Can we Measure Superflow on Quenching He4?

4 Pages·0.13 MB·English

by  E. Kavoussanaki

Can we Measure Superflow on Quenching 4He?
E. Kavoussanakiand R. J. Rivers
Blackett Laboratory, Imperial College, London SW7 2BZ
Although ξ (t) diverges at t = 0 this is not the case
eq
for the true correlation length ξ(t), which can only grow
Zurekhasprovidedasimplepicturefortheonset oftheλ-
so far in a finite time. Initially, for t < 0, when we
transition in 4He, not currently supported by vortex density
are far from the transition, we can assume that the field
experiments. However, we argue that the seemingly similar
correlationlengthξ(t)tracksξ (t)approximately. How-
argument by Zurek that superflow in an annulusof 4Heat a eq
ever, as we get closer to the transition ξ (t) begins to
quenchwill be measurable is still valid. eq
increase arbitrarily fast. As a crude upper bound, the
9
9 true correlation length fails to keep up with ξeq(t) by
9 the time t¯ at which ξ is growing at the speed of
eq
−
1 sound c(t) = ξ (t)/τ(t), which determines the rate at
eq
n As the early universe cooled it underwent a series of which the order-parameter can change. The condition
a phasetransitions,whoseinhomogeneitieshaveobservable dξeq(t)/dt=c(t) is satisfiedatt= t¯, where t¯=√τQτ0,
J consequences. Tounderstandhowsuchtransitionsoccur with corresponding correlationleng−th

9 it is necessary to go beyond the methods of equilibrium
2 thermalfield theory that identified the transitionsin the τ 1/4
first instance. ξ¯=ξeq(−t¯)=ξ0 τQ . (3)
1 (cid:18) 0(cid:19)
In practice,we often know remarkablylittle about the
v
dynamics of quantum field theories. A simple question After this time it is assumed that the relaxation time is
8
4 to ask is the following: In principle, the field correlation so long that ξ(t) is essentially frozen in at ξ¯until time
3 length diverges at a continuous transition. In reality, it t +t¯, when it sets the scale for the onset of the broken
≈
1 doesnot. Whathappens? Usingsimplecausalarguments phase.
0 Kibble [1,2] made estimates of this early field ordering, Aconcreterealisationofhowthefreezingsetsinispro-
9
because of the implications for astrophysics. vided by the time-dependent Landau-Ginzburg (TDLG)
9
Therearegreatdifficultiesinconvertingpredictionsfor equation for F of (1) [4],
/
t the early universeinto experimentalobservations. Zurek
a
1∂φ δF
m suggested [3] that similar arguments were applicable to a = +η , (4)
a
- condensed matter systems for which direct experiments Γ ∂t −δφa
d could be performed. In particular, for 4He he argued
n that the measurementof superflow ata quenchprovided for φ = (φ1 +iφ2)/√2, where ηa is Gaussian noise. We
o a simple testof these ideas. We presenta briefsummary can show self-consistently [5] that, for the relevant time-
c of his argument. interval t¯ t t¯the self-interaction term can be ne-
v: Assume that the dynamics of the 4He lambda- glected(−β =≤0),≤wherebyasimplecalculationfindsξ ξ¯
≈
i transition can be derived from an explicitly time- inthisinterval,aspredicted. Itthushappensthat,atthe
X
dependent Landau-Ginzburg free energy of the form onset of the phase transition, the field fluctuations are
r approximately Gaussian. The field phases eiθ(r), where
a
¯h2 1 φ(r)= φ(r)eiθ(r), are then correlatedon the same scale
F(T)= d3x φ2+α(T)φ2+ β φ4 , (1) | |
as the fields.
