Anomalous transport on the lattice

1. Anomalous transport on the lattice PavelBuividovich (Regensburg)

2. Chiral Magnetic Effect [Kharzeev, Warringa, Fukushima] Chiral Separation Effect [Son, Zhitnitsky] Chiral Vortical Effect [Erdmengeret al., Banerjee et al.] Anomalous transport: CME, CSE, CVE

3. Dissipative transport • (conductivity, viscosity) • No ground state • T-noninvariant(but CP) • Spectral function = anti-Hermitean part of retarded correlator • Work is performed • Dissipation of energy • First k → 0, then w → 0 • Anomalous transport • (CME, CSE, CVE) • Ground state • T-invariant (but not CP!!!) • Spectral function = Hermitean part of retarded correlator • No work is performed • No dissipation of energy • First w → 0, then k → 0 T-invariance and absence of dissipation

4. Folklore on CME & CSE: • Transport coefficients are RELATED to anomalies (axial and gravitational) • and thus protected from: • perturbative corrections • IR effects (mass etc.) • Check these statements as applied to the lattice. • What is measurable? • How should one measure? • What should one measure? • Does it make sense at all? If Anomalous transport: CME, CSE, CVE

5. CME with overlap fermions ρ = 1.0, m = 0.05

6. CME with overlap fermions ρ = 1.4, m = 0.01

7. CME with overlap fermions ρ = 1.4, m = 0.05

8. Staggered fermions [G. Endrodi] Bulk definition of μ5 !!! Around 20% deviation

9. CLAIM: constant magnetic field in finite volume is NOT a small perturbation “triangle diagram” argument invalid (Flux is quantized, 0 → 1 is not a perturbation, just like an instanton number) CME: “Background field” method More advanced argument: in a finite volume Solution: hide extra flux in the delta-function Fermions don’t note this singularity if Flux quantization!

10. Partition function of Dirac fermions in a finiteEuclidean box • Anti-periodic BC in time direction, periodic BC in spatial directions • Gauge field A3=θ – source for the current • Magnetic field in XY plane • Chiral chemical potential μ5in the bulk CME in constant field: analytics Dirac operator:

11. Creation/annihilation operators in magnetic field: Now go to the Landau-level basis: CME in constant field: analytics (topological) zero modes Higher Landau levels

12. Dirac operator in the basis of LLL states: Vector current: Closer look at CME: LLL dominance Prefactor comes from LL degeneracy Only LLL contribution is nonzero!!!

13. Polarization tensor in 2D: Proper regularization (vector current conserved): [Chen,hep-th/9902199] Final answer: Dimensional reduction: 2D axial anomaly • Value at k0=0, k3=0: NOT DEFINED • (without IR regulator) • First k3→ 0, then k0→ 0 • Otherwise zero

14. μ5is not a physical quantity, just Lagrange multiplier Chirality n5 is (in principle) observable Express everything in terms of n5 To linear order in μ5: Singularities of Π33cancel !!! Note: no non-renormalization for two loops or higher and no dimensional reduction due to 4D gluons!!! Chirality n5vsμ5

15. Anomalous current-current correlators: CME and CSE in linear response theory Chiral Separation and Chiral Magnetic Conductivities:

16. Integrate out time-like loop momentum Relation to canonical formalism Chiral Separation: finite-temperature regularization Virtual photon hits out fermions out of Fermi sphere…

17. Chiral Separation Conductivity: analytic result Singularity

18. Decomposition of propagators and polarization tensor in terms of chiral states Chiral Magnetic Conductivity: finite-temperature regularization s,s’ – chiralities

19. Careful regularization required!!! Chiral Magnetic Conductivity: finite-temperature regularization • Individual contributions of chiral states are divergent • The total is finite and coincides with CSE (upon μV→ μA) • Unusual role of the Dirac mass • non-Fermi-liquid behavior?

20. Pauli-Villars regularization (not chiral, but simple and works well) Chiral Magnetic Conductivity: still regularization

21. Pauli-Villars: zero at small momenta • -μA k3/(2 π2) at large momenta • Pauli-Villars is not chiral (but anomaly is reproduced!!!) • Might not be a suitable regularization • Individual regularization in each chiral sector? • Let us try overlap fermions • Advanced development of PV regularization • = Infinite tower of PV regulators [Frolov, Slavnov, Neuberger, 1993] • Suitable to define Weyl fermions non-perturbatively Chiral Magnetic Conductivity: regularization

22. Lattice chiral symmetry: Lüscher transformations Overlap fermions These factors encode the anomaly (nontrivial Jacobian)

23. First consider coupling to axial gauge field: Assume local invariance under modified chiral transformations [Kikukawa, Yamada, hep-lat/9808026]: Dirac operator with axial gauge fields Require (Integrable) equation for Dov!!!

