1. Pavel Buividovich (Regensburg)

2. Collective motion of chiral fermions High-energy physics: High-energy physics: Quark-gluon plasma Quark-gluon plasma Hadronic matter Hadronic matter Leptons/neutrinos in Early Universe Leptons/neutrinos in Early Universe Condensed matter physics: Condensed matter physics: Weyl semimetals Weyl semimetals Topological insulators Topological insulators Liquid Helium [G. Volovik] Liquid Helium [G. Volovik]

3. No mass term for Weyl fermions Weyl points survive ChSB!!! [Pyrochlore iridate]

4. Momentum shift of Weyl points: Anomalous Hall Effect Energy shift of Weyl points: Chiral Magnetic Effect [Experiment ZrTe 5 : 1412.6543] Also: Chiral Vortical Effect, Axial Magnetic Effect… Chiral Magnetic Conductivity and Kubo relations T-invariace Ground-state transport??? MEM Bloch theorem?

5. Expand current-current correlators in μ A : VVA correlators in some special kinematics!!! The only scale is µ k3 >> µ !!!

6. Difference between the gauge-invariant and non-invariant results: “surface” Chern-Simons term

7. 4 independent form-factors 4 independent form-factors Only wL is constrained by axial WIs Only wL is constrained by axial WIs [M. Knecht et al., hep-ph/0311100]

8. CME: p = (0,0,0,k3), q=(0,0,0,-k3), µ=1, ν=2, ρ=0 IR SINGULARITY Regularization: p = k + ε/2, q = -k+ε/2 ε – “momentum” of chiral chemical potential Time-dependent chemical potential: No ground state!!!

9. Spatially modulated chiral chemical potential By virtue of Bose symmetry, only w (+) (k 2,k 2,0) Transverse form-factor Not fixed by the anomaly [Buividovich 1312.1843]

10. In addition to anomaly non-renormalization, new (perturbative!!!) non-renormalization theorems [M. Knecht et al., hep-ph/0311100] [A. Vainstein, hep-ph/0212231]: Valid only for massless fermions!!

11. Special limit: p 2 =q 2 Six equations for four unknowns… Solution: Might be subject to NP corrections due to ChSB!!!

12. Anomalous transport coefficients: Related to axial anomaly Related to axial anomaly Do not receive corrections IF Do not receive corrections IF Screening length finite [Jensen,Banerjee,…] Screening length finite [Jensen,Banerjee,…] Well-defined Fermi-surface [Son, Stephanov…] Well-defined Fermi-surface [Son, Stephanov…] No Abelian gauge fields [Jensen,Kovtun…] No Abelian gauge fields [Jensen,Kovtun…] In Weyl semimetals with μ A / induced mass: Screening length is zero ( Goldstones?) Screening length is zero ( Goldstones?) Electric charges STRONGLY interact Electric charges STRONGLY interact Non-Fermi-liquid [Buividovich’13] Non-Fermi-liquid [Buividovich’13]

13. Time-reversal breaking WSM: Axion strings [Wang, Zhang’13] Axion strings [Wang, Zhang’13] RG analysis: Spatially modulated RG analysis: Spatially modulated chiral condensate [Maciejko, Nandkishore’13] chiral condensate [Maciejko, Nandkishore’13] Spontaneous Parity Breaking [Sekine, Nomura’13] Spontaneous Parity Breaking [Sekine, Nomura’13] Parity-breaking WSM: not so clean and not well studied… Only PNJL/ σ-model QCD studies Chiral chemical potential μ A : Chiral chemical potential μ A : Dynamics!!! Dynamics!!! Circularly polarized laser Circularly polarized laser … But also decays dynamically??? … But also decays dynamically??? [Akamatsu,Yamamoto,…] [Akamatsu,Yamamoto,…] [Fukushima, Ruggieri, Gatto’11]

14. Dynamical equilibrium / Slow decay

15. Lattice Dirac fermions with contact interactions Lattice Dirac Hamiltonian V>0, like charges repel Suzuki-Trotter decomposition Hubbard-Stratonovich transformation

16. Taking everything together… Partition function of free fermions with one-particle hamiltonian Action of the Hubbard field Possible homogeneous condensates (assume unbroken Lorentz symmetry)

17. Externalperturbationchange the condensate

18. Vector meson propagator CME response: Meson mixing with μ A (k z ≠ 0) ρ-mesons Pseudovector mesons

20. Green = μ A k/(2 π 2 ) “Conserved” currents!!!

26. One flavor of Wilson-Dirac fermions One flavor of Wilson-Dirac fermions Instantaneous interactions (relevant for condmat) Instantaneous interactions (relevant for condmat) Time-reversal invariance: no magnetic interactions Time-reversal invariance: no magnetic interactions Kramers degeneracy in spectrum: Complex conjugate pairs Complex conjugate pairs Paired real eigenvalues Paired real eigenvalues External magnetic field causes sign problem! External magnetic field causes sign problem! Determinant is always positive!!! Determinant is always positive!!! Chiral chemical potential: still T-invariance!!! Chiral chemical potential: still T-invariance!!! Simulations possible with Rational HMC Simulations possible with Rational HMC

27. Wilson-Dirac with chiral chemical potential: No chiral symmetry No chiral symmetry No unique way to introduce μ A No unique way to introduce μ A Save as many symmetries as possible [Yamamoto‘10] Save as many symmetries as possible [Yamamoto‘10] Counting Zitterbewegung, not worldline wrapping

28. In many physically interesting situations, anomalous transport coefficients receive nontrivial corrections due to interactions CME and chiral imbalance strongly enhanced if chiral symmetry or parity are spontaneously broken should be easier to observe in experiment Parity-breaking Weyl semimetals can be simulated using Rational HMC algorithm In many physically interesting situations, anomalous transport coefficients receive nontrivial corrections due to interactions CME and chiral imbalance strongly enhanced if chiral symmetry or parity are spontaneously broken should be easier to observe in experiment Parity-breaking Weyl semimetals can be simulated using Rational HMC algorithm

29. Dynamical stability of chirally imbalanced matter? “Chiral plasma instability” scenario? [Akamatsu, Yamamoto’12, Zamaklar’11] Real-time dynamics of “chirality pumping”? Effect of boundaries? Chirally symmetric lattice fermions with chiral chemical potential [See also the poster by Matthias Puhr] Dynamical stability of chirally imbalanced matter? “Chiral plasma instability” scenario? [Akamatsu, Yamamoto’12, Zamaklar’11] Real-time dynamics of “chirality pumping”? Effect of boundaries? Chirally symmetric lattice fermions with chiral chemical potential [See also the poster by Matthias Puhr]