G.L.J.A. Rikken

Laboratoire National des Champs Magnétiques Intenses,

CNRS/INSA/UGA/UPS, Toulouse & Grenoble, France

GDR Chirafun, Rennes 26/5/2016

Magneto-chiral anisotropy;

what has been done, what could be done

“Il n’y a pas de choses simples, mais il y a une manière simple

de voir les choses.”

Paul Valéry

Summary

- Symmetry and all that

- Magneto-chiral effects

history

in optics

in transport

in mechanics

- Summary and conclusion

Symmetry

Continuous symmetries (Noether theorem):

Translational time invariance tt+t Energy conservation

Translational invariance rr+r Momentum conservation

Rotational invariance φ φ+φ Angular momentum cons.

CPT symmetries:Charge conjugation C: q -q (matteranti-matter)Parity P: r-r (~ take mirror image)Time reversal T: t-t (~ play movie backwards)

We will assume physics to be invariant under C and P and T;valid for: electrodynamics and (quantum) mechanics, invalid for: nuclear processes (only CּPּT invariance)

Recipe for the use of C-P-T symmetries:

If an effect is described by ( , , ,..)X f Y z t

l.h.s. should transform identically separately under C and P and T

Note:

1) A symmetry argument does not say whether an effect exists, but only if

it is forbidden or not

2) CPT symmetry arguments only apply to deterministic descriptions

then the r.h.s. and

The CPT symmetry of a physical quantity; example of the magnetic field

Parity reversal P: r-r, tt, q q so BB

Charge conjugation C: rr, tt, q -q so B-B

Time reversal T: rr, t-t, q q so B-B

Magnetic field is a time-odd pseudo-vector

2

0

1 2

4 r

I rB

qNt

rI I

B

r

2

0

2

4

qN t

r

rr

B

CPT properties of common physical quantities

C P T

electrical current I - - -electric field E - - +magnetic field B - + -voltage V - + +momentum p + - -force F + - +angular momentum L + + -mass m + + +

Application to magneto-optics

Michael Faraday

L

B

k

e

e'

VBL

Faraday effect (1846)

The coupling between magnetic field and light was the key element in

the development of the theory of electromagnetism.

CPT symmetry and magneto-optics

Perturbation approach:

Symmetry:

0 0 1 0 2 0 0, , , , ....f f P B B E E B E B B E

C P T

P - - +

B - + -

E - - +

0 1 0 2 0 0 3 0 0 ....t

EP B E B B B E B E B

2

1 0 0i O B E B E

Simplest symmetry allowed form from P and T invariance:

0 0, 1 4ij ij ij ijk ki e B BDielectric constant:

odd under CC invariance:

Levi-Civitta

Maxwell:

Faraday geometry:

Ansatz:

Solution:

2

2c t

EE

( ) expE z i t ikz E x

0 k B z

exp cos sinz E i t ikz kz kz E x y

L

B

k

e

e'

Faraday effect (1846)

V = Verdet constant

0VB L

V k

Verdet constant is the only linear coupling

between light and magnetic field in matter.

k kB with

Chirality

ir = hand)

Definition: A system is called chiral when it exists in two forms

(enantiomers) that can only be interconverted by a parity operation

Examples:

DextroLaevo

Note: Biochemistry is ‘homochiral’, so chirality is vital

Optical properties of chiral systems

2

1 2( , ) ( ) ( ) ( ) ...k k k

spatial dispersion (non-local optical response):

in non-chiral systems: 1( ) 0

in chiral systems: 1 1 1( ) 0, ( ) ( )D L

Lk

e

e'

X L

X

optical activity

(in absorption: circular dichroism)

Magneto-chiral effect, some history

Pasteur tried crystallisation in magnetic field to select chirality (1855)

Curie established the symmetry of chirality and magnetic field (1894)

Groenewege; the first (implicit) prediction of the magneto-chiral effect in

optical properties (1962)

Baranova (1977), Wagnière (1982) and Barron (1984): theory of optical

magneto-chiral effect

Magneto-optical properties of chiral systems

Breaking of time reversal symmetry by a magnetic field leads

to the Faraday effect.

Breaking of parity symmetry by chirality leads to optical activity.

Breaking both symmetries leads to an additional new effect:

magnetochiral anisotropy.

