A rich variety of Fermi systems condense by forming bound pairs, including
high temperature [1] and heavy fermion [2] superconductors, Sr2RuO4 [3], cold
atomic gases [4], and superfluid 3He [5]. Some of these form exotic quantum
states having non-zero orbital angular momentum. We have discovered, in the
case of 3He, that anisotropic disorder, engineered from highly porous silica
aerogel, stabilizes a chiral superfluid state that otherwise would not exist.
Additionally, we find that the chiral axis of this state can be uniquely
oriented with the application of a magnetic field perpendicular to the aerogel
anisotropy axis. At suffciently low temperature we observe a sharp transition
from a uniformly oriented chiral state to a disordered structure consistent
with locally ordered domains, contrary to expectations for a superfluid glass
phase [6].Comment: 6 pages, 4 figure, and Supplementary Informatio

AP Mackenzie

BH Moon

C Chin

C. A. Collett

D Rainer

D Vollhardt

DD Osheroff

DG Ferguson

DT Sprague

E Collin

EV Thuneberg

GE Volovik

J Elbs

J Pollanen

J. A. Sauls

J. I. A. Li

J. Pollanen

JA Sauls

JR Kirtley

JV Porto

K Matsumoto

RG Bennett

RH Heffner

RM Scanlan

T Tsuneto

VV Dmitriev

W. J. Gannon

W. P. Halperin

WF Brinkman

Y Dalichaouch

YM Bunkov

English

arXiv

A rich variety of Fermi systems condense by forming bound

pairs, including high-temperature and heavy-fermion superconductors,

Sr_2RuO_4 (ref. 3), cold atomic gases and superfluid

^3He (ref. 5). Some of these form exotic quantum states with

non-zero orbital angular momentum. We have discovered, in

the case of ^3He, that anisotropic disorder, engineered from

highly porous silica aerogel, stabilizes a chiral superfluid state

that otherwise would not exist. Furthermore, we find that the

chiral axis of this state can be uniquely oriented with the

application of a magnetic field perpendicular to the aerogel

anisotropy axis. At sufficiently low temperature we observe

a sharp transition from a uniformly oriented chiral state to a

disordered structure consistent with locally ordered domains,

contrary to expectations for a superfluid glass phase

Pollanen, J.

Li, J. I. A.

Collett, C. A.

Gannon, W. J.

Halperin, W. P.

Sauls, J. A.

Caltech Authors - Main

New chiral phases of superfluid ^3He stabilized by

anisotropic silica aerogel

Crossref

Nature Physics

New chiral phases of superfluid 3He stabilized by anisotropic silica aerogel

© 2012 Macmillan Publishers Limited.  All rights reserved. 
 
SUPPLEMENTARY INFORMATION
DOI: 10.1038/NPHYS2220
NATURE PHYSICS | www.nature.com/naturephysics 1
New chiral phases of superfluid 3He stabilized by
anisotropic silica aerogel
Supplementary Information:
“New Chiral Phases of Superfluid 3He Stabilized by Anisotropic
Silica Aerogel”
J. Pollanen, J.I.A. Li, C.A. Collett, W.J. Gannon, W.P. Halperin,∗ and J.A. Sauls
Department of Physics and Astronomy,
Northwestern University, Evanston, Illinois 60208, USA
(Dated: January 31, 2012)
∗ Correspondence should be addressed to W.P.H.: w-halperin@northwestern.edu
1
© 2012 Macmillan Publishers Limited.  All rights reserved. 
 
SI.1. GINZBURG-LANDAU THEORY
The Ginzburg-Landau (GL) free-energy functional for p-wave, spin-triplet pairing of 3He
quasiparticles in a homogeneous isotropic medium is defined in Ref. [SI1] in terms of the
3× 3 matrix order parameter, Aµj, which transforms as a vector under spin rotations with
respect to index µ, separately as a vector under orbital rotations with respect to index j,
∆ΩGL[A] =
∫
V
d3r
{
α¯(T ) TrAA† + β¯1 | TrAAtr |2 +β¯2(TrAA†)2
+β¯3 TrAA
tr(AAtr)∗ + β¯4 Tr (AA†)2 + β¯5 TrAA†(AA†)∗
}
. (SI1)
The material coefficient, α¯(T ) ' (T − Tca)α¯′, depends on the single spin density of states
at the Fermi surface, NF , the pairing interaction, and the effects of scattering by the ran-
dom impurity potential, all of which determine the transition temperature, Tca. The prime
denotes differentiation with respect to temperature. The fourth-order material parameters,
{β¯i | i = 1 . . . 5}, which are also sensitive to impurity scattering, determine condensation
energy and the structure and residual symmetry of the superfluid condensate.
The homogeneous isotropic scattering model (HISM) of Thuneberg et al. [SI1] was ex-
tended by Sauls and Sharma in Ref. [SI2] to incorporate the effect of correlations between
silica strands and clusters. In this model, the aerogel correlation length, ξa, limits the pair-
ing transition for small superfluid coherence lengths, ~vF/2pikBTca < ξa (high pressures),
while the mean free path, λ, determines the transition when the superfluid coherence length
is larger than the aerogel correlation length. This condition is achieved at low pressures,
and always in the limit Tca → 0. Indeed this limit can be used to determine λ from the
critical pressure at which Tca(Pc) = 0. The two pair-breaking length scales define an ef-
fective pair-breaking parameter, x˜ = x/(1 + ζ2a/x), ζa = ξa/λ where x = ~vF/2pikBTλ is
the depairing parameter in the HISM [SI3] and vF is the pressure dependent Fermi velocity
of 3He [SI4]. The superfluid transition temperature in aerogel, Tca, is determined by the
condition α¯(Tca) = 0 in the following equation,
α¯(T ) =
1
3
NF
[
ln(T/Tc)− 2
∞∑
n=0
(
1
2n+ 1 + x˜
− 1
2n+ 1
)]
, (SI2)
where Tc is the transition temperature of pure
3He. In the limit x˜ = 0, Eq. SI1 reduces to
the case for pure 3He, namely α = (1/3)NF ln(T/Tc).
2
© 2012 Macmillan Publishers Limited.  All rights reserved. 
 
