Part II. p-orbital physics in optical lattices

1
Nematic Electron States in Orbital Band Systems
Congjun Wu, UCSD
Collaborator Wei-cheng Lee, UCSD
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.
Feb, 2009, KITP, poster
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.

3
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.

Grigera et. al., Science 306, 1154 (2004)
4
Resistance anomaly

• Very pure samples enhanced electron scattering
  between two meta-magnetic transitions below 1K.

• Phase diagram for the resistance anomaly region.

• A reasonable explanation domain formation.

Grigera et. al., Science 306, 1154 (2004)
5

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.

Grigera et. al., Science 306, 1154 (2004)
6
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)
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).
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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.

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Anisotropy liquid crystalline order

• Classic liquid crystal LCD.

Nematic phase rotational anisotropic but
translational invariant.
isotropic phase
nematic phase

• Quantum version of liquid crystal nematic
  electron liquid.

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

• The existence of Fermi surface. Electrons close
  to Fermi surface are important.

• Landau parameter in the l-th partial wave
  channel

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Pomeranchuk instability criterion

• Fermi surface elastic membrane.
• Stability

• Surface tension vanishes at

I. Pomeranchuk
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Spin-dependent Pomeranchuk instabilities

• Unconventional magnetism --- particle-hole
  channel analogy of unconventional
  superconductivity.

• Isotropic phases --- b-phases v.s.
  He3-B phase
• Anisotropic phases --- a-phases v.s.
  He3-A phase

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).
C. Wu and S. C. Zhang, PRL 93, 36403 (2004) C.
Wu, K. Sun, E. Fradkin, and S. C. Zhang, PRB 75,
115103(2007)
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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,p), (p,0).

• As the B-field increases, the Fermi surface (FS)
  of the majority spin expands and approaches the
  vHS.

• The 1st meta-magnetic transition the FS of
  the majority spin is distorted to cover one of
  vHs along the x and y directions.

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 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.

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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.

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Proposed solution

• The key bands are two quasi-one dimensional
  bands of dxz and dyz .

• The major difference of electron structures
  between Sr3Ru2O7 and Sr2RuO4 is the large bilayer
  splitting of these two bands.

• Similar proposal has also been made by S. Raghu,
  S. Kivelson et al., arXiv/0902.1336.

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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

• The Landau interaction acquires an angular form
  factor as.

• 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.

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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.

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Ginzburg-Landau Analysis
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.

• 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

• For simplicity, we only keep the bilayer bonding
  bands of dxz and dyz.

Fermi Surface in 2D Brillouin Zone
New eigen basis has internal d-wave like form
factors which could project a pure s-wave
interaction to d-wave channel!!!
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Microscopic Model

• Band Hamiltonian s-bonding , p-bonding
  , next-
• nearest-neighbour hoppings

• Hybridized eigenbasis.

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van Hove Singularity of density of states
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Mean-Field Solution based on the multiband
Hubbard model

• Competing orders magnetization, charge/spin
  nematic orders near the van Hove singularity.

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Phase diagram v.s. the magnetic field

• Metamagnetism induced by the DOS Van Hove
  singularity.
• Nematic ordering as orbital ordering.

metamagnetic transitions
nematic ordering for FS of majority spins
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Improvement compared to previous works

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

• 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

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