c© Copyright by Geoffrey Lee Warner, 2004

LITTLE BIG BANGS IN HELIUM-THREE

BY

GEOFFREY LEE WARNER

B.S., Massachusetts Institute of Technology, 1997M.S., University of Illinois at Urbana-Champaign, 2001

DISSERTATION

Submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Physics

in the Graduate College of theUniversity of Illinois at Urbana-Champaign, 2004

Urbana, Illinois

Abstract

With an eye toward the interpretation of so-called ‘cosmological’ experiments performed on the

low temperature phases of3He, in which regions of the superfluid are destroyed by local heating

with neutron radiation, we have studied the properties of degenerate Fermi systems under certain

extreme non-equilibrium conditions.

First, we consider the dynamical evolution of localized hot spots in ultralow temperature Fermi

liquids. Within a model calculation, it is found that such perturbations do not relax into a hydrody-

namic profile at long times, as might be expected of thermal hot spots under less exotic conditions.

Instead, the hot spot expands outward into a shell moving away from its center at a velocity com-

parable to the Fermi velocityvF , which is consistent with the ‘baked Alaska’ hypothesis proposed

earlier by Leggett as a possible solution to the riddle ofB-phase nucleation in3He.

Second, we examine the behavior of a Fermi gas subjected to uniform variations of an attrac-

tive BCS interaction parameterλ. In 3He the quenches induced by the rapid cooling of the hot

spots back through the transition may lead to the formation of vortex loops via the Kibble-Zurek

mechanism. A consideration of the free energy available in the quenched region for the production

of such vortices reveals that the Kibble-Zurek scaling law gives at best a lower bound on the defect

spacing. Further, for quenches that fall far outside the Ginzburg-Landau regime, the dynamics of

the pair subspace, as initiated by quantum fluctuations, tends irreversibly to a self-driven steady

state with a gap∆∞ = εC(e2/N(0)λ − 1)−1/2. In weak coupling this is half the equilibrium BCS

result, the extra energy being taken up by complex collective motion of the pairs.

iii

To my mother.

iv

Acknowledgments

My advisor, Tony Leggett, is by temperament the most honest and hard-working researcher I’ve

ever met. I have learned a lot from his example, principally how to sidestep the many opportunities

for self-delusion that research affords. I am immensely grateful to him for his patient indulgence

of my many questions over the years, and for his (unusually) open mind. Prof. James Wolfe has

provided some much-needed advice at various points during my time here, and I would like to

thank him for his generosity and continuing solicitude. I would like to thank Paul Goldbart for his

careful reading of the manuscript and for his many useful criticisms.

Among the graduate students I would like first and foremost to acknowledge Bill McMahon,

a close friend and ‘longtime cohort in physics’, to use his words. I have enjoyed innumerable

conversations with my officemate Vladimir Lukic; I’m glad we got assigned to the same office,

since otherwise I might never have gotten to know him. Joseph Jun is a kindred spirit and I will

miss him as we both embark on our respective postdoctoral journeys. Also, he was most helpful

with the numerical integrations and some of the figures in this thesis. Finally, I would like to

mention some of the other friends I’ve made while here at Illinois: Vivek Aji, Sahel Ashhab, Steve

Clayton, Parag Ghosh, Tereza Neocleous, William Tucker, Julian Velev, and Sahng-Kyoon Yoo.

I’m also grateful to Amalia Miller for her love and support throughout the period during which

the work appearing in this thesis was performed.

Finally, I would like to acknowledge my sources of financial support; namely, NSF Grant No.

DMR99-86199, and also the many Departmental TA-ships I have held while a graduate student.

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Contents

Chapter 1 General Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Nucleation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2

1.1.1 The Baked Alaska Scenario . . . . . . . . . . . . . . . . . . . . . . . . .41.2 ‘Cosmology’ in3He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 The Quenching Experiments . . . . . . . . . . . . . . . . . . . . . . . . .101.2.2 Some Comments on the Interpretation of these Experiments . . . . . . . .14

Chapter 2 Quasiparticle Diffusion in a Fermi Liquid . . . . . . . . . . . . . . . . . . . 182.1 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .202.2 A Random Walk on the Fermi Surface . . . . . . . . . . . . . . . . . . . . . . . .222.3 The Model and Associated Formalism . . . . . . . . . . . . . . . . . . . . . . . .26

2.3.1 Ordinary Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .302.3.2 Anomalous Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .332.3.3 Energy Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .352.3.4 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36

2.4 Connection With the Real Problem in3He . . . . . . . . . . . . . . . . . . . . . . 382.4.1 Breakdown of the Diffusion Picture . . . . . . . . . . . . . . . . . . . . .392.4.2 Estimation ofτcross . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41

Chapter 3 A Question of Quenching . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .433.2 Thermodynamics of a Shallow Quench . . . . . . . . . . . . . . . . . . . . . . . .463.3 Quenching and Order Parameter Dynamics . . . . . . . . . . . . . . . . . . . . .51

3.3.1 The Pseudospin Representation, and Initial Conditions . . . . . . . . . . .523.3.2 The Semiclassical Equations of Motion . . . . . . . . . . . . . . . . . . .573.3.3 Early- and Long-Time Behavior of Solutions . . . . . . . . . . . . . . . .593.3.4 Numerical Integration of the Semiclassical Equations . . . . . . . . . . . .623.3.5 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64

Chapter 4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

vi

Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

vii

List of Figures

1.1 Phase diagram for3He at low temperatures. . . . . . . . . . . . . . . . . . . . . .21.2 Typical baked Alaska event. The shaded region corresponds to effective tempera-

turesTeff > Tc. Initially there is a compact ‘hot spot’ which subsequently evolvesinto a shell moving ballistically outward at approximately the Fermi velocity. . . .6

1.3 Domain structure immediately after the quench. Arrows represent theU(1) gaugefield local to that domain. Dashed circle indicates a possible vortex. . . . . . . . .9

1.4 Zurek’s proposed experiment using4He confined to a narrow annular region andsubsequently quenched into the superfluid state. Different regions around the ringadopt independent orientations of theU(1) gauge field. . . . . . . . . . . . . . . .11

1.5 Schematic depiction of two competing models of inhomogeneous quenching. In(a) the diffusion of heat away from the hot spot leads to a contraction of the lo-cus of constant temperature atTc faster than the surrounding superfluid phase canpropagate inwards. In (b) the effective temperature profile is non-monotonic witha shell moving out ballistically from the interior at roughly the Fermi velocity. Theinterior is thus shielded from the outside phase. . . . . . . . . . . . . . . . . . . .17

2.1 Diagram of the cascade process. Dashed line corresponds to the single ‘particle’we are tracing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27

2.2 Schematic depiction ofν as a function ofζ for increasingN . In the limitN →∞,the curve develops a singularity atζ = 1. . . . . . . . . . . . . . . . . . . . . . . . 37

3.1 Adiabatic vs. finite-time quenches. a) For adiabatic passage through the transi-tion curve, the slope of the isentrope jumps discontinuously from zero at criticalcoupling. Thus the superfluid emerges at a steadily increasing temperature. b) Fora finite-time passage, the system falls out of equilibrium upon crossing the firstthick dashed line; this delays the slope discontinuity until the system intersects thedashed line on the other side. . . . . . . . . . . . . . . . . . . . . . . . . . . . .47

3.2 Schematic illustration of two possible mean-field configurations of the pseudospinsystem. The solid line indicates thex axis. a) Immediately following the quench,the pseudospins form a domain wall at the Fermi surface, which is energeticallycostly in the presence of the an off-diagonal field. This points in the positivexdirection. b) The BCS solution, which rotates individual pseudospins into theireffective fields, thereby lowering the energy. The domain wall has been ‘softened’as a consequence. This state is not dynamically accessible to the quenched state ina), since it has a lower energy and the semiclassical equations conserve this. . . . .53

viii

3.3 Gap dynamics for1000 pairs following a weak fluctuation (∼ 10−7) for a couplingparameterN(0)λ = 5. Curve appears black because of the high frequency of theoscillations on this scale. However the envelope is clearly discernible and showsthe amplitude of the oscillations decreases monotonically during the approach tothe asymptotic steady state. Inset depicts a close-up on the curve, making the gaposcillations visible. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63

1 Diagram for the calculation of the superfluid velocity on a point of the ring due toall of the other elements of the ring. . . . . . . . . . . . . . . . . . . . . . . . . .74

2 Energy of a vortex ring as a function of its radiusR0 in the presence of counterflow.76

ix

Chapter 1

General Overview

There can be no question of the profound influence exerted by the study of low temperature3He

on the field of condensed matter physics. From Landau’s Fermi liquid theory and the quasiparticle

concept to the nature of anisotropic superfluidity, many of the guiding principles of the field were

developed explicitly in answer to the fundamental questions posed by this quantum liquid. In

recent years this influence has widened considerably, and is being felt in some rather unlikely

communities: cosmology, usually connected with very large-scale phenomena, and elementary

particle physics, the study of the ultra-small. That this substance could inspire such interest outside

of its nominal field stems in part from the intricacy of the various broken-symmetry phases it

exhibits at low temperature, into which some researchers have read a number of analogies with field

theories of the fundamental particles and gravity [Volovik (2003)]. It also results from a change in

the scope of phenomena being studied, which now includes those properties that manifest under

extreme non-equilibrium conditions.

This thesis will focus on a particular group of experiments, in which the superfluid phases of

3He are bombarded with one or another kind of ionizing radiation [Schiffer (1993); Schiffer and

Osheroff (1995); Bauerle et al. (1995); Ruutu et al. (1995)]. In this connection there have emerged

two distinct but strongly overlapping domains of research. The first of these became apparent

shortly after the initial experimental discovery of theA andB superfluid phases in 1972, which we

will call the ‘nucleation problem’. The second, ‘cosmological problem’ was initiated by Kibble in

1976 and is in fact the primary reason for the recent cross-disciplinary upsurge of interest in3He.

1

Figure 1.1: Phase diagram for3He at low temperatures.

1.1 The Nucleation Problem

To begin, let us first consider the nucleation puzzle, which is perhaps best understood in terms of

the low temperature3He phase diagram depicted in Figure 1.1. For pressures between 22 and 34

bars at zero magnetic field, as one lowers the temperature from aboveTc ∼ 2mK the normal Fermi

liquid makes a second-order transition to the superfluidA phase. Continuing along this path one

eventually encounters a temperatureTAB(P ), below which theB phase has the lower free energy

and is therefore the thermodynamically stable phase. TheA-B transition is however first-order,

and we cannot get between the two phases continuously; experimentally theA phase has been

observed to persist on timescales of minutes to hours well beyondTAB, a phenomenon known as

‘supercooling’. In practice of course it always eventually makes the transition, so (discounting

spinodal decomposition, see [Leggett and Yip (1989)]) there must be a process whereby a bubble

of B phase forms inside the metastableA phase and subsequently expands to fill the whole liquid

volume.

2

It is logical to consider first the possibility that such bubbles form spontaneously via thermal

fluctuations of the metastable state. This is the so-called Cahn-Hilliard theory of homogeneous nu-

cleation [Lifshitz and Pitaevskii (1981)], which predicts that the rate associated with the nucleation

of a viable bubble of the new phase should be proportional toω0 exp(−∆Fmax

kBT). ∆Fmax is the free

energy barrier that must be surmounted by thermal fluctuations, and can be calculated in terms of

the total free energy of the bubble;ω0 is an extensive quantity known as the ‘attempt frequency’.

For a bubble of radiusR, ∆Fmax is made up, first, of the bulk free energy difference between the

two phasesFB − FA, and second, of the surface energy at the interface between them, that is, the

energy cost associated with the interfacial region where the liquid must interpolate between the

order parameters on either side. If we write the associated surface energy per unit area asσAB,

the total free energy takes the formFb = 4πR2σAB + 43πR3(FB − FA) , which has a maximum

at a critical radiusRc = 2σAB

(FA−FB). This represents a cutoff below which a bubble is crushed by

the surface energy, and above which the bulk free energy difference can drive the bubble to ex-

pand throughout the liquid. Substituting back into our expression for the free energy, this implies

a barrier height∆Fmax = 163π

σ3AB

(FA−FB)2which forT nearTAB is enormous,∼ 106kBT (a typical

barrier being on the order of tens ofkBT ). Even granting an unphysically high estimate for the

attempt frequency, this precludes by many orders of magnitude any possibility that the nucleation

occurs through spontaneous thermal fluctuations in the metastable state. That the rate should be

so miniscule constitutes something of a mystery, since experimentally theB phase is known to

nucleate fairly easily over timescales on the order of minutes to hours.

Calculations of the quantum tunnelling of aB-phase bubble through the free energy barrier

have similarly failed to produce reasonable nucleation rates, as have considerations of certain

extrinsic effects associated with sample-cell surface irregularities on the barrier height [Leggett

(1984)]. It would appear that whatever the actual mechanism, it cannot be intrinsic to the liquid

itself; otherwise we have to explain away a lifetime which is astronomically huge. Since the sim-

plest theories have been excluded, one is forced to ponder more exotic, extrinsic mechanisms for

the nucleation of theB phase.

3

1.1.1 The Baked Alaska Scenario

One such mechanism, the ‘baked Alaska model’ proposed by Leggett (1984), explains the ob-

served nucleation events as resulting from the passage of∼ 2GeV muons cosmic ray muons (or

other ionizing radiation) through the sample. Such a particle experiences a series of ionizing col-

lisions with the3He atoms, which process liberates a number of so-called ”δ-electrons”, each with

energies typically∼ hundreds of eV. These interact with the liquid through a combination of elas-

tic and inelastic (ionization or excitation) collision events distributed over a space of dimension

R ∼10−8 m and a time∼5×10−7 s, creating a ‘hot spot’ or ‘fireball’ composed of highly excited

quasiparticles. Since the mean distance between such ionization events (∼ .1mm) greatly exceeds

the dimensions of the initial fireball, the processes triggered by individualδ-electrons should be

regarded as independent.

The formation of hot spots byδ-electrons in the heavier4He isotope has been discussed in some

detail by Tenner (1963), in the course of explaining the operation of bubble chambers. Since the

interactions involved are electronic in origin, it is unlikely that a direct carry-over of the relevant

numbers to3He would incur much, if any, error. We emphasize here that there are many aspects of

the energy deposition process that are not well understood; however, any attempt on the author’s

part to justify the details would take us far beyond the scope of this thesis. At any rate we re-

quire only the order of magnitude estimates of the time and length scales involved. Given the high

energy density of the initial hot spot we cannot rigorously exclude the possibility of more ‘col-

lective’ responses such as shock waves, but again such considerations would take us too far afield

and we shall restrict ourselves to an exploration of effects associated only with the quasiparticle

excitations.

The question of what happens next has generated considerable controversy [Leggett (1984);

Leggett and Yip (1989); Bunkov and Timofeevskaya (1999); Bunkov (2000); Warner and Leggett

(2003)]. During the earliest stages of hot spot evolution, energetic (kT ε εF ) quasiparticles

are quite densely distributed, so that interquasiparticle scattering dominates the dynamics. How-

4

ever, once the quasiparticles escape the hot localized region their density drops very rapidly and

the scattering mode crosses over into particle-hole pair creation. The mean free path for energetic

quasiparticles undergoing such collisions varies asε−2; if on average they produce particles each

with roughly 13

the energy of the incoming particle, the mean free path is enhanced by a factor of

9. This makes it plausible that the initial fireball will, at least for short times, evolve into a shell

of ballistic quasiparticles moving at∼ vF more or less radially outward from the center of the hot

spot.

By virtue of the nonthermal distribution of energetic quasiparticles that make up the hot spot,

the phase space necessary for the formation and condensation of Cooper pairs will be largely un-

available. This ‘blocking’ effect destroys the superfluid state, so that the hot spot interior might

best be described as a highly excited Fermi liquid. This further suggests that the appropriate effec-

tive ‘temperature’ of the shell should be defined in terms of the self-consistency or gap equation,

kBTeff (r, t) = 1.14εC exp(−∫ εC

−εC[1−2n(ε,r,t)

2ε]) which is a functional of the distribution function

and might therefore be expected to show similar spatial features. The effective temperature dis-

tribution is therefore radially non-monotonic, quite different from the near-equilibrium diffusive

behavior one might expect.

If correct, this model sidesteps the principal constraints on nucleation by (1) destroying the

A-phase within a region of dimensionR0, (2) allowing the interior to cool quickly (on a timescale

∼ RvF

) through the phase transition, and thereby giving theB-phase a chance to form first, and (3)

isolating the nascentB-phase in the interior (in a sense, allowing it to ‘incubate’). This prevents

the surroundingA-phase from propagating inward and crushing theB-phase before it can get a

foothold. Of course, this whole scenario hinges on whether or not the shell can survive at least out

to the critical radius.

