Twofold spontaneous symmetry breaking in the heavy-fermion superconductor UPt3.
ABSTRACT The field-orientation dependent thermal conductivity of the heavy-fermion superconductor UPt3 was measured down to very low temperatures and under magnetic fields throughout the distinct superconducting phases: B and C phases. In the C phase, a striking twofold oscillation of the thermal conductivity within the basal plane is resolved reflecting the superconducting gap structure with a line of node along the a axis. Moreover, we find an abrupt vanishing of the oscillation across a transition to the B phase, as a clear indication of a change of gap symmetries. We also identify extra two line nodes below and above the equator in both B and C phases. From these results together with the symmetry consideration, the gap function of UPt3 is determined as a E(1u) representation characterized by a combination of two line nodes at the tropics and point nodes at the poles.
arXiv:1107.3082v1 [cond-mat.supr-con] 15 Jul 2011
Twofold spontaneous symmetry breaking in a heavy fermion superconductor UPt3
Y. Machida,1A. Itoh,1Y. So,1K. Izawa,1Y. Haga,2E. Yamamoto,2
N. Kimura,3Y. Onuki,2,4Y. Tsutsumi,5and K. Machida5
1Department of Physics, Tokyo Institute of Technology, Meguro 152-8551, Japan
2Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan
3Department of Physics, Tohoku University, Sendai 980-8577, Japan
4Department of Physics, Osaka University, Toyonaka 560-0043, Japan
5Department of Physics, Okayama University, Okayama 700-8530, Japan
(Dated: July 18, 2011)
The field-orientation dependent thermal conductivity of the heavy-fermion superconductor UPt3
was measured down to very low temperatures and under magnetic fields throughout three distinct
superconducting phases: A, B, and C phases. In the C phase, a striking twofold oscillation of the
thermal conductivity within the basal plane is resolved reflecting the superconducting gap structure
with a line of node along the a axis. Moreover, we find an abrupt vanishing of the oscillation across
a transition to the B phase, as a clear indication of a change of gap symmetries. We also identify
extra two line nodes below and above the equator in both B and C phases. From these results
together with the symmetry consideration, the gap function of UPt3 is conclusively determined as a
E1u representation characterized by a combination of two line nodes at the tropics and point nodes
at the poles.
PACS numbers: 74.20.Rp, 74.25.fc, 74.70.Tx
Spontaneous symmetry breaking is one of the funda-
mental paradigms encompassing from condensed matter
physics to high energy physics, constituting the founda-
tion of modern physics. This paradigm is crucial some-
times because it can give a handle to discover some un-
known exotic ordered phase. This is particularly true
when broken symmetry is extremely low, that is, the
“residual symmetry” is so small, one may effectively and
self-evidently narrows down possible ordered phase to
Understanding the unconventional superconductivity,
in which electron pairs are formed without phonon, has
been a challenge. Part of the problem in uncovering the
mechanism is that little is known about the pairing sym-
metry. The heavy-fermion superconductor UPt3 is one
of the examples whose pairing symmetries are as yet to
be clarified. The most intriguing feature of this material
is the existence of a multiple phase diagram; UPt3 un-
dergoes a double superconducting transition at the upper
critical temperature T+
c∼ 540 mK into the A phase and
at the lower critical temperature T−
B phase . In addition, the third (C) phase is stabi-
lized at low temperatures under high magnetic fields .
A crucial role of a weak antiferromagnetic order below
TN∼ 5 K for the phase multiplicity is indicated by the
pressure studies . Power law dependence of the ther-
modynamic and transport quantities reveal the presence
of nodes in the superconducting gap [4–7]. Moreover, a
possibility of an odd-parity pairing is inferred from the
nuclear magnetic resonance studies of the Knight shift 
and is supported theoretically  by eliminating the sin-
glet even parity scenario.
