unplugged. And many of them are so inured, so hopelessly dependent on the system, that they will fight to protect it.
You have to understand, most of these people are not ready to be unplugged. And many of them are so inured, so hopelessly dependent on the system, that they will fight to protect it.
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Main /
SymmetricHubbardModelConverting a wavefunction from an {$SO(4)$} basis to {$U(1)\times U(1)$} requires a few steps. A sample script to do most of the work is here: #!/bin/bash if [ $# -lt 2 ] ; then echo "usage: so4tou1 <wavefunction> <quantumnumbers>" echo "quantum numbers are Qz,Sz where Qz = (N-L)/2" exit 1 fi mp-wigner-eckart $1 $1.u1 "N:U(1),Sz:U(1)" $2 mp-scale-basis N 2 $1.u1 mp-shift-basis N 1 $1.u1 mp-reorder-basis 0,2,3,1 $1.u1 mp-normalize $1.u1 This script takes an input wavefunction that is assumed to have {$SO(4)$} symmetry, and writes an output wavefunction with {$U(1)\times U(1)$} symmetry, as the same filename with a '.u1' suffix. It also normalizes the wavefunction at the end. Due to the normalization conventions, the resulting wavefunction will have the norm decreased by {$\sqrt{(2q+1)(2s+1)}$}. This doesn't quite finish the task. There is a problem of sign conventions in the basis, that requires doing a unitary transformation on the wavefunction after the projection. The SO(4) basis is bipartite, and in effect the A sublattice (odd sites) uses the ordering up,down for the double occupied sites. ie, the signs of the {$C^{\dagger(A)}$} operator are set as {$C^{\dagger(A)}_{\uparrow}\mid\downarrow\rangle = \mid\uparrow\downarrow\rangle$}, and {$C^{\dagger(A)}_{\downarrow}\mid\uparrow\rangle = -\mid\uparrow\downarrow\rangle$}. On the B sublattice (even sites), it is the opposite, the ordering is down,up, giving {$C^{\dagger(B)}_{\uparrow}\mid\downarrow\rangle = -\mid\downarrow\uparrow\rangle$}, and {$C^{\dagger(B)}_{\downarrow}\mid\uparrow\rangle = \mid\downarrow\uparrow\rangle$}. This is inevitable, since without the alternation in basis the hopping matrix elements are not {$SO(4)$} symmetric. In the {$U(1)$} basis, there is no need to use a bipartite structure for the local basis, and the convention we have used is that all sites use the ordering {$\mid\uparrow\downarrow\rangle$}. This means that when we do the Wigner-Eckart projection of an {$SO(4)$} wavefunction, we also need to do a spatial reflection on the B sublattice to flip the sign of the {$\mid\downarrow\uparrow\rangle$} state into {$\mid\uparrow\downarrow\rangle$}. This is achieved with the new R_B operator. So, there is one more step to get a {$SO(4$}) wavefunction into the {$U(1)$} basis, mp-apply lattice:R_B input output (mp-apply works fine if the output file is the same as the input) This must be done on the {$U(1)$} wavefunction, as the spatial reflection operator violates {$SO(4)$} symmetry. This step could be included in the script above, but it would require another argument for the {$U(1)$} lattice file. An alternative choice would be to give the {$U(1)$} Hubbard basis the same bipartite structure as the {$SO(4)$} basis. But it is an unconventional choice of basis, and I don't think it is worth complicating the U(1) basis just for a minor simplication in {$SO(4) \rightarrow U(1)$} transformations. |