[PPT] - Chiral Anomaly Phenomena in Weyl Superconductors Satoshi Fujimoto PowerPoint Presentation

SLIDE 1

Satoshi Fujimoto Department of Materials Engineering Science, Osaka University

Chiral Anomaly Phenomena in Weyl Superconductors

SLIDE 2

Y. Ishihara

(Osaka Univ.)

Collaborators

T. Mizushima

(Osaka Univ.)

J. de Lisle

(Osaka Univ. )

T. Kobayashi

(Osaka Univ. )

SLIDE 3

Outline

Introduc-on ¡
Torsional ¡chiral ¡magne-c ¡effect ¡in ¡Weyl ¡superconductors ¡
Thermal ¡analogue ¡of ¡nega-ve ¡magnetoresis-vity ¡in ¡Weyl ¡

superconductors ¡

Chiral ¡spin-‑polariza-on ¡effect ¡in ¡a ¡ferromagne-c ¡Weyl ¡

superconductor ¡UCoGe ¡

3

SLIDE 4

qm = +1

qm = −1

b −b kx ky kz

Weyl superconductor due to broken TRS

characterized by monopole charge of point nodes of SC gap

σ : particle-hole space

Time-reversal symmetry breaking is necessary for Weyl SC

TRSB TRSB for by = 0 Weyl SC:

τz = ±1

chirality

τz = +1 τz = −1

∆ = 0 ∆ = 0 ∆ = 0

H = τzσ · k − σ · b

Weyl fermions = Bogoliubov quasiparticles from point-nodes

e.g. chiral px+ipy wave SC

qm = ±1

(sign of )

qm · Ωkk = qmδ(k b)

Berry curvature

SLIDE 5

Examples of Weyl superconductor/superfluid

Weyl points (monopole charge -1) (thermal) Anomalous Hall effect example ABM phase of He3

N.B. Weyl points have spin-degeneracy

∆↑↑

k = ∆↓↓ k = ∆(kx + iky)

(Volovik)

Weyl points(monopole charge +1) surface BZ

non-zero Chern number

c.f. H. Ikegami et al, Science (2013)

surface Majorana

SLIDE 6

example 2: URu2Si2

∆k = ∆kz(kx + iky)

Chiral dzx+idyz SC

Line nodes point nodes (linear dispersion)

N.B. Weyl points have spin-degeneracy

Thermal conductivity, specific heat, … (Kasahara et al.; Yano et al.)

Kerr effect (Kapitulnik’s group, 2015)
Giant Nernst effect due to chiral SC fluctuation

( exp. : Matsuda’s group, 2016 ; theory : H. Sumiyoshi and S. F. , 2015)

Examples of Weyl superconductor

SLIDE 7

example 3: UCoGe Ferromagnetic SC

non-unitary spin-triplet SC

d = (a1ka + ia2kb, a3kb + ia4ka, 0)

d d∗ = 0 ∆↑↑ = ∆↓↓

(Mineev, PRB66, 134504; Hattori, Tada et al., PRL108, 066403)

N.B. Weyl points have no spin-degeneracy !!

(Samsel-Czekala et al.)

point node Weyl point Genuine Weyl superconductor !!

Fermi surface for majority spin

Examples of Weyl superconductor

SLIDE 8

example 4: B phase of UPt3

Chiral f-wave SC ? (controversial)

N.B. Weyl points have spin-degeneracy

(Goswami-Nevidomskyy)

∆k = ∆kz(kx + iky)2

(Schemm et al.)

(b)

kz ky

Point node (k-quadratic) double-Weyl points (monopole charge +2) Point node (k-quadratic) double-Weyl points line node Majorana arc anomalous thermal Hall effect

(Huxley et al.)

B A C

(monopole charge -2)

Examples of Weyl superconductor

SLIDE 9

Examples of Weyl superconductor

example 5: U1-xThxBe13

VoLUME65,

NUMBER 22

PHYSICAL REVIEW LETTERS

26 NovEMaER 1990 Rauchschwalbe,

'

who studied a sample with x =0.033.

