QCD in strong magnetic field

Transcription

1 QCD in strong magnetic field M. N. Chernodub CNRS, University of Tours, France Plan maximum: 1. Link to experiment: Relevance to heavy-ion collisions [hot quark-gluon plasma and cold vacuum exposed to magnetic field] 2. Phase diagram of QCD in strong magnetic field [an unsolved (so far) mystery] 3. Nondissipative transport phenomena in QCD plasma [exotic interplay of QCD topology and strong magnetic field] Graz, November 2013

2 Magnetic field in heavy-ion collisions Noncentral heavy-ion collisions should produce magnetic field. B The magnetic field is directed out of the collision plane. The duration of the field's pulse is very short however (typically, 1 fm/c). Electromagnetism at work: The generated magnetic field is very strong!

3 What is «very strong» field? Typical values: Thinking human brain: Tesla Earth's magnetic field: 10-5 Tesla Refrigerator magnet: 10-3 Tesla Loudspeaker magnet: Levitating frogs: 10 Tesla Strongest field in Lab: 103 Tesla Typical neutron star: 106 Tesla Magnetar: Tesla Heavy-ion collisions: Tesla Early Universe: even (much) higher 1 Tesla IgNobel 2000 by A.Geim (got Nobel 2010 for graphene) Destructive explosion

4 How strong field is created? Estimations: ebmax (2...3) 1016 Tesla LHC LHC RHIC Ultraperipheral (b < 2R) W. T. Deng and X. G. Huang, Phys.Rev. C85 (2012) Conversion of units: 2 2 A. Bzdak and V. Skokov, Phys.Lett. B710 (2012) Vladimir Skokov, private communication. mp 0.02 GeV Tesla = Gauss

5 Physical environment Hot quark-gluon plasma B T ~ 200 MeV or K in strong magnetic field eb ~ ( ) GeV2 or B ~ T A natural question: What is the effect of the magnetic field on quark-gluon plasma? A simpler question: What is the phase diagram of QCD in a background of strong magnetic field?

6 Phase diagram of QCD with B = 0 1) Hot quark-gluon plasma phase and cold hadron phase constitute, basically, one single phase because they are separated by a nonsingular transition ( crossover ). 2) The color superconducting phases at high baryon chemical potential m were extensively studied theoretically [they are out of reach of both lattice simulations and Earth-based experiments] 3) The LHC and RHIC experiments probe low From a BNL webpage baryon density physics. One can safely take m = 0 in further discussions.

7 Finite-temperature structure of QCD (B = 0 & m = 0) QCD Lagrangian: Two most important physical phenomena in low-t phase: 1) Quark confinement: No quarks and gluons in the physical spectrum; The physical degrees of freedom are hadrons (mesons and baryons) 2) Chiral symmetry breaking: The source of hadron masses (mesons and baryons are massive)

8 Quark confinement (I): order parameter The relevant order parameter is the Polyakov loop: 1) defined at finite temperature T in the Euclidean space-time (after a Wick rotation, t t = - it = x4) 2) related to the free energy Fq of a single quark: 3) order parameter:

9 Quark confinement (II): lattice simulations The expectation value of the Polyakov loop vs. temperature Smooth transition (crossover) Confinement [hadron phase] Deconfinement [quark-gluon plasma] Reminder: B=0&m=0 [adapted from Borsanyi et al, JHEP 1009 (2010) 073, ArXiv: ]

10 Chiral symmetry breaking (I): The symmetry Left/right quarks, for each flavor f : ; projectors Lagrangian: If the quark masses are zero, M = 0, then the global internal continuous symmetries of QCD are as follows: with The global symmetry group:

11 Chiral symmetry breaking (II): order parameter The order parameter of the chiral symmetry is the chiral condensate: In the hadronic phase of QCD the chiral condensate is nonzero and Baryon symmetry (unbroken) the chiral symmetry subgroup is broken spontaneously: Axial symmetry (broken by an anomaly) so that the allowed transformations are as follows: and with

12 Chiral symmetry breaking (III): lattice results Chiral condensate vs. temperature (crossover transition): Chiral symmetry is restored [QGP] Chiral symmetry is broken [hadron phase] [adapted from Borsanyi et al, JHEP 1009 (2010) 073, ArXiv: ]

13 Physical picture at B = 0 0 T Restoration of the chiral symmetry Deconfinement transition What happens with this picture in strong magnetic field?