Extensive theoretical efforts have been devoted to ex-
plain these disparate experimental results [9–11]. Among
c ∼ 490 mK into the
them, the E2u scenario with a line node in the basal
plane and point nodes along the c axis has been regarded
as one of the promising candidates . Several exper-
imental results, such as the anisotropy of the thermal
conductivity  and the ultrasonic attenuation  as
well as the recent small-angle neutron scattering  and
the Josephson tunnel junction , have been claimed
to be compatible with this model. On the other hand,
there exist some controversies in explaining the following
experiments; 1) the spontaneous internal field due to the
broken time-reversal symmetry is most likely absent ,
2) the d-vector has two components in the B phase , 3)
a point where the three superconducting phases meet is a
tetracritical point . Moreover, to date no experimental
evidence for the gap structure of each phase associated
with the E2umodel has been provided. The pairing sym-
metry of UPt3, therefore, remains unclear.
One of the most conclusive ways to identify the pairing
symmetry is to elucidate the gap structure by the ther-
mal conductivity measurements with rotating magnetic
fields relative to the crystal axes deep inside the super-
conducting state. This technique has been successful to
probe the nodal gap structure of several unconventional
superconductors by virtue of its directional nature and
sensitivity to the delocalized quasiparticles . In this
paper, we present a decisive experiment of the angular
dependence of the thermal conductivity of UPt3reveal-
ing the spontaneous rotation symmetry lowering, namely
the unusual gap structure with a lower rotational sym-
metry than the crystal structure.
High quality single crystal of UPt3with the high resid-
ual resistivity ratio of 800 was grown by the Czochralski
pulling method in a tetra-arc furnace . We measured
the thermal conductivity along the hexagonal c axis (heat
current q ? c) on the sample with a rectangular shape (3
× 0.42 × 0.4 mm3). To apply the magnetic fields with
high accuracy relative to the crystal axes, we used a sys-
tem with two superconducting magnets generating the
fields in two mutually orthogonal directions. The mag-
nets are installed in a Dewar seating on a mechanical
rotating stage, enabling the continuous rotation of the
First, we begin with demonstrating that the thermal
conductivity (κ) well probes the superconducting quasi-
particle (QP) structures from its temperature (T) and
magnetic field (H) dependences.
hexagonal [¯12¯10], [¯1010], and  axes are denoted as
the a, b, and c axes, respectively. The inset of Fig. 1
shows the T dependence of κ(T)/T under zero field and
3 T along the b axis. With decreasing T, the zero-field
κ(T)/T shows a steep increase up to ∼ 0.3 K without
apparent anomalies at T+
ing, κ(T)/T considerably decreases due to a reduction
of the QP densities, and takes an extremely small value
at the lowest T ∼ T+
measurements . In the normal state (3T), κ(T)/T ap-
pears to continuously increase down to the lowest T. The
dashed line denotes κ(T)/T obtained from the normal-
state resistivity ρ(T) using the Wiedemann-Franz law,
κ(T)/T = L0/ρ(T) (L0: the Lorentz number). Impor-
tantly, we confirm that κ(T)/T is close to L0/ρ(T) at
low temperature T < 100 mK, indicating the dominant
electronic contribution in the heat transport.
T-range, the H dependence of the thermal conductivity
κ(H)/T at 55 mK shows a remarkable H-linear depen-
dence at low fields for both c and b directions (the main
panel of Fig. 1) in contradiction to the field-insensitive
behavior of fully gapped superconductors except in the
vicinity of Hc2, providing evidence for the nodal su-
perconductivity in UPt3.
In addition, we find distinct anomalies associated with
a transition from the B to C phase at HBC(open arrows).
The fact that the BC transition manifests by a sharp
change of the slope implies a suppression of one of the
degenerate order parameter components in the B phase.
This behavior can be more clearly resolved for the b axis.
The determined HBCtogether with Hc2denoted by the
solid arrows are summarized in Fig. 3(d) for H ? b. We
also note that a striking anisotropy is found in κ(H)/T
at 55 mK near Hc2: κ/T for H ? c shows a rapid increase
just below Hc2, while the one for H ? b linearly increases
up to Hc2, as similarly observed in Sr2RuO4 .
search of the relevance of this behavior to the odd-parity
superconductivity is a fascinating issue to be addressed.
