188
J.-P. Brison
6.8.2.3 E 1u Representation
However, NMR experiments seem to be in contradiction with this interpretation of
the crossing of the upper critical fields of UPt 3 . Indeed Knight-shift measurements,
which are the closest to a measure of the spin susceptibility in the superconducting
state, found no change in the superconducting state for fields applied along the a
or b directions in the C-phase, but also no change for field along the c-axis except
at very low field deep inside the B-phase [11]. Therefore, other models have been
proposed for UPt 3 , which are much closer to the situation of superfluid
3 He, with a
weak spin–orbit coupling allowing for a field reorientation of the d-vector as long
as the field is ‘strong enough’. Knight-shift measurements can tell nothing on the
gap nodes, but combining angle-dependent thermal conductivity measurements [10]
with the NMR result, a E 1u scenario has been proposed, predicting a non-chiral state
[d(k) ∝ (5 ˆ
k
2
c − 1)( ˆ
k a e b + ˆ
k b e c ) in the B-phase]. This scenario can also more easily
give account of some other features of the phase diagram (like the existence of a
tetracritical point in all field directions): see Fig. 6.5. For the different phases, in this
model, the d-vector would be as shown in Table 6.1.
So, as can be seen from Table 6.1, at low fields the d-vector does not depend on
the field orientation [check the A-phase and B-phase-(low H ) lines of the table]. In
the B-phase, for H c, there is a field-induced reorientation of the d-vector: at low
field, with d ∝ ( ˆ
k a e b + ˆ
k b e c ), the d-vector is not perpendicular to the c-axis (except
on the line ˆ
k b = 0) so there is a finite |S z = 0 component of the spin along the
field. This would be imposed by the orbital part of the wave function and spin–orbit
interaction or coupling to the small antiferromagnetic moments acting as a symmetry
breaking field. But for fields above 0.22 T, with d ∝ ( ˆ
k a e b + ˆ
k b e a ), the d-vector is
real and always perpendicular to the c-axis, so we know that it is equivalent to an
ESP state in that direction. Hence, the field-induced rotation of the d-vector. In
the high-field C-phase, where the Pauli limitation could be at play, we note that in
Table 6.1 the d-vector is always perpendicular to the field direction, so that again, it
is an ESP state explaining the observed absence of change of the Knight shift (but in
contradiction with the H c2 anisotropy). Note also that all these features are preserved
if in the B-phase, the d-vector is a complex combination of e b and e c , or e b and e a :
( ˆ
k a e b ± i ˆ
k b e c ); ( ˆ
k a e b ± i ˆ
k b e a ).
Then, the B-phase would be chiral (as in the E 2u model), but also non-unitary
(d(k) ∧ d
∗
(k) = 0, see Sect. 6.6.3). So was the original proposal in [11]. It is an interesting example of a non-unitary state with no global spin polarization, e.g. with d ∝
(5 ˆ
k
2
c − 1)( ˆ
k a e b + i ˆ
k b e a ), we derive from (6.25) and (6.31) that S(k) ∝ (5 ˆ
k
2
c −
1)
2 ˆ
k a ˆ
k b e c and
↑
(k) ∝ |(5 ˆ
k
2
c − 1)( ˆ
k a + ˆ
k b )|,
↓
(k) ∝ |(5 ˆ
k
2
c − 1)( ˆ
k a − ˆ
k b )|. So
indeed, averaging over the Fermi surface leads to no net spin and equal averaged
gap amplitudes for up-spins and down-spins, even though they are different for most
k of the Fermi surface.
J.-P. Brison
6.8.2.3 E 1u Representation
However, NMR experiments seem to be in contradiction with this interpretation of
the crossing of the upper critical fields of UPt 3 . Indeed Knight-shift measurements,
which are the closest to a measure of the spin susceptibility in the superconducting
state, found no change in the superconducting state for fields applied along the a
or b directions in the C-phase, but also no change for field along the c-axis except
at very low field deep inside the B-phase [11]. Therefore, other models have been
proposed for UPt 3 , which are much closer to the situation of superfluid
3 He, with a
weak spin–orbit coupling allowing for a field reorientation of the d-vector as long
as the field is ‘strong enough’. Knight-shift measurements can tell nothing on the
gap nodes, but combining angle-dependent thermal conductivity measurements [10]
with the NMR result, a E 1u scenario has been proposed, predicting a non-chiral state
[d(k) ∝ (5 ˆ
k
2
c − 1)( ˆ
k a e b + ˆ
k b e c ) in the B-phase]. This scenario can also more easily
give account of some other features of the phase diagram (like the existence of a
tetracritical point in all field directions): see Fig. 6.5. For the different phases, in this
model, the d-vector would be as shown in Table 6.1.
So, as can be seen from Table 6.1, at low fields the d-vector does not depend on
the field orientation [check the A-phase and B-phase-(low H ) lines of the table]. In
the B-phase, for H c, there is a field-induced reorientation of the d-vector: at low
field, with d ∝ ( ˆ
k a e b + ˆ
k b e c ), the d-vector is not perpendicular to the c-axis (except
on the line ˆ
k b = 0) so there is a finite |S z = 0 component of the spin along the
field. This would be imposed by the orbital part of the wave function and spin–orbit
interaction or coupling to the small antiferromagnetic moments acting as a symmetry
breaking field. But for fields above 0.22 T, with d ∝ ( ˆ
k a e b + ˆ
k b e a ), the d-vector is
real and always perpendicular to the c-axis, so we know that it is equivalent to an
ESP state in that direction. Hence, the field-induced rotation of the d-vector. In
the high-field C-phase, where the Pauli limitation could be at play, we note that in
Table 6.1 the d-vector is always perpendicular to the field direction, so that again, it
is an ESP state explaining the observed absence of change of the Knight shift (but in
contradiction with the H c2 anisotropy). Note also that all these features are preserved
if in the B-phase, the d-vector is a complex combination of e b and e c , or e b and e a :
( ˆ
k a e b ± i ˆ
k b e c ); ( ˆ
k a e b ± i ˆ
k b e a ).
Then, the B-phase would be chiral (as in the E 2u model), but also non-unitary
(d(k) ∧ d
∗
(k) = 0, see Sect. 6.6.3). So was the original proposal in [11]. It is an interesting example of a non-unitary state with no global spin polarization, e.g. with d ∝
(5 ˆ
k
2
c − 1)( ˆ
k a e b + i ˆ
k b e a ), we derive from (6.25) and (6.31) that S(k) ∝ (5 ˆ
k
2
c −
1)
2 ˆ
k a ˆ
k b e c and
↑
(k) ∝ |(5 ˆ
k
2
c − 1)( ˆ
k a + ˆ
k b )|,
↓
(k) ∝ |(5 ˆ
k
2
c − 1)( ˆ
k a − ˆ
k b )|. So
indeed, averaging over the Fermi surface leads to no net spin and equal averaged
gap amplitudes for up-spins and down-spins, even though they are different for most
k of the Fermi surface.