# 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:

$\rho_s = \cfrac{\partial^2}{\partial \theta^2}\cfrac{E_0(\theta)}{N}|_{\theta = 0}$

where $E_0$ is the ground state energy, $\theta$ 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:

$H_\mathrm{Heisenberg} = -J \sum_{} \left[S_i^z S_j^z + \cfrac{1}{2}(S_i^+ S_j^- + S_i^- S_j^+)\right]$

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

$S_i^+ \longrightarrow S_i^+e^{i \theta_i}$
$S_i^- \longrightarrow S_i^-e^{-i \theta_i}$

Plugging these back into the Heisenberg Hamiltonian:

$H(\theta_{ij}) = -J \sum_{} \left[S_i^z S_j^z + \cfrac{1}{2}(S_i^+e^{i\theta_i} S_j^-e^{-i\theta_j} + S_i^-e^{-i\theta_i} S_j^+e^{i\theta_j})\right]$

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

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

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

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

$E(\theta) - E(0) = N\rho_s\theta_x^2$

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

$\rho_s = \cfrac{1}{N} \left[ \cfrac{1}{2}\langle T_x \rangle + \sum_{\nu \neq 0} \cfrac{ | \langle 0| j_x^{(s)}|\nu\rangle |^2 }{E_{\nu} - E_0}\right]$