/ //( ) ( )( ( ), , ) ( ) D LD L D Lk B k kB B

- dependent on relative orientation of light and field

- independent of polarization

- enantioselective:D L

symmetry allowed expansion of dielectric constant for CPL:

MChA in luminescence

excitation

I RL IB

L

excitation

B RIIL

D

Observation of MChA in luminescence

+ -

L

F

B

MM

LA

S

AC

( ) ( )

( ) ( )

I Ig

I I

B k B k

B k B k

( ) ( ) ImI I B k B k

(nm)

590 600 610 620

g (

10-3

)-1,0

-0,5

0,0

0,5

1,0L-tfc

D-tfc B (T)0,0 0,5

g (

10- 3

)

-0,3

0,0

Luminescence of Eu(D/L)tfc3

G.L.J.A. Rikken and E. Raupach, Nature 390, 493 (1997)

B = 1 T

(nm)1000 1200 1400

A (

cm

-1)

0

5

10

15

A

MC

hA (

10-3

cm

-1)

-3

-2

-1

0

1

2

3D-crystal

L-crystal

B = 1 T

4 26 0NiSO H

(nm)690 695 700

gM

ChA(

10-5

)

-2,0

-1,0

0,0

1,0

2,0

A (

cm

-1)

3,0

3,5

4,0

4,5

5,0

5,5

6,0

D-Cr(ox)3

L-Cr(ox)3

B = 1 T

3( )Cr ox solution

Magneto-chiral anisotropy in absorption

Multichannel MChA spectrometer

Kopnov and Rikken, Rev. Sci. Instr. 85, 053106 (2014)

T-1

4 26 0NiSO H paramagnetic

MChA in the optical absorption of a chiral ferromagnet

C. Train et al, Nat. Mat. 7, 730 (2008)

MChA in Bragg scattering

(C.Koerdt, G. Duchs, G.L.J.A. Rikken, PRL 91, 073902 (2003))

Left

Right

Cholesteric LC( ) ( )

( ) ( )

T T

T T

B k B k

B k B k(for unpolarized light)

Strong enhancement!10 110moleculen T

Other MChA manifestations in optics observed:

-Magneto-chiral birefringence, Kleindienst et al (1998), Vallet et al (2001)

-Magneto-chiral dichroism in X ray absorption (2002)

-Magneto-chiral dichroism in microwave absorption (2015)

-Magneto-chiral dichroism in THz absorption (2014)

MChA in photochemistry

Optical activity ressembles the Faraday effect:

Magnetically induced enantiomeric excess in a chemical reaction

has been searched for since Pasteur, but is symmetry forbidden:

Lk

e

e'

X L X

L

B

k

e

e'

VBL

V k

achiral starting product D product

achiral starting product L product

B

B

assume:

Pparity operation:conflicting, so

nonsense!

Magnetic fields in chiral photochemistry

Magnetically induced enantiomeric exces in photochemistry is

symmetry allowed:

achiral starting product D product

achiral starting product L product

k·B

-k·B

assume:

Pparity operation:symmetry

allowed!

photochemical reaction rate absorption k·B

Magnetochiral anisotropy can drive a photochemical

reaction into enantiomeric excess.

Explanation for the homochirality of Life?

Magneto-chiral anisotropy in photochemistry

Photoresolution of Cr-oxalate:

spontaneous spontaneous

light absorption light absorption

In a parallel magnetic field, one enantiomer will absorb more

unpolarized light and will therefore decompose more rapidly.

Enantioselective magneto-chiral photochemistry

LA

PEM

Ti-sapphire

GreNe

B

B (T)

-10 -5 0 5 10 15

ee

(1

0-4

)-2

-1

0

1

k B

k // B

= 695.5 nm

G.L.J.A. Rikken and E. Raupach, Nature 405, 932 (2000)

MChA microscopic theory

Barron & Vbranchich 1984:

Harmonic oscillator theory MChA (Donaire, Rikken, Van Tiggelen 2013)

22 2 2 2 2 2

0( ) ( )2 2

x y z

p qH µ x y z Cxyx

µ µ r p B

Harmonic osc. Chiral Zeeman

0

2/

0

0

23

10

D L

MChA E ORn Bq c

Optical rotation

MChA

Molecule HO theory Experiment

limonene 2.6 3.2

tartaric acid 0.12 0.90

proline 0.4 2.7

Units 10-10 T-1

OR

Density fonctional theory MChA (Coriani, Rizzo, Rikken 2016)

methyloxirane O

6 1/ 5 10 T Experimental challenge!

MChA “wave” effects still to be observed:

-Magneto-chiral deflection of light

-Magneto-chiral anisotropy in light diffusion

-Magneto-chiral anisotropy in ultra-sound propagation

-Magneto-chiral enantio-selectivity in NMR/EPR

-Magneto-chiral anisotropy in magnon/helicon/plasmon propagation

Magneto-chiral deflection of light

Achiral

medium

Unpolarized light beam

•right-handed

•left-handed

B

kiik EEε

π

ω

π

c

kBES

16Re

8Poynting vector:

0a

πγBθ

λdeflection:

/ / /

0 0 0( , , ) ( ) ( ) ( ) ( )D L D L D Lk B k B k B

aθ

Chiral

medium

Unpolarized light beam

•right-handed

•left-handed

B

0MChA

π Bθ

λ

deflection:

MChAθ

Observed: Rikken et al 1996 Never observed

Two cases:

0

0 0

MChA in light diffusion

inI

B

2*

0out

in

IT g VBl

I

outI

A. Sparenberg, et al PRL 79, 757 (1997)

Achiral medium:

Chiral medium:

2

* *

0out

in

IT f kBl g VBl

I

Never observed

Magneto-chiral enantio-selectivity in NMR/EPR

NMR/EPR is based on magnetic circular dichroism of a spin system and

NMR/EPR in a chiral spin system should show magneto-chiral anisotropy, in

absorption and in emission, and therefore enantio-selectivity.

