21/04/23 1

Solitons: from kinks to magnetic

monopoles

Richard MacKenzie

Département de Physique, U. Montréal

U. Miss, 23 March, 2010

21/04/23 2

Outline1. 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

21/04/23 3

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 propagateswithout dispersion

21/04/23 4

History of solitionsFirst 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.

21/04/23 5

Soliton on the Scott Russell Aqueduc, Edinburgh, 12 July, 1995

21/04/23 6

2. Solitons in everyday life

21/04/23 7

Bicycle rack solitons

A topological problem encountered in the parking of bicycles

21/04/23 8

Bicyce rack: a non-artist’s misconception:

21/04/23 9

Formation of a bicycle soliton

21/04/23 10

Two «ground states»:

21/04/23 11

Another example inspired by

everyday life:

21/04/23 12

3. Solitons in conducting polymers

21/04/23 13

Two ground states:

«Peierls Instability»

21/04/23 14

Mathematical description

Order parameter:

«dimerisation» : φn=(-)nun→φ(x)

where un=displacement of nth C atom

Energy:

E=∑n{k(φn-φn-1)2+V(φn)} →

∫dx{(dφ/dx)2+V(φ)}

21/04/23 15

Potential energy density:

21/04/23 16

Soliton («kink»):

|←several C atoms→|

21/04/23 17

4. A topological detourTwo 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

21/04/23 18

Nontopological vs topological

21/04/23 19

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

21/04/23 20

Examples of SSB Ferromagnetism: spontaneous appearance of

magnetization

Broken symmetry: spin rotation SO(3)→SO(2)Order parameter: magnetizationSpace of ground states: S2 (surface of a sphere)

21/04/23 21

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: S1 (circle)

21/04/23 22

Electro-weak symmetry in the Standard Model of particle physics:

Order parameter: Higgs field (two complex fields (φ1,φ2) )

Broken symmetry: «rotations» of the Higgs field: SU(2)xU(1) → U(1) (electromag.)

Space of ground states: S3

| φ1|2 + |φ2|2 = v2

21/04/23 23

Grand Unified Theories (GUTs) Chiral symmetry Supersymmetry Family replication symmetry (?) Lorentz group? etc

Other examples of SSB:

21/04/23 24

5. Solitons in 2d: SC vortexOrder 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=∫d2x V(φ(x))

For this to be finite, V(φ)→0 for r→∞ :

|φ|→v -- but phase is arbitrary.

21/04/23 25

Here are two possibilities:

trivial (vacuum) topologically nontrivial

21/04/23 26

Can we make the soliton go away?

Mathematical description: finite energy ↔ map from spatial infinity to the space of ground states: S1→ S1. Take-home buzz word: homotopy π1(S

1)=Z

21/04/23 27

And the gradient energy…?

«It can be shown» that for the soliton,

∫d2x (∂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.

21/04/23 28

Unexpected bonus: soliton contains magnetic flux

21/04/23 29

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!!!)

21/04/23 30

6. Solitons in 3d: magnetic monopoles

Toy model : triplet of real fields (φ1,φ2,φ3) symmetric under rotations SO(3) SSB : SO(3)→SO(2)~U(1) Space of ground states : S2

symmetry gauged : non-abelian gauge theory (unbroken symmetry: U(1) e.m.)

21/04/23 31

Existence of solitons (3d) follows from the topological result : π2(S2)=Z

Soliton:

21/04/23 32

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.

21/04/23 33

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?

21/04/23 34

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.

21/04/23 35

(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)

21/04/23 36

Depending on the details of the SSB, many possibilities exist :

21/04/23 37

Subsequent cooling : topological defects become regions of overdensity → seeds for the formation of structure in the Universe

21/04/23 38

(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.

21/04/23 39

Toy model:

Field giving rise to solitons : φ

Fields of ordinary matter : ψ,ξ,…

21/04/23 40

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»

Thank you!

21/04/23 41