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arXiv:1107.3082v1 [cond-mat.supr-con] 15 Jul 2011

APS/123-QED

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

identify.

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 [1]. In addition, the third (C) phase is stabi-

lized at low temperatures under high magnetic fields [2].

A crucial role of a weak antiferromagnetic order below

TN∼ 5 K for the phase multiplicity is indicated by the

pressure studies [3]. 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 [8]

and is supported theoretically [9] 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 [12]. Several exper-

imental results, such as the anisotropy of the thermal

conductivity [13] and the ultrasonic attenuation [14] as

well as the recent small-angle neutron scattering [15] and

the Josephson tunnel junction [16], 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 [17],

2) the d-vector has two components in the B phase [8], 3)

a point where the three superconducting phases meet is a

tetracritical point [2]. 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 [18]. 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 [19]. We measured

the thermal conductivity along the hexagonal c axis (heat

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2

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

magnetic fields.

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 [0001] 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+

c

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 [7]. 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[20], 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 [21].

search of the relevance of this behavior to the odd-parity

superconductivity is a fascinating issue to be addressed.

From now on, the

and T−

c. On further cool-

c/20, consistent with the previous

In this

A

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,

0

2

4

6

8

10

12

κ /T (W/K2m)

55 mK

H // c

HBC

Hc2

µ0H (T)

410mK

55 mK

200mK

300mK

H // b

0

2

4

6

8

00.2 0.4

T (K)

0.6 0.8

3 T

0 T

H // b

Tc

κ /T (W/K2m)

L0/ρ

0123

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 [22]. 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[12] 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

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3

q

H

φφ φ

a

b

c

(d)

H // b

1.1

H // aH // b

0.9

1

|µ0H| = 3.0 T

θ = 90 deg.

T = 50 mK

normal phase

0.4

0.5

κ (φ)/κn

C phase

1.0 T

0.2

0.3

-90090

φ (deg.)

B phase

0.5 T

0

1

2

3

00.2 0.40.6

H/Hc2

0.811.2

|C2φ/κn| (%)

B phaseC phase

normal

state

(a)

(b)

(c)

(e)

θ θ

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

0.5

H // c

H // a

H // b

cq

HH H H H

H

θ

b

a

(a)

|µ0H| = 1.5 T

T = 50 mK

(b)

1.0 T

(c)

0.5 T

1 % of κn

κ (θ)/κn

θ (deg.)

0.3

0.4

C phase

0.2

0.3

0.1

0.2

090 180

B phase

-0.01

0

0.01

090180

θ (deg.)

∆κ (θ)/κn

normal phase

(d)

0

1

2

3

0 0.20.40.6

µ0H (T)

C

B

A

H // b

T (K)

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

shown.

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.

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4

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 [8],

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 [12].

By lifting the doubly degeneracy, the order parameter

for the C phase is given by kbˆ c(5k2

kbˆ a(5k2

c−1) for H ? c, respectively. In the same manner,

kaˆb(5k2

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 [9] has failed to describe, i.e., the absence of the

internal magnetic field [17], the two-component d-vector

for the B phase [8], and the tetracritical point in the

phase diagram [2].

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 [25] 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

331122

(κ (θ, φ = 90) - κ (θ, φ = 0))/ κn

(Nantinode - Nnode)/Nn

θ (deg.)

-0.05

0

0.05

0

090180

C phase

3

1

2

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.

c− 1)),

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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