Table Of ContentQUANTUM LIQUIDS
Lectures presented at the International School of
Low Temperature Physics, Erice, Italy, June 11-25, 1977
Edited by
J. RUVALDS
Department of Physics
University of Virginia
and
T. REGGE
Insitutefor Advanced Study
Princeton
1978
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Library of Congress Cataloging in Publication Data
International School of Low Temperature Physics
"Ettore Majorana," 1st, Erice, Italy, 1977.
Quantum liquids.
Includes index.
1. Liquid helium—Congresses. I. Ruvalds, J.
II. Regge, Tullio. III. Ettore Majorana International
Centre for Scientific Culture. IV. Title.
QC145.1+5. H4I57 1977 5**61.751 77-1903*+
ISBN 0-U1+1+-85117-8
PRINTED IN THE NETHERLANDS
FOREWORD
The first Course of the International School of Low Temperature
Physics "Ettore Majorana" was attended by forty seven scientists from
twelve countries. This Course was held in Erice during June 11-25,
1977 .
The School was sponsored by the Italian Ministry of Public Edu
cation, the Italian Ministry of Scientific and Technological Research,
the Italian National Research Council, and the Sicilian Regional
Government.
The countries represented at the School were: England, Finland,
France, the Federal Republic of Germany, Hungary, Israel, Italy, the
Netherlands, Poland, Scotland, Sweden, and the United States.
Quantum liquids have been the objects of intensive study as a
consequence of their superfluid properties at very low temperatures.
The recent discovery of various superfluid phases in 3He has generated
considerable research involving new cryogenic techniques at ultralow
temperatures. The exciting development in the study of superfluid
phases of the 3He system, which obeys Fermi statistics, form a primary
basis for the present school. In addition, new developments in the
understanding of quasipartic1es and their interactions in the Bose
4He system are emphasized, along with recent work on the 3Ηε-4Ηε
mixtures.
Originally the International School of Low Temperature Physics
was inspired by Professor T. Regge, Director of the School, and
Professor A. Zichichi, Director of the Ettore Majorana Center. Their
guidance and enthusiastic support of the Course on Quantum Liquids
were greatly appreciated by all participants.
Erice is a charming village at the summit of a mountain over-
looking the Mediterranean Sea. I t has a superb cultural heritage
and presents a magnificent settin g for a Scientific School. We are
especially grateful for the warm hospitality of our hosts in Sicily,
and the supreme contributions of Dr. A. Gabriele and Pinola Savelli
to the administration of the Scho ol
The scientific content of the Course benefited from the guidance
and encouragement of the organizing committee consisting of: P. W.
Anderson, R. A. Cowley, 0. V. Lounasma, R. Richardson, and J. Ruvalds.
Preparation of the manuscrips for the lectures of the Course
involved the efficient, patient, and conscientious services of the
following secretaries: M. Borsuk, B. Dexter, M. Engelhardt, L,
Heidenstrom, L. Lucky, A. Pamzarino, Ε. Rose, and D. Schaefer; and
we are especially grateful to Mary-Jo Harrington for the expert and
timely help with the final stages of the manuscript.
J. Ruvalds
Director of the Course
ν
INTERNATIONAL SCHOOL OF LOW TEMPERATURE PHYSICS
"ETTORE MAJORANA" CENTRE FOR SCIENTIFIC CULTURE
ERICE - ITALY - JUNE 11-25, 1977
1 . R. Pucci 11 . H. Kleinert 21 . A. Molinari 3 1 . D. Mermin
2 . Z. Galasiewicz 1 2 . Α. Zawadowski 22 . T. Regge 3 2 · R. Cowley
3 . I. Galasiewicz 1 3 . υ. Lombardο 2 3 . W. Alberico 3 3 . M. Wartak
1+ . A. Tartaglia 1 «t . 0. Avenel 2 4 . G. Coops 3 4 . J. Gspann
5 . J. Ruvalds 1 5 . R. Perzynski 2 5 . L. Reatto 3 5 . A. Leggett
6 . W. Saam 1 G . F. Moss 2 6 . D. Einzel 3 6 . M. Salomaa
7 . A. B. Brojeny 1 7 . Ν. Galasiewicz 27 . P. Hilton 3 7 . I. Tut to
8 . M. Deutsch 1 8 . G. Ahlers 28 . T. Greytak 3 8 . A. Szprynger
9 . R. Richardson 1 9 . Α. Cooper 29 . K. Skold
1 0 . E. Flynt 2 0 . G. Frossati 3 0 . A. Ahonen
ix
Quantum Liquids, J.Ruvalds and T. Regge, eds.