We now discuss

ur conclusions

from these data, ad- dressing first the nature

f the overall

T-x phase

diagram. There is no evidence of a second phase transition at

low

fields below

T,. i

in pure

UBet3 from either

the

pSR data or the H, i(T) data.

This contradicts the data

f Ref. 14, where

a small deviation from the T dependence

in pure UBei3 below 0.56 K was claimed

as evidence for a second phase, but supports previous specific- heat results.

'

The fact that

wt.

* see an increase in the

pSR

linewidth below

T,. ~ and

two diA'erent

quadratic temperature dependences

in H, i(T) only for x =0.0193,

0.0245, and 0.0355, but

not

for x =0.000, 0.0100, or

0.0600,

is a clear

indication that there are magnetic correlations

nly in regions of the phase diagram

where a second specific-heat peak has been

bserved

(0.019(x

(0.043). '

The previous suggestion' that

the superconducting transition

at T,i=0.86 K in UBetq

is to be

associated

with the lower transitions

at T,.2 for 0.019~ x

(0.043 is also ruled out by our data, because the latter

phase transitions exhibit magnetic

correlations

while the

former does not. Within uncertainties

in the Th concen-

tration

f about 0.005, the T-x phase

diagram

is con-

structed approximately as shown

in Fig. 3, augmented

by

specific-heat data for other Th concentrations. The dashed

lines may not be absolutely

vertical as drawn, but steep phase boundaries near x =0.019 and 0.043 are sug- gested

by the present

data and the specific-heat data

in

0.9—

V

~ 0.5—

I-

0.0 0.0

2.0

I 1 I I 1 I

MAGNETlC

I I I I I I I I I I I I I I j I

4.0

X ('/0)

FIG. 3. Phase diagram

for Ul —,

Th, Bei&. Open symbols

are

frOm thiS wOrk.

SquareS,

T,

I frOm g.,; CirCleS, T, 1 frOm mag-

netization

M(H); inverted

triangles, T, ~ from kink

in H, I(T-).

The solid

upright triangles

are T,

1 and

T, ~ from specific heat

in Ref. 17. The symbol (&) at x =0.043 indicates

a merging

f

T,

1

and

T, , as described

in

Ref. 17.

T,

1 =0.39

K for

x =0.0600 was determined

resistively.

TABLE II. The x dependence

f o, and [H,'i]

~-' at T=O in

U I —,

Th, Be I ~. x ( k)

x/1. 93

;.(x)/o, (1.93)

I,' (. )/,' ( . 3)l"-

1.93 2.45

3.55

1.00 1.27 1.84 1.00 1.11 ~ 0.06 1.31 ~ 0.07 1.00 1.14+ 0.07 1.21 + 0.07

Ref. 17.

We now discuss the nature

f the

phase below

T,, ~. The fact that

within

errors the transitions at T,.~ begin

and terminate

n the line of superconducting

phase transitions

at T,

~ means

that the order parameters for the

two phases must be strongly

coupled. This could denote

a purely

antiferromagnetic

(AFM) phase coexisting

with and coupled to superconductivity

(hypothesis

I), or a sin-

gle complex superconducting

rder

parameter

with difI'erent

symmetry-group representations and a magnetic (time-reversal-violating) ground

state (hypothesis

II).

Ultrasonic-attenuation data are consistent

with hy-

pothesis I. We note that local moments

n the Th sites'

would give rise to a dipolar linewidth

, (0) proportional

to x,

'

which is not seen (Table II). The fact that the magnetism

appears abruptly at x =0.019, and not con-

tinuously with x &0.019, is further evidence

against local Th moments; this is consistent

with either hypothesis.

Hypothesis

II is supported

by the fact that

T, ~ & T,.i. . That is, the Fermi surface

is largely

consumed

by the su-

perconducting transition

at

T, i.

Thus the large

b-

served

specific-heat jump

AC~ at T, ~ (comparable

to that at T, i) would be very surprising for a purely

AFM

phase, and would require an exceptional enhancement

f

the density of states near the zeros of the superconduct-

ing gap to account

for the large

AC&.

