Unconventional magnetism and spontaneous spin-orbit ordering · Jan 18, 2017, UCSD Congjun Wu...

Jan 18, 2017, UCSD

Congjun Wu (Univ. California, San Diego)

Unconventional magnetism and spontaneous spin-orbit ordering

Ref. 1) C. Wu and S. C. Zhang, PRL 93, 36403 (2004);

2) C. Wu, K. Sun, E. Fradkin, and S. C. Zhang, PRB.75, 115103 (2007).

3) Y. Li, C. Wu, PRB 85, 205126 (2012).

2fk

1fk

s

s

2

Collaborators

• K. Sun, UIUC (now at U. Michigan)

• S. C. Zhang, Stanford.

• E. Fradkin, UIUC.

Thanks to J. Hirsch, M. Beasley, A. L. Fetter, S. Kivelson,

J. Zaanen, S. Das Sarma,, A. J. Leggett, L. Balents for

stimulating discussions.

• W. C. Lee, UCSD (now at Binghamton, SUNY)

• Y. Li, UCSD (now at Johns Hopkins)

• D. Arovas, UCSD

• C Xu and S. L. Xu, UCSD

3

Introduction

spin-orbit coupling

unconventional magnetism

unconventional superconductivity

nematicelectron liquid

with spin

unconventional symmetries

E. C. Stoner

E

k

• Stoner criterion:

Itinerant FM: Quantum origin!

12

EE1

2

Q: How does spin-independentinteraction induce spin polarization?

10UN

U: interaction strength

Electrons with parallel spins avoid each

other to reduce repulsion.

5

Δ

k

k

Itinerant ferromagnetism: s-wave

• Spin rotational symmetry is broken.

• cf. Conventional superconductivity.

s-wave: gap function invariant over the Fermi surface.

• Orbital rotational symmetry is NOTbroken: spin polarizes along a fixed direction.

s

s

6

cf. Unconventional superconductivity

• High partial wave pairing symmetries (e.g. p, d-wave …).

• p-wave: Sr2RuO4, 3He-A and B.

• d-wave: high Tc cuprates.

22 yxd

++

-

-k

k

D. J. Van Harlingen, Rev. Mod. Phys. 67, 515 (1995); C. C. Tsuei et al., Rev. Mod. Phys. 72, 969 (2000).

New states of matter: unconventional magnetism!

• High partial wave channel magnetism (e.g. p, d-wave…) .

• Multi-polar spin distribution over the Fermi surface.

isotropic p-wave state

2fk

1fk

k

k

anisotropic p-wave state

s

sk

k

spin flips the sign as .kk

spin-split state by J. E. Hirsch, PRB 41, 6820 (1990); PRB 41, 6828 (1990).

8

Spin-orbit coupling

unconventional magnetism

unconventional superconductivity

anisotropic electron liquid

Introduction: electron spin liquid-crystal

s

s

9

Anisotropy: liquid crystalline order

• Classic liquid crystal.

isotropic phase

nematic phase

• Quantum version of liquid crystal: nematic electron liquid.

Nematic phase: rotational anisotropic but translational invariant.

Fermi surface anisotropic distortions

S. Kivelson, et al, Nature 393, 550 (1998); V. Oganesyan, et al., PRB 64,195109 (2001).

10

Nematic electron liquid in 2D GaAs/AlGaAs at high B fields

M. M. Fogler, et al, PRL 76 ,499 (1996), PRB 54, 1853 (1996); E. Fradkin et al, PRB 59, 8065 (1999), PRL 84, 1982 (2000).

M. P. Lilly et al., PRL 82, 394 (1999)

11S. A. Grigera et al., Science 306 1154, (2004). R. A. Borzi et al.. Science express (2006).

Nematic electron liquid in Sr3Ru2O7 at high B fields

• Quasi-2D system; resistivity anisotropy at 7~8 Tesla.

• Fermi surface nematic distortions.

I

I

12

Anisotropic unconventional magnetism: spin liquid-crystal phases!

• Both orbital and spin rotational symmetries are broken.

anisotropic p-wave magnetic phase

• p-wave distortion of the Fermi surface.

spin-split state by J. E. Hirsch, PRB 41, 6820 (1990); PRB 41, 6828 (1990).

