Unconventional magnetism and spontaneous spin-orbit ordering · Jan 18, 2017, UCSD Congjun Wu...
Transcript of Unconventional magnetism and spontaneous spin-orbit ordering · Jan 18, 2017, UCSD Congjun Wu...
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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
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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
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Introduction
spin-orbit coupling
unconventional magnetism
unconventional superconductivity
nematicelectron liquid
with spin
unconventional symmetries
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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.
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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
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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).
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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).
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Spin-orbit coupling
unconventional magnetism
unconventional superconductivity
anisotropic electron liquid
Introduction: electron spin liquid-crystal
s
s
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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).
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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)
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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
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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
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Spin-orbit coupling
unconventional magnetism
unconventional superconductivity
anisotropic electron liquid
Introduction: dynamic generation of spin-orbit coupling
s
s
2fk
1fk
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• 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.
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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
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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)
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Summary of the introduction
spin-orbit coupling
electron liquid crystal with spin
unconventional magnetism
unconventional superconductivity
s
s
2fk
1fk
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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.
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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
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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
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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
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• 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)(
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• : 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
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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!
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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
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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
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2D d-wave and -phases
-phase -phase: w=2 -phase: w=-2
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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
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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.
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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
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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.
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Magnetic dipoles: from classic to quantum
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• 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
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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).
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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).
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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
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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.
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• 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
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• 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)
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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)
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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).
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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.
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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).
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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.
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Summary
spin-orbit coupling
electron liquid crystal with spin
unconventional magnetism
unconventional superconductivity
s
s
-phase
2fk
1fk
-phase