Hence the con- nectedness

f the phase

diagram and the large specific jump at T, 2( T,

i to an AFM ground

state are proper- ties unlike

those observed

in other small-moment

heavy-

electron magnets

Rather

thi. s feature
f (U,Th)Bei3 is

similar

to

the two superconducting specific-heat anomalies

in UPt~, and is also consistent

with hypothesis II.

We note that both H, i(0) and a, (0) increase

with x

for 0.019~ x(0.043. The increase of H,'i (0) (Table I)

with x must be due to an increase in n, (0) or a decrease

in m*. As argued

previously an AFM transition

at T, ~

would

be expected to decrease m* because magnetic

r-

der

tends to suppress the f-moment spin fluctuations, which contribute greatly

to m*. Unfortunately,

it is not possible

to predict

how much m should

change

with x below

T,.2 without a detailed and believable microscopic theory.

If n, (0) increases

with x the correlation between

H;i(0) and a, (0) could

be explained under hypothesis

II, because

some models

for time-reversal-violating superconducting states predict orbital currents generated

by inhomogeneities

f the order parameter

produced

by

electron scattering

from nonmagnetic impurities.

This

2818

(Heffner et al.(1990)) (Shimizu et al.(2017))

− − te, d(k) = l1+il2 = ˆ xkx+ˆ yky+2 ˆ zkz

l1(k) = √ 3(ˆ xkx− ˆ yky) d l2(k) = 2ˆ zkz − ˆ xkx− ˆ yky, a

2

possible d-vector

dd parity pairing state ?

Eu representation (cubic symmetry) non-unitary state Weyl SC !!

(Mizushima and Nitta (2017))

Tc x (%)

degenerate

}

TRSB

SLIDE 10

Chiral Anomaly of Weyl semimetal

k E L R

jµ

5 = jµ L − jµ R

violation of conservation law of axial current

Leff = θe2 2πhE · B

∂μjμ

5 ¼

¼ q3 4π2 ~ E · ~ B:

e3

θ(r,t) = 2b · r − 2b0t, where 2b

momentum

and 2b0 is

Anomalous Hall effect Chiral magnetic effect

J = e2 πhb × E J = e2b0 πh B

µ = 0, 1, 2, 3

(t, x, y, z) negative magnetoresistance σzz ∝ B2τ

∝ Bτ

small B large B

(Nielsen,Ninomiya, Burkov, Son, Spivak)

E B

SLIDE 11

Chiral Anomaly of Weyl superconductor

Weyl fermions in p-h space Axial current does not couple to E and B Chiral anomaly due to geometrical distortion (gravitational fields) : torsion

∂μjμ

5 ¼

¼ q 32π2l2 ϵμνρσðηabTa

μνTb ρσ − 2Rab;μνea ρeb σÞ

ð

ρσ þ

q 192π2 ϵμνρσ 1 4 Rab

μνRcd ρσηadηbc

e e

Rab

µν

: Riemann curvature

: non-universal cutoff T 0

0j

T i

0j

: temperature gradient : spatial rotation e.g.

(Shitade,Bradlyn,Read,Gromov,Abanov)

T i

jk

i, j, k = x, y, z

: vortex, topological texture of SC order parameter (l-vector, d-vector)

(Parrikar, Hughes, Leigh,Shopurian, Ryu)

1 1

jµ

5 = vLnL − vRnR

: vielbein

ea

µ

SLIDE 12

Emergent “gauge field” and “magnetic field” due to torsion

pµ → eµ

apµ

spatial inhomogeneity:

Spatial distortion

emergent “U(1) gauge field” Aa Emergent “magnetic field”:

Bµ = µνλ 2 T a

νλpa

Torsion: depending on momentum !