From now on, the
c.On further cool-
c/20, consistent with the previous
Next, to shed light on the nodal topology in the su-
perconducting phases, we concentrate on the angular de-
pendence of κ. The most significant effect on the thermal
transport for nodal superconductors in the mixed state
comes from the Doppler shift of the QP energy spectrum,
κ /T (W/K2m)
H // c
H // b
H // b
κ /T (W/K2m)
FIG. 1: (color online). Magnetic field dependence of the ther-
mal conductivity κ(H)/T along the c and b axes at various
temperatures. The open and closed arrows represent the B →
C transitions HBC and the upper critical fields Hc2, respec-
tively. Inset: temperature dependence of κ(T)/T under zero
field and at 3 T for H ? b. The dashed line shows κ/T = L0/ρ
(L0: the Lorentz number) obtained from the normal-state re-
sistivity ρ using the Wiedemann-Franz law.
E(p) → E(p) − vs· p, in the circulating supercurrent
flow vs.This effect becomes important at such posi-
tions where the gap becomes smaller than the Doppler
shift term (∆ < vs· p). The maximal magnitude of the
Doppler shift strongly depends on the angle between the
node direction and H, giving rise to the oscillation of
the density of states (DOS). Consequently, κ attains the
maximum (minimum) value when H is directed to the
antinodal (nodal) directions . Figure 2 shows κ(φ)
normalized by the normal state value κn as a function
of the azimuthal angle φ at 50 mK (∼ T+
= (a) 3.0 T, (b) 1.0 T, and (c) 0.5 T, respectively. The
data are taken in rotating H after field cooling at φ =
-70◦, and κnis measured at 50 mK above Hc2for H ? b.
In the normal state (3.0 T) and the B phase (0.5 T), we
find no φ-dependence within experimental error.
c/10) at |µ0H|
By contrast, what is remarkably is that κ(φ) exhibits a
distinct twofold oscillation with a minimum at φ = 0◦in
the C phase (1.0 T). The open circles are obtained under
field cooling condition at each angle. The data obtained
by different procedures of field cooling coincide well with
each other, indicating negligibly small effect of the vor-
tex pinning. Strikingly, since the twofold symmetry is
lower than the hexagonal crystal structure, the in-plane
anisotropy of the Fermi surface and Hc2 is immedi-
ately ruled out as the origin. As shown by the solid lines,
κ(φ) can be decomposed into two terms; κ(φ) = κ0+κ2φ,
where κ0is a φ-independent term and κ2φ= C2φcos2φ
is a twofold component. Figure 2(e) shows the ampli-
tude of the twofold component |C2φ/κn| as a function of
H // b
H // aH // b
|µ0H| = 3.0 T
θ = 90 deg.
T = 50 mK
0 0.20.4 0.6
B phaseC phase
1 % of κn
FIG. 2: (color online). Angular variation of the thermal con-
ductivity κ(φ) normalized by κnat 50 mK as a function of the
azimuthal angle φ for |µ0H| = (a) 3.0 T, (b) 1.0 T, and (c)
0.5 T, respectively. κ(φ) is measured with rotating H within
the ab plane (the polar angle θ = 90◦) as schematically shown
in (d), where φ and θ are measured from the a and c axes,
respectively, and q is injected along the c axis. The solid lines
show the twofold component in κ(φ)/κn. The open circles
represent κ(φ)/κn at 1 T obtained under the field cooling
condition at every angle. (e) Field variation of the twofold
amplitude |C2φ/κn| at 50 mK at θ = 90◦(solid circles) and
63◦(open circle), respectively.
H/Hc2, where Hc2= 2.6 T for H ? b. It can be clearly
seen that |C2φ/κn| suddenly appears to be finite ∼ 3%
in the C phase, implying a change of the gap symmetries
across the BC transition that is of second order. We note
that |C2φ/κn| obtained by rotating H conically around
the c axis at fixed θ = 63◦is same order of magnitude
with the values at θ = 90◦as denoted by an open circle
in Fig. 2(e).
