Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 Hexagonal structure of baby...

Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008

Hexagonal structure of baby skyrmion lattices

Itay Hen,Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University

based on: Nonlinearity 21, 399 (2008) Phys. Rev. D 77, 054009 (2008)

joint work with Marek Karliner


baby Skyrme models – an overview solitonic solutions: baby skyrmions in flat space

lattice structure of baby skyrmions: motivation method solutions semi–analytical approach

summary and further remarks

outline


baby skyrme models


the model: 2D (baby) skyrmions

the “baby Skyrme model” is a nonlinear field theory in (2+1)D (Leese et al., 1990)

admits solitonic solutions with conserved topological charges

it is a 2D analogue of the Skyrme model in three spatial dimensions (Skyrme, 1962) which: is a low-energy effective theory of hadrons solitonic solutions are identified with

nucleons topological charge is identified with baryon

number baby model serves as a toy model for the 3D

one has applications in condensed matter physics,

specifically in quantum Hall ferromagnets

a charge-three baby skyrmion


target space: a triplet of scalar fields, ,subject to the constraint , i.e., .

base space: 22S

1 2 3, ,

2 2S S

base space

target space

1 2S



these maps may be classified into homotopy classes (topological sectors)

fields within each class are assigned a conserved topological charge

the charge takes on integer values

fields of different sectors cannot be continuously transformed to one another

2 2S S

2 2B S



the Lagrangian density of the model is comprised of

a kinetic term

this is the O(3) sigma model - analytic lump solutions which are unstable, as this model is conformally invariant



a kinetic term a Skyrme

term

introduces scale but is not enough, solutions inflate indefinitely




a kinetic term a Skyrme

term

stabilizes the solutions and topological solitons emerge

a potential term




the minimal energy configurations within each topological sector are called “baby skyrmions”

energy obeys the inequality , saturated only in the pure O(3) case

static solutions are obtained by minimizing the energy functional within each topological sector

4BE B



the potential term may be chosen almost arbitrarily

but must vanish at infinity for a given vacuum field value in order to ensure finite energy solutions

the vacuum is normally taken to be several potential terms have been studied in great

detail: the “old” model, with the holomorphic model, with the “double vacuum” model, with …

(0)2 23(1 ) (1 )

(0) 0,0,1

(0)2 2 2 231 ( ) 1

(0)2 4 2 43(1 ) (1 )

U



“old” baby Skyrme model (Piette et al.,

1995)

the one-skyrmion – the minimal energy configuration in the charge-one sector – is a lump configuration

(0)2( ) (1 )U

energy density plot contour plot



1995)

the two-skyrmion is a ring-like configuration:“skyrmions on top of each other”


(0)2( ) (1 )U



1995)

the three-skyrmion is more structured: partially overlapping skyrmions


(0)2( ) (1 )U



1995)

the four-skyrmion


(0)2( ) (1 )U



1995)

the five-skyrmion


(0)2( ) (1 )U


holomorphic baby Skyrme model (Leese et al., 90)

analytic stable solution in the charge-one sector no stable configurations in higher-charge sectors

charge one solution- stable charge two solution- unstable

(0)2 4( ) (1 )U


interpolating the two models (Hen & Karliner, 2008)

in order to appreciate the differences between the “old” model and the holomorphic one,

we studied the one-parametric family of potentials

which interpolates the two models:

s=1 corresponds to the “old” model s=4 corresponds to the holomorphic one

(0)2( ) (1 ) 0 4sU s

(0)2( ) (1 )sU 0 4s with


energy of solutions attains a minimum for some s in all sectors stable multi-skyrmions exist only below s ≈2 s serves as a control parameter for the repulsion/attraction between skyrmions

holomorphic

energy (per charge) of solutions as a function of s :


(0)2( ) (1 ) 0 4sU s

“old”


lattice structure of baby skyrmions


the (3+1)D Skyrme model is a low-energy effective theory of hadrons

its solitonic solutions are identified with nucleons

the topological charge is identified with baryon number

it can thus be used to study the structure of nuclear matter at high densities

motivation: crystalline structure of nucleons

energy density iso-surfaces of 3D skyrmions (Houghton et al., 1998)


Klebanov (1985) – cubic lattice – energy 1.08 per baryon

Goldhaber & Manton (1987) and Jackson & Verbbarschot (1988) – body centered cubic lattice

Battye & Sutcliffe (1998) – hexagonal lattice – energy 1.076 per baryon

Castillejo et al. (1987) and Kugler & Shtrikman (1988) – face centered cubic lattice – energy 1.036 per baryon

what is the true minimal energy configuration?