B

+

-excitation

chiral sample

detection up

detection down

0

Possible MChA spin echo experiment

MChA signal

MChA in non-linear optics

- Inverse magneto-chiral anisotropy

- Magnetic field induced second harmonic generation

Inverse optical magneto-chiral anisotropy

/ / /D L D L D LIMChAM k E E I k

Inverse Faraday effect: magnetization induced by circularly polarized light in all media

*IFEM VE E VI k

(Van der Ziel, 1965)

Inverse magneto-chiral anisotropy: magnetization induced by unpolarized light in chiral media

Not observed

(difficult to suppres IFE)

Magnetic field induced second harmonic generation

Electric field induced second harmonic generation (EFISHG):

2

2 2 20 0P E E E I E I in all media

2 2

2 / 2 /0 0

D L D LM M

EP B E I B I

t

Only In chiral media

Never observed

Magnetic field induced second harmonic generation (MFISHG):

Ab initio DFT calculations MFISHG (Rizzo, Rikken, Mathevet 2016)

d33 = 1 fm/V at 50 T

Experimental challenge!

E PJ

t t

so

2 / /0 0

D L D LM M M

EP B E J B E

t

MFISHG is electrical MChA at optical frequencies!

Simple picture MFISHG

Magneto-chiral anisotropy in electrical transport

Conjecture: magneto-chiral anisotropy exists in all momentum transport

through chiral media, both ballistic and diffusive.

Possible manisfestations:

Thermal conductivity, molecular difffusion,…..

/ 2 /

0( , ) (1 )D L D LR R B B I B IElectrical resistance:

/ //

0

( , ) ( , )2

D L D LD LR R

R

B I B IB I linear MR!

Onsager’s relation and magneto-chiral anisotropy

i ij jS FGeneralized diffusive transport:

Onsager ’s relation: *ij ji (* = time reversal)

In a magnetic field:

2( ) ( ) ( ) 1 ...ij ji B ii iiB B B

In a chiral medium in a magnetic field:

/ /

/ / 2

( , ) ( , )

( , ) 1 ..

D L D Lij ji

D L D L B

ii ii

k B k B

k B B k

So there is no symmetry argument against MChA in the two

terminal resistance of a chiral conductor.

B (T)

-0,6 -0,4 -0,2 0,0 0,2 0,4 0,6

R(B

) (m

)

-4

-2

0

2

4

T = 77 KBi wire

D-helix

L-helix

I (A)

0,00 0,05 0,10 0,15 0,20 0,25

R

(m

)

0,0

2,5

5,0

7,5

10,0 B = -0,63 TT = 77 K

Observation of MChA in electrical resistance I; self field

« Theory »:

2/

0( , ) 1D L

ext ext sfR R B I B B I

2

/ 2 /

0 01 1 2 ..D L D L

ext ext extR B I R B B I

I

V

B

eMChA!

Explanation: scattering by

screw dislocations

Observation of MChA in electrical resistance II; scattering

G.L.J.A. Rikken, J. Foelling and P. Wyder, PRL 87, 236602 (2001)

B (T)0,0 0,2 0,4 0,6

R

(B)

-

R(-

B)

(

)

-8

-4

0

4

8

D-torsion

annealed

L-torsion

I

V

B

Observation of MChA in electrical resistance III

Carbon nanotube

I (nA)0 250 500

R

(B,I)

()

0

4

8

B = 16 T

B = 8 T

B (T)-20 -10 0 10 20

R

(B)

(

)

-5,0

-4,5

-4,0

-3,5

-3,0

-2,5T = 4.2 KI = 500 nA

V. Krstic et al, J. Chem. Phys. 117,

11315 (2002)

nqJ v

-Of the 230 possible crystal space groups, 65 are chiral

-So there should be plenty of bulk chiral metals

- However, apart from Xray crystallography, no sign of chirality in 3D metals

has been observed

-Difficult to obtain a chiral metal of given handedness

-Solution: organic 3D metal, built from chiral molecules

[(X,X)-DM-EDT-TTF]2ClO4

Observation of MChA in electrical resistance IV; bulk

Result eMChA in bulk metal

SS RR

Bi SWCNT [DM-EDT-TTF]2ClO4

(T-2) 102 10–4 < 10–3

(T–1 A–1) 10–3 103 10–2

(m2 T–1 A–1) 10–8 10–14 10–10

Comparison eMChA

F. Pop, G. Rikken et al, Nature Comm. 2014

The effect exists in 3D, microscopic explanation?

