API

Skyrmion(lp::Int64, ls::Float64)
Skyrmion([lpx,lpy,lpx], [lsx,lsy,lsz])

Create a skyrme field with lp lattice points and ls lattice spacing.

Optional arguments

• mpi = 0.0: sets the pion mass for this Skyrme field

Grid([lpx,lpy,lpx], [lsx,lsy,lsz], boundary_conditions::String)

Create a Grid field with lp lattice points and ls lattice spacing.

Optional arguments

• boundary_conditions

set_lattice!(skyrmion, lp = [lpx, lpy, lpz], ls = [lsx, lsy, lsz])

Sets the underlying lattice to one with lpxxlpyxlpz points and lsxxlsyxlsz spacings, and reinterpolates skyrmion on the new grid.

set_dirichlet!(skyrmion::Skyrmion)

Sets the skyrmion to have periodic boundary conditions.

set_dirichlet!(skyrmion::Skyrmion)

Sets the skyrmion to have Dirichlet boundary conditions.

set_periodic!(skyrmion::Skyrmion)

Sets the skyrmion to have periodic boundary conditions.

set_mpi!(skyrmion::Skyrmion, mpi)

Set the pion mass of skyrmion to mpi.

set_physical!(skyrmion::Skyrmion, is_physical; Fpi=Fpi, ee=ee)

Sets skyrmion to use physical units, when is_physical is true. Also used to turn off physical units by setting is_physical=false

set_Fpi!(skyrmion::Skyrmion, Fpi)

Sets the pion decay constant of skyrmion to Fpi.

set_ee!(skyrmion::Skyrmion, ee)

Sets the Skyrme coupling constant of skyrmion to ee.

get_grid(skyrmion::Skyrmion)

Returns an array of 3D arrays [x, y, z], which can be used in integrals.

check_if_normalised(skyrmion)

Check if skyrmion is normalised.

Throws an error if any point is not normalised

get_field(skyrmion::Skyrmion)

Returns an array of pion fields [π1, π2, π3, π0], which can be used in integrals.

make_rational_map!(skyrmion, prof, pfn, qfn; kwargs... )

Writes a rational map skyrmion in to skyrmion. The rational map is given by the polynomials R(z) = p(z)/q(z) and the profile f(r).

If no f is given, the function will find an OK approximation for the profile.

Optional arguments

• X=[0.0,0.0,0.0]: translate the initial skyrmion by X
• iTH = 0.0: isorotate by initial skyrmion by iTH
• i_n = 0.0: isorotate initial skyrmion around i_n
• jTH = 0.0: isorotate by initial skyrmion by jTH
• j_n = 0.0: isorotate initial skyrmion around j_n

make_RM_product!(skyrmion, X_list)

Makes a product approximation of many rational map skyrmions, determined through the list X_list. The final field is written into skyrmion.

The formatting of the list is as follow: X_list = [ data_1, data_2, data_3, ... ] where data_1 = [ p(z), q(z), f(r), X, θiso, n_iso, θrot, n_rot ]

See also [product]

Example of list

p1(z) = z; q1(z) = 1; f1(r) = 4*atan(exp(-r));
p2(z) = z^2; q2(z) = 1; f2(r) = 4*atan(exp(-0.7*r));
X_list = [ [ p1, q1, f1, [0.0,0.0,1.5], 0.0, [0.0,0.0,1.0], 0.0, [0.0,0.0,1.0] ], [ p2, q2, f2, [0.0,0.0,-1.5], pi, [1.0,0.0,0.0], 0.0, [0.0,0.0,1.0] ] ]

Technical details

The product is taken pairwise in order. E.g. for a list of 3 skyrmions, we first calculate the symmetrised product of the first and second skyrmions then calculate the symmtrised product with the third skyrmion. Hence the final solution is not symmetric under permutations.

make_ADHM!(skyrmion, L, M )

Writes an ADHM skyrmion in to skyrmion. The ADHM data is given by L and M. L and M should be given by B and BxB arrays of Quaternions, from the GLMakie package.

Example of data

B=2

L = [ Quaternion(0.0,0.0,0.0,0.0) for a in 1:B ]
M = [ Quaternion(0.0,0.0,0.0,0.0) for a in 1:B, b in 1:B ]

L[1] = Quaternion(0.0, 0.0, 0.0, sqrt(2.0))
L[2] = Quaternion(0.0, 0.0, sqrt(2.0), 0.0)

M[1,1] = Quaternion(1.0, 0.0, 0.0, 0.0)
M[1,2] = Quaternion(0.0, 1.0, 0.0, 0.0)
M[2,1] = Quaternion(0.0, 1.0, 0.0, 0.0)
M[2,2] = Quaternion(-1.0, 0.0, 0.0, 0.0)

product_approx!(skyrmion1,skyrmion2)

Makes the symmetrised product approximation of skyrmion1 and skyrmion2. The output is written in to skyrmion1. The returned field is normalised.

