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![Page 1: Nematic Electron States in Orbital Band Systems Congjun Wu, UCSD Collaborator: Wei-cheng Lee, UCSD Feb, 2009, KITP, poster Reference: W. C. Lee and C.](https://reader035.fdocuments.us/reader035/viewer/2022062715/56649d815503460f94a66ad1/html5/thumbnails/1.jpg)
Nematic Electron States in Orbital Band Systems
Congjun Wu, UCSD
Collaborator: Wei-cheng Lee, UCSD
Feb, 2009, KITP, poster
Reference: W. C. Lee and C. Wu, arXiv/0902.1337
Another independent work by:
S. Raghu, A. Paramekanti, E.-A. Kim, R.A. Borzi, S. Grigera, A. P. Mackenzie, S. A. Kivelson, arXiv/0902.1336
Thanks to X. Dai, E. Fradkin, S. Kivelson, Y. B. Kim, H. Y. Kee, S. C. Zhang.
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Outline
• Experimental results: metamagnetism and nematic ordering in the bilayer Sr3Ru2O7.
• Nematic electron states – Pomeranchuk instabilities.
• Nematic electron states based on quasi-one dimensional bands (dxz and dyz ) and their hybridization.
• Ginzburg-Landau analysis and microscopic theory.
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Metamagnetism in Sr3Ru2O7
• Bilayer ruthenates.• Meta-magnetic transitions; peaks of the real part of magnetic susceptibility.
• Dissipative peaks develop in the imaginary part of magnetic susceptibility for H//c at 7.8T and 8.1T.
Re
Grigera et. al., Science 306, 1154 (2004)3
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Resistance anomaly
• Very pure samples: enhanced electron scattering between two meta-magnetic transitions below 1K.
• A reasonable explanation: domain formation.
• Phase diagram for the resistance anomaly region.
Grigera et. al., Science 306, 1154 (2004)4
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Grigera et. al., Science 306, 1154 (2004)
A promising mechanism: Pomeranchuk instability!
• A new phase: Fermi surface nematic distortion.
• Resistivity anomaly arises from the domain formation due to two different patterns of the nematic states.
• Resistivity anomaly disappears as B titles from the c-axis, i.e., it is sensitive to the orientation of B-field.
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Further evidence: anisotropic electron liquid
• As the B-field is tilted away from c-axis, large resistivity anisotropy is observed in the anomalous region for the in-plane transport.
Borzi et. al., Science 315, 214 (2007) 6
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Similarity to the nematic electron liquid state 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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Important observation
• Metamagnetic transitions and the nematic ordering is NOT observed in the single layer compound, Sr2RuO4, in high magnetic fields.
• What is the driving force for the formation of nematic states?
• It is natural to expect that the difference between electronic structures in the bilayer and single layer compounds in the key reason for the nematic behavior in Sr3Ru2O7.
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Outline
• Experimental results: metamagnetism and nematic ordering in the bilayer Sr3Ru2O7.
• Nematic electron states – Pomeranchuk instabilities.
• Nematic electron states based on quasi-one dimensional bands (dxz and dyz ) and their hybridization.
• Ginzburg-Landau analysis and the microscopic theory.
![Page 10: Nematic Electron States in Orbital Band Systems Congjun Wu, UCSD Collaborator: Wei-cheng Lee, UCSD Feb, 2009, KITP, poster Reference: W. C. Lee and C.](https://reader035.fdocuments.us/reader035/viewer/2022062715/56649d815503460f94a66ad1/html5/thumbnails/10.jpg)
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Anisotropy: liquid crystalline order
• Classic liquid crystal: LCD.
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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Landau Fermi liquid (FL) theory
L. Landau
• Landau parameter in the l-th partial wave channel:
DOS:0,
0, NfNF as
las
l
1p
2p
• The existence of Fermi surface. Electrons close to Fermi surface are important.• Interaction functions (no SO coupling):
)ˆ,ˆ(
)ˆ,ˆ()ˆ,ˆ(
21
2121,
ppf
ppfppfa
s density
spin
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Pomeranchuk instability criterion
• Surface tension vanishes at:
)12(, lF asl
I. Pomeranchuk
ln
• Fermi surface: elastic membrane.
