Currents

Currents tell you how conserved quantities flow from place to place.

Baryon number current

tdg.observable.current.current(ensemble)

The lattice-exact baryon number current \(ML^2 j^i_x\) so that \(\nabla \cdot j_x = - \partial_t \texttt{n}\) where the divergence is lattice-exact so that the total divergence is zero by periodic boundary conditions and conservation of baryon number.

Configurations slowest, then space x, then direction i.

The expectation value is 0 by translational and rotational invariance. But, more is true: the real part is zero configuration-by-configuration, and the imaginary part sums to zero.

tdg.observable.current.current_squared(ensemble)

The local square of the dimensionless current \(\tilde{\jmath}^2_x\), with a dot product of the vector indices.

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

tdg.observable.current.current_current(ensemble)

The convolution of the dimensionless current with itself \((\tilde{\jmath}^i*\tilde{\jmath}^i)_r\), with a dot product of the vector indices.

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

We are interested in low-energy current correlations. We define moments \(W\),

\[W_n(k) = \int d^2r\; e^{-ikr} |r|^n J(r).\]

where \(J(r) = \frac{1}{L^2} \int d^2 x [ J(x,x-r) = j^a(x) j^b(x-r) \delta_{ab} ]\) is

\[\lim_{\Delta x \rightarrow 0} \left(\frac{1}{ML^2 \Delta x}\right)^2 \left[\tilde{J}_r = \texttt{current_current} \right] \rightarrow J(r).\]

\(W_n(k)\) has dimensions \([M^{-2} L^{-(4-n)}]\).

So, \(M^2 L^{4-n} W_n(k)\) 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.current.w0_by_kF4(ensemble)

\(\frac{M^2 W_0(k=0)}{k_F^4}\)

tdg.observable.current.w2_by_kF2(ensemble)

\(\frac{M^2 W_2(k=0)}{k_F^2}\)

tdg.observable.current.w4(ensemble)

\(M^2 W_4(k=0)\)