The Critical Scaling of the
Helicity Modulus of the O(3)
classical Heisenberg ferromagnet

Robert G. Brown

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

Model
Classical Heisenberg ferromagnet (CHF) (the model on a 3 simple cubic lattice with periodic boundary conditions) in zero external field:

Goal
To compute and measure'' (with Monte Carlo) the critical exponents of the model, in particular the critical exponent of the Helicity Modulus .

Methods
• 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.

Review of Theory

• Landau potential for a continuous ferromagnetic model is:
 (1)

where are the cartesian components of the coarse grain block spins.

• 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)

(with phenomenological parameter , the spin wave stiffness''.

• 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)

with
 (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.

But...

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

 (9)

where is the change in internal energy caused by twisting the boundary conditions through the angle with either helicity. From this obvious substitutions yield:
 (10)

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

 (11)

with the critical exponent
 (12)

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)

or
 (14)

The enthalpy helicity should thus diverge at .

It is easy to show that:

 (15) (16) (17)

where the second step uses hyperscaling'' (widely believed but by no means proven for this model) to eliminate for . With this we can compute and given and possibly check hyperscaling.

Measuring with Monte Carlo

• 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.

Results

Best result to date:

Conclusions

• 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.