In reply to this hypothesis, Schiffer (1993) and Schiffer and Osheroff (1995) undertook to

investigate the effects of various kinds of ionizing radiation on the statistics ofB-phase nucleation

for different sectors of the phase diagram near the melting curve (corresponding to pressures of

about 34.2 bars). Their apparatus consisted of four fused silica cylindrical tubes, smooth down

5

Figure 1.2: Typical baked Alaska event. The shaded region corresponds to effective temperaturesTeff > Tc. Initially there is a compact ‘hot spot’ which subsequently evolves into a shell movingballistically outward at approximately the Fermi velocity.

to 100 A; this is on the order of the coherence length and much smaller thanRc, making the

likelihood that the observed nucleation events resulted from the action of surface defects tolerably

small. Their measurements were of three kinds: the first without any external source of radiation

(except, presumably, for the background cosmic ray muons), the next with 1.17 and 1.33 MeV

gamma ray photons from a60Co source, and the last with thermal neutrons, which interact with

the helium nuclei via ‘charge-exchange’ reactions of the form3He + n−→ 3H + H + 0.764 MeV.

In each case, the time taken forB-phase nucleation was measured at temperatures between .9 and

1.33 mK, and for typical fields on the order of mT.

The results demonstrate unequivocally that the process ofB-phase nucleation from the super-

cooledA-phase is by nature stochastic. In each case, at fixed temperature and field the nucleation

times varied according to a universal statistical law: the number of samplesN that failed to make

the transition after a timet was well fit by a distribution of the formN = N0e− t

τ , whereN0 is the

number of identical samples prepared att = 0, andτ is the average measured value of theA-phase

lifetime in the limitN0 → ∞. τ itself was then measured as a function of temperature for each

6

kind of radiation. The functional form ofτ(T ) appeared to be universal for all radiation types up

to a constant multiplicative factor.

That the statistics appear so similar is something of a mystery, given the fact that neutrons

interact quite differently with helium than do gamma rays or high energy muons. A very detailed

account of these differences, and a comparison of the experimental statistics with the theoretical

predictions of the baked Alaska model, may be found in Leggett (2002) and Schiffer et al. (1995);

we note one of the more salient points here. Following the charge-exchange reaction of the incident

neutron with one of the helium nuclei, a proton and tritium nucleus are released with energies 573

keV and 191 keV respectively. The energy deposited by the proton is largely concentrated in

a narrow cylindrical region of radius∼ 100 angstroms about a track of length∼ 80µm. The

triton behaves similarly but with a much shorter range. This is clearly far from the more spherical

distributions expected of theδ-electrons produced by gamma rays or muons, and it is at present

unclear whether this circumstance should lead to substantial differences in the nucleation statistics.

There is some probability that an electron with energy at least comparable to that of a typicalδ

electron can be produced by the proton; if so, then perhaps the relevant secondaries are not the

protons and tritons themselves but again the energetic electrons produced along the way. In that

case there would be no qualitative differences between the various methods of inducing nucleation,

but the evidence so far is inconclusive.

In Chapter 2 we address the question of whether the baked Alaska profile represents a plausible

non-equilibrium structure following local heating in a degenerate Fermi liquid. It is demonstrated

within a model calculation that such structures are inevitable consequences of the unusual scatter-

ing modes at low temperature. Moreover it will be seen that despite what might be expected on the

basis of more conventional transport problems, such structures do not represent transient solutions

of the Boltzmann equation; that is, to the extent that the unusual scattering modes discussed above

are the only ones present, the baked Alaska profile persists all the way into the long-time limit.

Ordinarily such structures would be expected to damp away, leaving behind a monotonic Gaus-

sian profile in its wake–this is the so-called ‘hydrodynamic limit’ discussed by Palmeri [Palmeri

7

(1989)]. The absence of any such limit in the case of quasiparticle scattering in a very degenerate

Fermi liquid is the central result of Chapter 2.

1.2 ‘Cosmology’ in 3He

Interest in3He under virtually identical nonequilibrium conditions has also developed along a quite

different track. It has long been hypothesized that the vacuum immediately following the Big Bang

was in a state of high symmetry, and that the subsequent expansion and cooling of the universe led

to a series of symmetry-breaking phase transitions in which the four fundamental forces as we

now know them split apart. Kibble in 1976 advanced the hypothesis that this process, if it indeed

occurred, might possibly have left behind some unusual remnants, traces of this ‘false vacuum’ in

the form of topological defects (such as cosmic strings, magnetic monopoles, etc.). He postulated

that, at least for Lagrangians with a continuous symmetry (e.g. global U(1) gauge fields), as

the effective temperature dropped and the symmetry was broken, regions of the early universe

separated by spacelike intervals would be unable to communicate their choice of phase to each

other, leading to a honeycomb-like domain structure like that depicted in Figure 1.3. In the absence

of global symmetry breaking fields, each domain acquires an independent, randomly oriented order

parameter after passing locally through the transition. At the junction of three or more domains, a

filament of normal fluid could be trapped; if tracing theU(1) phases of these domains on a loop

enclosing the filament yields a quantum of circulation, then the filament is in fact a vortex. Globally

this effect leads to a tangled network of vortex loops in the new phase. Cosmologically speaking

these are called cosmic strings, and by virtue of their great mass, would likely have influenced a

number of large scale phenomena of interest, including the mass distribution of the universe and

the cosmic microwave background.

Of course, by itself the Kibble hypothesis is useless without an estimate of the underlying

length scale that determines the domain sizes and hence the inter-defect spacing. Kibble took this

to correspond with the size for which equilibrium fluctuations of the order parameter forT < Tc

8

Figure 1.3: Domain structure immediately after the quench. Arrows represent theU(1) gauge fieldlocal to that domain. Dashed circle indicates a possible vortex.

can just surmount the central hump of the ‘Mexican hat’ potential∆V (T ) of a Ginzburg-Landau

Hamiltonian in which the state spontaneously breaks aU(1) gauge symmetry.∆V (T ) can also be

interpreted as the difference in the free energy density between the normal and symmetry-broken

phases. If one writes the equilibrium correlation length as a function of temperatureξeq(T ), the

above condition is equivalent to the requirement that the size of thermal fluctuations be comparable

to that of the central hump, i.e.∆V (T )ξ3eq(T ) ' kT . Since∆V (T ) andξeq(T ) are determined

entirely by the equilibrium state of the system, the above condition implicitly defines a temperature

TK , below which thermal fluctuations across the hump occur infrequently and the order parameter

points in a more or less fixed direction within a volume∼ ξ3eq(T ). This, Kibble argued, must set

the scale for the domains and hence the initial intervortex distance, at least for aU(1) theory. If

true, similar considerations should apply to the defect densities appropriate to a ‘quench’ of the

early universe through one of the postulated vacuum transitions.

Subsequently Zurek (1985) noted that under the actual conditions of such a quench it would

be inconsistent to base estimates of the domain sizes on purely equilibrium or near-equilibrium

9

considerations. Taking superfluid4He as his model system, he examined the case of a quench

linear in time nearTλ of the formT (t) = Tλ(1 − tτQ

), whereτQ is the timescale for the quench.

Writing the reduced temperature asε = tτQ

= (Tλ − T )/Tλ, and taking account of the fact that

both the equilibrium correlation lengthξ and dynamical relaxation timeτ diverge within Ginzburg-

Landau theory as

ξ = ξ0|ε|−12 (1.1)

and

τ = τ0|ε|−1 (1.2)

(whereξ0 and τ0 are theirT = 0 values) we see that, so long as the rate at whichξ changes

never exceeds the characteristic velocity of the mediumc = ξ/τ , the system remains close to

equilibrium. However there comes a time−t before the transition whendξdt

becomes comparable

to c, at which point the system can no longer accommodate changes inξ and the system departs

from equilibrium. The value ofξ at−t is therefore ‘frozen in’ until a timet = t after the transition.

Thus, from the conditiondξ

dt

∣∣∣∣t=t

' ξ(t)

τ(t)(1.3)

we find that (dropping factors of order unity)t =√τ0τQ, which immediately yields the Zurek

dynamicallength scaleξ(t) = ξZ = ξ0(τQ

τ0)

14 . Unlike the original Kibble picture, the domain sizes

in this model (and hence the initial intervortex spacing) are strong functions of the quench rate.

Assuming that each domain contributes, on average, a lengthξZ of vortex line to the total tangle,

the initial defect density can be estimated asξ−2Z .

1.2.1 The Quenching Experiments

Zurek’s modification of the original Kibble proposal by itself represents a substantial contribution

to the problem of topological defect formation in the early universe, but he amplified the argument

still further by suggesting experimental tests in condensed matter systems. He made a strong case

10

Figure 1.4: Zurek’s proposed experiment using4He confined to a narrow annular region and sub-sequently quenched into the superfluid state. Different regions around the ring adopt independentorientations of theU(1) gauge field.

in particular for4He, wherein the superfluid transition has been associated with a broken global

U(1) gauge symmetry. By confining a quantity of the liquid in its normal state to an annular geom-

etry of width smaller than the coherence length, and then quickly reducing the pressure uniformly

through the lambda-line, he argued that the resulting superfluid state should,a la Kibble, consist

of a number of domains along the circumference, each with a randomU(1) phase of the order

parameter. This leads to a statistical mismatch in the phase around the loop, and hence also to

metastable superflow. The magnitude and direction of this would of course vary from run to run

of the experiment, but its detection would provide strong evidence of something like the Kibble

mechanism operating in a real physical system.

As a test of this proposal, Dodd et al. (1998) at the University of Lancaster designed a cylindri-

cal container fitted with a movable bellows, allowing for a quick axial expansion with a minimum

of fluid flow parallel to the cylinder walls. As the container was only simply connected, any in-

duced vortices had to be measured directly using second-sound attenuation. Their experiments

ultimately yielded a null result, but even if vorticity had been detected it would be difficult, in prin-

11

ciple, to exclude other, probably hydrodynamic, mechanisms for their production, such as fluid

rubbing up against the walls.

Further, in a related experiment by Carmi and Polturak (1999) on thin samples of the cuprate

compound YBCO, no vortex lines were detected followingτQ ∼5 sec quenches through the su-

perconducting transition temperature. Thus, as far as homogeneous quenching is concerned, the

experimental results do not look promising. However they do not rigorously exclude some vari-

ant of the Kibble-Zurek hypothesis, since the only quantitative output of the theory is a scaling

relationship with a prefactor that is presumed to be the zero temperature correlation length but

could in principle deviate from this, possibly by orders of magnitude. There have also been some

confusing papers [Karra and Rivers (1998)] claiming to have explained the failure of at least the

4He experiment as being due to an insufficiently rapid quench. Whatever the ultimate outcome, the

experiments performed to date have been unable to confirm the Kibble-Zurek scenario, and indeed

seem rather to signify against it.

Following the Lancaster attempts, a pair of complementary experiments were undertaken [Bauerle

et al. (1995); Ruutu et al. (1995)] to test the possibility of topological defect formation during inho-

mogeneous quenching, this time in low temperature3He. As in the Stanford experiments described

above, samples of the superfluid phases were irradiated with thermal neutrons, some of them ul-

timately producing hot spots at effective temperatures somewhat higher thanTc. The diffusion

of quasiparticles away from the hot spot induces a quench back through the critical temperature,

opening up the possibility for defect formation via the Kibble mechanism. There are a number of

reasons for pursuing this line of experimental inquiry. First, as the superfluid states of3He exhibit

more complex symmetries than that of4He, one expects their quenching properties can be related

more straightforwardly to those of the vacuum in the early universe [Volovik (2003)]. Second,

since it is now well accepted that some variant of the Big Bang scenario holds, the localized heat

deposition process initiated by neutrons is more directly applicable to cosmological questions than

any homogeneous quenching experiment. Finally, the induced quenches occur much more rapidly

than those achievable through mechanical expansion.

12

To begin with the Grenoble group [Bauerle et al. (1995)], the experiment is essentially calori-

metric in nature. The apparatus consisted of a stationary cryostat in which a specially designed

vibrating wire resonator was used to measure the energy spectrum of thermal excitations for about

60 seconds following each nuclear event. For pressures from 0 to 19.4 bars and very low tem-

peratures∼ .1Tc they found that the spectra were shifted to lower energies, amounting to a∼100

keV deficit in the 764 keV released by the nuclear reaction. About half of this was interpreted as

having been lost due to the scintillation of ultraviolet photons, a process known to occur in4He

following ionizing collisions. These pass freely through the liquid without being reabsorbed. The

remaining energy was accounted for as the result of vortex formation during the rapid cooling of

the fireball. Equating the unexplained portion of the deficit with the vortex energy per unit length

∼ 12ρs(h/m

2) log(R/a), they claim a defect density commensurate with the Zurek prediction.

By contrast the Helsinki group [Ruutu et al. (1995)] used a rotating cryostat and worked at

temperatures and pressures close toTc(P ) below the polycritical point. Moreover, they were able to

measure individual vortices using an NMR technique, and thus did not resort to indirect inference

(as by calorimetry). In equilibrium the normal fluid rotates together with the cryostat, and because

of their mutual friction, so too do any vortex loops created by the hot spot. In the rest frame of

the normal fluid there is thus a superfluid ‘counterflow’ through the vortex loops which forces the

expansion of those above a critical radiusRc, and the contraction of the rest. Those that do expand

eventually line up along the rotation axis of the cryostat, by which point they have grown large

enough to be counted individually using NMR. Since the size of the vortices which ultimately

expand is a function of the counterflow velocityv = |vs − vn|, this technique can be used to probe

the scale distributionn of the initial vortex tangle, simply by varying the angular velocity of the

cryostat.

This distribution, for scales at whichξZ D, whereD is the dimension of a vortex loop,

should be independent ofξZ . Thus the functional form forn(D)dD, the number of vortex loops per

unit volume betweenD andD + dD, should be scale-invariant. Dimensional analysis thus implies

that this should take the formdn = 2CdDD4 , where C is a dimensionless constant and the factor

13

of 2 is conventional. Computer simulations of the Kibble-Zurek scenario on a lattice [Vachaspati

and Vilenkin (1984)] reproduce this scaling with a high degree of accuracy (except for very large

length scales, which appear to make a disproportionately large contribution). Combining these

results with the critical radius for vortex expansion in the counterflowv, one sees that the number

of vortex lines generated varies with the counterflow velocityv as (for more details about this

formula, and about the physics of the Helsinki experiment in general, see Appendix B)

N (v) =πC

9[(v

vcn

)3 − 1] (1.4)

wherevcn is the lowest critical velocity, coming from the finite size of the initial bubble. The

Helsinki data appear to confirm this formula rather well.

1.2.2 Some Comments on the Interpretation of these Experiments

In this section, we express some concerns with the prevailing interpretation of these experiments,

which seems largely based on an uncritical acceptance of the Kibble-Zurek argument, or perhaps

an unwillingness to consider alternative mechanisms. Further, and this applies also to the nucle-

ation experiments described above, there is lingering controversy surrounding the baked Alaska

theory of the evolution of initially localized hot spots. This appears to be founded on the belief

[Bunkov and Timofeevskaya (1998, 1999)] that the mere fact of collisions in the hot spot somehow

implies the validity of local hydrodynamic concepts, and in particular that the deviation in effective

temperature has at all times a monotonic profile of the Gaussian form. We raise here a number of

issues, some of which will be addressed by this thesis; the remainder await future work.

First, a rather unsettling question arises from even a cursory examination of these experiments,

namely: Why has the Kibble-Zurek mechanism seemingly failed in the case of homogeneous

quenching in4He, while for the highly inhomogeneous quenches induced in3He we have (at least

in the case of the Helsinki experiments) unambiguous proof of vorticity? As previously remarked,

there has been at least one attempt [Karra and Rivers (1998)] to answer this directly, but from

14

the author’s point of view there has been no convincing or decisive argument that manages to

preserve the integrity of the Kibble-Zurek theory while simultaneously accounting for the apparent

dichotomy between the two varieties of experiment.

It is also unclear exactly what role rotation plays in the Helsinki experiments. In fact, the

presence of a rotation-induced counterflow has inspired some disagreement as to the origin of the

observed vortices. On the one hand, Kibble and Volovik (1997) argue from the single component

time-dependent Ginzburg Landau (TDGL) equation that the original Kibble mechanism, suitably

modified, can still be made to account for all vortex production. Applying a local hydrodynamic

description to the hot spot, i.e. assuming that the deviation in effective temperature follows a spa-

tial Gaussian profile, they show that the surface of constant temperature atTc collapses inward

much more quickly than the surrounding superfluid order parameter can propagate. Thus the inter-

mediate region has been effectively quenched through the transition while (at least approximately)

isolated from its immediate environment, and should manifest a domain structure in accordance

with the Kibble-Zurek scenario. In this interpretation the counterflow induced by rotation plays no

role except to expand already existing vortex loops to measurable proportions. However, Aronson

et al. (1999) contend on the basis of simulations and analytic calculations from the same TDGL

equations, that the presence of counterflow near the normal-superfluid ‘interface’ induces an insta-

bility towards the spontaneous expulsion of vortex loops from the hot spot. Although they cannot

exclude the Kibble-Zurek mechanism completely, the dominant vortex production rate in their

simulations is that coming from counterflow.

