# Spin stiffness

The spin stiffness or spin rigidity or helicity modulus or the "superfluid density" (for bosons the superfluid density is proportional to the spin stiffness) is a constant which represents the change in the ground state energy of a spin system as a result of introducing a slow in plane twist of the spins. The importance of this constant is in its use as an indicator of quantum phase transitions—specifically in models with metal-insulator transitions such as Mott insulators. It is also related to other topological invariants such as the Berry phase and Chern numbers as in the Quantum hall effect.

## Mathematically

Mathematically it can be defined by the following equation:

${\displaystyle \rho _{s}={\cfrac {\partial ^{2}}{\partial \theta ^{2}}}{\cfrac {E_{0}(\theta )}{N}}|_{\theta =0}}$

where ${\displaystyle E_{0}}$ is the ground state energy, ${\displaystyle \theta }$ is the twisting angle, and N is the number of lattice sites.

## Spin stiffness of the Heisenberg model

Start off with the simple Heisenberg spin Hamiltonian:

${\displaystyle H_{\mathrm {Heisenberg} }=-J\sum _{}\left[S_{i}^{z}S_{j}^{z}+{\cfrac {1}{2}}(S_{i}^{+}S_{j}^{-}+S_{i}^{-}S_{j}^{+})\right]}$

Now we introduce a rotation in the system at site i by an angle θi around the z-axis:

${\displaystyle S_{i}^{+}\longrightarrow S_{i}^{+}e^{i\theta _{i}}}$
${\displaystyle S_{i}^{-}\longrightarrow S_{i}^{-}e^{-i\theta _{i}}}$

Plugging these back into the Heisenberg Hamiltonian:

${\displaystyle H(\theta _{ij})=-J\sum _{}\left[S_{i}^{z}S_{j}^{z}+{\cfrac {1}{2}}(S_{i}^{+}e^{i\theta _{i}}S_{j}^{-}e^{-i\theta _{j}}+S_{i}^{-}e^{-i\theta _{i}}S_{j}^{+}e^{i\theta _{j}})\right]}$

now let θij = θi - θj and expand around θij = 0 via a MacLaurin expansion only keeping terms up to second order in θij

${\displaystyle H\approx H_{\mathrm {Heisenberg} }-J\sum _{}\left[\theta _{ij}J_{ij}^{(s)}-{\cfrac {1}{2}}\theta _{ij}^{2}T_{ij}^{(s)}\right]}$

where the first term is independent of θ and the second term is a perturbation for small θ.

${\displaystyle J_{ij}^{s}={\cfrac {i}{2}}(S_{i}^{+}S_{j}^{-}-S_{i}^{-}S_{j}^{+})}$ is the z-component of the spin current operator
${\displaystyle T_{ij}={\cfrac {1}{2}}(S_{i}^{+}S_{j}^{-}+S_{i}^{-}S_{j}^{+})}$ 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

${\displaystyle E(\theta )-E(0)=N\rho _{s}\theta _{x}^{2}}$

then for small θx and with the help of second order perturbation theory we get:

${\displaystyle \rho _{s}={\cfrac {1}{N}}\left[{\cfrac {1}{2}}\langle T_{x}\rangle +\sum _{\nu \neq 0}{\cfrac {|\langle 0|j_{x}^{(s)}|\nu \rangle |^{2}}{E_{\nu }-E_{0}}}\right]}$