2m|∇ | | | 4 | |
Z (cid:18) (cid:19) Consider a closedpath in the bulk superfluid with cir-
in which α(T) vanishes at the critical temperature T . cumference C ξ(t). Naively, the number of ’regions’
c ≫
Explicitly, let us assume the mean-field result α(T) = through which this path passes in which the phase is
α ǫ(T ), where ǫ = (T/T 1), remains valid as T/T correlated is = O(C/ξ(t)). Assuming an independent
0 c c − c N
varies with time t. In particular, we first take α(t) = choiceofphaseineach’region’,ther.m.sphasedifference
α(T(t)) = α t/τ in the vicinity of T . Then the fun- along the path is
0 Q c
−
damentallengthandtimescalesξ andt aregivenfrom
0 0
Eq.1 as ξ02 = h¯2/2mα0 and τ0 = h¯/α0. It follows that ∆θC ≈√N =O( C/ξ(t)). (5)
the equilibrium correlationlength ξ (t)=ξ (T(t)) and
eq eq p
the relaxation time τ(t) diverge at T , which we take to If we now consider a quench in an annular container
c
be when t vanishes, as of similar circumference C of superfluid 4He and radius
l C, Zurek suggested that the phase locked in is also
t −1/2 t −1 gi≪venbyEq.5,withξ¯ofEq.3. Sincethephasegradientis
ξeq(t)=ξ0 , τ(t)=τ0 . (2) directly proportionalto the superflow velocitywe expect
τ τ
(cid:12) Q(cid:12) (cid:12) Q(cid:12) a flow after the quench with r.m.s velocity
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
1
¯h 1 (∆θ )2 = d2x d2y (x y,t), (12)
∆v =O msCξ¯ . (6) C −Zx6∈S Zy∈S C | − |
(cid:18) (cid:19)
where x and y are in the plane of S, and
providedl=O(ξ¯). Although in bulk fluid this superflow
will disperse, if it is constrained to a narrow annulus it 1 ∂ f′2(r,t)
(r,t)= . (13)
should persist, and although not large is measurable. C r∂r 1 f2(r,t)
In addition to this experiment, Zurek also suggested (cid:18) − (cid:19)
thatthesamecorrelationlengthξ¯shouldcharacterisethe SinceG(r,t)isshort-ranged (r,t)isshort-rangedalso.
C
separation of vortices in a quench. In an earlier paper With x outside S, and y inside S, all the contribution
[5] one of us showed that this is too simple. Causality to (∆θ )2 comes fromthe vicinity of the boundary ofS,
C
argumentsarenotenough,andwhethervorticesformon rather than the whole area. That is, if we removed all
this scale is alsodeterminedby the thermalactivationof fluid except for a strip from the neighbourhood of the
the Ginzberg regime, in which all 4He experiments take contourC wewouldstillhavethe sameresult. Thissup-
place. Experimentally, this seems to be the case [7]. ports the assertion by Zurek that the correlation length
Our aim in this paper is to see whether thermal fluc- forphasevariationinbulkfluidisalsoappropriateforan-
tuations interferewiththe predictionEq.6,forwhichex- nular flow. The purpose of the annulus (more exactly, a
periments have yet to be performed. Again consider a circular capillary of circumference C with radius l C)
≪
circularpathin the bulk fluid(in the 1-2plane), circum- is to stop this flow dissipating into the bulk fluid.
ference C, the boundary of a surface S. For given field More precisely, suppose that C ξ(t). Then, if we
≫
configurationsφ (x)the phasechangeθ alongthepath take the width 2l of the strip around the contour to be
a C
can be expressed as the surface integral largerthanthecorrelationlengthof (r,t),Eq.12canbe
C
written as
θC =2π d2x ρ(x), (7) ∞
Zx∈S (∆θC)2 2C drr2 (r,t). (14)
≈− C
Z0
where the topological density ρ(x) is given by
ThelineardependenceonC ispurelyaresultofGaussian
ρ(x)=δ2[φ(x)]ǫ ∂ φ (x)∂ φ (x), i,j =1,2 (8) fluctuations.