The discussion of Kibble and Volovik (1997) has also sparked vigorous debate [Bunkov and

Timofeevskaya (1998); Schiffer et al. (1999); Bunkov and Timofeevskaya (1999); Bunkov (2000)]

about the nature of inhomogeneous quenching in real3He. Bunkov and Timofeevskaya (1998) try

to expand on the original Kibble-Volovik argument by numerical simulation of the full 18 com-

ponent TDGL equation; however, they restrict their quench model to only one spatial dimension.

They nevertheless claim support for the Kibble-Volovik arguments from local hydrodynamics, the

only caveat being that the superfluid domains left behind by the contracting hot spot should be

15

roughly evenly divided between theA- andB-phases. Together with the baked Alaska scenario

discussed above, this gives us essentially two competing models of inhomogeneous quenching, as

depicted in figure 1.5.

Taken literally, the possibility of a roughly equal number of seeds of theA andB phases,

regardless of the question of hydrodynamics, leads us to conclude that the Kibble-Zurek mecha-

nism is fundamentally incompatible with the observedB-phase nucleation from the supercooled

A-phase. In the nucleation problem, it is the competition between surface and bulk energies that

determines the critical radius, and in Bunkov and Timofeevskaya (1999) it is claimed that a large

number ofA andB phase bubbles would be formed within the critical radius for either model of

quenching. If true, then it is highly improbable that a sufficiently large nucleus ofB phase could

ever form, and an independent mechanism forB-phase nucleation would have to be sought. While

we cannot, of course, strictly exclude such a possibility, it seems extremely unlikely given the fact

that neutron radiation has indeed been shown to enhance the rate ofB-phase nucleation.

Lastly, the Helsinki experiments rely on the scale distributionn(D) of vortex loops as discussed

in the previous section. The agreement of the experimental data with theory is impressive, but it

must be emphasized that this distribution is based on a dimensional argument assuming scale

invariance aboveξZ , and thus it does not seem altogether reasonable to suppose the data of the

Helsinki group really excludes any particular mechanism for vortex production. In fact, such

an argument suggests that we should expect similar data irrespective of the detailed mechanism,

provided we look at sufficiently large scales.

In terms of the Grenoble experiments, one obvious weakness of the calorimetric method is

the great difficulty of excluding all alternative energy sinks other than vortices. As pointed out

by Leggett (2002), one possible sink, known to be produced by ionizing radiation in4He, are the

so-called molecular excimers. These consist of two quasibound helium atoms, one of them in an

excited state. Some of these have lifetimes of at least 10-15 seconds, and could be responsible for

the observed energy deficit.

16

Figure 1.5: Schematic depiction of two competing models of inhomogeneous quenching. In (a) thediffusion of heat away from the hot spot leads to a contraction of the locus of constant temperatureat Tc faster than the surrounding superfluid phase can propagate inwards. In (b) the effectivetemperature profile is non-monotonic with a shell moving out ballistically from the interior atroughly the Fermi velocity. The interior is thus shielded from the outside phase.

17

Chapter 2

Quasiparticle Diffusion in a Fermi Liquid

In a Fermi liquid at temperatures sufficiently low compared to the energies of excited quasipar-

ticles, the decay of these quasiparticles occurs principally through the scattering of additional

particle-hole pairs out of the ground state [Nozieres and Pines (1990)]. To leading order each

collision produces only one extra pair, at a rate proportional toε2k, whereεk is the energy of the

incident quasiparticle as measured from the Fermi surface. A cascade of pair creation events is

thus initiated whereby one excitation multiplies itself over time into a large number (3N ) of lower

energy ones. Assuming that, on average, the energy of these excitations divides evenly down the

cascade, the mean free path increases by a factor∼ 9 at each level. On these intuitive grounds it was

postulated [Leggett (1984)] that the evolution of localized ‘hot spots’, associated with the action of

ionizing radiation on the supercooledA-phase of3He, should exhibit strongly non-hydrodynamic

behavior. In particular, it was argued that the distribution function in real space should develop a

shell structure moving at or somewhat belowvF away from the center of the hot spot. Some in-

vestigators [Bunkov and Timofeevskaya (1999); Bunkov (2000)] have questioned the plausibility

of this ‘baked Alaska hypothesis’, favoring instead a more conventional interpretation [Kibble and

Volovik (1997)] founded on local hydrodynamic concepts.

It is therefore of some interest whether hydrodynamic principles can be applied when the parti-

cle diffusion is governed by this kind of scattering. After all, in the various approaches to ordinary

diffusion one usually makes the assumption that the diffusing particles lose all memory of their

past trajectory after only a finite number of collisions, resulting in the characteristically Gaussian

18

distribution at long times. Here this is not the case, since after each collision the phase-space avail-

able for all subsequent scattering has irreversibly contracted. There is thus a ‘piling up’ and not

an ‘averaging out’ of this effect, and the distribution that emerges, even after a large number of

collisions, must reflect this.

These considerations apply when the quasiparticles in question are sufficiently dilute that colli-

sions between them may be neglected. Within the relevant3He experiments, this condition almost

certainly fails during the early-time regime immediately subsequent to the energy deposition in the

liquid. The hot spot effective temperature is initially so high that the mean-free path of the inte-

rior quasiparticles is on the order of the dimension of the hot spot itself [Leggett and Yip (1989);

Leggett (2002)]. We are led therefore to divide the problem into two stages, which shall be treated

separately: first, the early-time dynamics determined by interquasiparticle scattering, and second,

the later dilute stage dominated by the cascade effect. In the discussion of the first stage we shall

be focussed on an estimation of the crossover time to a non-monotonic distribution, in order to

assess the relevance of the baked Alaska concept to the real experiments.

Our approach to the long-time cascade problem consists in building up a single-particle Green’s

function designed to mimic the essential scattering physics in a tractable way. This is then reduced

to an effective kernel valid in the long wavelength, low frequency limit appropriate to a considera-

tion of local hydrodynamics. The basic parameter of our theory is the average multiplication factor

ζ of the mean free path at each level of the cascade. In addition we shall find another parameter,

ν, emerging in the long time limit which controls a scaling law|r − r′| ∼ (t − t′)ν relating the

root-mean-squared displacement of the particle with time. The principal results of this Chapter

are the following: for theζ > 1 case of interest,ν = 1, and there is no transition to diffusive

long-time behavior, this being inhibited by the phase-space memory. Further, the effective Green’s

function possesses a shell structure moving away from the origin at a renormalized Fermi velocity

vF . These properties are, of course, exactly those required by the baked Alaska hypothesis. Fi-

nally, it will be seen that despite the creation of additional particle-hole pairs during the cascade,

the associated quasiparticle density is in fact always decreasing. This ensures the self-consistency

19

of the assumption that pair creation indeed dominates its long-time evolution.

We model the early-time regime as much as possible within the constraints of local hydrody-

namics, in which the effects of collisions are systematically overrepresented. The inadequacy of

this description will be demonstrated by the divergence of a particular hydrodynamic parameter.

This enables us to set an upper limit on the crossover time to non-monotonic behavior which is

somewhat less thanRc

vF, whereRc is the critical radius above which aB-phase bubble expands

spontaneously in the presence of the bulkA phase.

2.1 Previous Work

In principle, a quasiclassical description of anomalous quasiparticle diffusion should be sought in

solutions of the full Fermi liquid Boltzmann equation,

∂np(r, t)

∂t+∇rnp(r, t) · ∇pεp(r, t)−∇pnp(r, t) · ∇rεp(r, t) = I[np] (2.1)

for initial conditionsn0(p, r) determined, presumably, by solving the separate (and enormously

complicated) problem of energy deposition byδ-electrons discussed in Chapter 1. Heren is the

Fermi distribution function, andε are quasiparticle energies.I[n] is the two-body collision integral

for particles in a degenerate Fermi liquid, which takes the rather unwieldy form (neglecting for

now the effects of spin) [Baym and Pethick (1991)]:

I[np1 ] =1

V 2

∫ ∫ ∫d3p2d

3p3d3p4W (1, 2; 3, 4)δ(p1 + p2 − p3 − p4)δ(ε1 + ε2 − ε3 − ε4)

× [n3n4(1− n1)(1− n2)− n1n2(1− n3)(1− n4)]. (2.2)

HereW is a transition rate. The important thing to note about this integral is the phase space

factor; the overall scattering rate depends very strongly on the distribution of quasiparticles, and in

particular on the proximity of the given particle to the Fermi surface.

20

Palmeri (1990, 1989) was the first to make a serious attempt at a solution of the baked Alaska

problem by means of the Boltzmann equation; he was, however, confronted with enormous practi-

cal difficulties which forced him into compromises on each of a number of fronts. First, ignoring

Fermi-liquid ‘molecular fields’, and expandingnp(r, t) for ‘small’ deviations from equilibrium,

the Boltzmann equation simplifies considerably:

(∂

∂t+ vF · ∇)δnp(r, t) = I[δnp(r, t)] (2.3)

wherevF is the Fermi velocity. Palmeri was able to make progress analytically by substituting

a new collision integral forI[δn], in which each particle, instead of colliding with others of like

kind, encounters a quasicontinous background of fixed scatterers whose only effect is an inter-

mittent randomization of its direction of motion. This concession allows for a complete solution

of the associated Boltzmann equation by means of integral transform techniques. Together with

results from extensive numerical work, this led him to conclude that a system described by such

an equation will manifest transient shell structures in all cases, except those for which the size of

the initial hot spot is comparable to or exceeds that of the mean free path for scattering against the

background.

While interesting in their own right, these results would seem to imply little of direct import

to the corresponding situation in low temperature3He. In particular, they ignore the effects of

interquasiparticle scattering, which must surely dominate the hot spot dynamics for some brief

interval before it has expanded to the point that the associated scattering rate becomes negligible.

Further, because the scattering background is fixed the only allowed collisions are elastic, while for

highly energetic quasiparticles at sufficiently low density in a Fermi liquid like3He, one expects

instead a cascade of ‘downscattering transitions’, accompanied by particle-hole pair creation at

each step. Inspection of the collision integral reveals that the resulting distribution function will

be strongly influenced by the redistribution of energy among the daughter particles, which, by

lowering the average energy per particle, leads to a rapid increase in their mean free paths. In

21

his own defense, Palmeri notes that in most transport phenomena studied to date, the qualitative

features appear to depend only on the fact of collisions, and not their microscopic properties.

Taken at face value, this philosophy implies that any shell structures that do arise as solutions to

the Boltzmann equation must be transients, which in the course of time inevitably succumb to

damping. Those pieces that do survive into the asymptotic regime are thus the ‘hydrodynamic’

solutions, and can show no signature traces of the initial distribution or its early dynamics.

If this philosophy is correct, then the presence of collisions leads invariably to universal long-

time behavior, and the study of baked Alaskas reduces to the study of ‘intermediate time’ solutions

of the Boltzmann equation. Existing methods are notoriously ill-suited to this regime, which un-

like the hydrodynamic limit suffers from the absence of any simplifying assumptions. Under such

circumstances, one has no recourse but to brute force computation. This of course is a rather unsat-

isfying conclusion, for even if successful, such methods are unlikely to yield qualitative insights.

By considering a simple toy model, we shall argue in the next section that the flaw consists not in

the methods, but in the philosophy behind them; namely, that the microscopic nature of collisions

can indeed have drastic effects on the macroscopic properties of the resulting diffusion process. We

ultimately conclude that baked Alaskas are not ‘transient’ effects (in the usual sense of exponential

damping), but persistent structures from the very earliest times all the way into the long-time limit.

2.2 A Random Walk on the Fermi Surface

The full problem is sufficiently complicated that it will be useful to consider first a simplified

model for orientation, before embarking on the more ambitious synthesis attempted in section 2.3.

For this purpose, consider the following picture: a single particle executes ann-step random walk

in three-dimensional space; at each step it receives an isotropically distributed random kick in

a new direction with a new step-length equal toζ times the previous one. The step-length thus

evolves as a geometrical progression in powers ofζ, i.e. l → ζl → ζ2l, etc., withl the initial

step size. By neglecting fluctuations inn, ζ, andl, we hope to recover features bearing at least

22

a passing resemblance to the ‘correct’ ones. This visualization of the physical problem reduces

immediately to the mathematical problem of summingn randomly oriented vectors with lengths

l, ζl, ζ2l, · · · , ζnl.

The introduction ofζ is intended to mimic the peculiarε−2-dependence of the quasiparticle

lifetime, and is essential to the question of whether local hydrodynamic concepts can be taken to

apply to quasiparticle diffusion close to theT = 0 Fermi surface. Further, some extra insight might

be gained upon variation ofζ; in particular, we can compare then → ∞ limits of the three cases

ζ < 1, ζ = 1, andζ > 1.

It is clear from the above description that the distribution inr, the distance from the starting

point aftern steps, is given by a simple function of the form

Φn(r) =

∫· · ·

∫d3r0 · · · d3rnφ0(r0) · · ·φn(rn)δ(r−

∑k

rk) (2.4)

where, for isotropic scattering, theφ’s can be written as a product of an angular and a radial part:

φk(rk) = 14πr2

kρ( rk

ζkl). Since by assumption there are no fluctuations about the mean free path, the

radial part is justρ = δ(rk − ζkl).

Fourier transforming (2.4) we obtain

Φn(q) =n∏

k=0

∫d3rkφk(rk)e

−iq·rk

=n∏

k=0

∫ ∞

0

drkδ(rk − ζkl)

∫dΩk

4πe−iqrk cos θk

=n∏

k=0

1

qlζksin qlζk. (2.5)

Now, for n = 0, corresponding to the very earliest times, we invert the Fourier transform to

obtain

Φ0(r) =1

(2π)3

∫d3qe−iq·r sin ql

23

=1

(2π)2

∫ ∞

0

dqq2 1

iqr[eiqr − e−iqr]

1

2iql[eiql − e−iql]

=1

4(2π)2lr

∫ ∞

−∞dq[eiq(r−l) + e−iq(r−l) − eiq(r+l) − e−iq(r+l)]

=1

4πlr(δ(r − l)− δ(r + l)). (2.6)

Naturallyr, l > 0 so that the second term may be neglected; however it is important to note that

these ‘shadow’ solutions are a generic feature of the method and we shall encounter them again

later. ForvF t < l the above clearly translates to14πvF tr

δ(r − vF t) , which is non-monotonic and

ζ-independent, since no collisions have yet taken place. This is exactly the kind of distribution

we would expect at the earliest times. We must now determine whether similar results hold in the

opposite limit.

Going back to (2.5), let us choosen = N 1. Further, we make the provisional assumption

thatq−1 is much larger than the largest length scale in the problem, i.e.qlζk 1 for all k. We will

later relax this assumption in theζ > 1 case. Expanding the sine, we obtain

ΦN(q) =N∏

k=0

(1− 1

6(qlζk)2) → exp[−1

6(ql)2

∑k

ζ2k]. (2.7)

Fourier-inverting this, and completing the square inq gives

ΦN(r) = N exp[− r2

23l2

∑Nk=0 ζ

2k] (2.8)

whereN is a normalization factor. We identify 3 qualitatively distinct behaviors at long times and

large distances, corresponding toζ < 1, ζ = 1 andζ > 1. In all three cases the time afterN steps

is justt = lvF

∑k ζ

k.

Forζ < 1,∑

k ζ2k → (1−ζ2)−1, and the distribution tends to a time-independent function ofr

after a time l(1−ζ)vF

, while the average velocity of the particle goes to zero. It is basically confined

24

to a region of dimension(1− ζ)−1l. The distribution is

ΦN,ζ<1(r) = N exp[− r2

23(1− ζ2)−1l2

]. (2.9)

For ζ = 1, vF

lt =

∑k ζ

k =∑

k ζ2k = N , and the distribution becomes

ΦN,ζ=1(r, t) = N exp[− r2

23(vF l)t

] (2.10)

which is of the usual form for diffusion, where the root-mean-squared displacement from the origin

scales with time asr ∼√t.

For ζ > 1, which is the case of interest,∑

k ζ2k ∼ ζ2N for N large, andt ∼ l

vF ζN . The

distribution inr becomes

ΦN,ζ>1(r, t) = N exp[− r2

23(vF t)2

] (2.11)

which reveals the scaling lawr ∼ t. It would seem that quasiparticle diffusion in aT = 0 Fermi

liquid is somewhat unusual in the sense of maintaining ‘ballistic’ scaling even at the very longest

times. One difficulty with this result, however, is that the distribution remains peaked atr = 0.

This feature cannot really be trusted, since by assumption we have coarse-grained overall length

scales. In the first two cases this is justified by the existence of an absolute upper limit to the

intrinsic length scales of the random walk. Forζ > 1 however, there is no such limit.

