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

Vortex Pairs Spin wave excitations

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

=.25

Stiffness  F = (1/2)2

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

Bipartite Lattices J<0 J>0

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

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

HAFT Two chiralities Q -Q Q=(4/3,0)

A B C

A B C

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

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

Stiffness Use standard Monte Carlo methods to study response functions

Consistent with Tc=0

Finite Tc ?

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

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

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

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)

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

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