(Hughes, Leigh, Parrikar)

eµ

a ≈ δµ a − ∂uµ/∂xa

eµ

apµ ≈ pa − ∂uµ

∂xa pµ

SLIDE 13

Semiclassical EOM of Bogoliubov quasiparticles in Weyl SC dr dt = ∂Eps ∂p + ∂U(r) ∂r × Ωpps − (∂Eps ∂p · Ωpps)B

Berry curvature of Weyl band with chirality

Ωpps s = ±1 B = T µpµ emergent magnetic field due to torsion (T µ)ν = 1

2νλρT µ

λρ

dp dt = −∂U(r) ∂r + dr dt × B + (∂U(r) ∂r · B)Ωpps

Eps

band energy of Bogoliubov-Weyl quasiparticle

U(r)

applied potential, or gravitational potential

torsional chiral magnetic effect

(thermal) AHE chiral anomaly

Weyl points

dp dt

Weyl points

dp dt

qm = 1 qm = −1

SLIDE 14

Torsional chiral magnetic effect in Weyl superconductors

SLIDE 15

Spectrum ¡asymmetry ¡!!

rc

Screw ¡disloca-on ¡ + ¡ ¡an--‑disloca-on

Torsional Chiral Magnetic Effect in Weyl (semi)metals with Broken Time-Reversal Symmetry and Lattice Defect

H. Sumiyoshi and S. F.,

Phys.Rev. Lett.116, 166601 (2016)

Local ¡equilibrium ¡current ¡ ¡ flowing ¡along ¡ screw ¡disloca-on

B = T µpµ

not ¡breaking ¡Bloch ¡theorem ¡! Current ¡ distribu-on

SLIDE 16

)
,-‒,-‒,,,,.-‒.,3,,.,,3, ¡,,33,,,,,-‒,,

,-‒,3,,

)
!

" #

)
,-‒,-‒,,,,.-‒.,3,,.,,3, ¡,,33,,,,,-‒,,

,-‒,3,,

)
"

Torsional CME in Weyl superconductor

Vortex of l-vector of A phase of Helium 3 :

Anderson-Toulouse vortex
Mermin-Ho vortex

∆ = ∆0(m + in) · k

H = eµ

a a(pµ − pF µ)

= m × n

Spatially varying Weyl points (Skyrmion-like textures) chiral anomaly due to and (Volovik)

∂/∂t

SLIDE 17

)
,-‒,-‒,,,,.-‒.,3,,.,,3, ¡,,33,,,,,-‒,,

,-‒,3,,

)
!

" #

)
,-‒,-‒,,,,.-‒.,3,,.,,3, ¡,,33,,,,,-‒,,

,-‒,3,,

)
"

Torsional CME in Weyl superconductor

Vortex of l-vector of A phase of Helium 3 :

Anderson-Toulouse vortex
Mermin-Ho vortex

∆ = ∆0(m + in) · k

Torsion:

T 3

23 = 23 − 32

etc. Torsional chiral magnetic effect of mass current

T 3

31 = 31 − 13

= m × n

(T a)µ = µνλ 2 T a

νλ

J = vF Λ 2π2 T 3kF 3

Λ ∼ ∆ EF kF

: UV momentum cutoff

in-plane torsional magnetic field (parallel to xy-plane)

SLIDE 18

)
,-‒,-‒,,,,.-‒.,3,,.,,3, ¡,,33,,,,,-‒,,

,-‒,3,,

)
"
)
,-‒,-‒,,,,.-‒.,3,,.,,3, ¡,,33,,,,,-‒,,

,-‒,3,,

)
!

" #

Torsional CME in Weyl superconductor

Mermin-Ho vortex

T 3

23 = 23 − 32

current distribution

T 3

31 = 31 − 13

2

‑1

1

‑2

40

‑40

20

‑20

E/Δ

in-plane torsional magnetic field

Spectrum asymmetry

¡qθ: ¡azimuthal ¡momentum

Numerical Results from BdG equation

suppress backward scattering dissipationless current

J = vF Λ 2π2 T 3kF 3

(T a)µ = µνλ 2 T a

νλ

SLIDE 19

A Generalized Ginzburg-Landau Approach to the Superfluidity of Helium 3 529

and r = 1 - de 8 9 sech 2 89 [1 - qS@) is the "Yoshida function" generalized to p-wave pairingj, and E is the excitation energy : Ea(p) = ~2 + iA(/3)12 and we have assumed a unitary

solution. Here jo is the spin current in the ith direction for spins along the

direction, and is expressed in terms of number : The spin transported is this multiplied by the spin 89

f each particle.

