1. 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

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

3. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 baby skyrme models

4. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 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. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 target space: a triplet of scalar fields,, subject to the constraint, i.e.,. base space: base spacetarget space the model: 2D (baby) skyrmions

6. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 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 the model: 2D (baby) skyrmions

7. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 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 the model: 2D (baby) skyrmions

8. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 a kinetic term a Skyrme term introduces scale but is not enough, solutions inflate indefinitely the Lagrangian density of the model is comprised of the model: 2D (baby) skyrmions

9. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 a kinetic term a Skyrme term stabilizes the solutions and topological solitons emerge a potential term the Lagrangian density of the model is comprised of the model: 2D (baby) skyrmions

10. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008  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 the model: 2D (baby) skyrmions

11. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 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 … the model: 2D (baby) skyrmions

12. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 “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 plotcontour plot

13. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 “old” baby Skyrme model ( Piette et al., 1995 ) the two-skyrmion is a ring-like configuration: “skyrmions on top of each other” energy density plotcontour plot

14. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 “old” baby Skyrme model ( Piette et al., 1995 ) the three-skyrmion is more structured: partially overlapping skyrmions energy density plotcontour plot

15. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 “old” baby Skyrme model ( Piette et al., 1995 ) the four-skyrmion energy density plotcontour plot

16. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 “old” baby Skyrme model ( Piette et al., 1995 ) the five-skyrmion energy density plotcontour plot

17. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 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- stablecharge two solution- unstable

18. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 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. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 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 : interpolating the two models (Hen & Karliner, 2008) “ old ”

20. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 lattice structure of baby skyrmions

21. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 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)

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

23. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 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)

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

25. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 baby skyrmions inside parallelograms base space: parallelograms two-torus, periodic boundary conditions here too, topological solitons with integer charges emerge base spacetarget space

26. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 the model: 2D (baby) skyrmions kinetic term the energy functional to be minimized is Skyrme term “ old ” model potential term

27. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 we map the parallelograms into a two-torus L – length of one side sL – length of the other side – angle to the vertical the model: 2D (baby) skyrmions

28. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 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 the model: 2D (baby) skyrmions

29. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 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. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 energy density energy distribution of the charge-two skyrmion as a function of relaxation time: the relaxation method

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

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

34. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 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. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 the energy difference between the hexagonal lattice skyrmions and the square-cell skyrmions results: the general case: a “phase transition”

36. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 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 starting off with the same energy functional where

37. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008  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

38. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 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. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 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. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 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) interpolating the two models (Hen & Karliner, 2008)

41. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 double vacuum model (Weidig, 1999) ring-like solutions in all charge sectors: “skyrmions on top of each other” energy density plotcontour plot

42. Nonlinear Physics – Theory and Experiment V, Gallipoli, June 2008 there is an optimal density for which energy is minimal over all densities results: the general case: a “phase transition”