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, 246603 (2013). Boltzmann transport theory: Phys. Rev. B 89, 195137 (2014). A review paper will appear in Science and Technology of Advanced Materials (ISSN 1468-6996). Topological Fermi-liquid theory: Yong-Soo Jho and Ki-Seok Kim (in preparation).

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

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

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

Boltzmann transport theory with the Drude model in metals

Landau’s Fermi-liquid theory for metals

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

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

Weyl metal B

𝑩𝒆𝒓𝒓𝒚 𝒑𝒉𝒂𝒔𝒆

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

More than the Berry curvature

(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), …

Adler-Bell-Jackiw anomaly (1d)

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

How to generalize the Drude model for Weyl metal ?

M. A. Stephanov and Y. Yin, Phys. Rev. Lett. 109, 162001 (2012)

Adler-Bell-Jackiw anomaly (3d): Semi-classical regime Phys. Rev. B 88, 104412 (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).

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

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

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, 246603 (2013)

Longitudinal magnetoresistivity 3d weak antilocalization

Theoretical analysis L T

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

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

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).

Anomalous transport in Weyl metal

Longitudinal magneto-resistivity and longitudinal Hall coefficient in Weyl metal Ki-Seok Kim, Heon-Jung Kim, and M. Sasaki, Phys. Rev. B 89, 195137 (2014)

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

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?

M. A. Stephanov and Y. Yin, Phys. Rev. Lett. 109, 162001 (2012)

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

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

Longitudinal negative magneto-electrical resistivity from Kubo formula

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)

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′

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 ??