To further elucidate the gap symmetry, we present
the polar angle (θ) dependence of κ in Fig. 3, showing
κ(θ)/κn measured by rotating H within the ac plane
(green circles) and the bc plane (orange circles) at 50
mK at |µ0H| = (a) 1.5 T, (b) 1.0 T, and (c) 0.5 T. Here,
κnis measured at 50 mK above Hc2for H ? c. The dom-
inant twofold oscillation is found in all the fields with
maxima at θ = 90◦, which could be attribute to, such as
the Fermi surface and/or the gap topology or the differ-
ence in transport with H parallel to and normal to the
heat current q. Regardless of the origin, the fact that
κ(θ)/κn is maximized at θ = 90◦excludes an artificial
origin of the in-plane twofold oscillation in the C phase
due to a misalignment of H relative to q. We thus con-
clude that the in-plane twofold symmetry in the C phase
is a consequence of the node.
In the B phase (0.5 T), the two different scanning pro-
cedures within the ac and bc planes well converge with
each other, consistent with the φ-independence of κ. In
H // c
H // c
H // a
H // b
HH H H H
|µ0H| = 1.5 T
T = 50 mK
1 % of κn
H // b
FIG. 3: (color online). Angular variation of the thermal con-
ductivity κ(θ) normalized by κn at 50 mK as a function of
the polar angle θ for |µ0H| = (a) 1.5 T, (b) 1.0 T, and (c)
0.5 T, respectively. The κ(θ)/κn curves measured by rotating
H (inset of (a)) within the ac plane (green circles) and the
bc plane (orange circles) are simultaneously plotted. Inset of
(c): ∆κ(θ)/κn ≡ (κ(θ) − κ0− κ2θ)/κn vs θ plot at 50 mK at
0.5 T, where κ0 is a θ-independent term and κ2θ = C2θcos2θ
is a twofold component . (d) The phase diagram of UPt3
with the three superconducting phases, labelled A, B, and C,
for H ? b. The red and blue circles represent HBC and Hc2,
respectively, deduced from the present measurements. The
schematic shapes of the gap symmetries for each phase are
addition, we find extra two minima at θ = 20◦and 160◦.
By plotting ∆κ(θ)/κn≡ (κ(θ) − κ0− κ2θ)/κnvs θ after
the subtraction of κ0and κ2θ= C2θcos2θ, the minima
become clearly visible at 35◦and 155◦(Fig. 3(c), inset).
This double-minimum structure is also found in the C
phase (Fig. 3(a)). We infer that these minima are de-
rived from the two horizontal line nodes at the tropics as
discussed below. In contrast to the B phase, the two scan-
ning results do not coincide in the C phase (Fig. 3(a)); the
difference is diminished at the poles and maximized at θ
= 90◦, being consistent with the in-plane twofold sym-
metry. Moreover, a significant appearance of the twofold
symmetry across the BC transition can be seen at 1.0 T
(Fig. 3(b)), in which one experiences the BC (CB) transi-
tion twice by varying θ because of the anisotropy of HBC.
Indeed, the transitions occur at θ = 30◦and 150◦taking
distinct kinks. Remarkably, the difference between the
two scanning procedures becomes finite upon entering
the C phase, providing the compelling evidence for the
twofold symmetry of the gap structure in the C phase.
Moreover, the fact that |C2φ/κn| takes same order of the
magnitude at θ = 90◦and 63◦is in favor of a line node
along the a axis rather than the point nodes in the basal
plane. Notably, although a mechanism which fixes do-
mains is a puzzle, the in-plane twofold symmetry of κ(φ)
indicates a single superconducting domain.