V. Oganesyan, et al., PRB 64,195109 (2001). C. Wu et al., PRL 93, 36403 (2004); Varma et al., Phys. Rev. Lett. 96, 036405 (2006)

0cos1 k

k

ksn

• Spin dipole moment in momentum space (not in coordinate space).

ks

k

kx

yk

13

Spin-orbit coupling

unconventional magnetism

unconventional superconductivity

anisotropic electron liquid

Introduction: dynamic generation of spin-orbit coupling

s

s

2fk

1fk

14

• Conventional wisdom:

Unconventional magnetism: dynamic generation of spin-orbit (SO) coupling!

• New mechanism (many-body collective effect):

A single-body effect from the Dirac equation

• Advantages: tunable SO coupling by varying temperatures;

new types of SO coupling.

Generate SO coupling through unconventional magnetic phase transitions.

15

knHHk

MF

0

• Isotropic phase with SO coupling.

• No net spin-moment; spin dipole moment in momentum space.

The isotropic p-wave magnetic phase

• Helicity is a good quantum number.

k

|||| 21 nnn

C. Wu et al., PRL 93, 36403 (2004); C. Wu et al., PRB PRB.75, 115103

(2007). .

k

k

kk

k

k snsn sin,cos 21

1n

2n

x

y

kx

yk

ks

k

16

The subtle symmetry breaking pattern

• Independent orbital and spin rotational symmetries.

J

• is conserved , but are not separately conserved.

SL

,

SO coupling

ordered phase disordered phase

aF1

SLJ

• Relative spin-orbit symmetry breaking.

Leggett, Rev. Mod. Phys 47, 331 (1975)

17

Summary of the introduction

spin-orbit coupling

electron liquid crystal with spin

unconventional magnetism

unconventional superconductivity

s

s

2fk

1fk

18

Outline

• Introduction.

• Mechanism for unconventional magnetic phase transitions.

• Possible directions of experimental realization and detection methods.

- Fermi surface instability of the Pomeranchuk type.

- Mean field phase structures.

- Collective modes and neutron spectroscopy.

• Spin-orbit coupled Fermi liquid theory – magnetic dipolar.

19

Landau Fermi liquid (FL) theory

L. Landau

• Landau parameter in the l-th partial wave channel:

DOS:0

,

0

, NfNF as

l

as

l

1p

2p

• The existence of Fermi surface.

• Electrons close to Fermi surface are important.

• Interaction functions:

)ˆ,ˆ(

)ˆ,ˆ()ˆ,ˆ(

21

2121,

ppf

ppfppf

a

sdensity

spin

20

Pomeranchuk instability

• Surface tension vanishes at:

)12(, lF as

l

I. Pomeranchuk

ln

• Fermi surface: elastic membrane.

• Stability:

2,,

int

2,

)(12

)(

as

l

as

l

as

lK

nl

FE

nE

• Ferromagnetism: the channel.aF0

• Nematic electron liquid: the channel.sF2

21

aF1

phase

s

s

phase

s

2fk

1fk

s

s

s

• An analogy to superfluid 3He-B (isotropic) and A (anisotropic) phases.

Unconventional magnetism: Pomeranchuk instability in the spin channel

22

• 3He-B (isotropic) phase.

B

)(ˆ kd

Δ

(k )

A

kz

kx

ky

• 3He-A (anisotropic) phase.

cf. Superfluid 3He-B, A phases

A. J. Leggett, Rev. Mod. Phys 47, 331 (1975)

• p-wave triplet Cooper pairing.

)ˆˆ(ˆ)( yx kikdk

kkdk ˆ)(ˆ)(

• : Spin currents flowing along x and y-directions, or spin-dipole moments in momentum space.

The order parameters: the 2D p-wave channel

k

k

kkn cos1

k

k

kkn sin2

x

1n2n

y

• cf. Ferromagnetic order (s-wave): k

kks

• Arbitrary partial wave channels: spin-multipole moments.

kkkk

a

l llF sinsin;coscos:

k

kx

yk

aF1

Mean field theory and Ginzburg-Landau free energy

2

212

22

2

2

11

2

2

2

121 ||)|||(|)|||(|)0(),( nnvnnvnnrFnnF

• Symmetry constraints: rotation (spin, orbital), parity, time-reversal.

)(])sincos()()[( 21 knnkkH kk

k

MF

• The simplest non-s-wave exchange interaction:

q

a qnqnqnqnqfH )}()()(){( 22111int

aF1

||

2/1

2 1

10

a

a

F

FNr

21 aFinstability!