14 Flavor symmetry breaking by magnetic field It is an immediate effect due to the magnetic field background. Consider QCD with two lightest quarks (Nf =2): with Electromagnetic gauge field (fixed) Gluon field Electric charge operator (matrix) Explicit flavor breaking:

15 Magnetic catalysis at zero temperature (I) Enhancement of the chiral symmetry breaking at strong B S.P. Klevansky and R. H. Lemmer ('89); H. Suganuma and T. Tatsumi ('91) - effective models V. P. Gusynin, V. A. Miransky and I. A. Shovkovy ('94, '95, '96,...) real QCDxQED In strong magnetic field quarks and antiquarks pair more effectively! Why? Two reasons: 1) Dimensional reduction (3+1)D (1+1)D: In a very strong magnetic field the dynamics of electrically charged particles (quarks, in our case) becomes effectively one-dimensional, because the particles tend to move along the magnetic field only. 2) Quarks interact stronger in one spatial dimension: In (1+1)D an arbitrarily weakest interaction between two objects leads to pair formation. This fact: (i) follows from Quantum Mechanics; (ii) is known as a Cooper theorem in solid state physics.

16 Dimensional reduction (I): energy spectrum Energy of free relativistic fermion in strong magnetic field: momentum along the magnetic field axis nonnegative integer number projection of spin on the magnetic field axis Gaps between the levels are The infrared dynamics of a fermion in strong magnetic field is governed by the lowest Landau Level (LLL) with n = 0 and sz = sgn(q) ½: The theory in the infrared region becomes effectively one dimensional!

17 Dimensional reduction (II): phase space Integral over momentum is split into the sum over Landau levels: For the LLL: Degeneracy of the LLLs: Dimensional reduction in the phase space: (3+1)D phase element: (1+1)D phase element: Volume:

18 Dimensional reduction (III): visual illustration mesons spin=0 charge=0 negatively charged r meson, charge=-e, spin=+1 u magnetic field due to the flavor breaking these mesons are different! u d direction of the magnetic moments of the r mesons d d u spins of quarks and antiquarks d positively charged r meson, charge=+e, spin=+1 u gluons

19 Dimensional reduction (IV): lattice visualization Typical structure of fermionic modes on the lattice [from studies by Buividovich et al, Phys.Lett. B682 (2010) 484]

20 Coming back: Magnetic catalysis at T = 0 (II) 1) The quark-antiquark pairing due to gluon exchange is enhanced due to the dimensional reduction. 2) The pairing is more effective for u-quarks compared to d-quarks Attractive channel: spin-0 flavor-diagonal states Magnetic field: 1) enhances chiral symmetry breaking 2) breaks flavor symmetry eb (GeV2) [picture from Frasca, Ruggieri, Phys.Rev. D83 (2011) ]

21 Magnetic catalysis + flavor breaking at T = 0 u-quarks should become heavier than d-quarks Dynamical quark mass: with Based on truncation of Schwinger-Dyson (gap) equations in QCD in strong-field limit at large number of colors. [Miransky, Shovkovy, Phys. Rev. D 66, (2002)]

22 Magnetic catalysis (III) Numerical simulations of lattice QCD Change of the chiral condensate due to strong magnetic field Theoretical estimations Chiral models: [Shushpanov, Smilga, Phys.Lett. B402 (1997) 351] Nambu-Jona-Lasinio model: [Klevansky, Lemmer, Phys.Rev. D39 (1989) 3478] Notations: [G. Bali et al, Phys.Rev. D86 (2012) ]

