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.

1

2

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.

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

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

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.

5

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

7

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)

8

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.

9

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.

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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

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.

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

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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.

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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19

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.

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

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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.

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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

van Hove Singularity of density of states

)3.0,0.0,145.0,0.1()'',',,( 21 tttt

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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.

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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.

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Improvement compared to previous works

• Conventional interactions of the Hubbard type are sufficient to result in the nematic ordering.

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• 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.

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.

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Angle-dependence of the ab-plane resistivity

aa bb

Borzi et. al., Science 315, 214 (2007)