International Symposium in Honour of Professor Nambu for the 10th
Anniversary of his Nobel Prize in Physics“From spontaneous symmetry breaking
to quantum turbulence
W F Vinen
School of Physics and Astronomy, University of Birmingham

Introductory Comments
A great honour to be asked to contribute to this symposium.
Professor Nambu’s view that condensed matter physics.is less attractive
than nuclear and particle physics.
Professor Nambu’s Nobel Prize was for his contributions to anunderstanding of spontaneous symmetry breaking in sub-atomic
physics, but his interest in symmetry breaking was probably generated
by an interest in symmetry breaking in condensed matter physics, and inparticular in superconductivity,
My aim: an elementary talk that connects a simple but quite subtle formof symmetry breaking - the transition to superfluidity in liquid 4He with an area of research that is very active here in OCU, namely quantum
turbulence.
Transitions to turbulence also involve symmetry breaking, as I shallmention briefly.. I shall not refer to the complicated case of superfluidity inliquid 3He, in spite of its having been much studied at OCU. But I shallrefer briefly to superconductivity and cold superfluid gases.

Superfluidity and Bose condensation
Superfluidity is always associated with some form of Bose condensation.
Superfluidity arises from what used to be called an “ordering process”.
Bose condensation was originally seen as an ordering in momentum spacerather than an ordering in real space.
An inadequate view of Bose condensation. Putting a macroscopic
number of identical Bosons into a single quantum state leads to theformation of a coherent matter wave (cf a coherent electromagnetic wave).
This coherent matter wave leads to superfluidity.
In practice Bose condensation occurs in systems of interacting particles.
What is the effect of this interaction? It leads to some virtual excitation out of
the condensate, but the condensate remains.
A formal description is in terms of the form of the single-particle density
matrix (Penrose and Onsager)    
where Ψ  ≡  → Ψ ∗  Ψ as −  → ∞is a “classical” phase-coherent matter field.

The superfluid order parameter, superfluid velocity,and supercurrentsΨ  is the complex order parameter for the superfluid transition, called
the “condensate wave function”. It is not a quantum-mechanical wave
function in the ordinary sense, rather a classical field. The ordering impliedby the existence of this field is often called Off-Diagonal Long Range Order.
In general Ψ  = Ψ    
where  = ℏ⁄   is the local
velocity with which the condensate is moving.  is the superfluidvelocity..
This superfluid velocity must be irrotational (curl  = 0) and thesuperfluid circulation must be quantizedΓ = ∮  .  = !", where " = ℎ⁄ is the quantum of circulation.
But does Bose condensation necessarily lead to superfluidity?.

Spontaneous symmetry breaking

Superfluidity depends on the existence of coherent matter wave in the fluid,with its well-defined phase. Changing that phase alters the state of thesuperfluid.. In other words it is the loss of global U(1) phase rotation
symmetry that characterizes the superfluid transition. (Gauge symmetry)
Observation of the phase: coupled superfluids.
Number-phase uncertainty.

The stability of superflow, the excitation spectrum, thenormal fluid and the condensate fractions
Is 4He a superfluid? Is frictionless superflow stable, or at least metastable?
The ideal Bose-condensed gas is not a superfluid because its flow is
unstable against single particle excitation, with spectrum ' =  ⁄2.
Interactions change the spectrum, leading to only phonons at the lowest
energies: ' = ), where c is the velocity of sound. These low-lying
phonon states are the Nambu-Goldstone modes associated with the phasetransition.
At $ = 0 this phonon spectrum ensures that % = %. (& = %  )(Landau)

At a finite T thermal excitation of phonons (+ rotons) leads to a “normalfluid”. density %* .. As $ → $+ %* → % (% → 0) .
The superfluid density must not be confused with the condensate fraction.
These ideas apply to any simple Bose superfluid: liquid 4He, and Bose
condensed cold atomic gases