23 Summary: basic chiral properties for T = 0 1) Explicit breaking of flavor (u d) 2) Dimensional reduction (3+1)D (1+1)D 3) Enhancement of chiral symmetry breaking ( magnetic catalysis )

24 Quark (de)confinement: energy arguments (I) Free energy arguments. 1) Consider lightest excitations: pions and quarks. 2) Physical degrees of freedom: Confinement (cold) phase pions; Deconfinement (hot) phase quarks. 3) Compare corresponding free energies and find which phase is better. Pion vacuum is diamagnetic. It does not like the magnetic field because the free energy of pions gets increased in magnetic field. Quark vacuum is paramagnetic. It likes the magnetic field because the free energy of quarks gets decreased in magnetic field. Qualitative conclusion: magnetic field should lead to the deconfinement! [the first attempt was done by Agasian and Fedorov, Phys.Lett. B663 (2008) 445] Warning 1: here we ignore the dynamics of gluons which are not electrically charged anyway Warning 2: this is the first serious attempt to get the phase diagram. To be critically revised...

25 Quark (de)confinement: energy arguments (II) Energy of free relativistic particle in strong magnetic field: momentum along the magnetic field axis nonnegative integer number Diamagnetism of the bosonic gas: Paramagnetism of the fermionic gas: [Pauli paramagnetism, spin polarization] projection of spin on the magnetic field axis

26 Quark (de)confinement: energy arguments (III) Phase diagram, based on energy arguments [Agasian and Fedorov, Phys.Lett. B663 (2008) 445] First order line Second order endpoint Crossover line Warning 3: The confinement phase was recently (2013) shown to be paramagnetic. Warning 4: The phase diagram was recently (2012) shown to be different.

27 Chiral restoration: magnetic catalysis arguments 1) T=0 experience: magnetic field enhances the chiral condensate 2) B=0 experience: Thermal fluctuations destroy the chiral condensate 3) Immediate conclusion: The critical temperature of the chiral phase transition is an increasing function of the magnetic field.

28 Naive phase diagram of QCD in magnetic field Additional arguments were used: 1) At B = 0 the chiral and deconfinement transitions almost coincide; 2) Both these transitions are crossovers 3) Magnetic field may enhance the strength of the (chiral) transition Enhancement of the chiral transition: [Mizher, Fraga, Phys.Rev. D78 (2008) ] Warning 5: the arguments based on the magnetic catalysis contradict the recent (2012) lattice data; this phase diagram is not correct.

29 s-model coupled to quarks and to Polyakov loop (I) Full Lagrangian: Quarks: Polyakov loop (confining properties) Mesons: Chiral + Confining + Electromagnetic properties

30 s-model coupled to quarks and to Polyakov loop (II) Polyakov-loop Lagrangian: Polyakov loop: Potential comes from the phenomenology (describes well a finite-temperature Polyakov potential and thermodynamics in pure Yang-Mills theory without quarks): - critical temperature in pure SU(3) gauge theory

31 s-model coupled to quarks and to Polyakov loop (III) Confining properties (no magnetic field) Confinement Deconfinement In the deconfinement phase the center symmetry is spontaneously broken:

32 s-model coupled to quarks and to Polyakov loop (IV) Mean-field approximation. Free energy of the system: Meson fields Polyakov loop Quark contribution Hadron phase Crossover QGP phase

33 Explicit breaking of the center symmetry due to magnetic field [A.Mizher, E.Fraga, M.Ch., Phys.Rev. D82 (2010) ] Confinement Deconfinement Compare with with

34 s-model coupled to quarks and to Polyakov loop (V) An issue with the vacuum corrections: effect of magnetic field on the T = 0 ground state (Euler-Heisenberg energy): with the constituent quark mass and The effective potential for the s field without vacuum corrections with vacuum corrections

