Baby Skyrmions in AdS 3 and Extensions to (3 + 1) Dimensions Durham University Work in collaboration with Matthew Elliot-Ripley June 26, 2015
Introduction to baby Skyrmions in flat space and AdS 3 Discuss the form that various numerical solutions take Propose an approximation to predict the form of solutions for higher charges Consider what the results in AdS 3 suggest about rational maps in AdS 4 Consider new results for the full Skyrme model
Topological solitons are stable, particle-like solutions to field theories, defined by their topological charge. Examples include kinks, sigma-model lumps, monopoles, vortices, Yang-Mills instantons and Skyrmions The Skyrme model is a non-linear theory of pions that has soliton solutions that can be interpreted as baryons. Obtained as a low-energy effective field theory for QCD in the large colour limit. The baby Skyrme theory is the (2 + 1) analogue of this theory. It appears in certain condensed matter systems such as ferromagnetic quantum Hall systems and chiral ferromagnets.
The baby Skyrme model is a non-linear modified sigma model in (2 + 1)-dimensions, on a general Lorentzian manifold M with metric ds 2 = g µν dx µ dx ν the energy is, ( ) 1 E = 2 φ φ + κ2 2 ( φ i φ) ( φ g i φ) d 2 x ( 1 + 2 iφ i φ + κ2 4 ( iφ j φ) ( i φ j φ ) ) g + m 2 (1 φ 3 ) d 2 x, φ is a unit vector with the natural boundary conditions φ (x) (0, 0, 1) as x. φ : S 2 S 2 The topological charge or Baryon number is given by: B = 1 φ ( 1 φ 2 φ) d 2 ( x Z = π 2 S 2 ) (2) 4π (1)
Derricks (non-existence) theorem - a stationary point of the energy should be stable against spatial rescalings. E(x) E(µx) = µ 2 d E 2 + µ 4 d E 4 + 1 µ d E 0 (3) d = 2 m 0 d = 3 m = 0 is stable Baby Skyrmion scale is given to be µ κ m.
Solutions for M = R 2,1 : Figure: B = 1, 2 Energy density plots. B = 1 and B = 2 solutions take the maximal O(2) symmetry, so can be reduced to a 1-dim problem to minimise some profile function f (r), with boundary conditions f (0) = π and f ( ) = 0. φ = (sin f (r) cos θ, sin f (r) sin θ, cos f (r)) (4) f (r) A r e mr (5)
Higher charge baby Skyrmion solutions: Figure: B = 6 Energy density contour plots, left is local minima, right is the global minima. Figure: Minimal energy solutions are baby Skyrmion chains.
Interested in AdS 3 though ( ) 1 + r ds 2 2 2 = 1 r 2 dt 2 + 4L2 ( dr 2 (1 r 2 ) 2 + r 2 dθ 2). (6) Constant time slices are hyperbolic space. Will not give similar solutions to hyperbolic space due to warp factor. AdS/CFT correspondence allows the investigation of strongly coupled theories by studying classical solutions in the bulk. ρ = 2 tanh 1 r (7)
Solutions for M = AdS 3 : Figure: B = 1 Energy density contour plots κ = 0.1 and m = 0, coloured by energy density, φ 1, φ 2, φ 3. B = 1 solution takes the maximal O(2) symmetry, so can be reduced to a 1-dim problem to minimise some profile function f (ρ), φ = (sin f (ρ) cos θ, sin f (ρ) sin θ, cos f (ρ)) (8) M = R 2,1 (Massive) f (r) A r e mr M = AdS 3 (Massless) f (ρ) Ae 2ρ/L
-0.6-0.7-0.8 scale, log µ(ρ) -0.9-1 -1.1-1.2-1.3-1.4-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 log κ/l Numerical y = x/2-1 Figure: The scale (size) µ of profile functions for B = 1 with m = 0, measured to be the radius ρ at which f (ρ) = π 2 for various values of κ L. The dotted green line is a line with gradient 0.5, showing for small scale the size µ κ/l.
Numerical solutions were found by running full dynamics using a 4th-order Runga-Kutta technique. The kinetic energy was then cut at predefined intervals or if the potential increases. Derivatives were 4th order finite difference approximations Parameters were set to be κ = 0.01 and m = 0, though qualitatively the results remain the same for all. Topological charge remained an integer value to 5 significant figures, which indicates the accuracy of the results.
Numerical solutions from running full dynamics and cutting the kinetic energy at intervals. κ = 0.01 and m = 0 B = 1 B = 2 B = 3 B = 4 B = 5 B = 6 B = 7 B = 8 Figure: Energy density contour plots for B = 1 8, κ = 0.1 and m = 0. Coloured by the φ 3 field, single soliton positions can be identified (φ 3 = 1) as the dark blue points.
