**Acknowledgements: This work was done with my good friend and colleague,
Dr. Mikael Ciftan. We gratefully acknowledge the support of the Army
Research Office.
**

- Importance Sampling Monte Carlo (heat bath) ``with a twist'' to
get
at high precision for
.
- Finite size scaling used to get critical exponents at (accepted value, ).
- Helicity studied by freezing and twisting the (previously
periodic) boundary conditions in the (X,Y,1) plane.

**
**

- Landau potential for a continuous ferromagnetic model is:

(1) - Define the block spin
in terms of its mean
value (the order parameter) plus a fluctuation:

(2) - Further decompose the fluctuation into a longitudinal and
transverse piece:

(3) - Derive the following general form for the free energy in terms of
the transverse coarse grained spin fluctuation gradient
:

(4) - One can relate a state of uniform twist angle to the gradient of
the transverse spin fluctuation via
. Substituting and differentiating to find the free
energy density, one obtains the following two relations:

(5)

(6) - In Landau theory, approximately constant so
as from below. In detailed treatment one gets
corrections:

(7) - Finally, to use finite-size scaling theory (FSST) to extract the
critical exponent, we must substitute
, or

(8)

**In the last expression,
.
term from clearly dominant ( is very small,
, for this model, while
and
). The helicity modulus should vanish sharply near
according to Landau theory.
**

**
**

**We cannot directly measure the free energy density . We can directly measure the enthalpy density . Following an identical
argument:
**

(9) |

(10) |

**With a page or two of algebra we can show that:
**

(11) |

**This is what we wish to measure, in part to invert this equation
and deduce the values of and .
**

**
**

**Note that as before, if we make the finite size scaling hypothesis we
will actually measure:
**

(13) |

(14) |

**The enthalpy helicity should thus diverge at .
**

**It is easy to show that:
**

(15) | |||

(16) | |||

(17) |

**
**

- Calculations were performed on several generations of ``brahma'' (our beowulf compute cluster, also ganesh and rama).
- Heat bath only (cluster method a bit difficult if boundary layers are ``frozen'').
- Equilibrate lattice with periodic boundary conditions.
- ``Freeze'' (x,y,1) layer of spins.
- Rotate (x,y,1) spins through angle and store them in
(x,y,L+1) layer (replacing PBC's in z-direction with frozen
*twisted*PBC's). - Re-equilibrate only the (x,y,2) to (x,y,L) spins with the heat bath (with PBC's in the x and y directions).
- Sample
- Repeat (easiest to restore PBC's, re-equilibrate, repeat).
- Sweep angles , at .
- Fit where .
- Fit
- Obtain , from hyperscaling.

**
**

**
**

**
**

**
**

- The
*only*direct measurement of this quantity to date. -
. This is quite large compared to most other Monte Carlo
results (which tend to yield
) but is not
inconsistent with the most recent renormalization predictions.
- The hyperscaling relation itself then yields
.
This is a weakly singular quantity and is
*very difficult to measure*. This is a major motivation of this work. - For this particular talk, we emphasize that there are easily more
than 30 ``GFLOP-years'' of effort in this result (whatever you consider
a GFLOP to be). (32x400x3 = 38400) + (16x1300x2 = 41600) + (32x1600x1 =
51200) = 131.200 GHz-years, supercomputing indeed. Impossible without
the beowulf/cluster model.

**
**