35 s-model coupled to quarks and to Polyakov loop (VI) The phase diagram(s): 1) at small B: a crossover transition (at a very small region) 2) first order transitions elsewhere 3) chiral and deconfinement split 4) both critical Tc raise with B Warning 6: the diagram without corrections is qualitatively correct, but there is a mismatch in the transitions' order. 1) at small B: a crossover transition (at a very small region) 2) first order transitions elsewhere 3) chiral and deconf. Coincide 4) both critical critical Tc decrease with B

36 Nambu-Jona-Lasinio model The phase diagram(s): 2-point + 4-point terms 2-point + 4-point + 8-point terms [Gatto, Ruggieri, Phys. Rev. D 83, (2011)] Warning 8: both diagrams do not agree with the results of lattice simulations.

37 Lattice QCD: inverse magnetic catalysis Surprise 1: At finite temperature the chiral condensate decreases with magnetic field ( inverse magnetic catalysis )! [G. S. Bali et al., JHEP 1202, 044 (2012)]

38 Lattice QCD: finite-t phase diagram [G. S. Bali et al., JHEP 1202, 044 (2012)] Surprise 2: Transitions converge, not split. Surprise 3: No enhancement of the transition (weak crossover remains weak crossover) Note: shown maximal field strengths Bmax are too modest according to new estimations (2013).

39 Magnetic susceptibility Magnetization: Magnetic susceptibility*: F is the free energy density. Paramagnetic: - likes magnetic field (energy is lowered) Diamagnetic: - dislikes magnetic field (energy is increased) *) assuming linear behavior surely valid for a weak magnetic field but may also be valid for stronger fields if magnetization is linear in B

40 Magnetic susceptibility of quark-gluon plasma (QGP) and hot vacuum for weak magnetic fields At weak magnetic fields both hot vacuum and QGP are paramagnetic! [Bonati, D Elia, Mariti, Negro, Sanfilippo, Phys. Rev. Lett. 111, (2013)] [more detailed study by the same Authors in arxiv: ] Strong paramagnet: 10 times more paramagnetic compared to ordinary materials. May lead to paramagnetic squeezing of the QGP fireball [Bali, Bruckmann, Endro di, Scha fer, arxiv: ]

41 Magnetic susceptibility of cold vacuum at strong magnetic field At strong field the vacuum becomes paramagnetic as well: Do we understand the paramagnetic behavior? 1) Hot vacuum and QGP. YES: the paramagnetism of emerging quarks dominates over the diamagnetism of pions. 2) Cold vacuum at strong magnetic fields:? - next (the results at weak fields agree with Hadron Resonance Gas model: [Endro di, JHEP 1304 (2013) 023]) [Bali, Bruckmann, Gruber, Endro di, Scha fer, JHEP 1304 (2013) 130]

42 Phase diagram continue (T, B) phase diagram paramagnetic region (T, m) phase diagram

43 Possible superconducting phase at strong field In a background of strong enough magnetic field the vacuum may become a superconductor. The superconductivity emerges in empty space. Literally, nothing becomes a superconductor. This claim seemingly contradicts textbooks which state that: 1. Superconductor is a material (= a form of matter, not an empty space) 2. Weak magnetic fields are suppressed by superconductivity 3. Strong magnetic fields destroy superconductivity [M. Ch., PRD82 (2010) ; PRL 106 (2011) ]

44 General features of superconducting state 1. spontaneously emerges above the critical magnetic field or Bc 1016 Tesla = 1020 Gauss ebc mr 31 mp 0.6 GeV 2. usual Meissner effect does not exist 3. perfect conductor (= zero DC resistance) in one spatial dimension (along the axis of the magnetic field). 4. No superconductivity in other (perpendicular) directions 5. Hyperbolic metamaterial (Smolyaninov, 2011 ): has a negative refraction index ( perfect lens ). 6. Strong paramagnet (contrary to a perfect diamagnetism of ordinary superconductors).

