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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