©North-Holland Publishing Company (1978)
CRITICAL PHENOMENA AND THE SUPERFLUID TRANSITION IN 4He
Guenter Ahlers
Bell Laboratories
Murray Hill, New Jersey 07974, USA
I. Introduction
One of the major advances in condensed matter physics in recent
decades has been the development of the renormalization group theory
(RGT) of critical phenomena. [1] It is believed that this theory
provides an exact description of the singularities which occur near
critical points, and therefore it should also provide a solution to
the problem of the phase transition at T^ in superfluid helium. The
claim that the RGT is an exact theory of the singularities which
occur in real physical systems is a strong one indeed; and it is now
up to the experimentalists to obtain data which can put the predic
tions of the theory to as detailed a test as the best laboratory
techniques permit.
For the purpose of obtaining quantitative measurements
extremely near a phase transition there is no other known system as
suitable as 4He. The sample is virtually free of the inhomogene-
ities which occur in many other systems in the form of impurities
and strains for instance. The effect upon the transition of the
gravitational field in which we do our experiments is relatively
weak and well understood. [2] The material has reasonably short
thermal relaxation times [3] which makes it possible to obtain mea
surements at equilibrium on a reasonable time scale. The transition
occurs at a temperature at which the techniques of high resolution
thermometry are very advanced. The transition temperature can be
determined with very high resolution by noting the onset of thermal
resistance in the fluid. [3] We are able to study the transition as
a function of pressure and ^Ee concentration. Thus, we have an
entire plane of phase transitions at our disposal. And finally, the
nature of the transition is such that by symmetry experiments are
always done on the coexistence curve. Whereas some of these advan
tages prevail also for other systems, no other phase transition can
lay claim to all of them. We can therefore expect to be able to put
the theory to a more detailed test by studying ^He near T^ than we
can hope to do by work near other phase transitions. In the study
of the superfluid transition we thus have yet another example of
work on liquid helium having a profound bearing upon a problem in
condensed matter physics which, viewed superficially, is only
peripherally related to the physics of liquid helium.
The sharpness of the phase transition at Τχ was demonstrated
already nearly two decades ago by the very beautiful measurements of
the specific heat at saturated vapor pressure of Buckingham,
Fairbank, and Kellers. [4] These measurements, together with other
results [5-7] over different temperature ranges, are shown in
Fig. 1. They indicate that the specific heat still increases with
decreasing |τ-Τ^| even when the temperature is within a few μΚ of
the transition. The nearly linear dependence of the specific heat
upon log |Τ-Τχ| demonstrates that the specific heat exponent α is
rather close to zero, [8] and the near parallelity of the two
1
log 1 τ-τ|
10 λ
-5 -4 -3 -2 -1 Ο
80 h
60 h
•ν
40 h
20 h
a
3 • n\aadPD
-5 -4 -3 -2
log io|T/Tx-i|
Figure 1. The heat capacity of liquid He at
saturated vapor pressure as a func
tion of log IΤ/Τχ-1I. The upper
branch is for Τ < Τχ, and the lower
branch is for Τ > Τχ. The data are
from: open circles, Ref. 4; open
squares, Ref. 5; open triangles,
Ref. 6; solid circles, Ref. 7.
After Ref. 7.
branches for Τ > Τχ and Τ < Τχ shows that the amplitude ratio
A/A" [8,9] is close to unity. Although we now know that α differs
slightly from zero and that the specific heat remains finite at Τχ,
and that A/A" is not quite equal to unity (we shall return to these
questions in later sections of this talk), the early measurements
already provided a remarkably accurate description of a singularity
near Τχ. Only a great deal of very detailed work has been able to
improve upon them. Nonetheless, this additional quantitative work
has yielded qualitatively new results about the nature of phase
transitions.
I would like to review in this talk some of the experimental
Critical phenomena and the superfluid transition in ^He 3
progress which has been made in the study of properties near Τχ
since the pioneering work of Buckingham, Fairbank, and Kellers, and
I will attempt to describe the relation between the experimental
results and the predictions of the RGT. For this purpose, I shall
use freely the predictions of the theory; but I shall not discuss
the theory for its own sake. In order to keep this review of
reasonable length, I shall limit myself to pure ^He where the most
quantitative results have been obtained, and to those properties
which are most directly related to the predictions of the theory. I
shall omit transport properties because they have not yet been
determined as accurately as equilibrium properties. Essentially,
this talk is then a review of the experimental results for the
superfluid density and the specific heat near Τχ in 4He as a func
tion of pressure. These two properties make contact with the pre
dictions of the RGT in a number of different ways; and the experi
mental values of the parameters which describe them provide us with
a very detailed test of the theory. In most instances, experiment
is found to be consistent with theory. In one particular case,
however, a theoretical prediction seems contrary to the measurements.