B = 9 {1, 8} B = 10 {1, 9} B = 11 {2, 9} B = 12 {2, 10} B = 13 {2, 11} B = 14 {2, 12} B = 15 {3, 12} B = 16 {3, 13} B = 17 {4, 13} B = 18 {4, 14} B = 19 {6, 13} B = 20 {6, 14}
Ring popping:? B = 8 B = 26 B = 9 B = 27? Difficult to predict ring popping for higher charges, due to large numbers of local minima.
Similarities in form with circle packing problems within a minimal radius circle. Is there some parallels we can draw, to enable a prediction of popping charges. B = 1 B = 2 B = 3 B = 4 B = 5 B = 6 B = 7 B = 8 B = 9 B = 10
B = 1 B = 2 B = 3 B = 4 B = 5 B = 6 B = 7 B = 8 B = 9 B = 10 B = 11 B = 12 B = 13 B = 14 B = 15 B = 16 B = 17 B = 18 B = 19 B = 20
Need a better approximation! Point particle approximation seems sensible from the circle packings Need to approximate two terms: Gravitational affect from metric Interaction between individual solitons Assume point particles are minimal energy single solitons in AdS 3 translated using the hyperbolic translations (mobius transformations). ( 1 a 2) ( x 1 2x a + x 2) a x 1 2x a + a 2 x 2 (9)
Gravitational Potential: Approximate affect of metric using a potential term We will assume that our solutions have negligible velocity throughout. This gives the geodesic equations to be, r = 1 L 2 r ( r 2 + 1 ) and hence an approximate potential of, V ads = A [ r 2 L 2 2 + log ( r 2 1 ) ] 1 r 2 (10) (11)
10 gravitational potential, Φ(r) 8 6 4 2 0 0 0.2 0.4 0.6 0.8 1 r Numerical Analytic Figure: Numerical and analytical approximations for the point particle gravitational potential produced by the AdS metric. The analytic approximation is Φ(r) = A L 2 [ r 2 /2 + log ( r 2 1 )] where L = 1, A = 62.8 and has been fit to the numerical data. The numerical approximation is the energy for a singe soliton translated about the grid with the minimal energy subtracted off.
Interaction term: Want to model an inter-molecular interaction. Morse potential: V (ρ) = D e (1 exp ( a (ρ ρ e ))) 2 Need to introduce phase difference into the potential U χ (ρ) = D e (exp (2a (1 ρ/ρ e )) + 2 cos χ exp (a (1 ρ/ρ e ))). (12)
Phase differences are dependent on the rotation relative to the direction of the connecting geodesics. Figure: Solitons with their connecting geodesics. Top pair are in the maximally repulsive channel, with relative rotations of of χ = 0 and the bottom pair are in the maximally attractive channel with relative rotation χ = π. Their relative rotations in the embedded flat space are shown using both their colour and the arrow, where χ [π, π].
0.1 Interaction Potential, U χ (ρ) 0.05 0-0.05-0.1 0 1 2 3 4 5 U 0 (ρ) (Numerical) U 0 (ρ) (Analytic) ρ U π (ρ) (Numerical) U π (ρ) (Analytic) Figure: Numerical and analytical approximations for the point particle interaction potential U χ (ρ). The parameters are D = 0.83, ρ e = 0.7, a = 1.1. The parameters above have been fit to the numerical data for ρ > 2µ, where µ = ρ : f (ρ) = π/2. The numerical approximation was found by removing the approximated gravitational potentials and the single soliton energys and considering a static soliton pair.
Figure: Approximations for κ Baby = 0.1, Skyrmions m = in AdS 0 3 andl Extensions = 1. to (3 + 1) Dimensions B = 1 B = 2 B = 3 B = 4 B = 5 B = 6 B = 7 B = 8 B = 9 B = 10 B = 11 B = 12 B = 13 B = 14 B = 15 B = 16 B = 17 B = 18 B = 19 B = 20 B = 21 B = 22 B = 23 B = 24
What do the approximations get right? Transitions! Pop 1 Pop 2 B = 8 B = 9 B = 26 B = 27
What about those we cant find? Pop 3 Pop 4 B = 54 B = 55 B = 94 B = 95
What does it get wrong? Low charge inner-rings. B = 11 B = 12 B = 14 B = 15 B = 16
What does it get wrong? Ring deformations: B = 8
Sum up for AdS 3 Shown that massless solutions are possible, with the curvature of the space acting similarly to the mass. Shown solutions take the form of concentric rings. Proposed an approximation that predicts transitions between different forms.
Further work/questions for AdS 3 What happens for really high charge and hence for a lattice? What does this mean for the full (3 + 1) Skyrme model, and for monopoles, in AdS 4.