jk j 1 k 2
Insofaraswecanidentifythebulkcorrelationwiththe
where ǫ = ǫ =1, otherwise zero. annular correlation, instead of Eq.6, we have
12 21
−
The ensemble average ρ(x) is taken to be zero
t
h i
at all times t, guaranteed by taking φa(x) t = 0 = ¯h 1
h i ∆v = . (15)
φ (x)∂ φ (x) . Thatis,wequenchfromaninitialstate
h a j b it msCξs(t)
with no rotation. For the Gaussian fluctuations that are
relevant for the times of interest [5,6], all correlations
The step length ξ (t) is given by
s
are given in terms of the diagonal equal-time correlation
function G(r,t), defined by 1 ∞ f′2(r,t)
=2 dr . (16)
ξ (t) 1 f2(r,t)
φa(x)φb(0) t =δabG(r,t) r = x. (9) s Z0 −
h i | |
TherearetwoimportantdifferencesbetweenEq.15and
The correlation length ξ(t) is defined by G(r,t) =
Eq.6. The first is in the choice of time for which ∆v of
o(e−r/ξ(t)), for large r > ξ(t). The TDLG does not lead
Eq.15 is to be evaluated. In Eq.6 the time is the time
tosimpleexponentialbehaviour,butthereisnodifficulty t¯of freezing in of the field correlation. Since ξ(t) does
in defining ξ(t) in practice [5,6]. n−ot change much in the interval t¯< t < t¯we can as
The variance in the phase change around C, ∆θ is −
C well take t = 0. We shall argue below that for Eq.15 a
determined from
more appropriate time is the spinodal time t at which
sp
the transition has completed itself in the sense that the
(∆θC)2 =4π2 d2x d2y ρ(x)ρ(y) t. (10) fields have begun to populate the ground states.
h i
Zx∈S Zy∈S Secondly, a priorithere is no reasonto identify ξ (t )
s sp
with either ξ¯(or even ξ(t )). In particular, because ξ¯
The properties of densities for Gaussian fields have sp
in Eq.6 is defined from the large-distance behaviour of
been studied in detail [8,9]. Define f(r,t) by f(r,t) =
G(r,t), and thereby on the position of the nearest sin-
G(r,t)/G(0,t). On using the conservation of charge
gularity of G(k,t) in the k-plane, it does not depend on
the scale at which we observe the fluid. This is not the
d2x ρ(x)ρ(0) =0 (11)
h it case for ξs(t) which, from Eq.16, explores all distance
Z scales. Because of the fractal nature of the short wave-
it is not difficult to show, from the results of [8,9], that length fluctuations, ξ (t) will depend on how many are
s
∆θ satisfies included, i.e. the scale at which we look. If we quench
C
2
in an annular capillary of radius l much smaller than its time at which to look for superflow since, although the
circumference, we are, essentially, coarsegrainingto that field correlationlength ξ(t) may have frozen in by t=0,
scale. That is, the observed variance in the flux along the symmetry breaking has not begun.
the annulus is πl2∆v for ∆v averaged on a scale l. We Assuming the linearised [5] Eq.4 for small times t> 0
make the approximation that that is the major effect of we see that, as the unfreezing occurs, long wavelength
quenching in an annulus. This cannot be wholly true, modeswithk2 <t/τ growexponentiallyandsoonbegin
Q
but it is plausible if the annulus is not too narrow for to dominate the correlation functions. How long a time
boundary effects to be important. we have depends on the self-coupling β which, through
Provisionally we introduce a coarsegraining by hand, G , sets the shortest time scale. This is because, at the
1
modifying G(r,t) by damping short wavelengths O(l) as absolute latest, G must stop its exponential growth at
1
t=t ,when φ2 ,satisfies φ2 =α /β. Wefur-
sp h| | itsp h| | itsp 0
G(r,t;l)= d/3keik.xG(k,t)e−k2l2. (17) thersupposethattheeffectofthebackreactionthatstops
the growth initially freezes in any structure. In Fig.1 we
Z
also show ξmin(t;l) for t=3t¯and t=4t¯, increasing as t
We shall denote the value of ξ obtained from Eq.17 as s
s increases.
