YITP, Kyoto University Masatoshi Sato

Quantum Response of (Helical) Majorana Fermions in Topological Superconductors

YITP, Kyoto University Masatoshi Sato

Kobayashi-Yamakage-Tanaka-MS, arXiv:1812.01857(today)

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In collaboration with

•Shingo Kobayashi (Nagoya University)

• Ai Yamakage (Nagoya University)

• Yuansen Xiong (Nagoya University)

• Yukio Tanaka (Nagoya University)

A review paper on topological SCs with Yoichi Ando MS, Ando, Rep. Prog. Phys. 80, 076501 (17)

Outline

1. Motivation

2. Anisotropic magnetic response of helical MFs

3. Majorana multipole response of helical MFs

MS-Fujimoto, PR B79, 094504 (2009)

Mizushima-MS-Machida, PRL, 109, 165301 (2012)

Kobayashi-Yamakage-Tanaka-MS, arXiv:1812.01857

• S-wave superconducting state with Rashba SO + Zeeman field MS-Takahashi-Fujimoto (09), J. Sau et al (10)

Zeeman field MF

nanowire

Lutchyn et al (10), Oreg et al (10)

Motivation

MFs were originally proposed as elementary particles, but now we know that they can be emergent excitations in electron or atomic systems.

Majorana Fermions in S-wave SC

Condition for MF

• Dirac fermion + s-wave condensate MS(03), Fu-Kane (08)

Hsieh et al

These emergent MFs in condensed matter physics share some

properties with elementary Majorana particles in high energy physics

• Both obey the Dirac equation with self-charge-conjugation condition

• Zero modes exhibit non-Abalian anyon statistics

charge-conjugation

However, there is an essential difference

between them

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• Elementary MFs should respect CPT inv. since they should respect Lorentz inv.

• This means that elementary MFs are self-conjugate under CPT, not merely under C

CPT theorem

 CPT is a fundamental symmetry of relativistic QFT

 Any reasonable relativistic QFT is invariant under CPT

C: charge conjugation P: parity (inversion) T: time-reversal

This fundamental invariance of elementary MFs gives a strong constraint in electromagnetic

responses

Electromagnetic response of elementary MFs

electric charge magnetic dipole electric dipole toroidal moment

General form of one-particle EM-coupling for spin-1/2 relativistic fermions

• Charge neutral condition forMFs (F=0)

• Electro-magnetic dipole momenta of MFs vanish (M=E=0)

self-conjugation condition under CPT

Kayser-Goldhaber (83)

Elementary MFs only show moderate EM responses

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However, emergent MFs are not subject to such a strong constraint.

• Emergent MFs only have approximate Lorentz invariance.

• They are self-conjugate just under C, not under CPT.

A different scheme is needed to describe EM responses of emergent MFs

In this talk, I will present a general theory of EM

responses of emergent MFs in time-reversal invariant TSCs

Majorana multipole response in topological superconductors

Xiong-Yamakage-Kobayashi-MS-Tanaka, Crystal 2017, 7, 58 Kobayashi-Yamakage-Tanaka-MS, arXiv:1812.01857

Anisotropic magnetic response of MF

Helical Majorana fermions in TRI topological SCs show peculiar anisotropic magnetic response. MS-Fujimoto (09)

Chung-Zhang (09)

2dim p-wave Rashba noncentrosymmetric SC

MS-Fujimoto (09), Y. Tanaka et al (09)

Non-trivial Z₂ topological number

2dim time-reversal invariant helical SC

SC

Helical Majorana fermion

x

y k_x

Under Zeeman fields, the helical MF shows anisotropic response.

• Zeeman field along the edge

• Zeeman field normal to edge

Gap opens

due to TR breaking

No gap opens in spite of TR breaking

MS-Fujimoto (09)

SC x y

SC x y

k_x

k_x

A similar anisotropic magnetic response has been reported in 3dim time-reversal invariant SCs Chung-Zhang(09)

Shindou-Furusaki-Nagaosa( 10)

Helical Majorana surface state in ³He-B

3He-B

• MF behaves like Ising spin (=magnetic dipole)

• MF does not couple to magnetic fields parallel to the surface

Spin density op.

These anisotropic behaviors can be explained by crystalline sym.

Rashba SC

SC x

y 3He-B

3He-B

𝜋𝜋

• TRS can remain partially as magnetic symmetry.

• The remaining anti-unitary symmetry may stabilize gapless helical MFs under magnetic fields

TRS , mirror reflection magnetic two-fold rotationTRS, rotation (TRS+two-fold rotation) magnetic mirror reflection

(TRS+mirror reflection)

Mizushima-MS-Machida (12) Shiozaki-MS (14)

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Actually, one can define top. # by using these magnetic symmetries

symmetric momentum under mirror/C₂-rot.

magnetic mirror/ C₂-rot.

PHS BdG Hamiltonian

For Rashba SC For 3He-B

Therefore, the magnetic winding # naturally explain why helical MFs can stay gapless even under magnetic fields

spin-degeneracy spin-degeneracy

k_x

Question

1. How can we know magnetic response more systematically?

2. What determines the details of anisotropic behavior?

To address these questions, we develop a general theory of quantum response of MFs

Rashba

SC 3He-B Similar but different

anisotropic behavior

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Basic idea

Use the generalized index theorem to evaluate quantum operator Generalized index theorem MS-Tanaka-Yada-Yokoyama PRB (11)、

Xiong-Yamakage-Kobayashi-MS-Tanaka(17)

Gapless MF is an eigenstate of Γ_M For

For

Spin Structure

= Magnetic Response

How to evaluate quantum operator

First, perform mode expansion of quantum field,

Substituting this, we can extract the contribution of gapless MFs as Quantum op.

Using this expression, we derive symmetry constraints for gap function and quantum operator with nonzero value O

Nambu base

gapless MF hermitian

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First, sym. of the gap fun. should be selected to obtain nonzero w_M1D,

1d wind. # for mag. C₂ rot.

odd under C₂ (B rep) even under C₂ (A rep)

For Nambu space

particle

hole

stable MF

Particle and hole behave in the same way

compatible with PHS

For Nambu space

unstable MF

Particle and hole behave in a different manner

incompatible with PHS

helical MFs protected by mag. C₂

Thus, for magnetic operator , using the definition , we have

Second, the operator O should be the same representation as ρ^(ab)

even under C₂ (A rep)

For instance, from the index theorem

Thus, O should be even under magnetic CS

In Nambu space

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In this manner, we complete list of gap functions with nonzero w_M1D and the corresponding mag. multipoles for all surface point groups

magnetic octupole high spin Cooper pairs

3He-B

SC

Kobayashi-Yamakage-Tanaka-MS (18)

Our theory predicts magnetic octupole response in high spin TSC !!

Application to half-Heusler SCs

YPtBi

Butch et al (11) Brydon et al (16)

experiment theory

high-spin Cooper pair Our result for mag resp of MFs on [111]

octupole response

c.f) ³He-B

T_c=0.7K

proposed gap fn.

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Summary

1. In contrast to elementary Majorana particles, emergent MFs may exhibit richer magnetic structures.

2. We find a one-to-one correspondence between symmetry of Cooper pairs and rep. of magnetic response, which provides a novel way to identify unconventional SC.

3. Detection of magnetic octupole response of MFs is a direct

evidence of high spin topological superconductivity.