To get a more accurate reckoning of the long-wavelength behavior, consider the following: in

the Fourier inversion of (2.5), we treatr as a parameter that we can vary freely froml to ζN l. For

smallr the largest response comes from that sine factor which oscillates the slowest; however there

areN − 1 other such factors, and these oscillate far more rapidly, thus tending to average out this

response. Asr increases the number of quickly oscillating factors decreases, up untilr ∼ ζN l,

at which point only the last sine oscillates appreciably (especially in the large-ζ limit), the other

factors being roughly constant. This suggests that we retain the fastest oscillating factor in the

low-q expansion of equation (2.5), in order that the spatial structure of the coarse-grained (2.4)

25

reflect the influence of at least the longest intrinsic length-scale. This yields

ΦN(r) =

∫d3qΦN(q)e−iq·r

=1

(2π)2lrζ

∫ ∞

0

dq exp[−1

6q2l2

N−1∑k=0

ζ2k] sin(qr) sin(qlζN)

or

ΦN(r, t) =

√6

16π3/2vF t(exp[− (r − vF t)

2

2/3ζ−2(vF t)2]− exp[− (r + vF t)

2

2/3ζ−2(vF t)2]) (2.12)

where we have again made use of the constant velocity conditionlζN

vF∼ t. This distribution has the

two key baked Alaska properties of non-monotonicity and ballistic scaling. Further, upon compar-

ison of this expression with that forn = 0 (2.6), it is clear that in theN → ∞ limit the physical

plus ‘shadow’δ-functions have fattened into Gaussians. However the rate of broadening proves

insufficient to erase the signature non-monotonicity. We shall see this effect from a somewhat

different point of view later.

These results are highly suggestive but as noted above there are flaws in the model. First, we

have not accounted for fluctuations inN , the number of scattering events, having simply assumed

it to scale with time. Second, we have neglected completely any fluctuations in the lengths of

the vectors and the parametersζ at each step, which in reality are quite substantial. Finally, we

have neglected the growth in the number of quasiparticles that necessarily accompanies inelastic

scattering atT = 0. There is no particular reason,a priori, to expect that the qualitative features

discussed here will survive the incorporation of such effects into the model; nevertheless, they do

provide a useful map of the possible physical behaviors of interest.

2.3 The Model and Associated Formalism

In the above description of the problem it is clear that a large number of quasiparticles are produced

by the cascade effect. It would only obscure the essential physics if we tried from the beginning

26

Figure 2.1: Diagram of the cascade process. Dashed line corresponds to the single ‘particle’ weare tracing.

to keep track of all of them. Instead, we study the motion of a fictitious particle corresponding to

a single line traced down through the cascade, as depicted in 2.1. The problem is thereby reduced

to that of a single particle propagating in a continuous field of scatterers according to an infinite

sequence of scattering ratesΓk. The phase-space memory is incorporated by the parametrization

Γk = ζkΓ ≡ ζ−kΓ, whereΓ is the starting rate at the ‘top’ of the cascade. Naturallyζ itself is

subject to fluctuation, since the energy cannot really be expected to distribute itself evenly down

the cascade; we defer consideration of such fluctuations until later.

A mathematical description of this process may be written formally as a weighted sum over the

various trajectories available to the particle,

K(r− r′, t− t′) =∑N

KN(r− r′, t− t′)

=∑N

∫· · ·

∫ N∏k

d4rkWNδ(r− r′ −N∑k

rk)δ(t− t′ −N∑k

τk) (2.13)

27

whereWN = WN [rk, τk] is the ‘weight’ associated with a given configuration of collision

coordinatesrk, τk. Physically this represents the space-time probability density that a particle

starting fromr′ propagates tor in a time t − t′, having undergone an indeterminate number of

collisions in between. The (classical) trajectories intermediate to collisions are broken up into

intervalsτk = tk − tk−1 which fluctuate according to the scattering statistics. For convenience we

assume that the particle starts at the origin att = 0.

The structure of theN -point weight functionWN follows from simple physical arguments.

First, collisions randomize the initial velocity at each step; for the sake of argument we assume

isotropic scattering, affording us the greatest possible randomization at each collision and thereby

isolating the physical effects of phase-space memory from those due to any anisotropies in the true

scattering cross-section. An additional important virtue of this assumption is a decoupling of the

averages over ‘in’ and ‘out’ momenta at each collision.

In general, the classical kernel describing propagation fromrk−1 to rk in a timeτk for a given

initial velocity and a fixed collision rateΓk is a simple Green’s functionG(rk − rk−1, τk) of the

form

G(rk − rk−1, τk) = δ(rk − rk−1 −∫ τk

dτ ′kv(τ ′k)) exp[−Γkτk],

wherev(τ ′k) is given by the particle’s equations of motion. In the absence of external potentials,

G is translationally invariant, so thatrk − rk−1 → rk, and the quasiparticle trajectories reduce

to straight-line paths. To obtain the appropriate probability density we average over all possible

velocities subject to the constraint|vk| ∼ vF coming from degenerate Fermi statistics:

K0(rk, τk) =1

4πrkvF τkδ(rk − vF τk) exp[−Γkτk]. (2.14)

It should be noted that this propagator corresponds to the lowest order in perturbation theory, and

is therefore the general solution at very early times. Thus the true propagator exhibits shell-like

behavior in this limit regardless of the precise value ofζ.

The weight function is a probability densityvis-a-vis the measure∏

k d4rk ≡

∏k d

3rkdτk,

28

which, in addition to the statistical independence of individual scattering events implies thatWN is

just a product of propagatorsK0 connectingN collision coordinates, multiplied by the probability

per unit time for a collision at each point, i.e.:

WN = K0(r0, τ0)Γ0K0(r1, τ1) · · ·ΓN−1K0(rN , τN). (2.15)

Upon transformation to(q, ω), our formal solution (2.3) factorizes term by term into products

of basic kernels of the formK0(q, ω) ≡ 1qvF

tan−1( qvF

Γ−iω), allowing us to reorganize our expression

for the true kernelK as an expansion in powers ofζΓ:

K(q, ω) = K0(q, ω)

+ K0(q, ω)ζΓK0(ζq, ζω)

+ K0(q, ω)ζΓK0(ζq, ζω)ζΓK0(ζ2q, ζ2ω)

+ · · · (2.16)

We have used a scaling property ofK0, namely

ζ−k

qvF

tan−1(qvF

Γk − iω) = K0(ζ

kq, ζkω)

Inspection of the series (2.16) reveals thatK(q, ω) inherits similar scaling properties:

KΓ(ζNq, ζNω) = ζ−NKΓN(q, ω) (2.17)

where the subscripts (usually suppressed) indicate scattering rates at the top of the associated cas-

cade. This equation reflects a self-similarity property implicit to our model.

The series (2.16) may be written more compactly as a Dyson-like scattering equation

K(q, ω) =1

qvF

tan−1(qvF

Γ− iω)[1 + ζΓK(ζq, ζω)]. (2.18)

29

from which the series itself follows by iteration.

The expansion (2.16) is not perturbative in the usual sense that successive terms represent

ever smaller corrections to the lowest order one; hence the ‘convergence’ of the series is not an

altogether trivial question. However the important issue is not the convergence of the series as such,

but whether or not it conserves probability, that is, whether the kernelK is normalized. There is

then only a finite weight to be distributed among the various terms of the series, and as we shall see

in later sections this becomes concentrated at long times about only a very few of them. Summing

the series is thus a question of identifying the terms of greatest weight, and then representing them

in real space.

Let us now consider the overall normalization of (2.3). It may be demonstrated by taking

q → 0 (i.e. integrating over all space) in equation (2.18) and noting that, in order for the series

to be normalized,K(0, ω) cannot depend on the parameterζ, which must therefore drop out of

the equation forK(0, ω) . This can only occur ifK(0, ζω) = ζ−1K(0, ω); substitution of this

condition yields the self-consistent solutionK(0, ω) = iω

.

2.3.1 Ordinary Diffusion

In order to develop the physical principles necessary to the problem at hand, we consider first

a more familiar problem but from a new perspective. In particular we explore the behavior of

the kernelK in the ζ = 1 case describing continuous diffusion at constant velocity and with

constant scattering rate (the discrete case corresponds to the usual random walk problem). This

has of course been studied extensively in various contexts by a number of investigators; we make

specific reference only to the work of Palmeri [Palmeri (1990, 1989)] who considered the baked

Alaska problem from the perspective of a ‘Lorentz gas’ system consisting of an aggregate of non-

interacting fermions propagating in a background of fixed, isotropic scatterers. A simplification

arises from the fixed character of the scatterers, which renders all collisions elastic (hence theζ = 1

condition within the present formalism). He arrived at his solution by methods rather different from

ours, preferring direct integration of a Boltzmann equation to the sum-over-trajectories approach

30

developed here.

Settingζ = 1, the Dyson-like equation reduces immediately to an algebraic equation inK with

the solution

K(q, ω) =

1qvF

tan−1( qvF

Γ−iω)

1− ΓqvF

tan−1( qvF

Γ−iω)

which is in exact correspondence with Palmeri’s kernel. It should be noted that Palmeri was ap-

parently the first to derive this analytic solution. An effective kernel valid in the limit of long

wavelengths and low frequencies follows from this exact solution; as shown by Palmeri this is

nothing more than the familiar Gaussian 4π(2πDt)3/2 exp[− r2

6Dt], indicating the applicability of local

hydrodynamic concepts on long length scales for scattering of this type. HereD = 13v2

F Γ−1 is the

diffusion constant.

The extraction of this effective kernel by Palmeri is rather laborious and of course requires that

one have an exact solution to begin with. We now discuss an alternative method based on direct

consideration of the series (2.16), which is in general much easier to implement, and perhaps of

some interest even beyond the context of the present discussion.

From a physical standpoint, the derivation of the effective kernel ought not to require the sum-

mation of each and every term in perturbation theory. Our method is to delay summation of pertur-

bation theory until we are well into the asymptotic regimeΓt 1, at which point the structure of

the series simplifies enormously. In fact, at long times only a very few terms in the series should

be relevant: terms that deviate significantly from the expected number of collisions at a given

time will be exponentially small. To prove this, let us consider first only theN -th termKN(r, t)

(N 1) of the series for long wavelengths and low frequencies. In this limitKN factorizes into

separate space and time dependent partsKN → ΩN(t)QN(r) with

ΩN(t) = ΓN−1

∫dω

e−iωt

(Γ− iω)N

and

QN(r) =

∫d3qe−iq·r(1− 1

3v2

F Γ−2q2)N

31

→∫d3qe−iq·r−N

3Γ−2(qvF )2 .

It should be noted thatΩN(t) =∫dωe−iωt limq→0 KN(q, ω), so that the integrand is actually valid

for all ω and the limits of integration may be taken to be(−∞,∞). The same does not hold true

for the integrand in the second factor, which is strictly valid only forqvF , ω Γ. However the

swift convergence of the Gaussian integral renders the introduction of a cutoff superfluous.

The first factor may be evaluated by the method of contours in the standard way by enclosing

theN -th order pole atω = −iΓ; this gives usΩN(t) = (Γt)N−1

(N−1)!e−Γt, which will be recognized as

the Poisson distribution. We note that this could have been anticipated on purely physical grounds

by considering first the distribution in the number of collisions with respect to time.

This makes evident an important property of the long-time limit which essentially fixes a re-

lation betweenN and t and gives us a means of performing the required sum overN . ForN

large enough, the Poisson formula reduces via Stirling’s approximation toΩN(t) ∼ exp[(N −

1)(1 + log Γt) − (N − 1) log(N − 1) − Γt]. The argument is stationary under variations in the

number of collisions forN = Γt, which is the mean of the Poisson distribution. Fluctuations

δN = N − Γt N are therefore distributed according toexp[− (δN)2

Γt]; hence in the sum overN

only relatively few terms make a contribution. The fractional width falls off as(Γt)−12 , andK(r, t)

may be treated as a weighted sum over the very narrow distributionΩN . Thus, to a very good ap-

proximation, we may replace this sum by thatQN(r) corresponding to its mean value. This yields

for the effective kernel an expression of the form

K(r, t) ∼∫d3qe−iq·r− 1

3v2

F Γ−1tq2

=4π

(2πDt)3/2exp[− r2

6Dt] (2.19)

which is of course the same result as that obtained from the exact solution.

32

2.3.2 Anomalous Diffusion

We proceed conceptually along lines parallel to those of the preceding section. We will deviate

in certain mathematical particulars, however, in order to accommodate the scattering statistics

peculiar to this case. For example, the analogue ofΩN(t) is here somewhat complicated and less

obviously useful for finding that term in the series which is most probable at long times. Instead

we locate theN closest to the point where the ratioR = ΩN+1(t)

ΩN (t)crosses unity for a given interval,

t.

Keeping only the slowest decaying terms,R becomes

R ∼ exp[−(ζ−N−1 − ζ−N)Γt−N log ζ]. (2.20)

Setting this equal to1 is the same as finding the stationary point of an argument of the form

−ζ−NΓt−12N2 log ζ, which is precisely what we should have expected on the grounds that, roughly

speaking, the most probableN results from a competition between factors like∏N

k ζ−k ∼ ζ−

12N2

and the ever-slower decay of the diffusing particle. This yields, to leading order inN , the scaling

relationN ∼ log Γtlog ζ

. This is very close to the mean value given byΓt ∼ 1−ζN+1

1−ζ, the difference

arising from the skewness in the distribution of small fluctuationsPN(δN), which goes as

PN(δN) ∼

ζ−|δN | for δN > 0,

exp[−ζ |δN |] for δN < 0.

Hence the fractional width of the distribution at long times is exceedingly narrow, and we

may restrict our attention to the single termKN ; however, becauseζ > 1 the structure of such

a term is somewhat different from what we encountered in the case of ordinary diffusion in that

a large number of length and time scales appear in the various factors. Most of these may be

safely expanded for small(q, ω) without affecting the long-wavelength, long-time properties of

the resulting integral, but the factor representing variations on the largest such scales will be kept

33

intact. This is to ensure that we do not ‘wash away’ important details on these scales by looking

too coarsely at the distribution. This gives the expression

KN(r, t) ∼ Γ

(2π)4

∫ ∞

0

∫ ∞

−∞

∫ 1

−1

d3qdωdu

2

ie−iq·r−iω(t−Γ−1β1)− 13(qvF )2Γ−2β2

ω + qvFu+ iζ−NΓ(2.21)

where the parametersβk = Γ−1 1−ζkN

1−ζk have been introduced for compactness, and the last arctan-

gent factor has been rewritten as an integral over the parameteru. Because we have kept the last

factor there is no need to impose cutoffs, and the integrations are therefore benign:

K(r, t) ∼ ηe−ΓN (t−β1)

rvF (t− β1)β122

(exp[−(r − vF (t− β1))2

43v2

F Γ−1β2

]− exp[−(r + vF (t− β1))2

43v2

F Γ−1β2

]) (2.22)

Hereη is a dimensionless numerical constant fixed by normalization. The decay factor may be ig-

nored, being in fact irrelevant since the sharpness of the distribution inδN forces the identification

of N with its mean value as given by the relationt ∼ Γ−1 1−ζN+1

1−ζ. This further implies that the

parametersβ1 → (1− ζ−1)t andβ2 → Γζ2

ζ−1ζ+1

t2 for N, t large. These substitutions produce finally

the effective kernelK(r, t):

η

r(vF t)2(exp[− (r − vF t)

2

43

1ζ2−1

(vF t)2]− exp[− (r + vF t)

2

43

1ζ2−1

(vF t)2]). (2.23)

HerevF = (1− ζ−1)vF is a renormalized Fermi velocity. This propagator is independent ofΓ, the

scattering rate at the top of the cascade. Thus at long times the initial distribution of quasiparticle

energies becomes irrelevant to the spatial profile, which acquires the universal form (2.23). Some

technical points associated with the form of (2.23) are addressed in Appendix A.

Up till now we have neglected the influence of the many extra quasiparticles generated by

particle-hole pair scattering at each level of the cascade. This can be accounted for by multiplying

the kernel (2.23) by a growth factor∼ 3N(t), whereN(t) is the mean number of scattering events

occurring before timet. The important point is thatN scales only logarithmically with time, so

that3N(t) ∼ t1+α, with α < 1. Thus the quasiparticle density is everywhere a strictly decreasing

34

function of time. This validates the assumption of a ‘crossover point’ in the expansion of the initial

hot spot beyond which particle-hole pair creation indeed becomes the dominant scattering mode,

and thereby establishes thea priori self-consistency of the whole multiple scattering scenario.

2.3.3 Energy Fluctuations

Let us examine again the Dyson equation (2.18), upon which the baked Alaska (ν = 1) propagator

is based. We have assumed throughout that the daughter particles at each level of the cascade divide

the energy inherited from their ‘parents’ equally among themselves. This of course is untrue; the

most natural generalization of the single-particle equation (2.18) corresponding to such fluctuations

takes the form

K(q, ω) =1

qvF

tan−1(qvF

Γ− iω)[1 +

∫dζF (ζ)ζΓK(ζq, ζω)]. (2.24)

whereF is the appropriate distribution inζ. Such an equation has, unfortunately, proved quite

intractable, at least in the hands of the present author. It is perhaps more fruitful to consider whether

these fluctuations introduce any new qualitative features, or change those already established for

the baked Alaska propagator. We shall here argue that they do not.

