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Nematic Electron States in Orbital Band Systems

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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Nematic Electron States in Orbital Band Systems

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  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. 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).

  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.

  10. 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).

  11. 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:

  12. Nematic electron liquid: the channel. • Ferromagnetism: the channel. Pomeranchuk instability criterion • Fermi surface: elastic membrane. • Stability: • Surface tension vanishes at: I. Pomeranchuk

  13. 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)

  14. 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.

  15. 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.

  16. 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.

  17. 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.

  18. 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.

  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.

  20. 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

  21. 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!!!

  22. Microscopic Model • Band Hamiltonian: s-bonding , p-bonding , next- nearest-neighbour hoppings • Hybridized eigenbasis.

  23. van Hove Singularity of density of states

  24. Mean-Field Solution based on the multiband Hubbard model • Competing orders: magnetization, charge/spin nematic orders near the van Hove singularity.

  25. 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

  26. 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.

  27. 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.

  28. Angle-dependence of the ab-plane resistivity Borzi et. al., Science 315, 214 (2007)

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