ActionSusceptibility

[evanberkowitz]

We have the Action_Action two-point function which is $(-\kappa \partial_{\kappax})(-\kappa \partial{\kappa_y}) \log Z$ as a function of $\Delta x = x-y$, and the implementation properly subtracts the quantum-disconnected piece.

• By analogy with the Spin and Vortex we can presumably compute the ActionSusceptibility?
• At $\Delta x=0$ is it fair to call this ActionDensityVariance (by analogy with InternalEnergyDensityVariance #85)?

[alcherman]

yes this seems very reasonable

[evanberkowitz]

I guess we should think carefully about the powers of volume and if/where they are needed.

[evanberkowitz]

At $\Delta x=0$ Action_Action is more like ActionDensitySquared. The action density variance comes from averaging over $\Delta x$, so that both x and y in $\langle S_x S_y \rangle$ are averaged.

[alcherman]

Hmm, why wouldn't Action_Action at \Delta_x = 0 be <S^2> - ^2, which is the variance of S? Why would the variance come from averaging over x and y in Action_Action?

[evanberkowitz]

I think Action_Action is really ActionDensity_ActionDensity. If we really had the full volume^2 (all-to-all) correlator then summing over the two positions would sum the densities and give S^2. So to go down to S^2/Λ^2, divide by the volume two times.

But the Action_Action is already volume averaged on one location. Then so we should sum one coordinate and then divide by Λ; ie average over coordinates.

But it's certainly possible I missed some factor somewhere!

[evanberkowitz]

If Action_Action really is two densities then evaluated at Δx=0 gives the density squared.