To describe correctly the limit of vanishing magnetic field one needs to consider also asymmetric two-particle processes (center figure).

Changing the angle between spin-orbital axis and that of the magnetic field from 90

These results are described in
Spin-orbit induced spin-density wave in a quantum wire,
Jianmin Sun, Suhas Gangadharaiah and Oleg A. Starykh, Phys. Rev. Lett. **98**,
126408 (2007), and in Spin-orbital effects in magnetized quantum wires and spin chains,
Suhas Gangadharaiah, Jianmin Sun, and Oleg A. Starykh,
Phys. Rev. B **78**, 054436 (2008). The findings are different from the case
of staggered Dzyaloshinskii-Moriya interaction analyzed in an important work of
M. Oshikawa and I. Affleck, Phys. Rev. B 65, 134410 (2002).