42
41
40
39
38
37
36
35
120118116114112110108106
1.4
1.3
1.2
1.1
1.0
0.9
0.8
S
ta
n
d
a
rd
 D
e
v
ia
tio
n
 o
f  ξ
a  (n
m
)
λ (nm)
ξ a 
(n
m
)
λ = 113 nm
ξa = 39 nm
FIG. SI1. Determination of Scattering Parameters Aerogel correlation length (red circles)
and standard deviation of ξa as a function of λ (blue squares). The minimum in the standard
deviation corresponds to λ = 113 nm and ξa = 39 nm.
The correlation length is a pressure independent length scale of the aerogel microstruc-
ture. We have used Eq. 2 to fit our measurements of Tca(P ) in the stretched aerogel (Fig. 1c)
with the constraint that ξa be pressure independent. From this fit we find λ = 113 nm and
ξa = 39 nm. In Fig. SI1 we present the standard deviation of ξa(P ) as a function of λ.
Using these same values of λ and ξa we have calculated the square of the axial state order
parameter amplitude in aerogel from the GL theory,
∆2Aa0 = ∆
2
A0
Tca
Tc
α¯′(Tca)
α′(Tc)
βA
β¯A
(SI3)
In Eq. SI2 ∆2A0 is the value for the pure A-phase [SI5, SI6], and βA (β¯A) is the combination of
β-parameters appropriate to pure (aerogel) 3He-A, i.e. βA = β2+β4+β5 (β¯A = β¯2+ β¯4+ β¯5).
The pure β-parameters were taken from Ref. [SI7]. For calculating the aerogel β¯-parameters,
we have extended the results of Thuneberg et al. [SI1] for the HISM, to include the combined
effects of the aerogel correlation length and mean-free path using the correlated scattering
model of Ref. [SI2].
SI.2. NMR LINE SHAPE ANALYSIS
We have performed an analysis of the NMR line shape to obtain the distribution of the
chiral axis, ~`, in our stretched sample. A distribution in the direction of ~` relative to the
external magnetic field, ~H, will lead to a distribution in the intrinsic superfluid dipole field
3
© 2012 Macmillan Publishers Limited.  All rights reserved. 
 
and hence a distribution of frequency shifts in the NMR spectrum. If θ = cos−1(ˆ`· Hˆ) is
the angle between the chiral axis and the magnetic field, then for small tip angle, β, the
frequency shift in the A-phase reduces to [SI8],
∆ωA(θ) = − Ω
2
A
2ωL
cos(2θ). (SI4)
For example, if the chiral axis orientation is unique and the normal state lineshape is given
by Fn(ω − ωL) then the lineshape in the superfluid, Fs, is unchanged except for a constant
shift, i.e. Fs = Fn(ω − ωL −∆ωA) where ∆ωA is given by Eq. SI3. However, for an angular
distribution of ~`, P (θ), there is a distribution of frequency shifts, P (ω), determined by
Eq. SI3. This results in a more complex NMR lineshape that is given by the convolution
product of the normal state line and P (ω),
Fs(ω) =
∫
P (ω
′
)Fn
(
ω − ωL − ω′
)
dω
′
. (SI5)
Since the normal state and superfluid spectra are obtained experimentally, P (θ) can be
determined by fitting the spectrum in the superfluid phase with Eq. SI4.
In general an inversion problem, like this one, requires constraints which make it difficult
to determine the uniqueness of the distribution. We choose the model that the ~`-distribution
can be represented by a sum of gaussian functions with adjustable position, θi, width, wi,
and relative weight, Ai,
P (θ) =
∑
i
Aie
(
θ−θi
wi
)2
, (SI6)
and
∫
P (θ) dθ = 1. In Fig. 4a we present NMR spectra (bold green curves) and correspond-
ing fits (dashed black curves) for temperatures above (left panel) and below (right panel)
the disorder transition Td with the results for P (θ) shown in Fig. 4b. For Td < T < Tca
we find that the spectrum is well-represented with a distribution centered at θ = 90◦, cor-
responding to the dipole-locked configuration, with a spread of 17.8◦ ± 3.6◦. For T < Td,
we find that the ~`-distribution is composed of domains as described in the text. Approxi-
mately 1/3 of the sample remains with ~` perpendicular to ~H with a spread . 4◦ as shown
in Fig. SI2. where we present the squared residual of the fit as a function of the width of
the dipole-locked peak while keeping the other parameters fixed. The remaining ∼ 2/3 of
the superfluid has θ = 43.8◦ ± 0.1◦ with a width of 5.0◦ ± 0.8◦ as shown in Fig. SI3. This
distribution fits our spectrum well for T < Td. To check its uniqueness we have attempted
4
© 2012 Macmillan Publishers Limited.  All rights reserved. 
 