• Stability:
2,,
int
2,
)(12
)(
asl
asl
aslK
nl
FE
nE
• Ferromagnetism: the channel.
aF0
• Nematic electron liquid: the channel.
sF2
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Spin-dependent Pomeranchuk instabilities
phase
s
s
phase
s
2fk
1fk
s
s
s
• Unconventional magnetism --- particle-hole channel analogy of unconventional superconductivity.
• Isotropic phases --- -phases v.s. He3-B phase
Anisotropic phases --- -phases v.s. He3-A phase
C. Wu and S. C. Zhang, PRL 93, 36403 (2004); C. Wu, K. Sun, E. Fradkin, and S. C. Zhang, PRB 75, 115103(2007)
J. E. Hirsch, PRB 41, 6820 (1990); PRB 41, 6828 (1990).
V. Oganesyan, et al., PRB 64,195109 (2001); Varma et al., Phys. Rev. Lett. 96, 036405 (2006).
wave-p13
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Previous theory developed for Sr3Ru2O7
based on Pomeranchuk instability
• The two dimensional dxy-band with van-Hove singularity (vHS) near (0,), (,0).
H.-Y. Kee and Y.B. Kim, Phys. Rev. B 71, 184402 (2005); Yamase and Katanin, J. Phys. Soc. Jpn 76, 073706 (2007); C. Puetter et. al., Phys. Rev. B 76, 235112 (2007).
• The 1st meta-magnetic transition: the FS of the majority spin is distorted to cover one of vHs along the x and y directions.
• As the B-field increases, the Fermi surface (FS) of the majority spin expands and approaches the vHS.
• The 2nd transition: four-fold rotational symmetry is restored.
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Outline
• Experimental finding: metamagnetism and nematic states in the bilayer Sr3Ru2O7.
• Nematic electron states – Pomeranchuk instabilities.
• Nematic electron states based on quasi-one dimensional bands (dxz and dyz ) and their hybridization.
• Ginzburg-Landau analysis and the microscopic theory.
![Page 16: Nematic Electron States in Orbital Band Systems Congjun Wu, UCSD Collaborator: Wei-cheng Lee, UCSD Feb, 2009, KITP, poster Reference: W. C. Lee and C.](https://reader035.fdocuments.us/reader035/viewer/2022062715/56649d815503460f94a66ad1/html5/thumbnails/16.jpg)
Questions remained
• The t2g bands (dxy, dxz, dyz) are active: 4 electrons in the d shell per Ru atom.
• The dxy band structures in Sr3Ru2O7 and Sr2RuO4 are similar. Why the nematic behavior only exists in Sr3Ru2O7?• A large d-wave channel Landau interaction is required, while the Coulomb interaction is dominated in the s-wave channel.
xyd
16
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Proposed solution
• The key bands are two quasi-one dimensional bands of dxz and dyz .
• Similar proposal has also been made by S. Raghu, S. Kivelson et al., arXiv/0902.1336.
xzdyzd
• The major difference of electron structures between Sr3Ru2O7 and Sr2RuO4 is the large bilayer splitting of these two bands.
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Band hybridization enhanced Landau interaction in high partial-wave channels
• A heuristic example: a hybridized band Bloch wavefunction with internal orbital configuration as
)sin(cos)( yzpxzp
ipr ddep
)()](2cos1[)0(),( 21212
1
21 ppVqVppf pp
• Even V(p1-p2) is dominated by the s-wave component, the angular form factor shifts a significant part of the spectra weight into the d-wave channel.
• The Landau interaction acquires an angular form factor as.
xzdxzd
yzd
yzd
1p
2p
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Outline
• Experimental results: metamagnetism and nematic ordering in the bilayer Sr3Ru2O7.
• Nematic electron states – Pomeranchuk instabilities.
