202
J.-P. Brison
d =
1
ψ
⎛
⎝
↑
2
e
−2iϕ
− 1
−
i ↑
2
1 + e
−2iϕ
0
⎞
⎠ =
1
ψ
⎛
⎝
−ie
−iϕ
↑ sin ϕ
−ie
−iϕ
↑ cos ϕ
0
⎞
⎠ .
If we rotate this state by π/2 around a vector ˆ
=
⎛
⎝
cos θ
sin θ
0
⎞
⎠ in the (x, y)-plane, we
obtain from (6.11)
R (d) = (d · ˆ
) ˆ
+ ( ˆ
∧ d)
=
1
ψ
⎛
⎝
sin(θ + ϕ)e
−iϕ
↑ cos θ
sin(θ + ϕ)e
−iϕ
↑ sin θ
−ie
−iϕ
↑ cos(θ + ϕ)
⎞
⎠ .
So indeed, if we choose θ = −ϕ, we recover a pure |S z = 0 state. Note that it is the
phase between the ↑ and ↓ components of the order parameter, which determines
the precise direction of the required π/2 rotation.
References
1. A.J. Leggett, A theoretical description of the new phases of liquid 3 He. Rev. Mod. Phys. 47,
331 (1975). https://doi.org/10.1103/RevModPhys.47.331
2. V. Mineev, K. Samokhin, Introduction to Unconventional Superconductivity (Gordon and
Breach Science Publishers, Amsterdam, 1999)
3. N. Mermin, V. Ambegaokar, The order parameter in liquid 3 He, in Collective Properties of Physical Systems (Proceedings of the 24th Nobel Symposium, Lerum 1973), ed. by B. Lundqvist,
S. Lundqvist, and Runnström-Reio (Academic, New York, 1974), p. 97. https://doi.org/10.
1016/B978-0-12-460350-9.50019-9
4. C. Kallin, J. Berlinsky, Chiral superconductors. Rep. Prog. Phys. 79, 054502 (2016). https://
doi.org/10.1088/0034-4885/79/5/054502
5. G.E. Volovik, V.P. Mineev, Investigation of singularities in superfluid He 3 in liquid crystals by
the homotopic topology methods, Zh. Eksp. Teor. Fiz. 72, 2256 (1977). [Sov. Phys. JETP 45,
1186 (1977).]
6. R. Joynt, L. Taillefer, The superconducting phases of UPt 3 . Rev. Mod. Phys. 74, 235 (2002).
https://doi.org/10.1103/RevModPhys.74.235
7. J. Sauls, The order parameter for the superconducting phases of UPt 3 . Adv. Phys. 43, 113
(1994). https://doi.org/10.1080/00018739400101475
8. E.R. Schemm, W.J. Gannon, C.M. Wishne, W.P. Halperin, A. Kapitulnik, Observation of broken
time-reversal symmetry in the heavy-fermion superconductor UPt 3 . Science 345, 190 (2014).
https://doi.org/10.1126/science.1248552
9. B.S. Shivaram, T.F. Rosenbaum, D.G. Hinks, Unusual angular and temperature dependence
of the upper critical field in UPt 3 . Phys. Rev. Lett. 57, 1259 (1986). https://doi.org/10.1103/
PhysRevLett.57.1259
10. Y. Machida, A. Itoh, Y. So, K. Izawa, Y. Haga, E. Yamamoto, N. Kimura, Y. Onuki, Y. Tsutsumi,
K. Machida, Twofold spontaneous symmetry breaking in the heavy-fermion superconductor
UPt 3 . Phys. Rev. Lett. 108, 157002 (2012). https://doi.org/10.1103/PhysRevLett.108.157002
J.-P. Brison
d =
1
ψ
⎛
⎝
↑
2
e
−2iϕ
− 1
−
i ↑
2
1 + e
−2iϕ
0
⎞
⎠ =
1
ψ
⎛
⎝
−ie
−iϕ
↑ sin ϕ
−ie
−iϕ
↑ cos ϕ
0
⎞
⎠ .
If we rotate this state by π/2 around a vector ˆ
=
⎛
⎝
cos θ
sin θ
0
⎞
⎠ in the (x, y)-plane, we
obtain from (6.11)
R (d) = (d · ˆ
) ˆ
+ ( ˆ
∧ d)
=
1
ψ
⎛
⎝
sin(θ + ϕ)e
−iϕ
↑ cos θ
sin(θ + ϕ)e
−iϕ
↑ sin θ
−ie
−iϕ
↑ cos(θ + ϕ)
⎞
⎠ .
So indeed, if we choose θ = −ϕ, we recover a pure |S z = 0 state. Note that it is the
phase between the ↑ and ↓ components of the order parameter, which determines
the precise direction of the required π/2 rotation.
References
1. A.J. Leggett, A theoretical description of the new phases of liquid 3 He. Rev. Mod. Phys. 47,
331 (1975). https://doi.org/10.1103/RevModPhys.47.331
2. V. Mineev, K. Samokhin, Introduction to Unconventional Superconductivity (Gordon and
Breach Science Publishers, Amsterdam, 1999)
3. N. Mermin, V. Ambegaokar, The order parameter in liquid 3 He, in Collective Properties of Physical Systems (Proceedings of the 24th Nobel Symposium, Lerum 1973), ed. by B. Lundqvist,
S. Lundqvist, and Runnström-Reio (Academic, New York, 1974), p. 97. https://doi.org/10.
1016/B978-0-12-460350-9.50019-9
4. C. Kallin, J. Berlinsky, Chiral superconductors. Rep. Prog. Phys. 79, 054502 (2016). https://
doi.org/10.1088/0034-4885/79/5/054502
5. G.E. Volovik, V.P. Mineev, Investigation of singularities in superfluid He 3 in liquid crystals by
the homotopic topology methods, Zh. Eksp. Teor. Fiz. 72, 2256 (1977). [Sov. Phys. JETP 45,
1186 (1977).]
6. R. Joynt, L. Taillefer, The superconducting phases of UPt 3 . Rev. Mod. Phys. 74, 235 (2002).
https://doi.org/10.1103/RevModPhys.74.235
7. J. Sauls, The order parameter for the superconducting phases of UPt 3 . Adv. Phys. 43, 113
(1994). https://doi.org/10.1080/00018739400101475
8. E.R. Schemm, W.J. Gannon, C.M. Wishne, W.P. Halperin, A. Kapitulnik, Observation of broken
time-reversal symmetry in the heavy-fermion superconductor UPt 3 . Science 345, 190 (2014).
https://doi.org/10.1126/science.1248552
9. B.S. Shivaram, T.F. Rosenbaum, D.G. Hinks, Unusual angular and temperature dependence
of the upper critical field in UPt 3 . Phys. Rev. Lett. 57, 1259 (1986). https://doi.org/10.1103/
PhysRevLett.57.1259
10. Y. Machida, A. Itoh, Y. So, K. Izawa, Y. Haga, E. Yamamoto, N. Kimura, Y. Onuki, Y. Tsutsumi,
K. Machida, Twofold spontaneous symmetry breaking in the heavy-fermion superconductor
UPt 3 . Phys. Rev. Lett. 108, 157002 (2012). https://doi.org/10.1103/PhysRevLett.108.157002