Spin stiffness

The spin-stiffness or spin rigidity or helicity modulus or the "superfluid density" (depending on which reference one reads) is a constant which represents the change in the ground state energy of a spin system as a result of introducing a twist at the boundary of the system. The importance of this constant is in its use as an indicator of quantum phase transitions-- specifically in models with metal-insulator transition.

Mathematically

Mathematically it can be defined by the following equation:

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

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

See also: Heisenberg model

Start off with the simple Heisenberg spin Hamiltonian:

$H_{Heisenberg}=-J\sum _{<i,j>}[S_{i}^{z}S_{j}^{z}+{\cfrac {1}{2}}(S_{i}^{+}S_{j}^{-}+S_{i}^{-}S_{j}^{+})]$

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

$S_{i}^{+}\longrightarrow S_{i}^{+}e^{i\theta }$

$S_{i}^{-}\longrightarrow S_{i}^{-}e^{-i\theta }$

Plugging these back into the Heisenberg Hamiltonian:

$H=-J\sum _{<i,j>}[S_{i}^{z}S_{j}^{z}+{\cfrac {1}{2}}(S_{i}^{+}e^{i\theta _{i}}S_{j}^{-}e^{-i\theta _{j}}+S_{i}^{-}e^{-i\theta _{i}}S_{j}^{+}e^{i\theta _{j}})]$

now let θ_ij = θ_i - θ_j and expand around θ_ij = 0 via a MacLaurin expansion only keeping terms upto second order in θ_ij

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

where the first term is independent of θ and the second term is a pertubation 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, θ_x 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 θ_x and with the help of second order pertubation theory we get:

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

See Also

spin waves

References

• S.E. Krüger, R. Darradi, J. Richter, D.J.J. Farnell (2006). "Direct calculation of the spin stiffness of the spin-(1/2) Heisenberg antiferromagnet on square, triangular, and cubic lattices using the coupled-cluster method". Physical Review B. 73 (9): 094404. doi:10.1103/PhysRevB.73.094404.{{cite journal}}: CS1 maint: multiple names: authors list (link)
• J. Bonča, J.P. Rodriguez, J. Ferrer, K.S. Bedell (1994). "Direct calculation of spin stiffness for spin-1/2 Heisenberg models". Physical Review B. 50 (5): 3415–3418. doi:10.1103/PhysRevB.50.3415.{{cite journal}}: CS1 maint: multiple names: authors list (link)
• Einarsson, T.; Schulz, H. J. (1994). "Direct Calculation of the Spin Stiffness in the J₁−J₂ Heisenberg Antiferromagnet". arXiv:cond-mat/9410090v1. {{cite arXiv}}: |class= ignored (help); Cite has empty unknown parameters: |version= and |doi= (help)
• B.S. Shastry, B. Sutherland (1990). "Twisted boundary conditions and effective mass in Heisenberg–Ising and Hubbard rings". Physical Review Letters. 65 (2): 243–246. doi:10.1103/PhysRevLett.65.243.
• R.R.P. Singh, D.A. Huse (1989). "Microscopic calculation of the spin-stiffness constant for the spin-(1/2) square-lattice Heisenberg antiferromagnet". Physical Review Letters. 40 (10): 7247–7251. doi:10.1103/PhysRevB.40.7247.