Spin Stiffness of The Heisenberg Model
See also: Heisenberg modelStart off with the simple Heisenberg spin Hamiltonian:
Now we introduce a rotation in the system at site i by an angle θi around the z-axis:
Plugging these back into the Heisenberg Hamiltonian:
now let θij = θi - θj and expand around θij = 0 via a MacLaurin expansion only keeping terms up to second order in θij
where the first term is independent of θ and the second term is a perturbation for small θ.
- is the z-component of the spin current operator
- is the "spin kinetic energy"
Consider now the case of identical twists, θx only that exist along nearest neighbor bonds along the x-axis Then since the spin stiffness is related to the difference in the ground state energy by
then for small θx and with the help of second order perturbation theory we get:
Read more about this topic: Spin Stiffness
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