Superconductivity
Another superfluid system is the conduction electron gas in a superconductor.
Electrons are Fermions, so Bose condensation of electrons cannot occur.
But a form of Bose condensation of Cooper pairs can occur, leading to a twoparticle density matrix having a form similar to that of the single particle
density matrix in heliumψ ↑+ ( r1 )ψ ↓+ ( r1 )ψ ↓ (r2 )ψ ↑ ( r2 ) → Ψ* ( r1 ) Ψ ( r2 ) as r1 − r2 → ∞where Ψ ( r ) has long-range phase coherence. Again superconductivity
arises from broken gauge symmetry.
But there are important differences:• Electrons are charged:  = ℏ⁄   − ⁄ ,  .
• The Bose condensing particles are not present in the non-interacting
system, which is degenerate Fermi gas.
• The Cooper pairs are very large compared with the electron spacing.
• There is an energy gap in the spectrum of excitations.
Observation of transition from BCS condensation to ordinary Bose
condensation of particle pairs as interaction is increased.

Quantized vortices
A non-zero Γ can exist only in a multiply-connected volume.
A possible topological defect: a free vortex line (Onsager,Feynman): |Ψ| = 0 along a line in the fluid, a phase changeof 2π round the line, and hence a circulation of κ.
Core size determined by a coherence length.
In an uncharged superfluid velocity ~ 1⁄ for all r (BiotSavart). In a charged superfluid velocity is cut off at
distances greater than the penetration depth.
A quantized vortex is quite different from a vortex in aclassical fluid, which can decay by viscous dissipation..
Quantized vortices play an important role in superfluids::• They allow bulk rotation (or penetration of amagnetic field);• They allow dissipation of a superflow.
In an uncharged superfluid this dissipation is often associatedwith a irregular array of vortices: Quantum Turbulence.

κ = 2πℏ m

The stability of superflow; vortex nucleation; superfluid
turbulence
Can vortex motion lead to a decay of superfluid
flow? Not necessarily, because this motion isopposed by potential barrier (interaction with image)
Intrinsic nucleation of vortices
very difficult.
Therefore vortex nucleation isusually extrinsic and depends on
expansion of existing (remanent)
vortices.
Such expansion can often leadto a moving tangle of vortices:
i.e. a turbulent superfluid.
At the same time the normal fluid maybecome turbulent (in 4He it has a very smallviscosity), so we can have twointerpenetrating turbulent fluids.

Vortex dynamics
The weakly interacting Bose-condensed gas is described by a non-linear
Schrodinger equation (the Gross-Pitaevskii eqn), from which we candeduce that a vortex will behave classically except on scales
comparable with the coherence length. Assume that same applies tosuperfluid 4He.
At $ = 0 (no normal fluid) a vortex moves with the local fluid velocity, due
often to other vortices (Biot-Savart)...
At $ ≠ 0 there are excitations forming the normalfluid. These are scattered by a vortex
force of“mutual friction” between the two fluids. The vortex
responds according to the Magnus effect.
If two vortices come very close they canreconnect (a non-classical effect).
Reconnections are very important.

Classical turbulence
We can think of classical turbulence in terms oftwo problems.
• Understanding the initial instability of the(usually) laminar flow and how the corresponding
flow structures develop. They tend not to bechaotic and may or may not be time dependent..
They will always involve symmetry breaking...
• Understanding the maintenance and properties offully turbulent flows, which tend to be chaotic,
with some recovery of (statistical) symmetry..
Homogeneous
turbulence
produced byflow through agrid

Kármán vortexstreet

Superflow past an obstacle
Little experimental study of steady motion of an obstacle in a superfluid..
Much study of the drag on objects oscillating in a superfluid: typically anoscillating cylinder in the form of a vibrating wire..
Results often difficult to interpret:• initial instability due to remanent vortices with unknown geometry• flow patterns depend crucially on the boundary conditions.
• flow is complicated and cannot yet be visualized.
But some nice experiments have been done,
especially by Hideo Yano here at OCU.