ξ (t;l). Itpermitsanexpansionintermsofthemoments
s For4Hewithquenchesofmillisecondsthefieldmagni-
of G(k,t)e−k2l2, tude has grownto its equilibrium value before the scale-
dependence has stopped [5]. For vortex formation, for
∞
Gn(t;l)= dkk2nG(k,t)e−k2l2. (18) which the scale is O(ξ0), the thickness of a vortex, the
Z0 dependence of the density on scale makes the interpre-
tation of observations problematic. This is not the same
For small r it follows that f′2(r,t;l)/(1 f2(r,t;l))
− here. That the incoherent ξs depends on radius l is im-
material. The end result is that
G 3G G
= 2 1 3 2 r2+O(r4) . (19)
3G − 20G − 12G
1(cid:20) (cid:18) 2 1(cid:19) (cid:21) ¯h 1
∆v = . (22)
Although,forlarger,f′(r,t;l)2 =o(e−2r/ξ(t)),wefind msCξs(tsp;l)
that the bulk of the integral Eq.16 lies in the forward
We saw that the expressionEq.20for ξ assumed that
peak,andthatagoodupperboundforξ isgivenbyjust s
s 2l is larger than
integrating the quadratic term, whence
−1/2
1 1 = 4G2 3G3 G2 −1/2, (20) ξeff(t;l)= 230GG3 − 12GG2 . (23)
ξs(t;l) ≥ ξsmin(t;l) 9G1(cid:18)20G2 − 12G1(cid:19) (cid:18) 2 1(cid:19)
Otherwise the correlations in the bulk fluid from which
with the equality slightly overestimated. In units of ξ
0 wewanttoextractannularbehaviourareoflongerrange
and τ we have, in the linear regime [5],
0 than the annulus thickness. Numerically, we find that
Gn(t;l)≈ 2nI+n1/2 e(t/t¯)2 ∞dt′ [t′e+−(lt2′−/2t)]2n/+t¯21/2TT(t′), ξξeeffff((t0,,ll)) ≥=2l2lforvearlyl tac>cu0ra.teAlycarutdte w=ay0,tobauctcotmhaot-
Z0 c date this is to cut off the integral Eq.14. With a little
(21) effort, we see that the effect of this is that ξsmin(tsp,l) of
Eq.20 is replaced by
where I = dkk2ne−k2. The presence of the T(t′)/T
n 0 c ξmax(t ,l)=ξmin(t ,l)[1 (1 4l2/ξ (t ,l)2)3/2]−1,
termisareminderthatthestrengthofthenoiseη ispro- s sp s sp − − eff sp
R
portional to temperature. However, for the time scales (24)
O(t¯) τ of interest to us this ratio remains near to
Q
unity≪and we ignore it. For smallrelative times the inte- greater than ξmin(t ,l) and thereby reducing the flow
s sp
grandgetsalargecontributionfromtheultravioletcutoff velocityfornarrowerannuli. Thesearethedashedcurves
dependent lower endpoint, increasing as n increases. in Fig.1. The effect is largest for small radii l ξ¯, for
≤
IfwereturntotheLandau-GinzbergequationEq.4we whichthe approximationoftrying to readthe behaviour
find that φ2 α /β in the interval t¯ t t¯. of annular flow from bulk behaviour is most suspect. A
t 0
h| | i ≪ − ≤ ≤
Although the field has frozen in, the fluctuations have more realistic approach for such narrow capillaries is to
amplitudesthataremoreorlessuniformacrossallwave- treat the system as one-dimensional [3]. For this reason
lengths. As a result, what we see depends totally on wehaveonlyconsideredl ξ¯inFig.1. Wewouldexpect,
≥
the scale at which we look. Specifically, from Eq.21 from Eq.20, that ξ (t ;l) has an upper bound that lies
s sp
ξmin(0;l)=O(l), as shown in the lowest curve of Fig.1. somewhere between the curves.
