1. Weyl metal: What’s new beyond Landau’s Fermi liquid theory?
Experiment: Heon-Jung Kim (Daegu Univ.), Ki-Seok Kim (POSTECH),
J.-F. Wang (Huazhong Univ.), M. Sasaki (Yamagata Univ.),
N. Satoh (Iwaki Meisei Univ.), A. Ohnishi, M. Kitaura,
M. Yang, L. Li, Phys. Rev. Lett. 111, (2013).
Boltzmann transport theory: Phys. Rev. B 89, (2014).
A review paper will appear in Science and Technology of Advanced Materials (ISSN ).
Topological Fermi-liquid theory:
Yong-Soo Jho and Ki-Seok Kim (in preparation).

4. These are not science fictions any more.
𝜌 𝐵 − 𝜌 0 𝜌 0
These are not science fictions any more.

5. Weyl metal is a topological Fermi-liquid state, described by axion electrodynamics and thus, distinguished from Landau’s Fermi-liquid state.

6. 𝑭 𝒊 𝝈𝝈′
𝑯 𝒆𝒇𝒇 ( 𝒗 𝒊 𝝈𝝈′ , 𝑭 𝒊 𝝈𝝈′ ,…)
𝒗 𝒊 𝝈𝝈′

7. Boltzmann transport theory with the Drude model in metals

8. Landau’s Fermi-liquid theory for metals

9. Integer quantum Hall Semiconductor
Topological semiconductor
Topological metal
(Fermi liquid)
Metal
(Fermi liquid)

11. Topological Fermi-liquid theory = Landau’s Fermi-liquid theory
+ “topological” term
(Berry curvature + chiral anomaly)

12. Weyl metal B

14. 𝑩𝒆𝒓𝒓𝒚 𝒑𝒉𝒂𝒔𝒆

15. Two questions for understanding Weyl metal
Graphene (2d) vs. Weyl metal (3d)
Multiple Fermi surfaces (Fe-pnictides: non-topological) vs. Weyl metal (topological)

16. More than the Berry curvature

17. (Quantum) anomalies
From classical field theory (symmetry) to quantum field theory (anomaly): Regularization scheme consistent with symmetry
Heisenberg (Poisson  Lie) vs. Feynman (Path integral)
Renormalizability of quantum field theory
Anomalies associated with local (gauge) symmetries must be cancelled for consistency of quantum theory (standard model & string theory).
Anomalies associated with global symmetries give rise to fantastic physics.
Quantum number fractionalization in solitons (Goldstone-Wilczek currents), deconfined quantum criticality (emergent non-abelian chiral anomaly), gapless boundary states and anomalous (quantized) electrical & thermal (Hall) transport phenomena, …
Chiral anomaly (3+1, 1+1), parity anomaly (2+1), Witten anomaly (3+1), …

18. Adler-Bell-Jackiw anomaly (1d)

19. Adler-Bell-Jackiw anomaly (3d): Ultra-quantum limit

20. How to generalize the Drude model for Weyl metal ?

21. M. A. Stephanov and Y. Yin,
Phys. Rev. Lett. 109, (2012)

23. Adler-Bell-Jackiw anomaly (3d): Semi-classical regime
Phys. Rev. B 88, (2013)
Xiao, D., Chang, M.-C. & Niu, Q., Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 (2010);
Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H. & Ong, N. P. Anomalous Hall effect, Rev. Mod. Phys. 82, 1539 (2010).

26. More than the band structure : Chiral anomaly  Axion electrodynamics
F. Wilczek, Phys. Rev. Lett. 58, 1799 (1987)
Chiral magnetic effect
Negative longitudinal
magneto-resistivity

27. Observation of Weyl metal  Confirmation of axion electrodynamics: Negative longitudinal magneto-resistivity

28. Adler-Bell-Jackiw anomaly
right Weyl corn
left Weyl corn
Dirac corn
𝜌 𝐵 − 𝜌 0 𝜌 0
Adler-Bell-Jackiw anomaly
Phys. Rev. Lett. 111, (2013)

29. Longitudinal magnetoresistivity
3d weak antilocalization

30. Theoretical analysis L T

31. Chiral anomaly allows the dissipationless current channel between the paired Weyl points, reducing electrical resistivity along the direction of the applied magnetic field.

32. Toward the theoretical description
Diagrammatic
approach
Boltzmann-equation approach
Weak anti-localization O X
Axion electrodynamics

33. Boltzmann equation approach with weak anti-localization + Semi-classical equation of motion
Xiao, D., Chang, M.-C. & Niu, Q., Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 (2010);
Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H. & Ong, N. P. Anomalous Hall effect, Rev. Mod. Phys. 82, 1539 (2010).

34. Anomalous transport in Weyl metal

35. Longitudinal magneto-resistivity and longitudinal Hall coefficient in Weyl metal
Ki-Seok Kim, Heon-Jung Kim, and M. Sasaki, Phys. Rev. B 89, (2014)

39. Berry curvature + Axion electrodynamics
Can we derive these “effective” coupled Boltzmann equations from a relativistic matrix Boltzmann equation, taking its non-relativistic limit (𝜇≠0) ?
Berry curvature + Axion electrodynamics

41. Toward topological Fermi-liquid theory
How to introduce the momentum-space Berry connection into Landau’s Fermi-liquid theory?
How to encode axion electrodynamics (chiral anomaly) into Landau’s Fermi-liquid theory?

42. M. A. Stephanov and Y. Yin,
Phys. Rev. Lett. 109, (2012)

44. Yong-Soo Jho and Ki-Seok Kim
(in preparation)

45. Derivation of the topological Landau’s Fermi-liquid theory from QED4 under magnetic fields
Yong-Soo Jho and Ki-Seok Kim
(in preparation)

46. Longitudinal negative magneto-electrical resistivity from Kubo formula

47. Weyl metal = A topological Fermi-liquid state  A novel fixed point beyond Landau’s Fermi-liquid theory ??
Yong-Soo Jho and Ki-Seok Kim (in preparation)

48. 𝑭 𝒊 𝝈𝝈′ 𝑯 𝒆𝒇𝒇 ( 𝒗 𝒊 𝝈𝝈′ , 𝑭 𝒊 𝝈𝝈′ ,…) 𝒗 𝒊 𝝈𝝈′
Yong-Soo Jho and Ki-Seok Kim (in preparation)
𝑭 𝒊 𝝈𝝈′
𝑯 𝒆𝒇𝒇 ( 𝒗 𝒊 𝝈𝝈′ , 𝑭 𝒊 𝝈𝝈′ ,…)
𝒗 𝒊 𝝈𝝈′

49. Conclusion: Weyl metal  Topological Fermi liquid theory (Berry curvature + Axion electrodynamics)
Dirac metal + time reversal (or inversion) symmetry breaking  Weyl metal
Experimental observation: Negative longitudinal magneto-electrical resistivity
Boltzmann equation approach + semi-classical equations of motion (Berry curvature & Axion electrodynamics)
Thermal and thermoelectric responses  Observation of violation of the Wiedemann-Franz law ??
A novel fixed point beyond Landau’s Fermi-liquid theory ??