24. We require that Lüscher transformations generate gauge transformations of Aμ(x) Overlap fermions with axial gauge fields and chiral chemical potential Linear equations for “projected” overlap or propagator

25. Dirac operator with axial gauge field Overlap fermions with axial gauge fields and chiral chemical potential: solutions Dirac operator with chiral chemical potential • Only implicit definition (so far) • Hermitean kernel (at zero μV) = Sign() of what??? • Potentially, no sign problem in simulations

26. Current-current correlators on the lattice Consistent currents!Naturaldefinition:charge transfer Axial vertex:

27. In terms of kernel spectrum + derivatives of kernel: Numerically impossible for arbitrary background… Krylov subspace methods? Work in progress… Derivatives of overlap Derivatives of GW/inverse GW projection:

28. Chiral Separation Conductivity with overlap Good agreement with continuum expression

29. Chiral Separation Conductivity with overlap Small momenta/large size: continuum result

30. Chiral Magnetic Conductivity with overlap At small momenta, agreement with PV regularization

31. Chiral Magnetic Conductivity with overlap As lattice size increases, PV result is approached

32. Let us try “slightly’’ non-conserved current Role of conserved current Perfect agreement with “naive” unregularizedresult

33. Role of anomaly Replace Dov with projected operator Lüscher transformations Usual chiral rotations

34. Is it necessary to use overlap? Try Wilson-Dirac Agreement with PV, but large artifacts They might be crucial in interacting theories

35. Expand anomalous correlators in μV or μA: CME, CSE and axial anomaly VVA correlator in some special kinematics!!!

36. At fixed k3, the only scale is µ Use asymptotic expressions for k3 >> µ !!! Ward Identities fix large-momentum behavior CME, CSE and axial anomaly: IR vs UV • CME: -1 x classical result, CSE: zero!!!

37. General decomposition of VVA correlator • 4 independent form-factors • Only wL is constrained by axial WIs [M. Knechtet al., hep-ph/0311100]

38. CSE: p = (0,0,0,k3), q=0, µ=2, ν=0, ρ=1 ZERO 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: Anomalous correlatorsvs VVA correlator

39. Spatially modulated chiral chemical potential Anomalous correlatorsvs VVA correlator By virtue of Bose symmetry, only w(+)(k2,k2,0) Transverse form-factor Not fixed by the anomaly

40. In addition to anomaly non-renormalization, new (perturbative!!!) non-renormalization theorems [M. Knechtet al., hep-ph/0311100] [A. Vainstein, hep-ph/0212231]: CME and axial anomaly (continued) Valid only for massless QCD!!!

41. Special limit: p2=q2 Six equations for four unknowns… Solution: CME and axial anomaly (continued) Might be subject to NP corrections due to ChSB!!!

42. Linear response of currents to “slow” rotation: Chiral Vortical Effect In terms of correlators Subject to PT corrections!!!

43. A naïve method [Yamamoto, 1303.6292]: • Analytic continuation of rotating frame metric • Lattice simulations with distorted lattice • Physical interpretation is unclear!!! • By virtue of Hopf theorem: • only vortex-anti-vortex pairs allowed on torus!!! • More advanced method • [Landsteiner, Chernodub & ITEP Lattice, 1303.6266]: • Axial magnetic field = source for axial current • T0y = Energy flow along axial m.f. • Measure energy flow in the background axial • magnetic field Lattice studies of CVE

44. First lattice calculations: ~ 0.05 x theory prediction • Non-conserved energy-momentum tensor • Constant axial field not quite a linear response • For CME: completely wrong results!!!... • Is this also the case for CVE? • Check by a direct calculation, use free overlap Lattice studies of CVE

45. No holomorphic structure • No Landau Levels • Only the Lowest LL exists • Solution for finite volume? Constant axial magnetic field

46. Conserved lattice energy-momentum tensor: not known How the situation can be improved, probably? Momentum from shifted BC [H.Meyer, 1011.2727] ChiralVortical Effect from shifted boundary conditions We can get total conserved momentum = Momentum density =(?) Energy flow

47. Anomalous transport: very UV and IR sensitive • Non-renormalizability only in special kinematics • Proofs of exactness: some nontrivial assumptions • Fermi-surface • Quasi-particle excitations • Finite screening length • Hydro flow of chiral particles is a strange object • Field theory: NP corrections might appear, ChSB • Even Dirac massdestroys the usual Fermi surface • What happens to “chiral” Fermi surface and to anomalous transport in confinement regime? • What are the consequences of inexact ch.symm? Conclusions

48. Backup slides

49. Chemical potential for conserved charge (e.g. Q): Non-compact gauge transform In the action Via boundary conditions For anomalous charge: Chemical potential for anomalous charges General gauge transform BUT the current is not conserved!!! Chern-Simons current Topological charge density

50. Simplest method: introduce sources in the action • Constant magnetic field • Constant μ5 [Yamamoto, 1105.0385] • Constant axial magnetic field [ITEP Lattice, 1303.6266] • Rotating lattice??? [Yamamoto, 1303.6292] • “Advanced” method: • Measure spatial correlators • No analytic continuation necessary • Just Fourier transforms • BUT: More noise!!! • Conserved currents/ • Energy-momentum tensor • not known for overlap CME and CVE: lattice studies