crystalline structure in the 3D case:

motivation: crystalline structure of nucleons


the method: parallelogrammic unit cells

a two-skyrmion in a square cell

lattice structure in the 2D case: to date, only the “square-cell” configuration

has been studied breaking into half-skyrmions was observed

(Cova & Zakrzewski, 1997)


lattice structure in the 2D case: to date, only the “square-cell” configuration

has been studied breaking into half-skyrmions was observed

(Cova & Zakrzewski, 1997) a two-skyrmion

in a square cell so what is the lattice structure of baby skyrmions? the problem has to be solved numerically

the method: parallelogrammic unit cells

placing many skyrmions in a box – very difficult computationally (requires a very large grid)

instead, a charge-two skyrmion is placed in different parallelogrammic unit cells, and periodic boundary conditions are imposed

we find the parallelogram for which the skyrmion’s energy is minimal


baby skyrmions inside parallelograms

base space: parallelograms two-torus, periodic boundary conditions here too, topological solitons with integer charges

emerge

base space

target space



kinetic term

the energy functional to be minimized is

Skyrme term

“old” model potential term


we map the parallelograms into a two-torus

L – length of one side

sL – length of the other side

– angle to the vertical



the energy functional becomes kinetic term

potential term

Skyrme term

s, – the parallelogram parameters

B – the charge of the skyrmion

– the skyrmion density = , charge per unit-cell area

2/ cosB sL



the relaxation method

the minimal energy configuration is found by a full-field relaxation method on a 100 X 100 grid

a field triplet is defined at each point on the grid for each parallelogram, we start off with a certain

initial two-skyrmion configuration repeatedly modify the fields at random points on

the grid accept changes only if energy is decreased terminate when the minimum is reached verify results using a more complicated algorithm -

“simulated annealing” – based on slowly cooling down the system (Kirkpatrick et al., 1983)


energy density

energy distribution of the charge-two skyrmion as a function of relaxation time:

the relaxation method


results: the pure O(3) case: no favorable lattice

in the pure O(3) case, only kinetic term is present (both Skyrme and potential terms are omitted) this model has analytic solutions in terms of

Weierstrass elliptic functions (Cova & Zakrzewski, 1997) same energy for all parallelograms –

the minimal energy bound is reached4 8E B


results: the Skyrme case: hexagonal structure

in the “Skyrme case”, only the potential term is missing

the skyrmion expands and covers the whole parallelogram

minimal energy is obtained for the hexagonal lattice

1.587 8E 1.446 8E

1.433 8E

1.454 8E


the zero-energy loci (violet) resemble tightly-packed circles

eight high-density peaks (red): the skyrmion splits to quarter-skyrmions

energy density

results: the Skyrme case: hexagonal structure


results: the general case: a “phase transition”

even with the potential term present, minimal energy is obtained for the hexagonal lattice

this time, the skyrmion has a definite size

as density is increased the skyrmions fuse together

low density: a ring-like shape

medium density: two one-skyrmions

high density: quarter-skyrmions

1.080 8E 1.084 8E 1.229 8E


the energy difference between the hexagonal lattice skyrmions and the square-cell skyrmions



first, we minimize the energy with respect to parallelogram parameters s and

as a next step, we plug these expressions into the energy functional.

semi-analytical approach

0 , 0 , sin

yy xy

xx xx yy

E Es

s

d dij

i jx y

starting off with the same energy functional

where


we arrive at a reduced energy functional:

now that the s and minimization conditions are “built in”, we relax the system as before

the hexagonal structure is obtained once again in the general case, we can also eliminate

by using

ending up with:

semi-analytical approach

0

E


summary and further remarks

the hexagonal lattice generates the minimal energy configuration

what would happen in (3+1)D, where skyrmions correspond to real nucleons?

baby skyrmions arise in ferromagnetic quantum Hall systems where they appear as spin textures

it has been suggested that they order themselves in a hexagonal lattice

our results support this claim Walet & Weidig (2001)


Hexagonal structure of baby skyrmion lattices

Thank you!

based on: Nonlinearity 21, 399 (2008) Phys. Rev. D 77, 054009 (2008)

joint work with Marek Karliner


the potential parameter may be eliminated by rescaling the role of the Skyrme parameter :

the charge density of the three-skyrmion for different values (s =0.5)


(0)2( ) (1 ) 0 4sU s


double vacuum model (Weidig, 1999)

ring-like solutions in all charge sectors:“skyrmions on top of each other”


(0)2 2( ) 1 ( )U


there is an optimal density for which energy is minimal over all densities

1.080 8E


2 2( 0.03, 0.1)

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