First we must consider the statistics associated with the distribution of energy down the cascade.

For energetic (i.e.ε kT ) quasiparticles the energy dependence of the Fermi functions appearing

in the scattering rate is extremely weak, its variations being confined to a width∼ kT about the

Fermi surface. Therefore every configuration, in energy space, of the final-state quasiparticles

subsequent to a collision event are equally probable, and the probability that one of the final-state

particles of energyε scatters with a fractionf = εεi

of the energyεi of the incident quasiparticle

is determined entirely by the number of configurations available to the remaining two (labelled

f ′, f ′′), subject to the constraint of energy conservation. This gives

P (f) = 2

∫ 1

0

∫ 1

0

df ′df ′′δ(1− f − f ′ − f ′′) = 2(1− f) (2.25)

35

the factor of 2 arising from the possible interchange of the quasiparticles. This is a triangular

distribution weighted towardf = 0. The mean〈f〉 = 13

and the root-mean-squared deviation

〈(f − 13)2〉 1

2 = 16, indicating that, as far asζ is concerned, fluctuations from step to step in the

diffusion process will tend to favor larger rather than smallerζs.

In terms of the behavior of the baked Alaska kernel, its qualitative features should be insensitive

to these fluctuations, which can be seen by the following argument. Imagine that we cut off the

distribution inf at a pointfc < 1 somewhere close to1: the probability, for a given cascade, that a

large number offs fall inside this region is vanishingly small in the largeN limit. Now choose an

f ≡ f ′ such thatfc < f ′ < 1, and hence aζ ′ = 1f ′2

> 1. Becauseζ ′ > 1, replacingζ by ζ ′ in the

effective kernel will not change it qualitatively. Thus we do not expect fluctuations inζ to change

our conclusions regarding the existence of baked Alaska structures.

2.3.4 Interpretation

By the nature of the scaling laws emerging in the long-time limit we see that the most natural

definition of the exponentν characterizing these laws assumes the form

ν = limN→∞

log(∑N

k ζ2k)

12

log∑N

k ζk

(2.26)

which is a nonanalytic function ofζ with the following values:

ν =

0 for 0 < ζ < 1,

12

for ζ = 1,

1 for ζ > 1

(2.27)

These values correspond to three possible forms for the effective kernel emerging from solutions

of (2.18) under coarse-graining.

First, theν = 0 propagator, which is not derived here, is basically aδ-function situated atr;

36

Figure 2.2: Schematic depiction ofν as a function ofζ for increasingN . In the limitN →∞, thecurve develops a singularity atζ = 1.

the geometric growth of the scattering rate in this case prevents the diffusing particle from trav-

elling very far from its starting point, leading in the limit to a time-independent, highly localized

distribution.

The kernel forν = 12

is the usual type of propagation one expects when collisions are uncorre-

lated. Although we know from earlier sections that at early times this kernel exhibits a shell-like

structure, this is apparently completely washed out by collisions; no artifact of this early, ‘orga-

nized’ behavior remains at long times.

The last and most relevant propagator corresponds to the valueν = 1. We must first note that

the overall shape and scaling of this kernel are completely consistent with the baked Alaska idea.

Second, this kernel possesses the following rather remarkable property: we are able, merely by

varying parameters appearing in the kernel itself, to recover the kernel valid at very early times

0 < t Γ−1. In particular, taking the parameterζ → ∞ in equation (2.23) reproduces this early

time behavior by pushing the intervals between collisions to an arbitrarily large value. The same

can be achieved by treatingN as a continuous parameter in equation (2.22) and taking the limit

37

N → 0. This reversibility is not evident in either of the other two propagators, from which nothing

of the corresponding early-time behavior may be inferred. Further, in contrast to theν = 0, 12

cases where theζ-parameter drops out completely, here it remains, buried like a fossil in the long-

time distribution; even at long times the propagator retains a memory of the originalδ-function

pulse moving out from the origin (which is precisely the lowest order in perturbation theory).

Physically this effect can be traced back to the simple fact that in the course of propagation near

the Fermi surface, the mean number of collision events experienced by the particle grows only

logarithmically with time.

The appearance of three qualitatively different effective kernels under variations of the parame-

terζ suggests a strong affinity with the concept of phase transitions, the analogue of the thermody-

namic limit being〈N〉, t→∞. The Dyson equation (2.18) develops singularities which manifest

as distinct qualitative behaviors of the macroscopic diffusion process. Under this interpretation,

we are tempted to generalize our conclusions regarding these limiting behaviors. Consider the

following: at very early times the propagator exhibits the baked Alaska property regardless ofζ,

and continues to do so at long times only forζ > 1. Theabsenceof any ‘phase transition’ for

ζ > 1 means that the true solution at intermediate times must interpolate smoothly between the

rigorous early and long time limits. Indeed, as shown above they remain connected by variation of

a parameter. We can thus ascribe the baked Alaska property to theζ > 1 propagators atall times.

2.4 Connection With the Real Problem in3He

If particle-hole creation were in fact the only scattering mode relevant to the real problem ofB-

phase nucleation in3He, then the whole baked Alaska scenario would follow quite naturally from

the discussion in Section 2.3.2. However, for some period of time immediately following the

energy deposition process which produces the hot spot, its dynamics will be characterized instead

by strong inter-quasiparticle scattering.

This stage is clearly unstable to the physics of Section 2.3.2 in that ultimately the density of

38

the hot spot must decrease to the point where collisions among the constituent quasiparticles may

be neglected at all later times. This is because the scattering mode in question conserves both the

total number and energy of the associated quasiparticles, and must spread out as a function of time.

However, it is conceivable that the presence of this inter-quasiparticle scattering can delay the

crossover to non-monotonic behavior for a timeτcross which is long on the scale ofRc

vF. This would

mean that, although baked Alaskas might themselves correspond to real phenomena, they would

not be relevant to theB-phase nucleation problem.

In what follows we work as much as possible within the framework of local hydrodynamics, in

order to demonstrate that even within this picture (1) diffusive propagation is simply an inadequate

description, (2) if we violate this picture in a particular physically reasonable way we immedi-

ately obtain baked Alaska-like distributions, and (3) the time scale for the appearance of these

distributions is set, as it should be, byRc

vF.

2.4.1 Breakdown of the Diffusion Picture

If we are to take the concept of local hydrodynamics seriously, it ought to be possible to de-

fine an effective temperatureT (r, t) and energy densityE(r, t) which are governed by reasonable

differential equations. In the case of a highly degenerate Fermi liquid these two quantities are

not independent: the energy densityE ∼ (number of excited quasiparticles)×(mean energy of

excited quasiparticles)∝ T 2. We begin by defining [Vollhardt and Wolfle (1990)] the energy

currentjε(r, t) = −κT∇T (r, t) whereκT ≡ κT−1 is the thermal conductivity and the constant

κ = 13CN(T )TvF l(T ), withCN(T ) (∝ T ) the normal state heat capacity per unit volume andl(T )

(∝ T−2) the mean free path. We have explicitly neglected any convective currents in the liquid; i.e.,

we have assumed the dynamics is purely diffusive. Because of the relationship betweenE andT we

may rewrite the energy current entirely in terms ofE itself, so thatjε(r, t) = κ2E(r, t)−1∇E(r, t).

In the next step we impose the constraint of local energy conservation, or

∂tE(r, t) +∇ · jε(r, t) = 0.

39

This gives us the equation

∂tE(r, t) =κ

2∇2 log E(r, t) (2.28)

The highly singular nature of this equation (the current blows up wherever the energy density

vanishes) makes it very difficult to characterize its solutions; in order to give the local hydro-

dynamics picture the benefit of the doubt we ‘soften’ the singularity by removing the explicit

position-dependence of the diffusion coefficient, replacing it instead with a self-consistent value

corresponding to the maximum energy density at that time (i.e. a ‘local density approximation’).

In this way we systematicallyoverestimatethe influence of collisions on the energy density by

assuming a diffusion coefficient which is spatially uniform at all times. We rewrite the above

diffusion equation in the somewhat more conventional form

(∂t −D(t)∇2)E(r, t) = 0 (2.29)

whereD(t) ≡ κ2E(rmax(t),t)

. In three dimensions we can write down the spherically symmetric so-

lutions formally asE(r, t) = E(π1/2R(t))3

exp[−( rR(t)

)2] for a Gaussian source atr = 0 of dimension

R0 and total energyE. HereR(t)2 ≡∫ tdt′D(t′), and the form of the solution indicates that the

radius of maximum densityrmax = 0 at all times. This implies the self-consistency relation

∂tR2(t) =

π3/2κ

2ER(t)3, (2.30)

which in turn yields a width functionR(t) = R0(1 − π3/2κR0

4Et)−1. This diverges after a time

τdiv = 4π−3/2 EκR0

, which we can rewrite in somewhat more familiar terms by noting that, according

to the hydrodynamic definitions we have adopted,12CN(T0)T0R

30 ∼ E, whereT0 is the effective

temperature of the initial hot spot. Takingl(T0) ≡ l0, this gives an estimateτdiv ∼ (R0

l0)R0

vF.

40

2.4.2 Estimation ofτcross

We can interpret the breakdown of local hydrodynamics in this instance by looking again at the

results of Section 2.3.2. There the effective propagator (2.23) consists of a hybrid of quasi-diffusive

behavior in the relative coordinater − vF t with a rapid ‘drift’ of the peak of the distribution

away from the origin at a velocityvF . Physically, the diverging mean-free-path, combined with

expansion from a localized source, leads to strong radial correlations in the quasiparticle velocities,

which manifests in the propagator as a drift effect. This is absent in the usual local hydrodynamics

picture, in which the average velocity for a diffusing particle is zero for any sufficiently coarse-

grained subvolume of the system.

It is likely that in the present case the divergence ofR(t), which expresses a clear tendency for

the distribution to spread very rapidly away from the origin, results from our neglect of a corre-

sponding drift effect in equation (2.29). We have, after all, the same basic ingredients: a diverging

mean-free-path combined with radial expansion. Under this interpretation, the neglect of such a

drift term so constrains the possible solutions (by forcing them into a monotonic ‘straitjacket’), that

the rapid spreading tendency can manifest itself only in the unphysical divergence of a parameter

in the solution, in this caseR(t).

Consider now the fuller version of Equation (2.29) with a drift term added:

(∂t + vd(t) · ∇ −D(t)∇2)E(r, t) = 0. (2.31)

Although we have not provided any physical constraints on the detailed time-dependence ofvd(t),

we may nevertheless write down formally the spherically symmetric solutions of the above equa-

tion. These areηE

rrd(t)R(t)(exp[−(r − rd(t))

2

R(t)2]− exp[−(r + rd(t))

2

R(t)2)]) (2.32)

whererd(t) =∫ tdt′vd(t

′), andη is a dimensionless numerical constant. This bears a striking

resemblance to the effective kernel (2.23), though in this case the form of the solution is supposed

41

to be valid for all timest ≥ 0. Expanding this solution for smallr

E(r, t) =4ηE

R(t)3exp[−(

rd(t)

R(t))2]1 + (

2

3· rd(t)

2

R(t)4− 1

R(t)2)r2 +O(r4)

reveals two qualitative regimes for the evolution ofE(r, t) depending on the sign of the coefficient

of r2; i.e. monotonic at early times, then crossing over to non-monotonic behavior whenrd(t) ≥√32R(t).

The self-consistency relation during this monotonic phase is very simple:

∂tR2(t) =

π3/2κ

2ER(t)3 exp[(

rd(t)

R(t))2].

This is quite similar to the corresponding relation with drift neglected, except for the exponential

factor which varies between1 ande3/2. ThusR(t) will diverge also in this case at a timeτ ′div <

τdiv. The divergence is avoided, however, if the crossover to the non-monotonic solution occurs

sometime before this. Based on the analogy to the situation discussed in Section 2.3.2, there are

strong physical reasons for supposing this to be true. In this sense the precise time dependence

of vd(t) is irrelevant so long as it compensates the unphysical divergence that occurs when it is

strictly zero. This means in particular thatτcross < τ ′div, or τcross < τdiv. We therefore takeτdiv

as an absolute upper limit on the crossover time to a shell-like energy density, for anyphysical

definition of the drift velocityvd(t).

Based on the estimates ofR0

l0appearing in Leggett (2002), together with the fact thatRc ∼ 10R0

or greater, we see thatτcross ∼ Rc

vFat most. Of course, this estimate is based on calculations which

strongly overemphasize the influence of collisions at each step, and place no restrictions on the

magnitude ofvd(t) (which, strictly speaking, should be≤ vF ), so that we may reasonably expect

τcross to be somewhat less than this, probably significantly less.

42

Chapter 3

A Question of Quenching

3.1 Introduction

It is widely believed that the vacuum immediately following the Big Bang embarked from a state

of high symmetry through a series of symmetry-breaking phase transitions during the subsequent

expansion and cooling of the universe (for a review, see [Kibble (1996)]). For sufficiently rapid

expansion, the spatial extent of any order parameters emerging at such transitions would have been

limited by the causal horizon. On this basis, it was proposed that the early universe spontaneously

acquired a domain structure characterized by independently directed order parameters in each do-

main. The frustrated dynamics resulting from such a structure may have left behind measurable

traces in the form of topological defects.

If we imagine, with Kibble (1976), the simplest case corresponding to the breaking of a global

U(1) gauge symmetry, it is then clear, provided we treat the fields classically, that these domains

adopt uncorrelatedU(1) orientations. At junctions between three or more such domains it will

sometimes occur that theU(1) phase of the order parameter winds by2π about a filamentary region

corresponding to the symmetry-unbroken state. In light of the topological constraint of quantized

circulation in the new phase, this circumstance may be viewed as the ‘trapping’ of a vortex core by

the frustrated dynamics. According to this picture, the initial domain structure resolves itself very

quickly into a tangle of vortex loops moving in the background of the new phase.

As suggested by Zurek (1985), one can test this idea by looking for topological defects follow-

43

ing controlled quenches in condensed matter systems such as4He, which exhibits exactly the kind

of U(1) symmetry-breaking invoked by Kibble. Although subsequent tests [Dodd et al. (1998)] in

this system failed to show any vortex formation associated with uniform pressure quenches through

the lambda-line, the basic idea continues to motivate new experiments [Bauerle et al. (1995); Ruutu

et al. (1995); Carmi and Polturak (1999)].

So far only one testable, quantitative output of the Kibble hypothesis has emerged, namely the

expected defect spacingd after the quench. Zurek estimates [Zurek (1985)] that this should scale as

d ∼ ξ0(τQ

τ0)1/4, whereξ0 andτ0 are the equilibrium correlation length and relaxation time, respec-

tively, andτ−1Q is a constant quench rate. The argument rests on the generic phenomenon known

as ‘critical slowing-down’ near second order phase transitions, which for finite-time quenches will

leave the system with a ‘frozen’ value of the order parameter correlation length as it crosses the

critical line.

In this connection, we are interested principally in the interpretation of a certain group of

experiments [Ruutu et al. (1995); Bauerle et al. (1995)] on low temperature3He, in which samples

of the superfluidA- andB- phases are bombarded with neutrons. These trigger the production of

localized ‘hot spots’ with effective temperatures102− 103Tc. The detailed dynamical evolution of

these hot spots is a matter of some debate [Kibble and Volovik (1997); Leggett (1984); Leggett and

Yip (1989); Warner and Leggett (2003)], however it is generally agreed that they cool (presumably

by quasiparticle diffusion) quite rapidly on the scale of the quasiparticle scattering rate. This leads

to quench processes very like those envisioned by Kibble, and it has been argued [Kibble and

Volovik (1997)] that the induced vorticity, unambiguously observed in Ruutu et al. (1995), and

inferred on calorimetric grounds in Bauerle et al. (1995), is directly associated with the Kibble-

Zurek mechanism.

There are, however, a number of fundamental difficulties with this interpretation. The validity

of the time-dependent Ginzburg-Landau (TDGL) equation, upon which the Zurek estimates are

based, requires that the quasiparticle inelastic scattering rate (greatly) exceed the gap frequency

∆/~, a condition which in3He holds only over a rather narrow strip of width∼ 10−5kBTc about

44

the critical curve. As such, the dynamics in this region consists in the motion of an order parameter

which is strongly overdamped by frequent quasiparticle collisions. The Grenoble experiments

[Bauerle et al. (1995)] are performed at temperatures much lower thanTc, and the corresponding

quenches pass far outside this region (i.e. the reduced temperature at freezeout is much greater

than 10−5); one may therefore expect the order parameter to obey a collisionless analogue of

these equations in which the long-wavelength components, damped out nearTc, make a substantial

contribution. Furthermore, the TDGL equations fail explicitly to account for the conservation of

energy, which is likely to impose rather strong constraints on the dynamics regardless of proximity

to the transition.

