1. d=2 xy model
n=2
planar (xy) model consists of spins of unit magnitude that can point in any direction in the x-y plane
si,x= cos(i) si,y= sin(i)
xy simulation

2. Vortex Pairs
Spin wave
excitations

3. x-y model M=0 for all T >0 spin waves destroy long range order
spin correlations have a power law decay
spin system has a stiffness at low T
vortices are tightly bound pairs at low T
unbind at T=Tc for form a vortex plasma
spin stiffness vanishes at Tc

5. =.25

6. Stiffness 
F = (1/2)2

8. Ferromagnetic Models
ferromagnetic models display phase transitions depending on the values of (n,d)
Ising model(n=1) need d>1
x-y model (n=2) need d>2 for long range order
however topological defects(vortices) unbind at a finite Tc and destroy the stiffness
antiferromagnetic models (J<0) may exhibit frustration

9. Bipartite Lattices
J<0
J>0

10. Triangular Lattice
J<0
J>0
Frustration => non-zero entropy at T=0

11. Classical Heisenberg Model n=3, d=2
in the ferromagnetic case (J>0), the order parameter is an 3-component vector
for bipartite lattices, the same is true for J<0
since n=3, we need d >2 to have Tc 0
for the triangular lattice the system is frustrated when J<0
energy wins at T=0 and ground state has spins arranged in a plane at 1200 to each other

12. HAFT
Two chiralities Q -Q
Q=(4/3,0)

13. A B C

14. A B C

15. HAFT
noncollinear arrangement of the spins on each triangle => 1200 structure
order parameter is no longer a vector
rather it is a tensor or rotation matrix
system of interacting rigid bodies
low energy excitations correspond to rotations about three principal axes
=> both in plane and out of plane twists
=> three stiffnesses 2 3 1

16. Spin Stiffness    L
twist the spin system by an angle ij = (/L) (ij.) about the -direction in spin space
H changes to

17. Stiffness
Use standard Monte Carlo methods to study response functions

18. Consistent with Tc=0

19. Finite Tc ?

20. Overview
Extremely rapid crossover in the structure factor and correlation length near T~.3J
spin stiffnesses at low T vanish at large length scales => Tc= 0
may be related to disappearance of free vortices at a finite Tc

21. Kawamura and Miyashita(1984) pointed out that
the isotropic model has a topologically stable point defect A A
m=0
m=+1

22. Vorticity Stiffness
Strength of a vortex is characterized by the winding number `m`
energy cost proportional to m2 and ln(L/a)
a vorticity can be defined as the response of the spin system to an imposed twist by =2m about an axis perpendicular or parallel to the spin plane ^ i ri O L

23. Vorticity F is the free energy cost for an isolated vortex
V(L) = C + v ln(L/a)
v is the vorticity modulus
v = [V(L2)-V(L1)]/ln(L2/L1)

25. Summary
Heisenberg antiferromagnet on the triangular lattice is frustrated
order parameter is non-collinear
topological defects exist
rapid change in structure factor near T=.3J
spin wave stiffness is zero at all T>0
vorticity stiffness is finite at low T and disappears abruptly near T=.3J
consistent with a defect unbinding transition
situation is different from the xy model

26. Summary
For the xy model, the spin wave and vortex degrees of freedom are decoupled
two spin correlation function has power law decay below the transition and exponential decay above
stiffness and vorticity modulus behave identically
for the Heisenberg model, two spin correlations decay exponentially at all T>0 => stiffness is zero at large length scales
vortices unbind at a finite Tc and influence two spin correlations indirectly