Vorticity

Baryon Number Vorticity

tdg.observable.vorticity.vorticity(ensemble)

The lattice-exact curl of the baryon number current, \(\tilde{\omega}_x^i\).

Zero in expectation value by periodic boundary conditions.

Configurations slowest, then space x, then direction i (in 2 dimensions, a singleton dimension).

tdg.observable.vorticity.vorticity_squared(ensemble)

The square of the dimensionless vorticity \(\tilde{\omega}_x^2\).

Configurations slowest, then the spatial coordinate \(x\).

tdg.observable.vorticity.vorticity_vorticity(ensemble)

The (dimensionless) spatial convolution of the vorticity, \(\tilde{\Omega}_r = (\tilde{\omega}^i * \tilde{\omega}^i)_r\) summed over \(i\).

By periodic boundary conditions, should vanish when summed on the radial coordinate.

Configurations slowest, then the relative coordinate \(r\).

There is a great deal of subtlety involved in understanding the low-energy behavior of vortex correlations. Because by periodic boundary conditions \(\sum_r \texttt{vorticity_vorticity} = 0\) and its Fourier transform goes like momentum squared at high momentum, we need to study moments. We define moments \(B\),

\[B_n(k) = \int d^2r\; e^{-i k r} |r|^n \Omega(r)\]

\(B_n(k)\) has dimensions \([M^{-2} L^{-(6-n)}]\).

So, \(M^2 L^{6-n} B_n(k)\) is dimensionless. We can divide by appropriate powers of \(2\pi N = k_F^2 L^2\) to eliminate \(L\).

These have good continuum- and infinite-volume limits.

tdg.observable.vorticity.b2_by_kF4(ensemble)

\(M^2 B_2(k=0) / k_F^4\) which is a non-zero low-energy moment of \(\Omega\);

\[\]

tdg.observable.vorticity.b4_by_kF2(ensemble)

\(M^2 B_4(k=0) / k_F^2\).

tdg.observable.vorticity.b6(ensemble)

\(M^2 B_6(k=0)\).