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Properties

A skyrmion has many properties. In general, Skyrmions3D allows you to compute the integrand or integral of the property in question. For all the following examples, if you pass density=true you'll get the integrand. You can also get the nth moment by passing moment=n.

Suppose you have a $B=4$ cubic skyrmion

using Skyrmions3D
skyrmion = Skyrmion(40,0.2)
p4(z) = z^4 + 2*sqrt(3)*im*z^2 + 1
q4(z) = z^4 - 2*sqrt(3)*im*z^2 + 0
make_rational_map!(skyrmion, p4, q4)

Baryon number

Basic use:

Baryon(skyrmion)
>>> 3.968749058206481

Other examples:

baryon_density = Baryon(skyrmion, density=true)
rms_baryon = sqrt(Baryon(skyrmion, moment=2))
>>> 3.882435997469747

Energy

Basic use:

Energy(skyrmion)
>>> 5.371145445834469

Note: you can turn on physical units to get the energy in those units:

skyrmion.Fpi = 174
skyrmion.ee = 3
skyrmion.physical = true
Energy(skyrmion)
>>> (6149.285364807477, "MeV")

Note that the energy of the skyrmion is dependent both on its underlying pion field (which determines the energy density at a point) but also the grid, as the energy is calculated as the sum of the energy density over the grid. If the grid is not sufficiently large, the computed energy will be smaller than the 'true' value.

Currents

For currents, we use the function compute_current and pass a label. We list the currents and labels now:

  • Rotational moment of intertia, vMOI
  • Isorotational moment of intertia, uMOI
  • Mixed moment of intertia, wMOI
  • Iso-Axial, uAxial
  • Rotational-Axial, vAxial
  • Stress-tensor, stress
  • Noether-iso, NoetherIso
  • Noether-axial, NoetherAxial

Like the Energy and Baryon we can get the densities by passing density=true and the nth moment by passing moment=n. For instance, the Isorotational moment of inertia of the cubic rational map skyrmion is

U = compute_current(skyrmion, label="uMOI")

The density of the stress-tensor is given by

stress_density = compute_current(skyrmion, label="stress", density=true)

The second moment of the Noether-iso current is

noether_moment = compute_current(skyrmion, label="NoetherIso", density=false, moment=2)