xy simulation

1
d2 xy model
n2

• 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
(No Transcript)
3
x-y model

• M0 for all T gt0
• 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 TTc for form a vortex plasma
• spin stiffness vanishes at Tc

4
(No Transcript)
5
?.25
6
Stiffness
?
?F (1/2)??2
7
(No Transcript)
8
Ferromagnetic Models

• ferromagnetic models display phase transitions
  depending on the values of (n,d)
• Ising model(n1) need dgt1
• x-y model (n2) need dgt2 for long
  range order
• however topological defects(vortices) unbind at a
  finite Tc and destroy the stiffness
• antiferromagnetic models (Jlt0) may exhibit
  frustration

9
Bipartite Lattices
Jlt0
Jgt0
10
Triangular Lattice
Jlt0
Jgt0
Frustration gt non-zero entropy at T0
11
Classical Heisenberg Modeln3, d2

• in the ferromagnetic case (Jgt0), the order
  parameter is an 3-component vector
• for bipartite lattices, the same is true for Jlt0
• since n3, we need d gt2 to have Tc ?0
• for the triangular lattice the system is
  frustrated when Jlt0
• energy wins at T0 and ground state has spins
  arranged in a plane at 1200 to each other

12
HAFT
Two chiralities

13
(No Transcript)
14
(No Transcript)
15
HAFT

• noncollinear arrangement of the spins on each
  triangle gt 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
• gt both in plane and out of plane twists
• gt 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 Tc0
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 gt 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
m0
m1
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)

24
(No Transcript)
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 Tgt0
• 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 Tgt0 gt stiffness is
  zero at large length scales
• vortices unbind at a finite Tc and influence two
  spin correlations indirectly