Elsewhere in this volume, Tom Greytak discusses the very
beautiful Rayleigh and Brillouin scattering measurements which he
and his collaborators have performed near the λ-line in ^He under
pressure. These measurements were made at a non-zero wave vector k
which is determined by the light wave length and the scattering
angle. I want to discuss experiments for which k = 0 (or negligibly
small). In order to show how the information obtainable from these
two types of measurements differs, and indeed how these experiments
complement each other, let us consider Fig. 2 which is the
temperature-wave-vector plane. [10] However, as a measure of the
distance from T^ along the temperature axis it is convenient to use
the inverse of the correlation length ξ = Ct~v where t Ξ Τ/Τχ-1.
0
Then the locus of kC = 1 is described by the two straight lines
which originate at Τχ and make a 45° angle with the axes. These
lines divide the k - ξ-1 plane into three regions in which the
behavior of various properties of the system differs in an essential
way. Of importance for the present purpose is mainly the fact that
the singularities associated with the phase transition occur only at
the point ξ"1 = 0 (Τ = Τχ) and k = 0. Thus, the light scattering
experiments, which follow a path of constant k such as the dashed
Τ Γ1 OR Τ
χ
Figure 2. The wave-vector temperature plane.
4 Ah le r s
line in Fig. 2, by their very nature are never able to approach the
critical point very closely. Indeed, we have seen that the shape
and position of the peaks in the Brillouin spectrum vary continuous
ly and without any indication of a singularity as the temperature
crosses Τχ. This is what we expect for finite wave vector. On the
other hand, the measurements for k > 0 can of course yield very
valuable information about the dependence upon kξ of the behavior of
various properties, and Greytak has discussed this in detail. How
ever, if one wants to study the singularities which occur at the
critical point, one must follow an experimental path which
approaches k = 0 at T. One can do this either for kξ < 1 (regions
c
I and III) or for kξ > 1. Whereas both cases are of considerable
interest, at present there is experimental information only for the
case kC << 1 because the measurements were made with k essentially
equal to zero.
Before we discuss some of the measurements; we shall consider
in Sect. II what we can expect about the general nature of critical
singularities. This will give us a clue about the type of function
to which one should fit experimental data. In Sect. Ill we will
examine measurements of the superfluid density. These results are
sufficiently detailed and precise to give us experimental informa
tion about the general nature of critical singularities. In
Sections IV and V we will compare measurements of the thermal expan
sion coefficient (which give the behavior of the specific heat via
thermodynamics) with the superfluid density results and the predic
tions of the RGT. Several predictions of the theory are found to be
consistent with the data. Finally, in Sect. VI we will examine a
combination of parameters which the theory predicts to be pressure
independent and find that experiment seems inconsistent with the
prediction.
II. Singularities Near Critical Points
The superfluid transition which occurs at Τχ is an example of a
critical point. Near critical points, many properties of a system,
when considered for k = 0 and e.g. as a function of temperature,
have singularities. Usually, these singularities will manifest
themselves in a divergence either of the property in question or of
certain of its derivatives. One problem faced by the experimental
ist is to extract from his data relevant parameters which describe
these singularities. This task would be relatively easy if theo
rists could provide us with explicit predictions, say in the form of
a function with a small number of parameters, for the behavior of
the property in question. Unfortunately, this usually is not the
case, and theoretical predictions pertain only to the asymptotic
behavior, for instance in the limit as t vanishes. Experiments, of
course, are always done only for |t| > 0 because they have finite
resolution. For a comparison with theory, it is therefore necessary
to compare results for |t| > 0 with predictions which pertain as t
vanishes. In order to accomplish this, the data have to be extra
polated to t = 0 by fitting them for |t| > 0 to relatively simple
functions such as the so-called "power laws" which contain only a
few parameters. One then hopes that the parameters extracted in
this manner from the data will also describe the asymptotic behavior
of the actual property in question. Let us examine somewhat more
explicitly to what extent this is justified.
Most singular functions of an argument, say t, are asymptoti
cally proportional to fλ where λ is a parameter which is called a
critical exponent. [11] We express this asymptotic proportionality
by writing
Critical phenomena and the superfluid transition in ^He 5
f(t) * |t| \ (1)
One way of stating in more detail what we mean by Eq. 1 is to say
that the limit
,. ainf
lim
* ο d)lnltl <2>
t
exists and has the value -λ. For a large class of functions the
limit Eq. 2 does exist, and it is not unreasonable to assume that
most properties encountered near critical points can be described by
Eq. 1 (there are known exceptions, however). Equation 1 does not
help us in the interpretation of our data; but it suggests that the
simplest approximation that we can make for |t| > 0 is to assume
that f(t) is, to a sufficiently good approximation, proportional to
|t|-\ i.e.
f(t) ^ f(t)
x
with
f(t) = f|t|~A (3)
1 0
for sufficiently small but experimentally accessible values of |t|.