We discuss the order parameter symmetry of UPt3
within the triplet category. The present experiments in-
dicate (i) the line node along the a axis in the C phase,
(ii) the absence of in-plane gap anisotropy in the B phase,
and (iii) the two line nodes at the tropics in both B and
C phases. Taking into account all these results and the
d-vector configurations assigned by the Knight shift ,
the order parameter is unambiguously determined with
a form of (kaˆb + kbˆ c)(5k2
and ˆ c are unit vectors of the hexagonal axes representing
the directions of d-vectors. This state belongs to two-
dimensional E1urepresentation with the f-wave charac-
ter, the so-called planar state in triplet pairing in the D6h
hexagonal symmetry, and to degenerate Eustate for the
recent claimed D3dtrigonal symmetry [23, 24]. The gap
structure consists of the two horizontal line nodes at the
tropics (kc= ±1/√5, θ = 63◦and 117◦) and the point
nodes at the poles (ka = kb = 0). Note that although
the locations of the horizontal line nodes estimated by
assuming a spherical Fermi surface do not agree with the
observation (θ = 35◦and 155◦), it could be changed by
considering the realistic Fermi surface .
By lifting the doubly degeneracy, the order parameter
for the C phase is given by kbˆ c(5k2
c−1) for H ? c, respectively. In the same manner,
c− 1) state is readily assigned for the A phase.
The schematic shapes of the gap symmetries in the three
phases are shown in Fig. 3(d). We emphasize that this
state is compatible not only with the hybrid gap state
indicated by the several experiments [13, 14], in the sense
that the line and point nodes simultaneously exist, but
also with some experimental results for which the E2u
model  has failed to describe, i.e., the absence of the
internal magnetic field , the two-component d-vector
for the B phase , and the tetracritical point in the
phase diagram .
To further strengthen our identification, in particular
on the existence of the horizontal line nodes on the trop-
ics, we calculate the angle-resolved DOS by solving the
Eilenberger equation  for several possible gap func-
tions. We compare here putative three gap functions in
the C phase relative to the data in Fig. 4 where κ(θ)/κn
and the DOS differences along the vertical nodal and
antinodal θ-scannings are depicted.
structure characteristic in E2u and E1g whose origin
comes from the horizontal node on the equator is not sup-
ported by the data that are consistent with the present
E1uwith the horizontal nodes on the tropics. In view of
the Doppler shift idea mentioned above the QPs in the
horizontal node on the equator contribute more when the
field direction is away from θ = 90◦.
In summary, we find striking twofold oscillations in
angle-resolvedthermal conductivity measurements at low
temperatures in a strongly correlated heavy fermion su-
perconductor UPt3. This spontaneous symmetry low-
ering, which is the lowest possible rotational symmetry
c− 1) for the B phase, whereˆb
c− 1) for H ? ab and
The double peak
(κ (θ, φ = 90) - κ (θ, φ = 0))/ κn
(Nantinode - Nnode)/Nn
FIG. 4: (color online). θ-dependence of the thermal conduc-
tivity obtained by subtracting the green data from the orange
data in Fig. 3(a) (left axis) and the density of states difference
normalized at θ = 90◦(right axis, arbitrary scale) along the
vertical nodal and antinodal scannings for three possible gap
functions in the C phase: 1. The present E1u (kb(5k2
2. E1g (kbkc), 3. E2u (kakbkc). Those gap structures are
sketched in the inset.
breaking in hexagonal crystal fortuitously and effectively
narrows down the possible symmetry classes and leads us
to uniquely identify the pairing symmetry for each phase
in the multiple phase diagram. We conclude that the
realized pairing function is E1uwith the f-wave charac-
ter, i.e., the so-called planar state in the triplet pairing.
This state is analogous to the B phase in superfluid3He,
and obviously bears the Majorana zero mode at a sur-
face [26, 27], namely a topological superconductor that
is quite rare to find. Thus it is worth exploring further
to understand this interesting material as a new platform
for topological physics.
We acknowledge insightful discussions with T. Ohmi,
M. Ozaki, M. Ichioka, H. Kusunose, and K. Ueda.
This work is partially supported by grants-in-aid from
the Japan Society for the Promotion of Science; by
grants-in-aid for Scientific Research on Innovative Ar-
eas “Heavy Electrons” (20102006) from the Ministry
of Education, Culture, Sports, Science, and Technology
(MEXT), Japan; and by Global COE Program from the
MEXT through the Nanoscience and Quantum Physics
Project of the Tokyo Institute of Technology.
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