Localized objects in the Faddeev-Skyrme model
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1 Localized objects in the Faddeev-Skyrme model Department of Physics and Astronomy, University of Turku FIN Turku, Finland IWNMMP, Beijing, China, July 15-21,2009
2 Introduction Solitons: Topological solitons The model Numerical results 1: Knots Localized on the line (usually) or in the plane (lumps, dromions). Stability due to infinite number of conservation laws. Amenable to exact analytical treatment; completely integrable.
3 Introduction Solitons: Topological solitons The model Numerical results 1: Knots Localized on the line (usually) or in the plane (lumps, dromions). Stability due to infinite number of conservation laws. Amenable to exact analytical treatment; completely integrable. Topological solitons: Localized on the line (kinks), on the plane (vortices) or in 3D-space (Hopfions). Stability due to topological properties. Often can only solve numerically.
4 Introduction Solitons: Topological solitons The model Numerical results 1: Knots Localized on the line (usually) or in the plane (lumps, dromions). Stability due to infinite number of conservation laws. Amenable to exact analytical treatment; completely integrable. Topological solitons: Localized on the line (kinks), on the plane (vortices) or in 3D-space (Hopfions). Stability due to topological properties. Often can only solve numerically. In this talk: Review of work on Hopfions in, done in collaboration with P. Salo and J. Jäykkä.
5 Topology in R 3 : Hopfions Topological solitons The model Numerical results 1: Knots Carrier field: 3D unit vector field n in R 3, locally smooth. 3D-unit vectors can be represented by points on the surface of the sphere S 2. Asymptotically trivial: n(r) n, when r can compactify R 3 S 3.
6 Topology in R 3 : Hopfions Topological solitons The model Numerical results 1: Knots Carrier field: 3D unit vector field n in R 3, locally smooth. 3D-unit vectors can be represented by points on the surface of the sphere S 2. Asymptotically trivial: n(r) n, when r can compactify R 3 S 3. Therefore n : S 3 S 2. Such functions are characterized by the Hopf charge, i.e., by the homotopy class π 3 (S 2 ) = Z.
7 A concrete Hopfion Topological solitons The model Numerical results 1: Knots Example of vortex ring with Hopf charge 1: ( 4(2xz y(r 2 1)) n = (1 + r 2 ) 2, 4(2yz + x(r 2 1)) (1 + r 2 ) 2, 1 8(r 2 z 2 ) ) (1 + r 2 ) 2. where r 2 = x 2 + y 2 + z 2.
8 A concrete Hopfion Topological solitons The model Numerical results 1: Knots Example of vortex ring with Hopf charge 1: ( 4(2xz y(r 2 1)) n = (1 + r 2 ) 2, 4(2yz + x(r 2 1)) (1 + r 2 ) 2, 1 8(r 2 z 2 ) ) (1 + r 2 ) 2. where r 2 = x 2 + y 2 + z 2. Note that n = (0, 0, 1) at infinity (in any direction). n = (0, 0, 1) on the ring x 2 + y 2 = 1, z = 0 (vortex core).
9 A concrete Hopfion Topological solitons The model Numerical results 1: Knots Example of vortex ring with Hopf charge 1: ( 4(2xz y(r 2 1)) n = (1 + r 2 ) 2, 4(2yz + x(r 2 1)) (1 + r 2 ) 2, 1 8(r 2 z 2 ) ) (1 + r 2 ) 2. where r 2 = x 2 + y 2 + z 2. Note that n = (0, 0, 1) at infinity (in any direction). n = (0, 0, 1) on the ring x 2 + y 2 = 1, z = 0 (vortex core). Computing the Hopf charge: Given n : R 3 S 2 define F ij = ɛ abc n a i n b j n c. Given F ij construct A j so that F ij = i A j j A i, then Q = 1 16π 2 ɛ ijk A i F jk d 3 x.
10 Possible physical realization Topological solitons The model Numerical results 1: Knots
11 Topological solitons The model Numerical results 1: Knots In 1975 Faddeev proposed the Lagrangian (energy) [ ] E = ( i n) 2 + g Fij 2 d 3 x, F ij := n i n j n.
12 Topological solitons The model Numerical results 1: Knots In 1975 Faddeev proposed the Lagrangian (energy) [ ] E = ( i n) 2 + g Fij 2 d 3 x, F ij := n i n j n. Under the scaling r λr the integrated kinetic term scales as λ and the integrated F 2 term as λ 1. Therefore nontrivial configurations will attain some fixed size determined by the dimensional coupling constant g. (Virial theorem)
13 Topological solitons The model Numerical results 1: Knots In 1975 Faddeev proposed the Lagrangian (energy) [ ] E = ( i n) 2 + g Fij 2 d 3 x, F ij := n i n j n. Under the scaling r λr the integrated kinetic term scales as λ and the integrated F 2 term as λ 1. Therefore nontrivial configurations will attain some fixed size determined by the dimensional coupling constant g. (Virial theorem) Vakulenko and Kapitanskii (1979): a lower limit for the energy, E c Q 3 4, where c is some constant, and Q the Hopf charge. Similar upper bound has been derived recently by Lin and Yang.