In the following we attempt to build a coherent physical picture of these quench phenomena

by transplanting the relevant physics to the more tractable degenerate Fermi gas. In Section 3.2

we study the thermodynamics of a finite time quench from the normal to the superfluid state of

such a Fermi gas, with special attention given to the energy available for defect formation in this

system. We find that in order for the Kibble-Zurek scaling relation to hold, the conditions of the

quench must be such that the interior of the quenched region remains thermally isolated from its

environment during the entire process, and further that the system must cool all the way back to the

ambient temperature. Both of these assumptions are questionable in the case of neutron-irradiated

3He. Section 3.3 derives the zero-temperature dynamics that ensue from the sudden variation of the

interaction parameterλ from zero to some finite (attractive) value. For a uniform quench, quantum

fluctuations in the off-diagonal field avalanche into large oscillations of the order parameter, which

subsequently relax into a self-driven steady state. This state, while not the BCS groundstate, is

nevertheless characterized by a finite gap∆∞ = εC(e2/N(0)λ − 1)−1/2, with εC the usual BCS

cutoff parameter; in weak coupling this is half the BCS gap. The gap is smaller than that of BCS

because of residual collective motion of the pairs, which prevents full condensation.

45

3.2 Thermodynamics of a Shallow Quench

The two most prominent3He quench scenarios proposed thus far [Leggett and Yip (1989); Kibble

and Volovik (1997)], while differing in their approach to the question of energy transport away

from a hot spot, have in common the notion that the region left behind must evolve in effective

isolation from its immediate environment. Such a feature is a requirement of the condition that

the choice of order parameter inside the cooling hot spot be made independently of that in the

surrounding liquid. The spontaneous generation of vortices, if indeed it occurs, must therefore

draw its energy from within the quenched region itself. In this section we identify the source of

this energy and assess its consequences for the Kibble-Zurek scenario by a consideration of the

relevant thermodynamic functions.

In the standard, and experimentally usually most relevant, analysis [Lifshitz and Pitaevskii

(1998)] of Fermi gas-superfluid transitions, thermal contact with a reservoir is tacitly assumed.

Thus if the temperature is made to drop very slowly from the normal state through the transition,

the superfluid expels any condensation energy spontaneously. We will here take a different route,

first by treating the gas as thermally isolated, and second by tuning instead of temperature, the

attractive interaction parameter itself.

At first sight this might seem a poor model for the quenching of a3He hot spot. However, upon

closer inspection there is a close analogy between these two apparently dissimilar methods of

quenching to the superfluid state. This consists in a kind of ‘duality’ between the matrix elements

that are tuned, and the phase space available for the scattering of Cooper pairs which causes the

instability. Within a given hot spot, the distribution of excited quasiparticles acts to block this

phase space, which opens up very rapidly with cooling back to the ambient temperature. Thus

we surmise that the dynamical situation would be little changed if instead the matrix elements

themselves were varied suddenly at a given temperature. At any rate, excepting the speed with

which the quench is performed, one would not expect the physics to depend much on the detailed

manner in which the critical line is approached.

46

Figure 3.1: Adiabatic vs. finite-time quenches. a) For adiabatic passage through the transitioncurve, the slope of the isentrope jumps discontinuously from zero at critical coupling. Thus thesuperfluid emerges at a steadily increasing temperature. b) For a finite-time passage, the systemfalls out of equilibrium upon crossing the first thick dashed line; this delays the slope discontinuityuntil the system intersects the dashed line on the other side.

We proceed by considering a degenerate Fermi gas subject to the BCS reduced Hamiltonian:

H =∑k,α

(~2k2

2m− µ)c†k,αck,α − λ

∑k,k′

c†k↑c†k↓ck′↓ck′↑. (3.1)

We have kept only those terms associated with Cooper pair scattering, and neglect scattering away

from or into the pair subspace. Thus the system can never approach a true equilibrium; however,

at ultralow temperatures far from the transition the relaxation timescale is so long that this is a

reasonable first approximation. As discussed above, we shall assume that the couplingλ can be

tuned, as by a Feshbach resonance [Tiesinga et al. (1993)]; this opens up the very real possibility

for experimental investigation of quench phenomena along the lines of the present discussion.

Let us first explore the situation for slow variation of the interaction parameter. By taking

the entropyS = S(T, λ) and tracing the adiabat through the transition line (i.e. following the

curve for whichδS = ∂S∂T|λδT + ∂S

∂λ|T δλ = 0), we can determineδT as a function ofδλ. For

this purpose consider a close-up on a portion the phase diagram in theT − λ plane (Fig. 3.2),

whereTc(λ) = .31EF e− 2π2~2

mkF λ [Gor’kov and Melik-Barkhudarov (1961)] separates the normal and

superfluid phases. For the normal Fermi gas in equilibrium, the entropyS ∝ T for the low

temperatures of interest; thus for quasistatic variation ofλ the normal state remains at constant

temperature. Upon crossing the critical line, however, the resulting superfluid must emerge with a

slightly elevated temperatureT + δT to accommodate the energy of condensation. The possibility

arises that this energy will heat the nascent superfluid back into the normal state. However, this

is not a problem so long as the temperature increaseδT for a small variationδλ is smaller than

∂Tc

∂λδλ, which indeed proves to be the case.

47

Beginning with the combinatorial expression for the entropy of a Fermi system

S = −2kB

∑k

[fk ln fk + (1− fk) ln(1− fk)]

and changing the sums to integrals, we obtain, after some algebraic manipulation and integration

by parts, the following expression forS in the superconducting state:

S = 2k2BN(0)T

∫ ∞

−∞dε[(β∆)2 f

E− 2β ln(1− f)] (3.2)

whereN(0) is the Fermi surface density of states,∆ is the gap andf = 1eβE+1

withE =√ε2 + ∆2.

Following the mathematical treatment of superconducting thermodynamics developed by Muhlschlegel

(1959), we define the function

a(x) ≡ − 2

π

∫ ∞

−∞du ln(1 + e−π

√u2+x) + x(ln γ

√x− 1

2)− 1

3(3.3)

and its derivative

a′(x) =

∫ ∞

−∞dεf

E+ ln γ

√x (3.4)

where the dimensionless variablesu ≡ επβ and x ≡ ∆2

π2 β2, and γ ≈ 0.57 . . . is the Euler-

Mascheroni constant. Thus defined,a(x) anda′(x) are regular functions from which the loga-

rithmic singularities have been explicitly subtracted. A number of properties of these functions

are tabulated in Muhlschlegel (1959); for our purposes we need only thata(0) = a′(0) = 0 and

a′′(0) = 78ζ(3).

The entropy, as expressed in terms of these functions, becomes

S = 2π2k2BN(0)T [1 + 3(xa′(x)− a(x))− 3

2x] (3.5)

For a small incrementδλ beyond the critical curve, the proportionδTδTC

by which the superfluid must

heat itself to compensate the energy of condensation is simplyχ = limλ→λ+c[(− ∂S

∂Tc|T )/( ∂S

∂T|λ)].

48

The relevant derivatives are, from (3.5),

∂S

∂Tc

= 2π2k2BN(0)T [xa′′(x)− 1

2]∂x

∂Tc

and∂S

∂T=S

T+ 2π2k2

BN(0)T [xa′′(x)− 1

2]∂x

∂T

In theT → Tc limit, ∆2 ≈ 8π2k2B

7ζ(3)T 2

c (1− TTc

), so that

∂x

∂Tc

→ 8

7ζ(3)k2BTc

∂x

∂T→ − 8

7ζ(3)k2BTc

and we obtain

χ =1

1 + 7ζ(3)12

≈ .588 (3.6)

Thus, upon traversing the critical line, the superfluid must choose a ‘compromise temperature’

T ∗ = T + χ∂T∂λδλ intermediate between the new value ofTc and the temperatureT of the normal

liquid from which it started.

Now let us imagine (Fig. 3.2) a slight generalization of the preceding argument in whichλ is

made to vary at a constant finite rateτ−1Q through the transition, and define the associated ‘quench

parameter’ or ‘reduced coupling’ε = δλλc

= tτQ

, whereδλ = λc − λ is the deviation from critical

coupling. If, following Zurek, we allow that nearTc the correlation length and velocity assume the

scaling formsξ = ξ0ε−ν andu = u0ε

1−ν , and thus thatτ = τ0ε−1, then the system ‘freezes out’ at

a time t = τ(t) =√τ0τQ before emerging finally at the same temperature a distanceδλ(t) from

λc on the other side, which state presumably contains a number of trapped vortex loops.

From this picture it is clear that the energy required to create these loops must derive from

the free energy difference between the superfluid states atT andT ∗. This energy is evidently

an overestimate predicated on the total freeze-out of thermodynamic variables at−t. In reality we

49

might expectT to increase somewhat during this process; however the condition of total freeze-out

does allow us at least to put a rigorous upper bound on the defect density.

The relevant changes in the free energy density nearTc can be calculated from the Hellmann-

Feynman theorem for the thermodynamic potentialΩ

∂Ω

∂λ|T,V,µ = 〈∂H

∂λ〉

= −∑kk′

ukvkuk′vk′(1− 2fk)(1− 2fk′)

whereuk, vk are the usual BCS coefficients arising from the average. Since the gap∆ = λ∑

k ukvk(1−

2fk), we have∂Ω∂λ

= −∆2/λ2, or

Ωs − Ωn = −∫ λ

0

dλ′∆2

λ′2(3.7)

which is valid for any thermodynamic potential [Lifshitz and Pitaevskii (1998)]. For convenience

we choose the Helmholtz free energy at constant temperature, again using the near-TC expression

for ∆:∆Fs−n

V∼=

2π2

7ζ(3)N(0)k2

B(TC(λC + δλ)− T )2. (3.8)

If we start fromT in the normal state, then clearlyTC(λC + δλ) = T + ∂TC

∂λδλ.

The energy available to vortex production is thus∆E = ∆Fs−n(T ∗)−∆Fs−n(T ), or

∆E

V= αN(0)(kBT )2| ln kBT

.31EF

|4ε2 (3.9)

whereλC has been expressed as a function of the starting temperatureT , andα = χ(2 − χ) 2π2

7ζ(3)

is a numerical prefactor∼ 1.9. An estimate of the initial defect density requires a comparison of

(3.9) with the energy density of a particular vortex distribution, to which there are two principal

contributions: that associated with the suppression of∆ in the core, and the kinetic energy of su-

perflow about it. Only the latter can be expected to contribute substantially for distances somewhat

50

larger than the core dimension. For a vortex loop of radiusd this is εL ∼ 2π2ρs~2

m2d ln da, with a

the dimension of the core. The energy density of a vortex line per volumed3 is justεL/d3, which

gives a lower bound on the Zurek length scaled of the form

d ≥ ~m

(2π2ρS(T )

αN(0))1/2

√ln d/a

kBT | ln kBT.31EF

|2(τQτ0

)1/2 (3.10)

The superfluid density near the transition grows linearly asρS(T ) = 2ρ(1 − T/TC(λC + δλ)),

or, expressed in terms of more appropriate variables,ρS(ε, T ) = 2ρ ε

ε+| ln(kBT

.31EF|−1

, whereε

| ln( kBT.31EF

)|−1. The combination~m

( ρN(0)

)1/2 ∼ ~vF

kBT; hence up to logarithmic factors and dimen-

sionless constants of order 1 we have

d ≥ ~vF

kBT(τQτ0

)1/4. (3.11)

For quenches in3He near the transition curve this is of course the coherence lengthξ0. This implies

that for such quenches the Kibble-Zurek scaling law gives a lower bound to the defect spacing; it

requires that the quenched region does not expel any energy to its surroundings and that the tem-

perature of the liquid just after the quench drops all the way back to its original temperature. Both

of these assumptions are dubious, and their violation tends strongly to decrease the free energy

available to vortices. Further, for quenches induced in the superfluid at much lower temperatures,

the prefactor will be somewhat larger thanξ0, tending once again to suppress vortex production.

We are led to conclude that the Kibble mechanism depends rather precariously on the availabil-

ity of free energy in the quenched region, requiring perfect isolation and cooling, and reasonable

proximity to the critical line.

3.3 Quenching and Order Parameter Dynamics

In this section we develop a theory for the order parameter dynamics following a ‘deep’, uniform

quench through the superfluid transition in a Fermi gas. It is hoped that such considerations can

51

shed light on quenching phenomena induced in3He in the context of so-called cosmological ex-

periments. These involve the rapid decrease in effective temperature of a small region of the super-

fluid that has been heated back into the normal state by neutron irradiation and subsequently cools

through the diffusion (anomalous or otherwise [Warner and Leggett (2003); Kibble and Volovik

(1997)]) of quasiparticles. By ‘deep’ we mean that the variation in effective temperature away from

Tc spans a range of values far wider than that embraced by the usual Ginzburg-Landau theory.

From the above description it is clear that the neutron-induced quenches in3He occur under

highly inhomogeneous conditions; however we are here concerned with the spontaneous emer-

gence of a length scale which in this context has been predicted to be much smaller than the

dimension of the quench itself [Kibble and Volovik (1997)]. Within the time-dependent Ginzburg-

Landau picture, upon which the prevailing theory of these quench phenomena is founded, col-

lisions between thermal quasiparticles maintain local equilibrium and are therefore the dominant

timescale. The motion of the order parameter is overdamped, killing contributions from the longest

length scales; by contrast, in the deep quench scenario~∆

is the largest timescale in the problem,

and the long-wavelength fluctuations which ‘freeze out’ in the Kibble-Zurek picture may here play

a vital role. Therefore in what follows we focus exclusively on these long-wavelength dynamics. In

particular we show that the Fermi gas is unstable to even the smallest long-wavelength fluctuations,

and that the ensuing dynamics leads to a steady-state with a finite value of the gap.

3.3.1 The Pseudospin Representation, and Initial Conditions

Visualization of the order parameter dynamics following the sudden turn-on of a pairing potential

is perhaps the chief difficulty of our problem. This can be mitigated to a large extent by making

use of an analogy, first noticed by Anderson (1958), between a BCS system on the pair subspace

and a one-dimensional lattice of spins. That is, for each distinctk we can identify operators as

σz(k) ≡ 1 − nk − n−k, σ+(k) ≡ c†kc†−k, andσ−(k) ≡ c−kck, whereσ±(k) = 1

2(σx(k) ± iσy(k)),

and theσi are the2× 2 Pauli matrices. It is easily verified that these obey theSU(2) commutator

algebra[σi(k), σj(k′)] = iδk,k′εijlσl(k). Expressed in terms of pseudospin degrees of freedom, the

52

Figure 3.2: Schematic illustration of two possible mean-field configurations of the pseudospinsystem. The solid line indicates thex axis. a) Immediately following the quench, the pseudospinsform a domain wall at the Fermi surface, which is energetically costly in the presence of the anoff-diagonal field. This points in the positivex direction. b) The BCS solution, which rotatesindividual pseudospins into their effective fields, thereby lowering the energy. The domain wallhas been ‘softened’ as a consequence. This state is not dynamically accessible to the quenchedstate in a), since it has a lower energy and the semiclassical equations conserve this.

BCS reduced Hamiltonian (3.1) assumes the form

H = −∑

k

εkσz(k)− λ∑k 6=k′

σ−(k)σ+(k′). (3.12)

The normal state|N〉 corresponds in this description to a domain wall centered at the Fermi energy,

with occupied states below the Fermi surface pointing along the negative-z direction. This is

clearly degenerate with respect to global rotations about thez-axis; forλ > 0 the BCS groundstate

breaks this degeneracy and lowers the total energy by rotating the spins into the new effective field,

which interpolates smoothly between the up and down orientations at±εC . This is depicted in Fig.

3.2.

We begin by writing the appropriate operator equations of motion, which are gotten by the

53

usual Bloch relationsi~σi(k) = [σi(k), H]. These yield

σz(k) =−λ~(

∑k′ 6=k

σx(k′))σy(k) + σy(k)(

∑k′ 6=k

σx(k′))

−(∑k′ 6=k

σy(k′))σx(k)− σx(k)(

∑k′ 6=k

σy(k′))

σx(k) =εk~σy(k)−

λ

~σz(k)(

∑k′ 6=k

σy(k′)) + (

∑k′ 6=k

σy(k′))σz(k)

σy(k) =−εk~σx(k) +

λ

~σz(k)(

∑k′ 6=k

σx(k′)) + (

∑k′ 6=k

σx(k′))σz(k) (3.13)

It remains to consider the rather delicate question of initial conditions. A complete quantum-

mechanical description requires taking the expectation value of the above operator equations with

respect to|ψ(t = 0+)〉, the total wavefunction immediately following the variation ofλ from

zero to some finite value att = 0. In the sudden approximation this is of course just the normal

state, implying that the off-diagonal effective field∝ 〈∑

k σx(k)〉 vanishes, and hence that the

pseudospins persist indefinitely in their original configuration. Since we know the normal state to

be unstable forλ > 0, this cannot possibly be correct.