• Nematic electron states based on quasi-one dimensional bands (dxz and dyz ) and their hybridization.
• Ginzburg-Landau analysis and the microscopic theory.
![Page 20: Nematic Electron States in Orbital Band Systems Congjun Wu, UCSD Collaborator: Wei-cheng Lee, UCSD Feb, 2009, KITP, poster Reference: W. C. Lee and C.](https://reader035.fdocuments.us/reader035/viewer/2022062715/56649d815503460f94a66ad1/html5/thumbnails/20.jpg)
Ginzburg-Landau Analysis
spc
spspspspcccc
m
nnmg
ngnrngnr
hmmFF
)(
)(4242
m: magnetization; nc,sp: charge/spin nematic; h: B-field; g(m) odd function of m
required by time reversal symmetry.
• Metamagnetic transitions: common tangent lines of F(m) with slopes of h and h’.
scrrmg 4)]([ 2
h
h
• If g(m) is large between two metamagnetic transitions, it can drive the nematic ordering even with small positive values of rc,sp under the condition that
20
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Hybridization of dxz and dyz orbitals
New eigen basis has internal d-wave like form factors which could project a pure s-wave interaction to d-wave channel!!!
Hybridized
xzdxzd
yzd
yzd
Fermi Surface in 2D Brillouin Zone
• For simplicity, we only keep the bilayer bonding bands of dxz and dyz.
21
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Microscopic Model
• Band Hamiltonian: -bonding , -bonding , next-nearest-neighbour hoppings
yx
yxxyy
yxyxx
kkyzkxzkyzkyzykxzkxzxband
kktk
kktktktk
kktktktk
chddkddkddkH
coscos''4)(
coscos'4cos2cos2)(
coscos'4cos2cos2)(
..)()()(
//
//
,,,,,,,
)cos(cos
coscos42tan ,
cossin
sincos
//,
,
,
,
yx
yx
k
kyz
kxz
kk
kk
k
k
kk
kk
tt
td
d
• Hybridized eigenbasis.
t//ttt ,
22 )(4))()(()()(
2
1)( 22 kkkkkkE yxyx
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van Hove Singularity of density of states
)3.0,0.0,145.0,0.1()'',',,( 21 tttt
23
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Mean-Field Solution based on the multiband Hubbard model
yzxzaaa
iyzyzxzxz
yzxzi
yzxzi
yzxzyzxzai
aazeeman
ddhchidididid
niniSiSJininVininUHH
,
4
1
,,int
..)()()()(
(i)})()()({)()( )()(
42 ,
24 ,
42
2cos)(
,
222
JUV
UJVV
JUV
nVnVmVkE
nVnVmVH
spcm
kspspccm
kspspccmkkmf
)()( },{ ,)( 2
1
,
zSzSnnnnzSm yzxzspyzxzcyzxza
a
• Competing orders: magnetization, charge/spin nematic orders near the van Hove singularity.
24
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Phase diagram v.s. the magnetic field
metamagnetic transitions
nematic ordering for FS of majority spins
/// th
• Metamagnetism induced by the DOS Van Hove singularity.
• Nematic ordering as orbital ordering.
25
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Improvement compared to previous works
• Conventional interactions of the Hubbard type are sufficient to result in the nematic ordering.
26
• The interaction effect in the ferromagnetic channel is self-consistently taken into account. This narrows down the parameter regime of nematic ordering in agreement with experiments.
• The asymmetry between two magnetization jumps is because the asymmetric slopes of the DOS near the van-Hove singularity.
• To be investigated: the sensitivity of the nematic ordering to the orientation of the B-field; STM tunneling spectra; etc.
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Conclusion
xzd yzd
• Quasi-1D orbital bands provide a natural explanation for the nematic state observed in Sr3Ru2O7. • Orbital band hybridization provides a new mechanism for the nematic states.
27
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Angle-dependence of the ab-plane resistivity
aa bb
Borzi et. al., Science 315, 214 (2007)