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Spin stiffness

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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 transition.

Mathematically

Mathematically it can be defined by the following equation:

where is the ground state energy, 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:

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 upto second order in θij

where the first term is independent of θ and the second term is a pertubation 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 pertubation theory we get:

See Also

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)
  • T. Einarsson, H.J. Schulz (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: |doi= and |version= (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 B. 40 (10): 7247–7251. doi:10.1103/PhysRevB.40.7247.
  • R. G. Melko, A. W. Sandvik, and D. J. Scalapino1 (2004). "Two-dimensional quantum XY model with ring exchange and external field". Physical Review B. 69 (10): 100408–100412. doi:10.1103/PhysRevB.69.100408.{{cite journal}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)