The difficulty stems from the global gauge symmetry of the normal state, and can be eliminated

if we take as our initial wavefunction, instead of the normal state itself, its projection onto the

Hilbert space appropriate to a particular direction of symmetry breaking. In spirit this is not unlike

the problem of a single particle in the groundstate of an external potential which has been suddenly

varied; the first step towards the solution of its subsequent dynamics is the projection of this initial

state onto a basis appropriate to the new potential. In much the same way we generate the dynamics

for a ‘branch’ of the many-body wavefunction along a given direction of symmetry-breakingφ. As

we shall see, this makes sense when the number of pairsN participating in the wavefunction is

large, so that each directionφ represents a quasi-distinct Hilbert space, and the various branches

therefore evolve independently. Note in particular that negative eigenvalues of the off-diagonal

field operator along theφ direction belong properly to theφ + π sector, so that only positive

expectation values contribute toφ.

54

To illustrate the general idea let us consider the full manifold of states, which can be generated

by operating on the normal state ‘domain wall’ with appropriately defined spin-12

rotation opera-

tors. Let us fix thex, y axes and denote the various possible symmetry-breaking directions byφ,

the deviation fromx. In terms of these conventions our states become

|Ψ(θk, φk)〉 =∏

k

exp(− i2φkσz(k)) exp(− i

2θkσy(k))|N〉 (3.14)

Expanding the exponentials inσy, and then applying thez-rotation operators yields

=∏

k

(cosθk

2− eiφk sin

θk

2(iσy(k)))|N〉 (3.15)

where we have used the fact that the wavefunction is defined only up to an overall phase. This

coincides with the usual BCS wavefunction whenφk = φ for all k andθk = tan−1 ∆εk

, provided

∆ is chosen self-consistently to satisfy∆ = λ〈Ψ|(∑

k σx(k))|Ψ〉, whereσx(k) is thex-operator

rotated byφ.

To a good approximation we may restrict ourselves to states with particle-hole symmetry, so

thatθ(εk) = −θ(−εk) andφ(εk) = −φ(−εk) for all k. The direction of symmetry breaking for a

state of the form (3.15) obeying this symmetry is thusφ = 12N

∑k φk. The inner product between

any such state|Ψ(φ)〉 and a rotated version of itself|Ψ(φ′)〉, is easily calculated as〈Ψ(φ)|Ψ(φ′)〉 =

〈Ψ(φ)| exp(− i2δφ

∑k σz(k))|Ψ(φ)〉 = cos2N( δφ

2), which for largeN is practically zero. This

is because the two states in question can be connected only by the simultaneous rotation of a

macroscopic number of spins. The statement that states of differentφ belong to distinct Hilbert

spaces should therefore be interpreted in this sense.

For concreteness let us consider theφ = 0 direction, and single out the state|Ψx〉 = |Ψ(θk =

sgn(εk)π2, φk = 0)〉. A complete basis on the pair subspace forφ = 0 can be constructed by acting

on |Ψx〉with operators of the formexp( i2πσy(k

′)), making sure to rotate symmetric partners across

the Fermi surface to preserve particle-hole symmetry. Let us denote such a p-h symmetric rotation

55

by πk; the projection of the normal state onto theφ = 0 Hilbert space may therefore be expanded

as

Pφ=0|N〉 = (α2N +∑|k|<kF

αk2N−2πk + · · · )|Ψx〉

where, in each sum, we must be careful not to repeat indices. In this basis, all the coefficients equal

1/2N . The subscripts label the associated eigenvalues of the off-diagonal operator∑

k σx(k),

which are2m for m = 1, 3, . . . N whenN is odd, andm = 0, 2, . . . N for N even. Thus the

probability for a nonzero, positive eigenvalue follows a distribution of the form

P (m) =1

2N−1

N !

(N/2 +m/2)!(N/2−m/2)!,

yielding a small but finite expectation value for the off-diagonal field∼ λN− 12 along this direction.

States of the form (3.15) do not possess the full gauge symmetry of the Hamiltonian, and thus

do not conserve particle number. As mentioned above, our starting state, and hence the many-

body state that evolves from it, must preserve this symmetry. It can in fact be restored at any time

by taking the superposition12π

∫ 2π

0dφe−

i2φ

∑k σz(k)|ψS(θk, φk)〉, whereφ = 1

2N

∑k φk, which is

nothing but the usual prescription [Anderson (1958)] for ‘projecting out’ theN particle subspace

from a BCS wavefunction.

It is perhaps appropriate at this point to summarize the above discussion by appealing to a

simpler, more intuitive picture of the quantum fluctuations. This picture gives some insight into

the nature of these fluctuations without incurring the above mathematical complications. For a

quantum-mechanical spin-12

object in an eigenstate ofσz, there is a component in thex− y plane

associated with zero-point precession about thez-axis. Thus for a large group of such spins,

the collective zero-point motion consists of a superposition of all possible configurations of these

‘extra’ x − y components, the bulk of which largely cancel out. There will however be a small

minority which make an enormous contribution, so that there is a non-zero expectation value along

any given unit vector at the origin of thex − y plane sufficient to drive the system away from the

normal state in that direction.

56

3.3.2 The Semiclassical Equations of Motion

Having discussed the nature of quantum fluctuations in the immediate aftermath of the quench, let

us now consider the evolution of each ‘branch’ of the total wavefunction, arising from equations of

motion for the associated expectation values of the pseudospin operators. Such equations constitute

a semiclassical description of the pseudospins, expressing the self-consistent precession of each

operator’s ‘axis of quantization’.

All of the wavefunctions of interest consist of products of factors, one for eachk; this has the

advantage that expectation values of operator products break into products of expectation values,

provided the operators themselves are functions of distinctk. When the number of pairsN is large,

we may identify terms as

limN→∞

λ∑k 6=k′

〈σx(k)〉 = λ∑

k

〈σx(k, t)〉 = ∆,

∑k 6=k′〈σy(k)〉 → 0, where again the direction of symmetry-breaking points by convention along

thex-axis. In all subsequent equations we denote expectation values of the pseudospin operators

by 12〈σi〉 ≡ si. The semiclassical equations of motion are thus (dropping factors of~),

d

dtsx(k, t) = 2εksy(k, t)

d

dtsy(k, t) = 2∆sz(k, t)− 2εksx(k, t)

d

dtsz(k, t) = −2∆sy(k, t) (3.16)

and the assumption of particle-hole symmetry takes the form

sx(−εk, t) = sx(εk, t)

sy(−εk, t) = −sy(εk, t)

sz(−εk, t) = −sz(εk, t)

57

As a consequence, the direction of symmetry-breakingφ is a constant of the motion, in harmony

with our earlier quantum mechanical reasoning. Hence the dynamics may be interpreted in terms of

the evolution of each branch of the total wavefunction along independent semiclassical trajectories.

We note in passing that the equations of motion (3.16) may also be derived in terms of the

interaction of each pseudospin with a local effective field, writtenddts(k, t) = s(k, t) × Hk(t)

where the effective field takes the form

Hk(t) = 2εkz + 2∆x.

Since we are interested here only in the gap dynamics, it proves convenient to collapse the three

equations of motion into one forsx alone. Taking an extra time-derivative of thex-component, and

substituting into this the equation for they-component gives

d2

dt2sx(k, t) + 4ε2ksx(k, t) = 4εk∆sz(k, t). (3.17)

To eliminatesz from this equation, we must combine thex- andz-components of (3.16) and find

a way to expresssz in terms ofsx. Combining, we have

d

dtεksz(k, t) + ∆

d

dtsx(k, t) = 0. (3.18)

Integrating with respect to time (and doing thex-part by parts) we obtain:

sz(k, t) = sz(k, 0) +1

εk

∫ t

dt′sx(k, t′)∆− ∆

εksx(k, t). (3.19)

Noting thatsz(k, 0) = 12sgn(εk), we obtain the desired equation upon substitution into (3.17):

(d2

dt2+ 4(ε2k + ∆2))sx(k, t) = 4∆(

1

2|εk|+

∫ t

dt′sx(k, t′)∆). (3.20)

The left hand side is of course the harmonic oscillator equation, but with a time-dependent fre-

58

quency. The right hand side consists of a driving term proportional to∆ representing feedback

from the aggregate of pseudospins in the mean-field. Equation (3.20), together with the self con-

sistency relation

∆ = λ∑

k

sx(k, t), (3.21)

gives us the complete order-parameter dynamics at the semiclassical level.

There are two conserved quantities of note; in terms of the effective field, the total energy on

the pair subspace becomes

E = −2∑

k

εksz(k, t) + λ∑k′ 6=k

sx(k, t)sx(k′, t)

= −2∑

k

εksz(k, t) +1

2∆sx(k, t)

This means in particular that

d

dtE = −2

∑k

εkd

dtsz +

1

2∆d

dtsx +

1

2sxd

dt∆

Applying the equations of motion we find

d

dtE = −

∑k

sx(k, t)d

dt∆−∆

d

dtsx(k, t) = 0.

Hence the semiclassical trajectories following from (3.20), (3.21) are energy conserving. The

equations of motion also conserve mean particle number, which may be seen by summing overk

in the last equation of (3.16).

3.3.3 Early- and Long-Time Behavior of Solutions

Consider again a particular branch of the Fermi gas wavefunction. The normal ground state cor-

responds, on the semiclassical level, to a domain wall with all spins pointing along their local

effective fields. The system would persist in this state were it not for the presence of the small

59

quantum fluctuation att = 0, which drives the precession of each pseudospin at its natural fre-

quencyεk about a now slightly perturbed effective field. It is not difficult to see, on essentially

geometric grounds, that such precession will tend to reinforce the initial fluctuation, leading to the

growth of the off-diagonal field.

This can be demonstrated explicitly by solving the linearized version of Equation (3.20) valid

at very early times:

(d2

dt2+ 4ε2k)sx(k, t) = 2|εk|(∆QF + ∆(t))

The key point here is that∆QF , the initial quantum fluctuation in the gap, is determined at a

level independent of the semiclassical description; as such it is not subject to the self-consistency

condition and therefore constitutes an ‘external’ input to the semiclassical equations.∆(t) is the

self-consistent contribution defined by∆(t) = λ∑

k sx(k, t), and thus must satisfy the initial

conditions∆(0) = 0, ∆(0) = 0. Absent any singular terms in these linearized equations, their so-

lutions are regular functions of time and may be expanded aboutt = 0 assx(k, t) =∑∞

n=0 γn(k)tn;

hence∆(t) = λ∑

n=0(∑

k γn(k))tn. The initial conditions implyγ0(k), γ1(k) = 0. Substitution

into the linearized equations reveals a hierarchy of relations forn > 2 of the form

γn(k) =2|εk|

n(n− 1)

∑k′

γn−2(k′)− 4ε2k

n(n− 1)γn−2(k) (3.22)

and

γ2(k) = |εk|∆QF (3.23)

obtained by matching powers oft. These determine allγn(k) uniquely, and automatically satisfy

the self-consistency constraint. The first thing to note about these relations is that all oddn coeffi-

cients vanish sinceγ1 = 0, and second, that the resulting gap varies as∆(t) = λ∆QFN(0)ε2Ct2 +

O(t4). This clearly demonstrates the instability of the pseudospin system to even the smallest

fluctuations.

We can expect this growth to carry the system very quickly into the semiclassical regime where

60

∆QF

∆ 1. However, we must bear in mind that within this level of description,∆ is built up

from a large number of individual pseudospin oscillations spanning a quasicontinuous range of

frequencies. This suggests that at long times, even in the absence of dissipation, interference

between these various contributions will lead to a non-equilibrium steady-state characterized by a

static gap∆∞ left over from the early dynamics.

To establish the existence of such steady-state solutions, we must go back to the basic equations

of motion (3.16), and ask ourselves whether, by replacing∆ everywhere in these equations with

a static value∆∞ > 0, it is possible to find self-consistent solutions in thet → ∞ limit. Making

this substitution, and carrying through the steps (3.17)-(3.19) as before, we arrive at an equation of

motion forsx(k, t) of the form

(d2

dt2+ 4(ε2k + ∆2

∞))sx(k, t) = 2∆∞|εk| (3.24)

with general solutions

sx(k, t) =12|εk|∆∞

ε2k + ∆2∞

(1− sin(2√ε2k + ∆2

∞t+ δk)). (3.25)

The δk are arbitrary phase factors; since we are interested only in timest ∆−1∞ , ε−1

C , they are

irrelevant to the actual value of the gap, and are henceforth dropped.

Substitution of these solutions into the self-consistency equation (3.21) leads, when expressed

in terms of appropriate energy integrals, to the gap equation

1

N(0)λ=

1

2ln(1 + (εC/∆∞)2) + Si(2∆∞t)− Si(2

√ε2C + ∆2

∞t) (3.26)

where Si(x) is the sine integral. These are known to have the asymptotic property that Si(x) → π2

uniformly asx → ∞; thus a self-consistent solution emerges precisely in thet → ∞ limit of

61

interest for a value of the static gap

∆∞ = εC(e2

N(0)λ − 1)−1/2 (3.27)

In weak coupling this isεCe− 1

N(0)λ , or half the equilibrium result. Physically this solution represents

a collective state of motion, in which the pseudospins each precess on fixed cones about their static

local fields. The energy locked up in this motion cannot be expelled in the absence of collisions,

and the steady-state gap is thus somewhat less than the value it would have in equilibrium under

otherwise identical conditions.

The structure of the gap equation (3.26) suggests certain trends in the dynamical behavior,

which it may be useful to enumerate: first, that for weak coupling the gap oscillations are dom-

inated by the cutoffεC , and for strong coupling by the gap parameter itself; second, that the ap-

proach to the steady state is faster for deeper (i.e. largerλ) quenches; and third, that the ultimate

steady state to which the system tends does not depend on the size of the initial fluctuation. All

of these trends, in addition to the actual values of the steady-state gap, have been checked by

numerical integration.

3.3.4 Numerical Integration of the Semiclassical Equations

The preceding sections address both initial conditions and the physical state at asymptotically

long times, but not the intermediate behavior. In the absence of exact analytical expressions we

must resort to numerical integration of the semiclassical equations of motion (3.20), (3.21). These

have been performed over a wide range of interaction parameters and initial fluctuation sizes.

A typical output is shown in Fig. 3.3 corresponding to a valueN(0)λ = 5. To start things

off, we introduce a tiny off-diagonal field and evolve away from the normal state for a single

timestep; all subsequent evolution occurs self-consistently using the off-diagonal field produced

by the pseudospins themselves in the previous timestep.

The basic shape of the curve shown in in Fig. 3.3 appears to be generic; there are a large

62

Figure 3.3: Gap dynamics for1000 pairs following a weak fluctuation (∼ 10−7) for a couplingparameterN(0)λ = 5. Curve appears black because of the high frequency of the oscillations onthis scale. However the envelope is clearly discernible and shows the amplitude of the oscillationsdecreases monotonically during the approach to the asymptotic steady state. Inset depicts a close-up on the curve, making the gap oscillations visible.

63

number of high frequency oscillations and a slow decay to a constant value. All of the trends noted

in the previous section have been observed; particularly interesting is the fact that the long time

value of the gap corresponds rather well with Eq. (3.27) despite variation in the initial fluctuation

size over 7 orders of magnitude. The earlyt2 behavior can also be seen, which shows that the

early- and long-time behaviors derived above are indeed connected by exact solutions of Equation

(3.20).

3.3.5 Related Work

In the course of this work, the authors became aware of the related unpublished calculations of

Shumenko (1990) and more recently of Barankov et al. (2003) regarding quench-induced dynamics

in a BCS system. There are substantial qualitative differences between our own findings and

those of SBLS, despite the fact that both proceed from identical semiclassical equations of motion

(3.16). To clarify the situation, we offer here an interpretation of their results and an analysis of

the discrepancies.

Looking again at equations (3.20), (3.21), we notice that they describe what is essentially a

system of independent oscillators subject to a universal driving force, which force just happens

to be determined self-consistently by the oscillators themselves. For a linear, damped oscillator

driven harmonically and off resonance, the long-time solutions consist of pure oscillations at the

frequency of the driving force. This suggests by analogy that the semiclassical equations of motion

can exhibit solutions in which each pseudospin follows the motion of the gap, subject of course to

the constraint of overall self-consistency. Such solutions would be separable in energy and time:

sx(k, t) = Ak∆(t) (3.28)

which is precisely theansatzproposed by SBLS.

64

Substitution into Eqs. (3.20), (3.21) yields immediately

d2

dt2∆ + (4ε2k −

2|εk|Ak

)∆ + 2∆3 = 0 (3.29)

and

1− λ∑

k

Ak = 0. (3.30)

If we chooseAk such that4ε2k−2|εk|Ak

= C, a constant independent ofk, then each pseudospin obeys

the same equation of motion. Thus theansatz can indeed be made self-consistent. Multiplying

the first of these equations by∆ and integrating overt, we obtain

∆2 + (∆2 −∆20)

2 = Γ2

whereΓ ≡ ∆(0), ∆0 ≡ ∆(0), and we have chosenC = −2∆20, which yields for the self-

consistency equation

1− λ∑

k

|εk|2ε2 + ∆2

0

= 0.

