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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.

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slide1

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

outline
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 sr 3 ru 2 o 7
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)

resistance anomaly
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)

slide5

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)

slide6

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)

slide7

M. P. Lilly et al., PRL 82, 394 (1999)

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).

important observation
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.
outline1
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.
slide10

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).

landau fermi liquid fl theory

Interaction functions (no SO coupling):

density

spin

L. Landau

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:
slide12

Nematic electron liquid: the channel.

  • Ferromagnetism: the channel.

Pomeranchuk instability criterion

  • Fermi surface: elastic membrane.
  • Stability:
  • Surface tension vanishes at:

I. Pomeranchuk

slide13

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)

slide14

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.
outline2
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.
slide16

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.
slide17

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.
band hybridization enhanced landau interaction in high partial wave channels
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.
outline3
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
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
hybridization of d xz and d yz orbitals

Hybridized

Hybridization of dxzand 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!!!

microscopic model
Microscopic Model
  • Band Hamiltonian: s-bonding , p-bonding , next-

nearest-neighbour hoppings

  • Hybridized eigenbasis.
mean field solution based on the multiband hubbard model
Mean-Field Solution based on the multiband Hubbard model
  • Competing orders: magnetization, charge/spin nematic orders near the van Hove singularity.
phase diagram v s the magnetic field
Phase diagram v.s. the magnetic field
  • Metamagnetism induced by the DOS Van Hove singularity.
  • Nematic ordering as orbital ordering.

metamagnetictransitions

nematic ordering for FS of majority spins

improvement compared to previous works
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.
conclusion
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.
slide28

Angle-dependence of the ab-plane resistivity

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

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