s
If, as suggestedby Zurek, we take l =O(ξ¯) we recover Once l is very large, so that the power in the fluctu-
Eq.6 qualitatively, although a wider bore would give a ations is distributed strongly across all wavelengths we
correspondingly smaller flow. However, this is not the recover our earlier result, that ξ (t ;l)=O(l). In Fig.1
s sp
3
x
this corresponds to the curves becoming parallel as l in-
è
creases for fixed t. However, the change is sufficiently
x
slowthatannuli,significantlywiderthanξ¯,forwhichex-
periments are more accessible, will give almost the same 7
flow as narrower annuli. This would seem to extend the
6
original Zurek prediction of Eq.6 to thicker annuli, de-
spite our expectations for incoherent flow. However, we 5
stressagainthatcautionisnecessary,sinceintheapprox-
4
imation to characterise an annulus by a coarse-grained
ring without boundaries we have ignored effects in the 3
direction perpendicular to the annulus. In particular,
2
the circular cross-sectionof the tube has not been taken
into account. One consequence of this is that infinite 1
l
(non-selfintersecting) vortices in the bulk fluid have no
counterpart in an annulus. Removing such strings will 2 3 4 5 è
x
have an effect on ∆θ , since the typical fraction of vor-
C
tices in infinite vortices is at the level of 70%. However, FIG.1. ξsmin(t,l)of Eq.20(solid lines) plottedagainst l,in
atthespinodaltimethefluctuationsin4Hearerelatively units of ξ, for t=0, 3ξ and 4ξ. ξsmax(t,l) of Eq.24 (dashed
enhanced in the long wavelengths,and such an enhance- lines) plotted against l for t=0, 3ξ and 4ξ. In each case the
ment is known to reduce the amount of infinite vortices, higher lines correspond to higher values of time
perhaps to something nearer to 20%. The details of this
effect (being pursued elsewhere) are unclear but, for the
sakeofargumentwetakethepredictionsofthecurvesin
Fig.1 as a rough guide in the vicinity of their minima.
So far we have avoided the question as to which time
curves we should follow. This is because t itself de-
sp
pends on the scale l of the spatial volume for which the
field average achieves its ground state value. In practice [1] T.W.B. Kibble, J. Phys. A9, 1387 (1976).
variation is small, with t for 4He varying from about [2] T.W.B. Kibble, in Common Trends in Particle and Con-
3t¯to 4t¯as l varies fromsξp ξ¯to l = 10ξ¯. Since the densed Matter Physics, Physics Reports 67, 183 (1980).
0 ≪ [3] W.H. Zurek, Nature 317, 505 (1985), Acta Physica
curves for ξ (t ;l) lie so close to one another in Fig.1
once l 4ξ¯sthsep scale at which the coarse-grained field PolonicaB24,1301 (1993). Seealso W.H.Zurek,Physics
≥ Reports 276, Number4, November1996
beginstooccupythegroundstatesbecomeslargelyirrel-
[4] W.H. Zurek,Nature 382, 297 (1996),
evant.
Since ∆v only depends on ξ−1/2 it is not sensitive to P. Laguna and W.H. Zurek, Phys. Rev. Lett. 78, 2519
s (1997). A. Yatesand W.H.Zurek, hep-ph/9801223.
choice of l > 2ξ¯at the relevant t. Given all these ap-
[5] G. Karra and R. J. Rivers, Phys. Rev. Lett. 81, 3707
proximationsourfinalestimateis(inthecm/secunitsof
(1998).
Zurek [3])
[6] D. Ibaceta and E. Calzetta, hep-ph/9810301 (1998).
[7] P.V.E McClintoch et al., Phys. Rev. Letters 81, 3703
∆v 0.2(τQ[µs])−ν/4/ C[cm] (25) (1988).
≈
[8] B.I.Halperin,publishedinPhysicsofDefects,proceedings
for radii of 2ξ¯ 4ξ¯, τ of the ordper of milliseconds and
Q of Les Houches, Session XXXV 1980 NATO ASI, editors
−
C of the order of centimetres. ν =1/2 is the mean-field Balian, Kl´eman and Poirier (North-Holland Press, 1981)
critical exponent above. In principle ν should be renor- p.816.
malisedtoν =2/3,butthedifferenceto∆vissufficiently [9] F. Liu and G.F. Mazenko, Phys. Rev. B46, 5963 (1992).
smallthatweshallnotbother. Giventheuncertaintiesin
its derivation the result Eq.25 is indistinguishable from
Zurek’s [3] (with prefactor 0.4), but for the possibility
of using somewhat larger annuli. The agreement is, ul-
timately, one of dimensional analysis, but the coefficient
could not have been anticipated. How experiments can
beperformed,evenwiththewiderannulithatEq.25and
Fig.1 suggest, is another matter.
We thank Glykeria Karra, with whom some of this
work was done. This work is the result of a network
supported by the European Science Foundation .
*
4