Other MChA transport effects still to be observed:

-MChA in thermal conductivity

-Magneto-chiral induction:/D L B

Et

χ

-Inverse electrical magneto-chiral anisotropy: /D LM I χ

Magneto-mechanical effects

Magnetic alignment M B

Magnetic force M B

Magneto-striction 2B

Quadratic effects:

Linear effects:

Einstein-deHaas effect (1915): L M

Barnett effect (IEdHe, 1915): M L(Gyromagnetic effects)

Wiedemann effect (1858): chiral distorsion I B

Matteucci effect (IWe): I ~ chiral distorsion in B

Anything else?

(Magneto-chiral effects)

Magnetic torque M B

Additionnal symmetry allowed magneto-mechanical effects

Magneto-electric:

p E B

'M E p (Rontgen, 1888)

"P B p (Wilson, 1905)

Magneto-chiral:

/D Lp BD L with

/'D LM p (« inverse magneto-chiro-dynamical

effect », unobserved!)

(« magneto-chiro-dynamical

effect », unobserved!)

Relativistic effects

??

p E B

Mechanical magneto-electric anisotropy

Symmetry allowed:

Experiment on

gazes:

Sound pressure

acoustic

dp dEP = B

dt dt

Dimension must be C2s2kg-1 which is the dimension of electric polarizability

Mechanical magneto-electric anisotropy exists!

It is called the Abraham force in electrodynamics

and is a result of the Lorentz & Coulomb forces

Rikken et al, PRL 107, 170401 (2011)

The magneto-chirodynamical effect/D Lp B Part 1

-The effect is symmetry allowed but has no classical explanation

-Hypothesis: It results from coupling between linear and angular momentum

intrinsic in chiral wavefunctions

-Model: Free electron on a helix

2 a

B

2 a

B

p-1 p-1

y

z

x

φ

B

pitch: p = (2π|b|) -1

a a

Hamiltonian:

2 2 2 2 22

2 2 2 2 2ˆ

2 ( ) 2 8ext ext

q a q aH i B B

m a b m a b m

b

b

baN

np nz

)( 22,

2

, 2 22 ( )z n

e

en am

m N a b b

linear momentum:

orbital magnetic

moment

For a perfect typical CNT: v/B = 10-4 m/s/T

normalized eigenstates with

periodic boundary conditions:

n = ..,-1,0,1,2,,…

N number of turns NinbNn /exp2

1

Y

2 2

2 2e em b m bp m B

ea ea

so

(Krstic, Wagniere and Rikken 2004)

The magneto-chirodynamical effect/D Lp B

-The effect is symmetry allowed but has no classical explanation

-Hypothesis:

It results from “optical” MChA interacting with the quantum vacuum fluctuations

Model Hamiltonian of harmonic oscillator + QV: H = H0 + HEM + W with

/2

ln 19

D LOR N

E e

mp eB

m

(Donaire, Rikken, Van Tiggelen 2014, 2015)

2-octanol in 10 T: V = 0,6 nm/s

(experimental challenge!)

Fine structure constant

So purely QM

Part 2

The magneto-chirodynamical effect

/D Lp B

2/

2

D L

kin

N

BE

m

Where does the energy come from? Field or quantum vacuum?

Considerations:

-The momentum can only come from the QV, B has no momentum

-One can extract energy from the QV; cf. Casimir effect

Ekin is supplied by the source of the magnetic field, necessary to create the vacuum

corrections to the magnetization of the molecule ( “magnetic Lamb shift”)

Conjecture:

The second magneto-chirodynamical effect/ 2 2/D Lp B t

This effect is symmetry allowed, and has a classical explanation:

The electric displacement current in a dielectric also

carries mechanical momentum. Driving the displacement

current with an induced voltage gives

/ 2 20 /D Lp B t

0 /DJ E t

/ /D LE B t

D Dp J

Rikken and Van Tiggelen, Phys.Rev. Lett. 110, 194301 (2013)

The magneto-chirodynamical effect in other areas:

L R j B

Chiral magnetic effect for relativistic fermions:

e.g. Prog. Part. Nucl. Phys. 75, 133 (2014)

High energy physics:

Chiral magnetic effect in Weyl semimetals: Li et al, Nature Physics 2016

Condensed matter physics:

Chiral electronics ?!

Conclusion

Symmetry arguments are a powerful instrument to discover new effects, in

magnetic fields or elsewhere

There are still plenty of undiscovered magneto-chiral effects out there.