1. 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. 2 Outline Experimental results: metamagnetism and nematic ordering in the bilayer Sr 3 Ru 2 O 7. N ematic electron states – Pomeranchuk instabilities. Nematic electron states based on quasi-one dimensional bands (d xz and d yz ) and their hybridization. Ginzburg-Landau analysis and microscopic theory.

3. Metamagnetism in Sr 3 Ru 2 O 7 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) 3

4. 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

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. 5

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) 6

7. 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. 8 Important observation Metamagnetic transitions and the nematic ordering is NOT observed in the single layer compound, Sr 2 RuO 4, 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 Sr 3 Ru 2 O 7.

9. 9 Outline Experimental results: metamagnetism and nematic ordering in the bilayer Sr 3 Ru 2 O 7. N ematic electron states – Pomeranchuk instabilities. Nematic electron states based on quasi-one dimensional bands (d xz and d yz ) and their hybridization. Ginzburg-Landau analysis and the microscopic theory.

10. 10 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).

11. 11 Landau Fermi liquid (FL) theory L. Landau Landau parameter in the l-th partial wave channel: The existence of Fermi surface. Electrons close to Fermi surface are important. Interaction functions (no SO coupling): density spin

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

13. Spin-dependent Pomeranchuk instabilities 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). 13

14. 14 Previous theory developed for Sr 3 Ru 2 O 7 based on Pomeranchuk instability The two dimensional d xy -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 1 st 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 2 nd transition: four-fold rotational symmetry is restored.

15. 15 Outline Experimental finding: metamagnetism and nematic states in the bilayer Sr 3 Ru 2 O 7. N ematic electron states – Pomeranchuk instabilities. Nematic electron states based on quasi-one dimensional bands (d xz and d yz ) and their hybridization. Ginzburg-Landau analysis and the microscopic theory.

16. Questions remained The t 2g bands (d xy, d xz, d yz ) are active: 4 electrons in the d shell per Ru atom. The d xy band structures in Sr 3 Ru 2 O 7 and Sr 2 RuO 4 are similar. Why the nematic behavior only exists in Sr 3 Ru 2 O 7 ? A large d-wave channel Landau interaction is required, while the Coulomb interaction is dominated in the s-wave channel. 16

17. 17 Proposed solution The key bands are two quasi-one dimensional bands of d xz and d yz. Similar proposal has also been made by S. Raghu, S. Kivelson et al., arXiv/0902.1336. The major difference of electron structures between Sr 3 Ru 2 O 7 and Sr 2 RuO 4 is the large bilayer splitting of these two bands.

18. Band hybridization enhanced Landau interaction in high partial-wave channels A heuristic example: a hybridized band Bloch wavefunction with internal orbital configuration as Even V(p 1 -p 2 ) 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. 18

19. 19 Outline Experimental results: metamagnetism and nematic ordering in the bilayer Sr 3 Ru 2 O 7. N ematic electron states – Pomeranchuk instabilities. Nematic electron states based on quasi-one dimensional bands (d xz and d yz ) and their hybridization. Ginzburg-Landau analysis and the microscopic theory.

20. Ginzburg-Landau Analysis m: magnetization; n c, 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 r c,sp under the condition that 20

21. Hybridization of d xz and d yz orbitals New eigen basis has internal d-wave like form factors which could project a pure s-wave interaction to d-wave channel!!! Hybridized Fermi Surface in 2D Brillouin Zone For simplicity, we only keep the bilayer bonding bands of d xz and d yz. 21

22. Microscopic Model Band Hamiltonian:  -bonding,  -bonding, next- nearest-neighbour hoppings Hybridized eigenbasis. 22

23. van Hove Singularity of density of states 23

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

25. Phase diagram v.s. the magnetic field metamagnetic transitions nematic ordering for FS of majority spins Metamagnetism induced by the DOS Van Hove singularity. Nematic ordering as orbital ordering. 25

26. Improvement compared to previous works Conventional interactions of the Hubbard type are sufficient to result in the nematic ordering. 26 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 Sr 3 Ru 2 O 7. Orbital band hybridization provides a new mechanism for the nematic states. 27

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