45 Too strong critical magnetic field? ebc mr mp 0.6 GeV 2 Over-critical magnetic fields (of the strength B ~ Bc) may be generated in ultraperipheral heavy-ion collisions (duration is short, however detailed calculations are required) LHC ebc LHC ebc RHIC ultraperipheral W. T. Deng and X. G. Huang, Phys.Rev. C85 (2012) A. Bzdak and V. Skokov, Phys.Lett. B710 (2012) Vladimir Skokov, private communication.

46 Conventional BCS superconductivity 1) The Cooper pair is the relevant degree of freedom! 2) The electrons are bounded into the Cooper pairs by the (attractive) phonon exchange. Three basic ingredients:

47 The vacuum (T = 0) in strong magnetic field Ingredients needed for possible superconductivity: A. Presence of electric charges? Yes, we have them: there are virtual particles which may potentially become real (= pop up from the vacuum) and make the vacuum (super)conducting. B. Reduction to 1+1 dimensions? Yes, we have this phenomenon: in a very strong magnetic field the dynamics of electrically charged particles (quarks, in our case) becomes effectively one-dimensional, because the particles tend to move along the magnetic field only. C. Attractive interaction between the like-charged particles? Yes, we have it: the gluons provide attractive interaction between the quarks and antiquarks (qu=+2 e/3 and qd=+e/3)

48 Pairing of quarks in strong magnetic field Similar to the magnetic catalysis at T = 0 attractive channel: spin-0 flavor-diagonal states enhances chiral symmetry breaking 0 B This talk: ± attractive channel: spin-1 flavor-offdiagonal states (quantum numbers of r mesons) electrically charged condensates: lead to electromagnetic superconductivity 0 B Bc

49 Naïve qualitative picture of quark pairing in the electrically charged vector channel: r mesons - Energy of a relativistic particle in the external magnetic field Bext: momentum along the magnetic field axis nonnegative integer number projection of spin on the magnetic field axis (the external magnetic field is directed along the z-axis) - Masses of ρ mesons and pions in the external magnetic field becomes heavier becomes lighter - Masses of ρ mesons and pions:

50 Electrodynamics of ρ mesons = NSSM* *NSSM - naive simplest solvable model - Lagrangian (based on vector dominance models): Nonminimal coupling leads to g=2 - Tensor quantities - Covariant derivative - Kawarabayashi-SuzukiRiadzuddin-Fayyazuddin relation - Gauge invariance [D. Djukanovic, M. R. Schindler, J. Gegelia, S. Scherer, PRL (2005)]

51 Condensation of ρ mesons (mean-field) Mean-field: the ρ± mesons become condense at certain Bc masses in the external magnetic field Kinematical impossibility of dominant decay modes The pion becomes heavier while the r meson becomes lighter - The decay stops at certain value of the magnetic field Energy of the condensed state: Ginzburg-Landau potential for ordinary superconductivity:

52 Symmetries In terms of quarks, the state implies (the same structure of the condensates in the Nambu-Jona-Lasinio model) Abelian gauge symmetry Rotations around B-axis - The condensate locks rotations around field axis and gauge transformations: (similar to color-flavor locking in color superconductors at high quark density) No light goldstone boson: it is eaten by the electromagnetic gauge field! + Discussion in the literature [Y.Hidaka and A. Yamamoto, arxiv: ; Chuan Li and Qing Wang, PLB, arxiv: ; M. Ch., PRD, arxiv: ]

53 Condensates of ρ mesons, solutions Superconducting condensate (charged ρ condensate): Superfluid condensate (neutral ρ condensate)

54 Condensates of ρ mesons, solutions Superconducting condensate (charged rho mesons) Superfluid condensate B = 1.01 Bc (neutral rho mesons) New objects, topological vortices, made of the rho-condensates The phases of the rho-meson fields wind around vortex centers, at which the condensates vanish. similar results in holographic approaches by M. Ammon, Y. Bu, J. Erdmenger, P. Kerner, J. Shock, M. Strydom (2012)

