Hexagonal structure of baby skyrmion lattices

1. 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

2. outline • 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

3. baby skyrme models

4. 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

5. the model: 2D (baby) skyrmions • target space: a triplet of scalar fields, ,subject to the constraint , i.e., . • base space: base space target space

6. the model: 2D (baby) skyrmions • 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

7. the model: 2D (baby) skyrmions 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

8. the model: 2D (baby) skyrmions the Lagrangian density of the model is comprised of a kinetic term a Skyrme term introduces scale but is not enough, solutions inflate indefinitely

9. the model: 2D (baby) skyrmions the Lagrangian density of the model is comprised of a kinetic term a Skyrme term a potential term stabilizes the solutions and topological solitons emerge

10. the model: 2D (baby) skyrmions • static solutions are obtained by minimizing the energy functional within each topological sector • the minimal energy configurations within each topological sector are called “baby skyrmions” • energy obeys the inequality , saturated only in the pure O(3) case

11. the model: 2D (baby) skyrmions • 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 • …

12. “old” baby Skyrme model (Piette et al., 1995) the one-skyrmion – the minimal energy configuration in the charge-one sector – is a lump configuration energy density plot contour plot

13. “old” baby Skyrme model (Piette et al., 1995) the two-skyrmion is a ring-like configuration:“skyrmions on top of each other” energy density plot contour plot

14. “old” baby Skyrme model (Piette et al., 1995) the three-skyrmion is more structured: partially overlapping skyrmions energy density plot contour plot

15. “old” baby Skyrme model (Piette et al., 1995) the four-skyrmion energy density plot contour plot

16. “old” baby Skyrme model (Piette et al., 1995) the five-skyrmion energy density plot contour plot

17. 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

18. 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 with

19. interpolating the two models (Hen & Karliner, 2008) energy (per charge) of solutions as a function of s : “old” holomorphic • 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

20. lattice structure of baby skyrmions

21. motivation: crystalline structure of nucleons • 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 energy density iso-surfaces of 3D skyrmions (Houghton et al., 1998)

22. motivation: crystalline structure of nucleons crystalline structure in the 3D case: • 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?

23. the method: parallelogrammic unit cells 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

24. the method: parallelogrammic unit cells 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) • so what is the lattice structure of baby skyrmions? • the problem has to be solved numerically a two-skyrmion in a square cell • 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

25. baby skyrmions inside parallelograms • base space: parallelograms two-torus, • periodic boundary conditions • here too, topological solitons with integer charges emerge base space target space

26. the model: 2D (baby) skyrmions the energy functional to be minimized is kinetic term Skyrme term “old” model potential term

27. the model: 2D (baby) skyrmions we map the parallelograms into a two-torus L– length of one side sL– length of the other side – angle to the vertical

28. the model: 2D (baby) skyrmions the energy functional becomes kinetic term Skyrme term potential term s, – the parallelogram parameters B– the charge of the skyrmion – the skyrmion density = , charge per unit-cell area

29. 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)

30. the relaxation method energy distribution of the charge-two skyrmion as a function of relaxation time: energy density

31. 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 reached

32. 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 results: the Skyrme case: hexagonal structure

33. results: the Skyrme case: hexagonal structure • the zero-energy loci (violet) resemble tightly-packed circles • eight high-density peaks (red): the skyrmion splits to quarter-skyrmions energy density

34. 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

35. results: the general case: a “phase transition” the energy difference between the hexagonal lattice skyrmions and the square-cell skyrmions

36. semi-analytical approach starting off with the same energy functional • first, we minimize the energy with respect to parallelogram parameters s and • as a next step, we plug these expressions into the energy functional. where

37. semi-analytical approach • we arrive at a reduced energy functional: • now that the s andminimization 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:

38. 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)

39. 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

40. interpolating the two models (Hen & Karliner, 2008) • 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)

41. double vacuum model (Weidig, 1999) ring-like solutions in all charge sectors:“skyrmions on top of each other” energy density plot contour plot

42. results: the general case: a “phase transition” there is an optimal density for which energy is minimal over all densities