14 Topological solitons The model Numerical results 1: Knots Numerical studies of What is the minimum energy state for a given Hopf charge? Studied in by Gladikowski and Hellmund, Faddeev and Niemi, Battye and Sutcliffe, and Hietarinta and Salo.
15 Topological solitons The model Numerical results 1: Knots Numerical studies of What is the minimum energy state for a given Hopf charge? Studied in by Gladikowski and Hellmund, Faddeev and Niemi, Battye and Sutcliffe, and Hietarinta and Salo. Our work: Full 3D minimization using dissipative dynamics: n new = n old δ n(r) L. No assumptions on symmetry, on the contrary: Linked unknots of various charges.
16 Topological solitons The model Numerical results 1: Knots Numerical studies of What is the minimum energy state for a given Hopf charge? Studied in by Gladikowski and Hellmund, Faddeev and Niemi, Battye and Sutcliffe, and Hietarinta and Salo. Our work: Full 3D minimization using dissipative dynamics: n new = n old δ n(r) L. No assumptions on symmetry, on the contrary: Linked unknots of various charges. J. Hietarinta and P. Salo: Faddeev-Hopf knots: dynamics of linked unknots, Phys. Lett. B 451, (1999). J. Hietarinta and P. Salo: Ground state in the Faddeev-Skyrme model, Phys. Rev. D 62, (R) (2000).
17 How to visualize vector fields? Topological solitons The model Numerical results 1: Knots Cannot draw vectors at every point and flow lines do not make sense.
18 How to visualize vector fields? Topological solitons The model Numerical results 1: Knots Cannot draw vectors at every point and flow lines do not make sense. n is a point on the sphere S 2. There is one fixed direction, n = (0, 0, 1), the north pole. All other points are defined by latitude and longitude. Vortex core is where n = n (the south pole).
19 How to visualize vector fields? Topological solitons The model Numerical results 1: Knots Cannot draw vectors at every point and flow lines do not make sense. n is a point on the sphere S 2. There is one fixed direction, n = (0, 0, 1), the north pole. All other points are defined by latitude and longitude. Vortex core is where n = n (the south pole). Latitude is invariant under global gauge rotations that keep the north pole fixed, therefore we plot equilatitude surfaces (e.g., tubes around the core) defined by {x : n(x) n = c}.
20 How to visualize vector fields? Topological solitons The model Numerical results 1: Knots Cannot draw vectors at every point and flow lines do not make sense. n is a point on the sphere S 2. There is one fixed direction, n = (0, 0, 1), the north pole. All other points are defined by latitude and longitude. Vortex core is where n = n (the south pole). Latitude is invariant under global gauge rotations that keep the north pole fixed, therefore we plot equilatitude surfaces (e.g., tubes around the core) defined by {x : n(x) n = c}. Longitudes are represented by colors on the equilatitude surface (under a global gauge rotation only colors change): we paint the surfaces using longitudes.
21 Topological solitons The model Numerical results 1: Knots Isosurface n 3 = 0 (equator) for Q = 1, 2 Color order and handedness of twist determine Hopf charge. Inside the torus is the core, where n 3 = 1.
22 Evolution of linked unknots Topological solitons The model Numerical results 1: Knots Total charge = sum of individual charges + linking number
23 Energy evolution in minimization Topological solitons The model Numerical results 1: Knots
24 Evolution (1, 5) Topological solitons The model Numerical results 1: Knots
25 Evolution trefoil Topological solitons The model Numerical results 1: Knots
26 Vakulenko bound Topological solitons The model Numerical results 1: Knots E Q / (E 1 Q 3/4 ) Q
27 Topological solitons The model Numerical results 1: Knots Different and improved final states
28 Framed links and ribbon knots Framed links and ribbon knots Ribbon deformations The proper knot theoretical setting is to use framed links. Framing attached to a curve adds local information near the curve, e.g., twisting around it.
29 Framed links and ribbon knots Framed links and ribbon knots Ribbon deformations The proper knot theoretical setting is to use framed links. Framing attached to a curve adds local information near the curve, e.g., twisting around it. One way to describe framed links is to use directed ribbons, which are preimages of line segments.