What about the Skyrme model? Skyrme model takes the form of rational maps, hence multi-layered rational maps seem a sensible choice. Do we get more exotic solutions? How does it relate to monopoles?
The Skyrme model is a non-linear theory of pions that has soliton solutions that can be interpreted as baryons. Our theory is defined on a general Lorenzian M. E = { 1 12π 2 1 2 Tr ( R i R i) 1 16 Tr ( [R i, R j ] [ R i, R j]) +m 2 Tr (1 U) } gd 3 x where R i = ( i U) U is a current of the SU(2) matrix U (t, x) and g is the metric of our curved space.
U = σ + iπ τ where π = (π 1, π 2, π 3 ) is the triplet of pion fields, with constraint σ 2 + π π. We impose the natural boundary conditions U (x) 1 2 as x. U : S 3 S 3 The topological charge or Baryon number is given by: B = 1 24π 2 ɛ ijk Tr (R i R j R k ) d 3 x Z = π 3 (SU (2)) (13)
In flat space we get the following numercial solutions:
But we are interested in AdS 4 : ( ) 1 + r ds 2 2 2 = 1 r 2 dt 2 + 4L2 ( dr 2 (1 r 2 ) 2 + r 2 ( dθ 2 + sin 2 θdφ 2)). (14)
Rational map ansatz: [ ( if (ρ) 1 R 2 2 U (ρ, z) = exp R )] 1 + R 2 2R R 2 1 where F (0) = π and F ( ) = 0. (15) U (ρ, z) = exp ( if (ρ) ˆn R 1 (z) τ ) (16) R(z) is some rational map representing the angular dependence of the field configuration, z = tan (θ/2) exp (iφ), R (z) = p (z) q (z) where p and q are polynomials and (17) B = max (deg p, deg q) (18)
Substituting into the previous energy functional, E = 1 3π ( 1 + r 2 1 r 2 ) ( F 2 sinh2 (κρ) κ 2 + 2B ( F 2 + 1 ) + I κ2 sin 4 F sinh 2 (κρ) + 2m2 sinh2 (κρ) κ 2 (1 cosf ) where I is an integral to be minimised by choice of R: I = 1 ( 1 + z 2 4π 1 + R 2 ) dρ, (19) ) 4 dr 2idzd z dz (1 + z 2) (20) 2
B = 1 B = 2 B = 3 B = 4 B = 5 B = 6 B = 7 B = 8
B E G 1 1.8506 O(3) 2 4.0019 O(2) Z 2 3 6.2898 T d 4 8.3593 O h 5 11.1383 D 2d 6 13.7687 D 4d 7 15.9506 Y h 8 19.1875 D 6d 9 22.2371 D 4d 10 25.1997 D 4d 11 28.4272 D 3h 12 31.4320 T d Table: Rational map energies for single shell ansatz with the symmetry of the resulting field configuration for B = 1 12.
Results in AdS 3 suggest multi-shell rational maps may give lower energy solutions. For n-shells, the ansatz is now, exp ( if 1 (ρ) ˆn R 1 (z) τ ) 0 ρ ρ 1, exp ( if 2 (ρ) U (ρ, z) = ˆn R 2 (z) τ ) ρ 1 ρ ρ 2,. exp ( if n (ρ) ˆn R n(z) τ ) ρ n 1 ρ, where the profile has the fixed points F (ρ k ) = (n k)π, F (0) = nπ and F ( ) = 0. (21) B = n i=1 B i, where B i is the degree of the rational map in each sector i.
E = n i=1 ρi ρ i 1 E i dρ (22) Values for I i will not change, hence minimal energy will have the same rational maps. Full numerical results may not have a global minima with these rational maps, more symmetric maps may be a better choice. To find the minimal energy of the multi-shell rational map we minimise over the values of ρ i and the multiple profile functions. A gradient flow method was utilised for a 1-dimensional grid of 1001 points.
B single-shell multi-shell form 1 1.8506 1.8506 {1} 2 4.0019 4.0019 {2} 3 6.2898 6.2898 {3} 4 8.3593 8.3593 {4} 5 11.1383 11.1383 {5} 6 13.7687 13.7687 {6} 7 15.9506 15.9506 {7} 8 19.1875 19.1875 {8} 9 22.2371 22.2371 {9} 10 25.1997 25.1997 {10} 11 28.4272 28.2196 {1, 10} 12 31.4320 31.3806 {1, 11} 13 34.6596 34.3451 {1, 12} 14 38.4591 37.5331 {1, 13} 15 41.9903 40.9831 {2, 13} 16 45.4242 44.5797 {2, 14} 17 48.5426 48.1048 {2, 15} 18 52.7858 51.4764 {3, 15}
Final remarks: Full Skyrme model seems to have lower energies for multi-shell rational maps. What symmetries will the final numerical results take? Do monopoles take a similar form?