55 Anisotropic superconductivity (via an analogue of the London equations) - Apply a weak electric field E to an ordinary superconductor - Then one gets accelerating electric current along the electric field: [London equation] - In the QCDxQED vacuum, we get an accelerating electric current along the magnetic field B: ( ) Written for an electric current averaged over one elementary (unit) rho-vortex cell

56 Acoustic phonon spectrum Transverse phonons Longitudinal phonons Low-energy phonon spectrum:!!!

57 Normalized energy of the ρ meson condensate in the transverse plane. Check x-y slice at fixed time t and distance z V.Braguta et al, 2013 Instead of a regular lattice structure we see an irregular vortex pattern (liquid/glass?) The vortices move as we move the slice.

58 Condensate in the transverse plane: plane by plane

59 Normalized energy of the ρ meson condensate along the magnetic field. Check x-z slice at fixed time t and coordinate y Vortices are not straight and static: they are curvy moving one-dimensional (in 3d) structures.

60 Phase diagram current status

61 Effects of magnetic field: (temporary) conclusions More or less understood: 1) Explicit breaking of flavor (u d) 2) Dimensional reduction (3+1)D (1+1)D 3) Enhancement of chiral condensate ( magnetic catalysis ) at T = 0 4) Explicit breaking of the center group ( induced deconfinement ); Not well understood: 5) Inverse magnetic catalysis at T > 0; 6) Converging chiral and deconfinement transitions; 7) No enhancement of finite-t transition strength; 8) Strong paramagnetism both in weak magnetic fields (warm/hot plasma) and strong magnetic fields (cold vacuum) 9) Electromagnetic superconductivity of cold vacuum? Bad news: No solid/full picture so far Good news: No solid/full picture so far a very active (surging) area of research!

62 The Chiral Magnetic Effect (CME) Electric current is induced by applied magnetic field: Spatial inversion (x - x) symmetry (P-parity): Electric current is a vector (parity-even quantity); Magnetic field is a pseudovector (parity-odd quantity). Thus, the CME medium should be parity-odd! In other words, the spectrum of the medium which supports the CME should not be invariant under the spatial inversion transformation.

63 An example of a parity-odd system? Consider a massless fermion: P-parity Right handed Left handed The chirality (left/right) is a conserved number.

64 The CME in heavy-ion collisions (II) Visual picture: Red: momentum Blue: spin Electric charges: u-quark: q=+2e/3 d-quark: q= - e/3 P-invariant plasma of quarks (and gluons) Sphaleron (instanton) = topological charge Role of topology: ul ur dl dr P-odd plasma of quarks (and gluons)

65 The CME in heavy-ion collisions (II) Visual picture: Red: momentum Blue: spin Electric charges: u-quark: q=+2e/3 d-quark: q= - e/3 P-invariant plasma of quarks (and gluons) Sphaleron (instanton) = topological charge Role of topology: ul ur dl dr P-odd plasma of quarks (and gluons)

66 Why the CME is astonishing? 1) Because it gives an equilibrium dissipationless current! (similar to the electric current in superconductivity) 2) Because it exists in an interacting system: interactions cannot destroy the CME if they do not wash out the chiral imbalance. 3) Because the CME does not need a presence of a condensate. A) Thus, thermal fluctuations cannot destroy the CME. B) Thus, the dissipationless electric current may exist at high temperatures. BTW, the temperature of the quark-gluon plasma is of the order of 1012 Kelvin (200 MeV and more) 4) now, an analogue of the CME is under very active search in condensed matter physics (for example, in suggested Weyl semi-metals). Why? Because the CME is a very good alternative to the room-temperature superconductivity.