30 Computing the charge Framed links and ribbon knots Ribbon deformations For a ribbon define: twist = linking number of the ribbon core with a ribbon boundary, locally. writhe = signed crossover number of the ribbon core with itself. linking number = 1 2 (sum of signed crossings)
31 Computing the charge Framed links and ribbon knots Ribbon deformations For a ribbon define: twist = linking number of the ribbon core with a ribbon boundary, locally. writhe = signed crossover number of the ribbon core with itself. linking number = 1 2 (sum of signed crossings) The Hopf charge can be determined either by twist + writhe or linking number of the two ribbon boundaries, or linking number of the preimages of any pair of regular points.
32 Ribbon view, Q = 2 Framed links and ribbon knots Ribbon deformations Two ways to get charge 2: twice around small vs. large circle. The first one has twist = 1, writhe = 1, the second twist = 2, writhe = 0. Both have boundary linking number = 2.
33 Framed links and ribbon knots Ribbon deformations Example of ribbon deformation during minimization
34 Framed links and ribbon knots Ribbon deformations Close-up of the deformation process
35 Framed links and ribbon knots Ribbon deformations Diagrammatic rule for deformations Knot deformations correspond to ribbon deformations, e.g., crossing and breaking, but the Hopf charge will be conserved. Note that when considering equivalence of ribbon diagrams type I Reidemeister move is not valid:
36 What is different with vortices Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding do not allow 1-point compactification of R 3 S 3. Instead we have R 2 T if periodic in z-direction or T 3 if periodic in all directions. Topological conserved quantities studied by Pontrjagin in 1941, They are mainly related to vortex punctures of the periodic box.
37 Single twisted vortex Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding J. Hietarinta, J. Jäykkä and P. Salo: Dynamics of vortices and knots in, JHEP Proc.: PrHEP unesp2002/17 J. Hietarinta, J. Jäykkä and P. Salo: Relaxation of twisted vortices in the Faddeev-Skyrme model, Phys. Lett. A 321, (2004). Knotting as usual if tightly wound:
38 Vortex bunches and unwinding Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding If the vortices are close enough there can be bunching or in the fully periodic case, Hopfion unwinding. J. Jäykkä and J. Hietarinta: Unwinding in Hopfion vortex bunches, Phys. Rev. D 79, (2009).
39 Vortex bunches and unwinding Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding If the vortices are close enough there can be bunching or in the fully periodic case, Hopfion unwinding. J. Jäykkä and J. Hietarinta: Unwinding in Hopfion vortex bunches, Phys. Rev. D 79, (2009). In the following pictures the lines corresponding to two preimages have been plotted, one in blue and one in red.
40 Vortex bunches and unwinding Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding If the vortices are close enough there can be bunching or in the fully periodic case, Hopfion unwinding. J. Jäykkä and J. Hietarinta: Unwinding in Hopfion vortex bunches, Phys. Rev. D 79, (2009). In the following pictures the lines corresponding to two preimages have been plotted, one in blue and one in red. First a 3 3 section of the fully periodic case.
41 Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding
42 Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding
43 Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding
44 Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding
45 Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding
46 Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding
47 Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding
48 Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding
49 Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding
50 Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding
51 Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding 4 4
52 Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding 4 4
53 Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding 4 4
54 Conclusion Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding We have studied the minimum energy states and the deformation dynamics of Hopfions in.
55 Conclusion Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding We have studied the minimum energy states and the deformation dynamics of Hopfions in. The results follow closely the Vakulenko-Kapitanskii bound
56 Conclusion Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding We have studied the minimum energy states and the deformation dynamics of Hopfions in. The results follow closely the Vakulenko-Kapitanskii bound The trefoil knot is obtained at Q = 7 from various initial configurations.
57 Conclusion Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding We have studied the minimum energy states and the deformation dynamics of Hopfions in. The results follow closely the Vakulenko-Kapitanskii bound The trefoil knot is obtained at Q = 7 from various initial configurations. Tightly would vortices form knots in the middle.
58 Conclusion Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding We have studied the minimum energy states and the deformation dynamics of Hopfions in. The results follow closely the Vakulenko-Kapitanskii bound The trefoil knot is obtained at Q = 7 from various initial configurations. Tightly would vortices form knots in the middle. Closeup vortices form a bunch.
59 Conclusion Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding We have studied the minimum energy states and the deformation dynamics of Hopfions in. The results follow closely the Vakulenko-Kapitanskii bound The trefoil knot is obtained at Q = 7 from various initial configurations. Tightly would vortices form knots in the middle. Closeup vortices form a bunch. In the fully periodic case the vortices can unwind completely.
60 Conclusion Topology for vortices Numerical results 2: Hopfion vortices Process of unwinding We have studied the minimum energy states and the deformation dynamics of Hopfions in. The results follow closely the Vakulenko-Kapitanskii bound The trefoil knot is obtained at Q = 7 from various initial configurations. Tightly would vortices form knots in the middle. Closeup vortices form a bunch. In the fully periodic case the vortices can unwind completely. See also:
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