See also [product_approx]

product_approx(skyrmion1,skyrmion2) -> product_skyrmion

Returns the symmetrised product approximation of skyrmion1 and skyrmion2. The returned field is normalised.

See also [product_approx!]

translate_sk!(skyrmion,X)

Translates skyrmion by the 3-Vector X, e.g. X = [1.0, 0.0, 0.0]

See also [translate_sk]

translate_sk(skyrmion,X) -> translated_skyrmion

Returns skyrmion translated by 3-Vector X, e.g. X = [1.0, 0.0, 0.0]

See also [translate_sk!]

rotate_sk!(skyrmion,theta,n)

Rotates skyrmion by theta around the vector n. The given vector is automatically normalised.

See also [rotate_sk]

rotate_sk(skyrmion,theta,n) -> rotated_skyrmion

Returns skyrmion rotated by theta around the vector n. The given vector is automatically normalised.

See also [rotate_sk!]

isorotate_sk!(skyrmion,theta,n)

Isorotates skyrmion by theta around the vector n. The given vector is automatically normalised.

See also [isorotate_sk!]

isorotate_sk(skyrmion,theta,n) -> isorotated_skyrmion

Returns skyrmion isorotated by theta around the vector n. The given vector is automatically normalised.

See also [isorotate_sk!]

center_skyrmion(my_skyrmion)

Translates skyrmion' so that the center of mass is(0,0,0)'.

normer(skyrmion)

Returns normalised skyrmion.

See also [normer!]

normer!(skyrmion)

Normalises skyrmion.

See also [normer]

overview(skyrmion)

Displays an overview of skyrmion's properties.

Energy(skyrmion; density=false)

Compute energy of skyrmion.

Set 'density = true' to output the energy density and moment to n to calculate the nth moment of the energy density.

Baryon(skyrmion; density=false)

Compute baryon number of skyrmion.

Set 'density = true' to output the baryon density and moment to n to calculate the nth moment of the baryon density.

center_of_mass(skyrmion; density=false, moment=0)

Compute the center of mass of skyrmion, based on the energy density.

compute_current(skyrmion; label="uMOI", indices=[0,0], density = false, moment=0)

Compute a variety of currents in the Skyrme model, based on a skyrmion.

You can calculate specific indices using e.g. indices = [1,2]. If indices = [0,0], the function will calculate all indices. If density = false, the function will return the integrated densities, while density = true it will return the densities.

The possible currents are (currently):

• uMOI: the isospin moment of inertia.
• wMOI: the mixed moment of inertia.
• vMOI: the spin moment of inertia.
• uAxial: the u-axial moment of inertia.
• wAxial: the w-axial moment of inertia.
• NoetherIso: the Noether vector current.
• NoetherAxial: the Noether axial current.
• stress: the stress tensor.

rms_baryon(skyrmion)

Compute root mean square charge radius of a skyrmion, using the baryon density.

gradient_flow!(skyrmion; steps = n, tolerance = tol, dt=ls^2/80.0, checks = freq, print_stuff = true)

Applies a gradient flow to skyrmion with timestep dt, either for n steps or until the error falls below tol. The error is checked every checks steps.

See also [newton_flow!, arrested_newton_flow!]

newton_flow!(skyrmion; skyrmion_dot, steps = n, dt=ls^2/80.0, frequency_of_printing = freq, print_stuff = true)

Applies a newton flow to skyrmion whose initial time derivative field is skyrmion_dot with timestep dt, either for n steps or until the error falls below tol. The energy is checked every freq steps.

See also [gradient_flow!, arrested_newton_flow!]

arrested_newton_flow!(skyrmion; skyrmion_dot, steps = n, tolerance = tol, dt=ls^2/80.0, checks = freq, print_stuff = true)

Applies an arrested Newton flow to skyrmion whose initial time derivative field is skyrmion_dot with timestep dt, either for n steps or until the error falls below tol. The error is checked every checks steps.

See also [gradient_flow!, newton_flow!]

plot_overview(skyrmion)

Plots the pion fields and a baryon density of skyrmion.

plot_field!(skyrmion; component=3, iso_value=0.5 )

Plots an isosurface of constant value, skyrme_field[component] = iso_value

plot_baryon_density(skyrmion; iso_value = 0.5*(max(BD) - min(BD)), juggling = false, kwargs...)

Plots an isosurface of constant baryon density, with value iso_value, coloured to reveal the pion field structure, originally described in [].

Can use a juggling ball colouring scheme by setting juggling = true.

Optional argument

Can accept any arguments used in Axis3 from the Makie package. See more: [].

interactive_flow(my_skyrmion; iso_value=2.0, kwargs...)

Initialises a interface, used for dynamics. Requires GLMakie to be active. Once activated, can interactively apply a gradient flow or arrested newton flow to the initial skyrme field `my_skyrmion'.