w (degree)
θ = 90o (dipole locked peak)
S
q
u
a
re
d
 R
e
s
id
u
a
l 
(x
 1
0
6
 a
rb
. 
u
n
it
s
)
T < Td
14
13
12
11
10
14121086420
FIG. SI2. Squared residual versus the width of the dipole-locked peak for T < Td.
θ (degree) A (%) w (degree)
a b c
90
80
70
60
50
40
30
20
10
0
50484644424038
90
80
70
60
50
40
30
20
10
0
7068666462605856
20
18
16
14
12
10
14121086420
S
q
u
a
re
d
 R
e
s
id
u
a
l 
(x
 1
0
6
 a
rb
. 
u
n
it
s
) dipole-unlocked
peak position
dipole-unlocked
peak weight
dipole-unlocked
peak width
FIG. SI3. Squared residual versus the a) position, b) percent weight, and c) width of the dipole-
unlocked peak for T < Td.
to add an additional peak with variable relative weight, wv, at the position θ = 67
◦, with
a width of θ = 6◦ and we find that the best fit to the spectrum corresponds to wv = 0
as shown in Fig. SI4a. To emphasize this point, in Fig. SI4b we show the lineshape that
results for the case wv = 15% that clearly adds substantial weight in the spectrum which
is inconsistent with our experiment. We infer, therefore, that the disordered state has an
~`-distribution which is non-monotonic with three principal components as shown in Fig. 4b.
5
© 2012 Macmillan Publishers Limited.  All rights reserved. 
 
14
12
10
8
6
4
2
0
3020100
wv (%)
T < Td
peak at θ = 67o
S
q
u
a
re
d
 R
e
s
id
u
a
l 
(x
 1
0
7
) 30
25
20
15
10
5
0
N
M
R
 I
n
te
n
s
it
y
 (
a
rb
. 
u
n
it
s
)
-1.0 -0.5 0.0 0.5 1.0
Frequency Shift (kHz)
T < Td
Fit with wv =15%
 Tca < T
a b
35
FIG. SI4. Attempt to add an additional peak a) Squared residual versus the weight of an
additional peak at θ = 67◦. b) Fit (dashed black curve) to the NMR spectrum below Td (bold
green curve) produced with wv = 15%. The normal state line is presented as the fine red curve
and the fit is performed using Eq. SI4. Significant discrepancy between the best fit and the NMR
spectrum indicates that extra weight in the ~`-distribution between 44◦ and 90◦ can be ruled out.
[SI1] Thuneberg, E. V., Yip, S. K., Fogelstro¨m, M. & Sauls, J. A. Models for superfluid 3He in
aerogel. Phys. Rev. Lett. 80, 2861 (1998).
[SI2] Sauls, J. A. & Sharma, P. Impurity effects on the A1-A2 splitting of superfluid
3He in aerogel.
Phys. Rev. B 68, 224502 (2003).
[SI3] Our definition of x is a factor of 2 larger than that in Ref. [SI1].
[SI4] Halperin, W. P. & Varoquax, E. Helium Three (Elsevier, 1990).
[SI5] Schiffer, P. E. Studies of the Superfluid Phases of Helium Three and The Magnetization of
Thin Solid Films of Helium Three. Ph.D. thesis, Stanford University (1993).
[SI6] Rand, M. R. Nonlinear Spin Dynamics and Magnetic Field Distortion of the Superfluid
3He-B Order Parameter. Ph.D. thesis, Northwestern University (1996).
[SI7] Choi, H., Davis, J. P., Pollanen, J., Haard, T. M. & Halperin, W. P. Strong coupling
corrections to the Ginzburg-Landau theory of superfluid 3He. Phys. Rev. B 75, 174503
(2007).
[SI8] Bunkov, Y. M. & Volovik, G. E. On the possibility of the homogeneously precessing domain
in bulk 3He-A. Europhys. Lett. 21, 837–843 (1993).
6


New Chiral Phases of Superfluid 3He Stabilized by Anisotropic Silica
  Aerogel

New Chiral Phases of Superfluid 3He Stabilized by Anisotropic Silica Aerogel

Abstract

Similar works

Full text

Available Versions

Caltech Authors - Main

Crossref

Caltech Authors - Main