To the same order in Tc/EF: F ~ = T~ajb(VidJ(Vfl*~) + non-current-giving terms (5) The non-current-giving terms involve gradients of IA] 2 only, and so do not couple to the gauge field. They can be written in the form

1

~ Tiajb

ViVj(da~d~) Equations (3) and (4) are exactly the results that would be obtained from

Eq. (5) by the gauge arguments.

In the Ginzburg-Landau region the tensor T is isotropic and state

independent. The free energy is exactly of the form BBA suggested, with

7~(3) p

Ko = KA -- 80n2T2 m

and, to this order in Tc/EF, KL = 0. Away from T~, T remains isotropic for the BW phase, but for the ABM phase is invariant under rotations about the direction 7 only, and the BBA free energy no longer applies. The results will now be displayed for each phase in turn explicitly in terms of the angle variables.

6.1. The ABM Phase

The number current is 2~ C ~ curl ? jo = pO Vgb + 2m (6) where the (tensor) st~perfluid density and C O are given, in our chosen axes, by

pO L

so 0

p O 0 , CO ~P~l 1

po
)
,-‒,-‒,,,,.-‒.,3,,.,,3, ¡,,33,,,,,-‒,,

,-‒,3,,

)
"
)
,-‒,-‒,,,,.-‒.,3,,.,,3, ¡,,33,,,,,-‒,,

,-‒,3,,

)
!

" #

Comparison between torsional CME and supercurrent induced by l-vector textures

[Cross(1975) , Ishikawa et al.(1978), Mermin-Muzikar(1980)]

supercurrent : Weyl quasiparticle current (torsional CME) :

supercurrent Result of BdG eq. (total current) distance from the center of MH vortex current density T/Tc = 0.001 ∼ (supercurrent) × ∆ EF due to torsional CME !!

∼ J0 × ∆ EF

J = vF Λ 2π2 T 3kF 3

T 3 =

SLIDE 20

Thermal analogue of negative magnetoresistivity in Weyl superconductors

SLIDE 21

Thermal negative magnetoresistivity in Weyl superconductor with vortices

∆ = ∆0eiφ(kx + iky)

Vortex of chiral p-waveSC :

kx ky

∆ = 0 ∆ = 0

∆ = 0

kz

vortex line Torsion : T 2

12 = kF

∆0r

Emergent magnetic field due to torsion : Negative magnetoresistivity

f thermal current for JH B

Torsional CME does not occur T 3

µν = 0

= 0

However

(B)z = 1 r (ky cos φ − kx sin φ) J = vF Λ 2π2 T 3kF 3

r =

x2 + y2

B

JH

thermal current

SLIDE 22

∆ = ∆0eiφ(kx + iky)

Vortex of SC order : Torsion : T 2

12 = kF

∆0r

Emergent magnetic field due to torsion :

vortex line kx ky ∆ = 0

∆ = 0 ∆ = 0

kz thermal current

JH

Negative thermal magnetoresistivity

(B)z = 1 r (ky cos φ − kx sin φ)

Thermal negative magnetoresistivity in Weyl superconductor with vortices

: Berry curvature due to Weyl points

r =

x2 + y2

B

JH =

s=±1
k

(vps · Ω+

pps)2ε2 ps

∂f ∂εps

τps(T

T · B)B,

kk

Ωkks

SLIDE 23

Negative thermal magnetoresistivity Thermal conductivity (Born approx.)

