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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

Thanks to X. Dai, E. Fradkin, S. Kivelson, Y. B. Kim, H. Y. Kee, S. C. Zhang.

Feb, 2009, KITP, poster

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.

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

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)

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)

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)

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

- 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.

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.

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).

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:

Nematic electron liquid: the channel.

- Ferromagnetism: the channel.

Pomeranchuk instability criterion

- Fermi surface: elastic membrane.
- Stability:

- Surface tension vanishes at:

I. Pomeranchuk

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)

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.

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.

- 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.

- 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

- 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.

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

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 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

- Band Hamiltonian: s-bonding , p-bonding , next-

nearest-neighbour hoppings

- Hybridized eigenbasis.

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

- 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

- 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

- 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.

Angle-dependence of the ab-plane resistivity

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

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