Actually, of course, we can infer from Eq. 1 that
f(t) = f|t|"A[l+g(t)] (4)
0
where
lim g(t) = 0 . (5)
t -> 0
Using Eq. 3 is equivalent to saying that g(t) is negligible for
sufficiently small values of |t]. In general, one would expect this
to be the case for |t| << 1 if g(t) were a regular function of t;
but of course g(t) may itself be singular and in that case it could
be still large even for small |t|. In any event, until rather
recently Eq. 3 has been the basis of all data analysis near critical
points. We now know that its use may lead to erroneous conclusions
about the implications of experimental results with regard to the
nature of critical singularities. We shall discuss an explicit
example below in Sect. IV.
If we cannot use Eq. 3 as a useful approximation for the inter
pretation of experiments, we must make some simplifying assumption
about g(t) in Eq. 4. Here again the simplest, and probably valid,
assumption is that g(t) is asymptotically proportional to a power of
ItI, say
g(t) * tx . (6)
We can now make the next best approximation (after abandoning Eq. 3)
of assuming g(t) to be proportional to tx. Then we write
f(t) £ f(t)
2
with
6 Ahlers
f(t) = ft A[l+gtx] . (7)
2 Q Q
Although f (t) is only approximately equal to f(t) , it is a far
2
better approximation to f (t) than was f ]_ (t) .
In principle, one could of course carry this further and
include more and more terms in the fitting function. In practice,
Eq. 7 already contains more parameters than can be extracted from
most experiments with anything like usefully small statistical
errors, and only the most precise experimental data, taken at the
smallest possible values of |t|, warrant a comparison with Eq. 7.
One can hardly overstate the difficulties which are involved in
extracting meaningful parameters from a fit of experimental data to
Eq. 7. The function f2 obviously contains the parameters fg, gg, λ,
and x; but ÎQ, go, λ, and χ might assume different values for Τ > Τχ
and for Τ < Τχ. We therefore distinguish between the parameters on
the high temperature and low temperature sides of the transition by
using fg, go, λ", and x" for Τ < Τχ. In addition, Τχ must be deter
mined from the data unless it is known independently. For some
properties there might also be a regular background contribution to
the measurement which is not included in Eq. 7 and which might
require one or two more parameters. Thus, in principle we are faced
with having to extract a dozen or so parameters from a set of expe
rimental data. Even with the best measurements this would yield very-
large statistical errors for most of the parameters because of the
high statistical correlation between them. This difficulty is of
course the reason why in the past the much simpler function f^,
given by Eq. 3, has been used. We know now, however, that nature
does not always permit us this simplification, and find ourselves
forced to cope with the complexities of f2 (Eq. 7).
In practice, one normally finds that a fit even of very precise
and extensive data to Eq. 7 will yield parameter values with such
large statistical errors that a comparison with theoretical predic
tions becomes meaningless in the sense that the uncertainties in the
parameters are larger than the differences in the predictions for
different types of critical points. [14] In some of those cases, a
very fruitful compromise has been to use some of the available
theoretical predictions as constraints in the data analysis. This
has the effect of reducing the number of parameters to be determined
from experiment. Often, the results of the analysis can then be
used to compare with those additional theoretical predictions which
were not used in the data analysis. We shall see examples of this
procedure below, but let us make the idea somewhat more concrete at
this point. As an illustration, one might in the analysis of the
specific heat near Τχ assume, as predicted by theory, that the lead
ing exponent α (or a" for Τ < Τχ) is the same above and below Τχ. [12]
This reduces the number of parameters to be determined by one, and
results in a reduction of the statistical errors for all the para
meters. One can then still compare the value of a = a" with theoret
ical calculations. [13] In addition, one can examine the value of
α = α" for any dependence upon the pressure. Theory predicts α = α"
to be independent of pressure. [14] In the discussion of data in the
the rest of this paper, we shall when necessary, employ this
approach of using some of the theoretical predictions as input in
the data analysis, and of then comparing the results with the remain
ing theoretical predictions. The amount of theory going into the
analysis should of course be kept to a minimum. Occasionally, use
ful information can be obtained even without any theoretical input,
and in the following section we shall see an example of experiment
by itself giving significant information.
