1. 21/10/20151 Solitons: from kinks to magnetic monopoles Richard MacKenzie Département de Physique, U. Montréal U. Miss, 23 March, 2010

2. 21/10/20152 Outline 1. What is a soliton? Brief history of... 2. Solitons in everyday life 3. Example in 1 dimension : conducting polymers 4. Topological detour: spontaneous symmetry breaking 5. Solitons in 2d : superconducting vortices 6. Solitons in 3d : magnetic monopoles 7. A few applications

3. 21/10/20153 1. What is a soliton? Definition: a solution to a set of partial differential equations which is localized in space, which is either time dependent or does not change its form as a function of time. or A localized wave which propagates without dispersion

4. 21/10/20154 History of solitions First observation: John Scott Russell, 1834 I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped—not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well- defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation.

5. 21/10/20155 Soliton on the Scott Russell Aqueduc, Edinburgh, 12 July, 1995

6. 21/10/20156 2. Solitons in everyday life

7. 21/10/20157 Bicycle rack solitons A topological problem encountered in the parking of bicycles

8. 21/10/20158 Bicyce rack: a non-artist’s misconception:

9. 21/10/20159 Formation of a bicycle soliton

10. 21/10/201510 Two «ground states»:

11. 21/10/201511 Another example inspired by everyday life:

12. 21/10/201512 3. Solitons in conducting polymers

13. 21/10/201513 Two ground states: «Peierls Instability»

14. 21/10/201514 Mathematical description Order parameter: «dimerisation» : φ n =(-) n u n →φ(x) where u n =displacement of n th C atom Energy: E=∑ n {k(φ n -φ n-1 ) 2 +V(φ n )} → ∫dx{(dφ/dx) 2 +V(φ)}

15. 21/10/201515 Potential energy density:

16. 21/10/201516 Soliton («kink»): |←several C atoms→|

17. 21/10/201517 4. A topological detour Two classes of soliton: «Nontopological» soliton: owes its existence to nonlinearities which act against dispersion (e.g. waves observed by JSR) «Topological» soliton: owes its existence to a multiplicity of ground states, giving rise to topologically non-trivial field configurations

18. 21/10/201518 Nontopological vs topological

19. 21/10/201519 Multiplicity of ground states: A rare exception? No! It’s a fact of life with spontaneous symmetry breaking (SSB) Symmetry: φ→-φ V(-φ)=V(φ) mais φ= -v→φ=+v

20. 21/10/201520 Examples of SSB Ferromagnetism: spontaneous appearance of magnetization Broken symmetry: spin rotation SO(3)→SO(2) Order parameter: magnetization Space of ground states: S 2 (surface of a sphere)

21. 21/10/201521 Superconductivity (SC): Order parameter: wave function of Cooper pairs (superconducting electrons) – complex scalar field φ Broken symmetry: rotation of the phase of φ Space of ground states: S 1 (circle)

22. 21/10/201522 Electro-weak symmetry in the Standard Model of particle physics: Order parameter: Higgs field (two complex fields (φ 1,φ 2 ) ) Broken symmetry: «rotations» of the Higgs field: SU(2)xU(1) → U(1) (electromag.) Space of ground states: S 3 | φ 1 | 2 + |φ 2 | 2 = v 2

23. 21/10/201523 Grand Unified Theories (GUTs) Chiral symmetry Supersymmetry Family replication symmetry (?) Lorentz group? etc Other examples of SSB:

24. 21/10/201524 5. Solitons in 2d: SC vortex Order parameter: complex field φ How to look for solitons: study the space of finite- energy field configurations One contribution to the energy of a field configuration: potential energy U=∫d 2 x V(φ(x)) For this to be finite, V(φ)→0 for r→∞ : |φ|→v -- but phase is arbitrary.

25. 21/10/201525 Here are two possibilities: trivial (vacuum) topologically nontrivial

26. 21/10/201526 Can we make the soliton go away? Mathematical description: finite energy ↔ map from spatial infinity to the space of ground states: S 1 → S 1. Take-home buzz word: homotopy π 1 (S 1 )=Z

27. 21/10/201527 And the gradient energy…? «It can be shown» that for the soliton, ∫d 2 x (∂φ) 2 →∞ !!! Is all lost? No! Gauge symmetry comes to the rescue. If the U(1) symmetry is «gauged», ∂φ→(∂-ieA)φ A: vector potential (electromag.) For φ in the form of a soliton, we can find an A with which the gradient energy is finite.

28. 21/10/201528 Unexpected bonus: soliton contains magnetic flux

29. 21/10/201529 Abrikosov lattice: Type II superconductor in a sufficiently strong magnetic field: field penetrates by forming a lattice of vortices (Abrikosov 1957; Nobel Prize in Physics, 2003!!!)

30. 21/10/201530 6. Solitons in 3d: magnetic monopoles Toy model :  triplet of real fields (φ 1,φ 2,φ 3 )  symmetric under rotations SO(3)  SSB : SO(3)→SO(2)~U(1)  Space of ground states : S 2  symmetry gauged : non-abelian gauge theory (unbroken symmetry: U(1) e.m.)

31. 21/10/201531 Existence of solitons (3d) follows from the topological result : π 2 (S 2 )=Z Soliton:

32. 21/10/201532 As with the vortex, the gradient energy diverges. Again, this can be remedied via the (non- abelian) gauge field. This time, the gauge field describes a magnetic flux coming from the soliton: far from it, the magnetic field is a Coulomb field; the soliton is a magnetic monopole (« ’t Hooft-Polyakov monopole »). Does this occur in the Standard Model? No... but any GUT (or model which reduces to a GUT) has magnetic monopoles.

33. 21/10/201533 7. A few «applications» (a) Quantization of electric charge. Observational fact : The charge of every particle we know is a multiple of e (or e/3 with quarks) Question : WHY?

34. 21/10/201534 Dirac (1931) : If a magnetic monopole existed, the quantum mechanics of a charged particle in the presence of the monopole does not make sense unless the electric charge is quantized. (Thus, the fact that electric charge does seem to be quantized is suggestive of the existence of magnetic monopoles!) Modern context : In field theory, charge need not be quantized... but in GUTs, (i) electric charge is quantized; (ii) monopoles do exist.

35. 21/10/201535 (b) Cosmology, phase transitions, topological defects Following the Big Bang, the Universe expanded and cooled, causing (via SSB) phase transitions : eg SU(5) → SU(3)xSU(2)xU(1) → SU(3)xU(1)

36. 21/10/201536 Depending on the details of the SSB, many possibilities exist :

37. 21/10/201537 Subsequent cooling : topological defects become regions of overdensity → seeds for the formation of structure in the Universe

38. 21/10/201538 (c) Large extra dimensions – where are they? If the Universe is of dimension > 3+1, the extra dimensions could be small (Kaluza-Klein models, superstrings) or large (D-branes, Randall-Sundrum, etc) One way to reconcile the existence of large extra dimensions with our observed 3+1d Universe is if we live in the core of a 3- dimensional topological defect.

39. 21/10/201539 Toy model: Field giving rise to solitons : φ Fields of ordinary matter : ψ,ξ,…

40. 21/10/201540 8. What I said Pay attention to where you park your bicycle! Topological solitons ; importance of spontaneous symmetry breaking examples in 1d (polymers), 2d (vortices), 3d (magnetic monopoles) a couple of «applications»

41. Thank you! 21/10/201541