Best to avoid boundaries

Homogeneous turbulence in a classical fluid∂v
1+ (v ⋅ ∇)v = − ∇p + ν ∇2v
ρ∂tRichardson cascade
Energy is injected into
turbulent eddies at large
Reynolds number. It thendecays through non-linear
interactions in a cascadeof smaller and smaller
eddies, until the eddies
are so small that they
have a Reynolds number~ 1 and there is viscous
dissipation.

Kolmogorov
energy spectrum(Local interactions)Homogeneous turbulence in superfluid 4He at 1 = 2Compared with the classical case• No viscous dissipation.
• Polarization of vortices
large eddies• Quantized circulation
minimum eddy size
characterized by a length scale ℓ.
On large scales (>>ℓ): eddies containing many
quanta. Therefore classical behaviour? Energy
cascade with Kolmogorov spectrum? Experimental
evidence (Maurer & Tabeling – pressure fluctuations);
evidence from computer simulations (Tsubota)
On small scales (≤ ℓ): quantum dominated ((quantized vorticity;discrete vortex lines; vortex reconnections)), but apparently no
dissipation! Rules out Kolmogorov? Can ℓ get smaller and smaller?
No, because a decreasing ℓ implies an increasing velocity!
What happens to the Reynolds number Re= 34⁄5? Re→ ∞?
We might define a “quantum” Reynolds number Req = 34⁄". .
Req =1 corresponds to motion on scale ℓ. (dominated by quantum effects).
But dissipation??

Dissipation in homogeneous quantum turbulence
A moving (oscillating) vortex emits sound.
Vortex reconnections emit soundIn a cold Bose-condensed gas phononemission due to reconnections is sufficient
to terminate the Kolmogorov cascade.
But not in superfluid 4He
But reconnections also produce helicalwaves on vortices (Kelvin waves).
These Kelvin waves can build up in
amplitude, start interacting, generatingwaves of higher and higher frequency. A
process we call wave turbulence.
At a high enough frequency phonon emission
becomes sufficient to terminate the wave
turbulent spectrum.
All this is speculation. Exptal
confirmation a challenge

What happens at a finite T?

Forcing the two fluids to move at different velocities
Leads to the generation of selfsustaining small-scale turbulence insuperfluid component, with laminar flowof normal fluid.
The pioneering simulations of Schwarz,
and the important developments inthese simulations by Tsubota.

MFMF

Crucial role of reconnections.
More recent developments: large-scale,partially coupled turbulence in both fluids. Amajor challenge, for both experiment andtheory. (Coupled dynamics in numericalmodelling especially challenging- Tsubota et al)Why study quantum turbulence? The future.
New types of turbulence. Can we understand them? Can such
understanding enrich our understanding of turbulence in general?
Can the discrete structure of quantum turbulence make it easier to study
and understand than is the case with classical turbulence? For example
can we see and understand better the non-linear processes that lead toturbulent energy cascades? Can we understand intermittency better?
Can we look for and understand coherent structures?
This improved understanding will depend on our ability to see what isreally going on in quantum turbulence. This requires visualization ofthe flow. For many years such visualization eluded us. But recent
developments are changing this situation (Lathrop, Van Sciver, Wei Guo
Skrbek): finding tracer particles that track the motion of one or other of thetwo fluids (as in the discovery of large scale turbulence in counterflow).

Why study quantum turbulence? The future.
New types of turbulence. Can we understand them? Can such
understanding enrich our understanding of turbulence in general?
Can the discrete structure of quantum turbulence make it easier to study
and understand than is the case with classical turbulence? For example
can we see and understand better the non-linear processes that lead toturbulent energy cascades? Can we understand intermittency better?
Can we look for and understand coherent structures?

Exploiting these new techniques for visualizing flow patterns in a superfluid
promise to transform the subject. After 80 years superfluidity is still interesting.

Thank youNotes
Intermittency – inertial range decay into smaller and smaller eddies
may not fill space.
Very high Rn.
Cold gases. 3He. Mention at end.