25

phase:02 v

s

s

aF1

phase:02 v

s

2fk

1fk

s

s

s

and -phases (p-wave)

1n

2n

x

y

1n

x

y

arbitary||/||;// 1221 nnnn

||||and 2121 nnnn

The 𝛽-phases: vortices in momentum space

• Perform global spin rotations, A B C.

B

x1n2n

y

W=1

Rashba

W=-1

C

x1n2n

y

Dresselhaus

A

x1n2n

y

W=1

L. Fu’s(PRL2015): gyro ferroelectric muti-polar

27

2D d-wave and -phases

-phase -phase: w=2 -phase: w=-2

28

Anisotropic overdamping: The mode is maximally overdamped for q along the x-axis, and underdamped along the y-axis (l=1).

The -phases: orbital & spin channel Goldstone (GS) modes

• Orbital channel GS mode: FS oscillations (intra-band transition).

)}2cos1(2||

)({),(

2

0 q

f

a

l

FSqv

iF

qNqL

q

xk

yk

q

• Spin channel GS mode: “spin dipole” precession (spin flip transition).

Nearly isotropic, underdamped and linear dispersions at small q.

22

2 )(||

qF

na

l

iyx

q

xk

yk

q

The -phases: neutron spectra

k cos

k cos

• No elastic Bragg peaks.

),(Im , qs

q

)(),(Im 222

0, qqs Nq

𝐿 = (𝑛1× 𝜕𝑡 𝑛1 + 𝑛2 × 𝜕𝑡𝑛2) ∙ 𝑆→ 𝑛 (𝑆𝑦𝜕𝑡𝑛1𝑥 −𝑆𝑥𝜕𝑡𝑛1𝑦)

• 𝑛1,2 can couple with spin dynamically at 𝑻 < 𝑻𝒄 -- coupling

between GS modes and spin-waves (spin-flip channel).

• In-elastic: resonance peaksdevelop at T<Tc.

30

The -phases: GS modes

;;

);(2

1

12

12

z

y

z

x

yx

z

nOnO

nnO

• 3 branches of relative spin-orbit modes.

x

y

z• For , these modes are with linear dispersion relations, and underdamped at small q.

• Inelastic neutron spectra: GS modes also couple to spin-waves, and induce resonance peaks in both spin-flip and non-flip channels.

q

2l

Particle-hole continuum

q

qvs

31

Outline

• Introduction.

• Mechanism for unconventional magnetic phase transitions.

• Possible directions of experimental realization and detection methods.

- Fermi surface instability of the Pomeranchuk type.

- Mean field phase structures.

- Collective modes and neutron spectroscopy.

• Spin-orbit coupled Fermi liquid theory – magnetic dipolar interaction.

Magnetic dipoles: from classic to quantum

32

• Ferro-fluid: iron powders in oil.

• In solids, magnetic dipolar interaction ≪ Coulomb interaction.

𝑟𝑠 =𝑑

𝑎𝐵𝐸𝑚 =

𝜇𝐵2

𝑑3 =𝜆𝑐𝑚𝑝2

𝑎𝐵2

𝑅𝑦

𝑟𝑠3 =

𝛼2

𝑟𝑠2 𝐸𝑒𝑙 ≈

1.4

𝑟𝑠3 𝑚𝑒𝑉

Magnetic dipolar Fermi gases

𝒓 𝒓′

)]ˆ)(ˆ(3[)(

)(21213

2

rFrFFFr

grV

• SO coupling at the interaction level.

• Itinerant magnetic dipolar system: (161𝐷𝑦, 163𝐷𝑦) 10𝜇𝐵

%15f

d

E

EnKTcmn F 300,104 313

• SO coupled many-body physics (no Fermi surface splitting):

SO coupled Fermi liquid:

Weyl p-wave triplet pairing (L=S=J=1) Y. Li, C. Wu, Sci. Rep., 2,392 (2012).

Y. Li, C. Wu, PRB 85, 205126 (2012).

Spin-orbit (SO) coupled Fermi liquid theory

1k

2k

• Landau functions: SO harmonic partial-wave decomposition.