Thermal negative magnetoresistivity in Weyl superconductor with vortices

κ = κ0 + κB

= κBT

usual contribution due to torsion fields

∝ T ∝ 1 T

Chiral anomaly effect is enhanced as temperature is lowered However, Berry phase formula is not applicable to zero temperature limit, because of singularity at Weyl points !

at Weyl points

JH =

s=±1
k

(vps · Ω+

pps)2ε2 ps

∂f ∂εps

τps(T

T · B)B,

kk

∼ 1/|k − kF |2 k ∼ kF

SLIDE 24

Results from Keldysh formalism of Eilenberger equation

𝜗 Ƹ 𝜐𝑨 − ෠ ℎ, ො 𝑕 + 𝑗ℏ𝒒𝐺 ∙ 𝛼𝑆 ො 𝑕 = 𝑗 2 𝜶𝑺 ෠ ℎ ∙ 𝜶𝒒 ො 𝑕 − 𝜶𝒒෠ ℎ ∙ 𝜶𝑺 ො 𝑕 − 𝑗 2 𝜶𝑺 ො 𝑕 ∙ 𝜶𝑸෠ ℎ − 𝜶𝒒 ො 𝑕 ∙ 𝜶𝑺 ෠ ℎ 進捗報告・ Tanuma’s ・・ 𝐾0, 𝐾2

quantum corrections (torsional magnetic fields)

ˆ h = ˆ σ + ˆ ∆

ˆ σ ˆ ∆

: self-energy(Born approx.) : SC gap

We consider the case with single vortex line

ˆ g =

ˆ

gR ˆ gK ˆ gA

Quasi-classical

Green function

Thermal negative magnetoresistivity in Weyl superconductor with vortices

JH = N(0)

dp
dε

4πiεvtr[δˆ gK]

Heat current

SLIDE 25

Chiral spin-polarization effect in ferromagnetic Weyl superconductors

SLIDE 26

(Samsel-Czekala et al.)

point node Weyl point

Fermi surface for majority spin

Chiral Anomaly in Ferromagnetic Weyl superconductor

UCoGe Ferromagnetic SC

✦ strong Ising-type magnetic anisotropy ✦ close to quantum criticality

(TCurie ∼ 2.5 K a d TSC ∼ 0.6 K)

✦ non-unitary spin-triplet SC large longitudinal spin fluctuation

∆↑↑ = ∆↓↓

fluctuating Weyl point position

(NMR-exp, Hattori et al.)

SLIDE 27

Chiral Anomaly in Ferromagnetic Weyl superconductor

L = µνλξ 42 Aem5

µ

Aem

ν ξAem ξ

+ µνλξ 42 Aem5

µ

Aem5

ν

ξAem5

ξ

Effective Lagrangian for emergent electrodynamcis (Axion electrodynamcis)

Aem5

µ

Aem

µ

: emergent gauge field : emergent chiral gauge field

Aem5 kF = kF 0 + δkF δkF ∝ δm

position of Weyl points in k-space fluctuation due to ferromagnetic spin fluctuation

FM spin fluctuation couples to emergent electromagnetic fields !

(e.g. strain, twist distortion of lattice) (e.g. vortex of SC gap)

SLIDE 28

Chiral Spin-Polarization Effect in Ferromagnetic Weyl superconductor UCoGe

Effective Lagrangean

r, A5

σ = kF σ +

er vector, and δkF σ, w

n, δkF σ = (0, 0, cσδm)

longitudinal FM spin fluctuation along z-axis

δm

Sspin =

ω,q

iω vq − q2 − κ

δm(q, ω)δm(−q, −ω),

Total action of spin fluctuation

Lσ = Φem 2π Aem5

σ

· Bem Eem = Φem

Stot = Sspin +

drdt(L↑ + L↓)

Increase of spin magnetization !!

δm = c↑ + c↓ 4πκ ΦemBem

z

e.g. temperature gradient

SLIDE 29

SUMMARY

Torsional chiral magnetic effect in Weyl superconductors can

be induced by skyrmion-like vortex textures of SC order parameter

Negative thermal magnetoresistivity as a signature of chiral

anomaly is realized by a vortex in Weyl superconductors

In FM Weyl superconductors near FM quantum criticality,

chiral anomaly can be detected as the increase of spin magnetization due to torsional magnetic fields induced by, e.g., twist deformation of a sample around c-axis.