Matrix Product Toolkit | Tools / MpShiftBasis

This program shifts the quantum numbers of the local basis of a wavefunction. That is, it changes all quantum numbers {$x$} by {$x \leftarrow x' = x+g$} where {$g$} is a fixed input, and {$+$} here denotes the group operation. This is only well defined if the representation {$g$} has degree 1, and the reduced matrix elements of {$x$} and {$x'$} are the same. This is always true for abelian symmetries, but only sometimes true for non-abelian symmetries.

A typical use of this tool is to shift the quantum numbers by 0.5, to convert between a spin representation of {$S^z = \{-1/2, +1/2\}$} and a particle representation of {$N = \{0,1\}$}.

Matrix Product Toolkit version HEAD-0.7.4.0 (subversion tree rev 802)
Compiled on Dec 10 2007 at 11:50:09
usage: mp-shift-basis <quantum-number-name> <shift> <psi>

The quantum-number-name denotes one of the quantum numbers in the symmetry list (shown by mp-info).

Examples

Suppose we have a wavefunction psi of a spin-1/2 chain, with local basis

Basis has symmetry Sz:U(1), size = 2
   N  QuantumNumber
   0           -0.5
   1            0.5

To shift the quantum numbers to a particle representation (ie. spinless fermions), use

mp-shift Sz 0.5 psi

This gives a new local basis of

Basis has symmetry Sz:U(1), size = 2
   N  QuantumNumber
   0              0
   1              1

Notes

To be able to use the resulting wavefunction with lattice files defined for a particle number symmetry, one more step is needed to rename the Sz quantum number to N. This is done using mp-rename-symmetry.

This program applies the transformation to all sites of the lattice. It would be easy to apply the transformation to only a single site, or a selection - the only question is what format to use to specify this on the command line.

To see what the quantum numbers of the local basis are, use mp-info -b psi.

MpShiftBasis

Examples

Notes

See Also