As discussed by SBLS, these equations clearly demonstrate the existence of a class of periodic

solutions∆(t). While consistent mathematically, physically they leave out an equally important

class of solutions characterized by the response of individual pseudospins at their natural frequen-

cies, and the concomitant dephasing of the pseudospin orientations with time. The nonlinearity

of the semiclassical equations of motion, and the absence of substantial damping in the zero tem-

perature limit, both indicate that these solutions cannot be safely ignored. It is in fact extremely

plausible on physical grounds, and clear from the analytical and numerical results in previous sec-

tions, that dephasing will affect rather strongly both the frequency of the gap oscillations and the

nature of their long-time behavior, except perhaps for special choices of initial conditions. This

consideration acquires even greater urgency in the limit of weak coupling∆/εC 1, for which

case the majority of pseudospins precess very far from resonance. In such a circumstance it is very

65

difficult to see how the full pseudospin system, when subjected to a sudden perturbation of the type

described in Section 3.3.1, could respond with such a high degree of synchronicity. Indeed, the

early time solutions in Section 3.3 indicate otherwise, and appear to be inconsistent with those of

SBLS.

Thus it is reasonable to locate the essential differences between our own formulation of the

quenching problem, and that of SBLS, in their respective approaches to the question of initial

conditions. The former theory invokes latent quantum fluctuations of the normal state which, in

the wake of the quench, provide a ‘kick’ within each symmetry-broken Hilbert space. The sudden

appearance of a finite, if minute, off-diagonal field will of necessity introduce some small degree of

dephasing, leading inevitably to a non-equilibrium steady-state as described above. SBLS appear

to assume the complete absence of dephasing from the start. In the full space of possible initial

conditions this represents a rather special class, which leads us to conclude that the associated

periodic behavior of the gap, while intriguing, is unlikely to be observed. Finally, there is no

discussion by SBLS of the apparent breaking of gauge symmetry within their theory immediately

after the quench. Of course, the quench by itself cannot break this symmetry; the system evolves

simultaneously along each direction of symmetry breaking in such a manner that the overall gauge

symmetry remains unbroken.

66

Chapter 4

Concluding Remarks

In this thesis we have studied the effects of ionizing radiation on the low temperature phases of

3He, in particular, whether such radiation can induce the nucleation of theB from the supercooled

A-phase, and/or the formation of vortices via the Kibble-Zurek mechanism. To this end, we have

studied energy transport in a degenerate Fermi liquid as a means of understanding the dynamical

evolution of localized ‘hot spots’ created by such radiation. Further, we develop a theory for the

evolution of a BCS superfluid order parameter following the sudden variation of attractive coupling

in a Fermi gas, as a means to understanding quench phenomena induced by rapid energy transport

away from the hot spots.

On the transport side, we have argued that the baked Alaska scenario, as first described in

Leggett (1984) and discussed in somewhat greater detail in Leggett and Yip (1989), is not only

qualitatively plausible but even quantitatively accurate as a model of energy transport following

intense local heating in a normal Fermi liquid. First, a hydrodynamic description of the usual sort

breaks down when considering either inter-quasiparticle scattering or particle-hole pair creation

processes within their respective regions of applicability. In the latter case it is even possible to

find, within a simple model calculation, the effective propagator for single quasiparticles which

shows explicitly a non-monotonic shell structure moving out from the origin at a velocity∼ vF .

Although the single particle description upon which this is based does not conserve energy in

detail, it does in expectation value. It thus seems likely that at least the asymptotic solutions of the

‘true’ Boltzmann equation bear a strong resemblance to (2.23). Lastly, the time-scale for the onset

67

of baked Alaska behavior, which is set by the inter-quasiparticle scattering that dominates at early

times, is found to be sufficiently rapid for the range of values ofR0/l0 appropriate to liquid3He at

these (effective) temperatures.

As stated above, we have also studied the behavior of a Fermi gas following the sudden turn-on

of an attractive BCS interaction parameterλ, as a model for analogous process which are thought to

occur in the low temperature superfluid phases of3He upon exposure to neutron radiation. For val-

ues of the parameters appropriate to3He near the transition line, a study of the free energy available

for vortex generation in the wake of a quench reveals that the Kibble-Zurek scaling law gives only

a strict lower bound on the defect spacing. That is, the Kibble-Zurek law assumes that a maximum

quantity of free energy is available for vortices, a condition requiring that the quenched region

cools all the way back to its original temperature and does not expel any condensation energy to

its surroundings. Both of these requirements are questionable. Further, the reduced temperature at

freezeout far exceeds that at which the quasiparticle inelastic scattering rateτqp becomes compara-

ble to the gap frequency∆/~, a condition which should be well satisfied in the Ginzburg-Landau

regime on which the Kibble-Zurek scenario is based. This suggests that the dynamics following

such a quench must be treated, to a first approximation, in the absence of collisions. Under these

conditions, for a sudden turn-on of the interaction parameter we demonstrate an absolute instability

to quantum fluctuations already present in the normal state. These subsequently amplify into large,

semiclassical oscillations along each direction of symmetry-breaking which, even in the absence of

collisions, eventually settle due to cancellations among the various pair oscillations to a self-driven

steady state with gap∆∞ = εC(e2/N(0)λ − 1)−1/2.

Taken together, these results strongly favor the baked Alaska picture of nucleation, and tend

to disfavor the Kibble-Zurek mechanism. There are still many unanswered questions; for one, it

is unknown to what extent the actual dynamical evolution of the quenched region will violate the

Kibble-Zurek relation, either by releasing energy to its environment or increasing in temperature.

This must await future study. Further, it would be nice to understand some of the properties of the

self-driven steady state of Section 3.3, in particular, its response to weak damping. Finally, there

68

may perhaps be a better treatment of the ‘drift velocity’ proposed in Section 2.4; in the author’s

view the argument has strong physical appeal, but it is far from a proof.

69

Appendix A

In this appendix we address an important technical point associated with the mathematical form of

the asymptotic propagator (2.23) which, while not affecting the physics of Chapter 2, nevertheless

merits consideration as a separate mathematical problem. It concerns the question of how much of

the weight of the distribution (2.23) falls in the unphysical regionr > vF t. Define the ‘spillover

function’

s(t) = 4π

∫ ∞

vF t

drr2K(r, t) (1)

which measures the fraction of the total weight beyond this radius. SinceK is normalized,0 ≤

s(t) ≤ 1. Any physical asymptotic distribution, gotten by coarse-graining over a microscopic

scattering process, must satisfy the conditions(t) → 0 as t → ∞; that is, it must be possible

to makes(t) smaller than any preselected number by substituting sufficiently large values oft.

For example, ifK is the propagator corresponding to normal diffusion (2.3.1), we have, defining

x ≡ r2/6Dt

s(t) ∝∫ ∞

vF

(6D)1/2t1/2

dxx2e−x2

.

The lower bound diverges, so that the weight in the unphysical region tends to zero in the limit of

long times, as it should.

It is evident upon inspection of (2.23) that it can be written in the formK(r, t) = 1r(vF t)2

f(y),

wherey ≡ r/vF t. The spillover function becomes

s(t) = 4π

∫ ∞

vF /vF

dyyf(y)

70

implying that the spillover in the case of the baked Alaska propagator is time-independent. Taken

by itself this would appear to invalidate the baked Alaska propagator. However, the defect lies not

in the propagator itself but in our neglect of certain asymptotic corrections to its form, which of

course vanish in the long-time limit; in particular, to the exponent of the dimensionless combination

Γt appearing in the denominators of the arguments in (2.22). Asymptotically this approaches 2,

indicating ‘ballistic’ scaling, but at all finite times is actually2 − ε, whereε is some function of

time such thatε→ 0 ast→∞. This was illustrated previously in a schematic way by Fig.2.2, as

an aid to our discussion of the singularities that develop in the functionν(ζ) in the largeN limit.

In fact the propagator forζ > 1 represents a marginal case in the sense that for any constantε,

no matter how small,s(t) → 0 for long times. This is because for finiteε the lower bound in the

expression (4) becomes proportional to(Γt)ε. However, in the case of a time-dependentε things

are more complicated:ε must decay slowly enough (in fact slower than1ln Γt

) that the lower bound

still diverges. We must therefore calculate the actual time-dependence ofε in order to assess the

validity of expression (2.23).

To do this we must return to equation (2.20) in Chapter 2, which we used to locate the terms of

greatest weight in the expansion (2.16). SettingR ∼ 1 gives

ζ−N(ζ−1 − 1)Γt+N ln ζ = 0

or

Γt ∼ NζN ln ζ

1− ζ−1

and thus that

ln Γt ∼ N ln ζ + lnN + · · · (2)

To leading order we haveN ∼ ln Γtln ζ

, as before. However, we can iterate to find the next order

71

correction and thereby invert the expansion (2), which yields

N =ln Γt

ln ζ− ln ln Γt+ · · · (3)

The exponent can be found by looking at the long-time expansion of the expression

2ν =ln Γβ2

ln Γt→ 2N ln ζ

ln Γt+ · · · (4)

Substituting from our expansion forN we find that the exponent varies as

2ν = 2− ε = 2− ln ζln ln Γt

ln Γt+ · · · (5)

in the limit Γt 1. Thusε decays more slowly than1ln Γt

, ensuring that, indeed, the spillover tends

to zero as it should.

72

Appendix B

The Helsinki neutron experiments [Ruutu et al. (1995)] are conducted in a rotating cryostat, the

stated purpose of which is the expansion of vortex rings above a critical size determined by the

rotation velocity. In this appendix we detail some of the physics of vortex rings in the presence of

superfluid flow, in order to motivate and clarify the formula (1.4). A relevant discussion of these

topics may be found in Lifshitz and Pitaevskii (1998), Iordanskii (1965), and Kibble (1996).

A length of vortex filament gives rise to a superfluid velocityvs, whose ‘circulation’ is quan-

tized in units of~:∮

vs · dr = 2πn ~m

, where the integral encircles the filament. Consider a unit

vortex withn = 1, and defineκ ≡ ~m

. The expression for the circulation is formally analogous

to the magnetostatic expression for Ampere’s law,∮

B · dr = Ienc. This analogy enables us, in

principle, to compute the superfluid velocity at any point in the fluid due to the presence of an

arbitrarily shaped filament, by allowing us to write down what is essentially the Biot-Savart law

for vortices:

vs =κ

2

∫dl× r

r2(6)

Now consider a vortex ring of radiusR0, such as those produced by a hot spot. Let us calculate

the velocity of such a ring relative to that of the bulk superfluid. At zero temperature the ring gets

carried along by the superflow it induces; that is, the velocity of the ring as a whole and the local

velocity of superflow at each element of the ring are equal. We can calculate the velocity of an

element of the ring by summing the contributions to the superfluid velocity at that point from all

of the other elements, using our ‘Biot-Savart’ law. This point ‘P ’ can be chosen arbitrarily for a

circular ring, since the velocity is the same at all points of the ring by symmetry. The geometry is

73

Figure 1: Diagram for the calculation of the superfluid velocity on a point of the ring due to all ofthe other elements of the ring.

as given by the diagram 1.

From the diagram, we have|dl| = R0dθ, |dl× r| = R0 sin θ2dθ, andr = 2R0 sin θ

2, and hence

the velocity of the vortex ring is

vs =κ

4R0

∫ π

0

dθ

sin θ2

(7)

This integral is logarithmically divergent, the dominant contribution coming from the lower limit.

We cut this off forθ ∼ aR0 π, wherea is of atomic dimensions. Thereforevs = κ

2R0

∫a

R0

dθθ∼

κ2R0

ln R0

a.

The energyε of the ring is the energy of induced superflow, orε = 12ρs

∫d3rv2

s(r). To logarith-

mic accuracy this can be evaluated in cylindrical coordinates local to a given element of the ring

as 12ρs(2πR0)

∫ 2π

0dφ

∫ R0

ardr(κ

r)2 or

ε = 2ρsπ2κ2R0 ln

R0

a. (8)

The momentum of the ring can be found by the relationdεdp

= v, or p =∫ R0

0dεv

, wheredε =

74

2ρsπ2κ2 ln R0

adR0 up to terms of ordera

R0. This evaluates to

p = 2ρsπ2κR2

0. (9)

A vortex ring therefore constitutes a well defined excitation of the system with an energy-momentum

relationε(p) given by equations (8), (9).

At nonzero temperatures, there is friction between the filament and the normal component.

The vortex therefore comes to rest with respect to the normal fluid, and a mismatch develops

between the local superfluid velocity at the filament, and the velocity of the filament itself. Such a

mismatch induces a ‘counterflow’|vs−vn| = v of superfluid through the ring in its own rest frame,

but the energy and momentum relations, as they are based on the superflow induced by the ring

itself, remain unchanged in the rest frame of the bulk superfluid. Thus in considering the Helsinki

experiments, we must take account of the fact that, at least for rotation velocities below the critical

velocity at the walls of the container, at finite temperatures the normal component rotates with the

cryostat.

Let us look at the region local to the hot spot, and consider again a given vortex ring of radius

R0. In the rest frame of the ring, we must transform the energyε(p) of the superfluid due to the

vortex via the Galilean relation

ε′ = ε(p) + p · vs +1

2Mv2

s (10)

whereM is the mass of the superfluid. The last term is of course just the energy of the counterflow

in the absence of the vortex, so thechangein energy of the system due to the vortex ring itself in

this new frame is just

ε(p) + p · v (11)

where we have substituted the counterflow velocityv instead ofvs, since the two are equal in

this frame. The momentum of the ring and the velocity of counterflow are obviously oppositely

75

Figure 2: Energy of a vortex ring as a function of its radiusR0 in the presence of counterflow.

directed, thus we have

ε′ = ε− pv = 2ρsπ2κ(κR0 ln

R0

a− vR2

0) (12)

using equations (8), (9). TheR20 term wins at largeR0, and we get a curve of the form shown in

Figure 2. Thus, above a critical radiusRcrit ∼ κ2v

ln Rcrit

a(to logarithmic accuracy) determined

through the conditiondε′

dR0= 0, the ring expands spontaneously in the counterflow.

It is assumed by the Helsinki experimenters that the initial fireball evolves by diffusion, that

is, that the deviation of the effective temperature from the ambient temperatureT follows the law

(assuming spherical symmetry)

Teff (r, t)− T ≈ (E0

CV (4πDt)3/2) exp(− r2

4Dt) (13)

whereE0 is the initial energy that has been deposited into the liquid by the neutron,CV is the

heat capacity of the liquid, andD a diffusion constant. The radius of the bubble may be taken

to be the point where the effective temperature distribution above falls to1/e of its peak value,

i.e. Rb ∼ 4Dt. We can define the maximum radius of such a bubble as the value attained when

76

the total volume enclosed is at an effective temperature aboveTc, that is, for points such that

Teff (r, t)− T > Tc − T = E01

eπ3/2CV R3b, and hence

Rb ∼ (E0

CV Tc

)1/3(1− T

Tc

)−1/3. (14)

Clearly, 2Rb is the upper cutoff to the size distribution of the vortices. The lower cutoff is that

imposed, presumably, by the Zurek lengthξZ = ξ0(τQ

τ0)1/4. In between, at least for characteristic

dimensionsD >> ξZ , the size distributionn(D) (the number of vortices per unit volume of

dimensionD) can be expected to be scale invariant; as discussed in Chapter I this implies, by

dimensional analysis,

dn = 2CdD

D4. (15)

The vortices which escape the bubble have dimensions2Rcrit(v) < D < 2Rb, and their number is

N (v) = (4

3πR3

b)

∫ 2Rb

2Rcrit(v)

n(D)dD

1

9πC[(

Rb

Rcrit(v))3 − 1] (16)

The critical velocity above which vortices start to escape from the bubble is given byN (vcn) = 0;

henceRb = Rcrit(vcn) = κ2vcn

ln Rcrit

a, orvcn = κ4Rb ln Rcrit

a. For an arbitrary counterflow velocity

v, we have Rb

Rcrit= Rb/(

κ2v

ln Rcrit

a) = v

vcn. Thus the number of vortex loops which expand for a

given counterflowv varies as

N (v) =1

9πC[(

v

vcn

)2 − 1] (17)

as discussed previously. The Helsinki data appear to fit this formula quite well; one expects that

the result should break down for higher rotation speeds as one gets closer to the underlying length

scaleξZ , but this so far has not been seen.

77

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79

Vita

Geoffrey Lee Warner was born on January 28, 1975 in the State of Washington, U.S.A. He receivedhis bachelor’s degree in physics from the Massachusetts Institute of Technology in 1997.

80