),ˆ(),ˆ()ˆ,ˆ(4

2;;1;21,0

kYFkYkkfN

SLJJSLJJLSJJLLJJ

LSJJ zzz

z

z

)ˆ(1

kn )ˆ(

2kn

• Landau matrices: an eigenvalue <-1 Pomeranchuk instability

Y. Li, C. Wu, PRB 85, 205126 (2012).

• 𝐽 = 1−(odd parity), 𝐿 = 𝑆 = 1.

• Transfer SO coupling to the single particle level (Rashba like).

Topological SO zero sound

• Underdamped mode 𝑠 = 𝜔/(𝑣𝑓𝑞) > 1 ∶ sound velocity > Fermi velocity.

• SO coupled Fermi surface oscillations.

𝑢0: hedgehog distribution: 𝐽 = 0−; 𝑢1: longitudinal ferro: 𝐽 = 1+, and 𝑢0>𝑢1

35

𝑆 𝑟, 𝑘, 𝑡 = (𝑢0 𝑘 + 𝑢1 𝑞)𝑒

𝑖(𝑞∙𝑟−𝜔𝑡)

𝐹+ = 𝐹10;01 + 𝐹00;11 𝐹× = 𝐹10;01𝐹00;11

36

Outline

• Introduction.

• Mechanism for unconventional magnetic phase transitions.

• Possible directions of experimental realization and detection methods.

- Fermi surface instability of the Pomeranchuk type.

- Mean field phase structures.

- Collective modes and neutron spectroscopy.

• Spin-orbit coupled Fermi liquid theory – magnetic dipolar interaction.

37

• The driving force is still exchange interactions, but in non-s-wave channels.

s-wave p-wave d-wave

SC/SF Hg, 1911 3He, 1972 high Tc, 1986

magnetism Fe, ancient time ? ?

• Optimistically, unconventional magnets are probably not rare.

cf. Antiferromagnetic materials are actually very common in transition metal oxides. But they were not well-studied until neutron-scattering spectroscopy was available.

A natural generalization of ferromagnetism

38

• URu2Si2: hidden order behavior below 17.5 K.

T. T. M. Palstra et al., PRL 55, 2727 (1985); M. B. Maple et al, PRL 56, 185 (1986)

helicity order (the p-wave -phase);

Search for unconventional magnetism (I)

U

Si

RuVarma et al., Phys. Rev Lett. 96, 036405

(2006)

Search for unconventional magnetism (II)

• Sr3Ru2O7 in the external B field – Orbital-assisted unconventional meta-magnetic state.

xzdxz

d

yzd

yzd

21 pp

1p

2p

)(]2cos1[)0(),(212

1

21 21ppVqVppf

pp

)0(),(21

qVppf

W. C. Lee, C. Wu, PRB 80, 104438 (2009)

40

Science 332, 1410 (2011)

• Consistent with orbital ordering between dxz/dyz orbitals.

H. H. Hung, C. L. Song, Xi Chen, Xucun Ma, Q. K. Xue, C. Wu, Phys. Rev. B 85, 104510 (2012).

41

Detection (I): ARPES

• Angular Resolved Photo Emission Spectroscopy (ARPES).

• and -phases (dynamically generated spin-orbit coupling):

ARPES in spin-orbit coupling systems ( Bi/Ag surface), Ast et al., cond-mat/0509509.

band-splitting for two spin configurations.

The band-splitting is proportional to order parameter, thus is temperature and pressure dependent.

42

Detection (II): neutron scattering and transport

• Elastic neutron scattering: no Bragg peaks; Inelastic neutron scattering: resonance peaks below Tc.

• Transport properties.

-phases: Temperate dependent beat pattern in the Shubnikov -de Hass magneto-oscillations of r(B).

N. S. Averkiev et al., Solid State Comm. 133, 543 (2004).

Detection (III): transport properties

,xj

,xj

echj arg

2l

spinj

spinj

echj arg

𝑗𝑥𝑠𝑝𝑖𝑛

𝑗𝑦𝑠𝑝𝑖𝑛 ∝ 1

−1

𝑗𝑥𝑐ℎ𝑎𝑟𝑔𝑒

𝑗𝑦𝑐ℎ𝑎𝑟𝑔𝑒

,yj ,yj

• Spin current induced from charge current (d-wave). Their directions are symmetric about the x-axis.

44

Summary

spin-orbit coupling

electron liquid crystal with spin

unconventional magnetism

unconventional superconductivity

s

s

-phase

2fk

1fk

-phase