| 51 |
|
\end{cases} |
| 52 |
|
\end{equation} |
| 53 |
|
where $J$ has non zero value only when spins $s_n$ ($\vec s_n$) and |
| 54 |
< |
$s_{n'}$ ($\vec s_{n'}$) are the nearest neighbours. |
| 54 |
> |
$s_{n'}$ ($\vec s_{n'}$) are the nearest neighbours. When $J > 0$, the |
| 55 |
> |
spins prefer aligned with each other, and if $J < 0$, the spins want |
| 56 |
> |
to be anti-aligned. |
| 57 |
> |
|
| 58 |
|
\begin{figure} |
| 59 |
|
\centering |
| 60 |
< |
\includegraphics[width=\linewidth]{./figures/inFrustration.pdf} |
| 61 |
< |
\caption{Frustration on a triangular lattice, the spins are |
| 62 |
< |
represented by arrows. No matter which direction the spin on the top |
| 63 |
< |
of triangle points to, the Hamiltonain of the system goes up.} |
| 60 |
> |
\includegraphics[width=3in]{./figures/inFrustration.pdf} |
| 61 |
> |
\caption{Frustration on triangular lattice, the spins and dipoles are |
| 62 |
> |
represented by arrows. The multiple local minima of energy states |
| 63 |
> |
induce the frustration for spins and dipoles picking the directions.} |
| 64 |
|
\label{Infig:frustration} |
| 65 |
|
\end{figure} |
| 66 |
< |
Figure~\ref{Infig:frustration} shows an illustration of the |
| 67 |
< |
frustration on a triangular lattice. When $J < 0$, the spins want to |
| 68 |
< |
be anti-aligned, The direction of the spin on top of the triangle will |
| 69 |
< |
make the energy go up no matter which direction it picks, therefore |
| 70 |
< |
infinite possibilities for the packing of spins induce what is known |
| 71 |
< |
as ``complete regular frustration'' which leads to disordered low |
| 72 |
< |
temperature phases. |
| 66 |
> |
The spins in figure~\ref{Infig:frustration} shows an illustration of |
| 67 |
> |
the frustration for $J < 0$ on a triangular lattice. There are |
| 68 |
> |
multiple local minima energy states which are independent of the |
| 69 |
> |
direction of the spin on top of the triangle, therefore infinite |
| 70 |
> |
possibilities for the packing of spins induce what is known as |
| 71 |
> |
``complete regular frustration'' which leads to disordered low |
| 72 |
> |
temperature phases. The similarity goes to the dipoles on a hexagonal |
| 73 |
> |
lattice, which are shown by the dipoles in |
| 74 |
> |
figure~\ref{Infig:frustration}. In this circumstance, the dipoles want |
| 75 |
> |
to be aligned, however, due to the long wave fluctuation, at low |
| 76 |
> |
temperature, the aligned state becomes unstable, vortex is formed and |
| 77 |
> |
results in multiple local minima of energy states. The dipole on the |
| 78 |
> |
center of the hexagonal lattice is frustrated. |
| 79 |
|
|
| 80 |
|
The lack of translational degree of freedom in lattice models prevents |
| 81 |
|
their utilization in models for surface buckling which would |