67 Lattice simulations of the CME: a task (II) 4) Then, impose an external magnetic field. 5) According to the CME, the chirally-imbalanced quark ensembles will lead to the generation of electric current. [Depending on the sign of the topological charge, these currents may be positive or negative]. 6) Check the presence of the electric current numerically! Checked: [Buividovich et al, '09] [QCD was studied in the quenched limit (= backreaction of fermions on gluons was neglected ) because this feature (the presence of the vacuum quark loops) is not important for the CME]

68 Lattice simulations of the CME: currents (I) Typical density of the electric current eb = 0 eb = (780 MeV) eb = (500 MeV) 2 eb = (1.1 GeV) 2 2

69 Lattice simulations of the CME: currents (II) Typical density of the electric current in gradually increasing magnetic field. Each step in magnetic field is (350 MeV)2.

70 Lattice simulations of the CME: instanton An illustration: 1) take an instanton-like gluon configuration (= numerically obtained configuration of the non-abelian field of a unit topological charge ) 2) Apply the external magnetic field. 3) Check for the existence of the electric current. The induced current along the magnetic-field axis The (absence of induced) current in the transverse (perpendicular) directions

71 A list of anomalous non-dissipative effects (I) 1) Chiral Magnetic Effect the electric current is induced in the direction of the magnetic field due to the chiral imbalance: [all formulae are written for one flavor of fermions] 2) Chiral Vortical Effect the axial current is induced in the rotating quark-gluon plasma along the axis of rotation: Note: The axial current = the difference in currents of right-handed and left-handed quarks.

72 How to understand the Chiral Vortical Effect. Imagine massless fermions... Right handed Left handed in a rotating frame.

73 A list of anomalous non-dissipative effects (II) 3) Chiral Separation Effect the axial current is induced in the direction of the magnetic field: Notes: A) This effect is realized even if the plasma is chirally-trivial. B) is the the quark chemical potential (the plasma is dense). 4) Axial Magnetic Effect the energy flow is induced in the direction of the axial (=chiral) magnetic field: All these effects will become more complicated (richer) in a rotating, chirallyimbalanced dense plasma subjected to both the usual and axial magnetic fields.

74 The Axial Magnetic Effect For one quark's flavor: The energy flow is given by the off-diagonal component of the energy-momentum tensor The axial magnetic field is the magnetic field which acts on left-handed and right-handed quarks in the opposite way. It can be described by the axial field in the Lagrangian:

75 The Axial Magnetic Effect vs. the Chiral Vortical Effect For a many-flavor system: The Axial Magnetic Effect: The Chiral Vortical Effect: Their anomalous conductivities are the same: They have the same origin! (It is, BTW, quite impressive: the generation of the axial current in a rotating system is tightly related to the induction of the energy flow in a static system).

76 The Axial Magnetic Effect on the lattice Task: simulate lattice QCD in the background of the axial magnetic field and study the energy flow of fermions both in the direction of the field and in transverse direction. [QCD was studied in the quenched limit (= backreaction of fermions on gluon files was neglected ) because the vacuum fermion loops are not important for the AME] Results [Braguta et al., '13]: a) the effect was not observed in low-temperature phase, where the plasma is absent. This is explained by the quark confinement as the individual quarks do not exist in this phase. b) the effect is negligible at the deconfinement phase transition.

77 The Axial Magnetic Effect on the lattice c) the Axial Magnetic Effect is clearly seen in the high-temperature (deconfinement) phase: However, the linear coefficient is one order of magnitude smaller than the one predicted by the theory (a renormalisation of the effect?)

78 Summary The quark-gluon plasma exhibits various dissipationless transport phenomena in thermodynamic equilibrium: A) Chiral Magnetic Effect: generation of the electric current by magnetic field B) Axial Magnetic Effect: generation of the energy flow by axial magnetic field C) Chiral Vortical Effect: generation of the axial current in rotating plasma D) Chiral Separation Effect: generation of the axial current by magnetic field