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Chiral Heat Wave in chiral fluids (A review of “strange” anomalous transport effects in chiral media) M. N. Chernodub ● • • • • ● • • • • Centre national de la recherche scientifique (CNRS) University of Tours, France Topics (a review-style talk) Anomalous transport phenomena in chiral fluids and gases. Dissipationless transfer of charges and energy Chiral Magnetic Effect (in magnetic field at finite chiral density) Chiral Separation Effect (in magnetic field at finite fermion density) Chiral Vortical Effects (in rotating medium at finite densities and temperature) Transfer of energy (rotating hot system at finite densities and temperature) Anomalous chiral waves: Chiral Magnetic Wave Chiral Vortical Wave Chiral Alfvén Wave Chiral Heat Wave Dissipationless charge transfer: superconductivity The London equation: Wikipedia Ballistic acceleration of the electric current J by the electric field E. The electric current in a superconducting wire will flow forever! Properties: 1) no resistance = no dissipation; 2) superconducting condensate is needed! Alternative(s) to superconductivity? ● ● ● Superconductivity emerges due to the presence of a condensate of electrons pairs (the Cooper pairs). The Cooper pairs are quite fragile: thermal fluctuations destroy them quite effectively. That's why the roomtemperature superconductivity is not observed (yet). May dissipationless charge transfer exist in the absence of any kind of condensation? Do we have alternatives? Yes, we do. The simplest example is Chiral Magnetic Effect (CME) 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. Non-dissipative nature of the current ● Time inversion: Ohm conductivity (dissipative, loss of current): ● Superconductivity (non-dissipative, no losses): Chiral Magnetic Effect (non-dissipative, no losses): ● 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. How to make a parity-odd system out of fermions? Systems with equal number of left- and right-handed particles are P-invariant. Examples of P-invariant systems in the daily life: One right-handed particle and one left-handed particle with opposite momenta at IHEP, Beijing. However, if we have a different number of left- and right-handed particles (for example, in a gas) then this system is not P-invariant. If these particles are electrically charged, then we may expect that this system exhibits the CME: [A. Vilenkin, '80; K. Fukushima, D. E. Kharzeev, H. J. Warringa, '08; D. E. Kharzeev, L. D. McLerran and H. J. Warringa, '08] The CME in quark-gluon plasma 1) The quark-gluon plasma is created in heavy-ion collisions (RHIC and LHC). 2) Typical energies of u and d quarks are so high, that these quarks may be considered as massless fermions. 3) The imbalance in chirality is produced due to rapid topological transitions mediated by QCD sphalerons: a left quark may turn into a right quark (and vice versa) due the axial QCD anomaly (→ the CME is “anomalous”) 4) Noncentral collisions create huge magnetic field (1016 Tesla). The CME in heavy-ion collisions (I) The induced electric current in the quark-gluon plasma is proportional to the chiral chemical potential which controls imbalance between left-handed and right-handed quarks. [this formula is written for one fermion with electric charge e] The CME predicts that there should be an asymmetric flow of charged particles along the normal to the reaction plane. And such asymmetry was indeed observed both at RHIC (USA) and at LHC (Europe)! J, B 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) Chiral Magnetic Effect – experiment (Quark Gluon Plasma) ● Signatures of the Chiral Magnetic Effect in heavy-ion collisions …. Chiral Magnetic Effect – experiment (Solid State) ● Signatures of the Chiral Magnetic Effect in Dirac Semimetals ArXiv:1412.6543 Why the CME is interesting? 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 (in Dirac and Weyl semi-metals discovered this and last years). Why? Because the CME is a good alternative to the room-temperature superconductivity. Examples of anomalous transport 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 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 quark chemical potential (the plasma is dense) C) The axial (chiral) current = the difference “right current”–“left current”. Examples of anomalous transport effects (II) 3) Chiral Vortical Effect – the axial current is induced in the rotating quark-gluon plasma along the axis of rotation: angular velocity (vorticity) 4) Dissipationless energy transfer – the energy flow is generated along the axis of magnetic field: Energy current: Note: All these effects will become more complicated (much richer) in a rotating, chirally-imbalanced dense matter subjected to background magnetic fields. Chiral Waves ● ● The chiral waves are massless excitations in the fluid or plasma of chiral fermions. There are four types of waves (disclaimer: status of August 18, 2015) • • • • Chiral Magnetic Wave (Dec '10) – reviewed in details Chiral Vortical Wave (Apr '15) – reviewed Chiral Alfvén Wave (May '15) – not covered, but interesting Chiral Heat Wave (Aug '15, to appear) – discussed These sound-like excitations correspond to coherent propagation of charge density, axial (chiral) charge density, and energy density waves in different combinations. Chiral Magnetic Wave: currents and chemical potentials ● Chiral fluid/plasma (massless fermions of both chiralities) ● Currents: vector axial ● ● Right-handed and left-handed currents: Chemical potentials: vector axial Chiral Magnetic Wave: currents and density fluctuations ● Anomalous transport laws: chiral magnetic effect ● chiral separation effect Consider a neutral chiral system: → on average, all densities are zero → globally, all chemical potentials are zero … but local fluctuations of densities may exist: Magnetic Wave: couple densities and chemical potentials ● Anomalous transport laws: What are these chemical potentials? It is better to work with densities rather than with chemical potentials! For small perturbations: “susceptibility” (calculable) Chiral Magnetic Wave: couple currents and densities ● Combine: and with the result: Axial density induces vector current. Vector density induces axial current. Question: Are they married chained to each other? Chiral Magnetic Wave: two current-density relations ● Anomalous current-density relation: vector current axial density axial current vector density magnetic field B ● Usual current-density relation (conservation of charge): [We do not consider external electric field so that we have no anomaly for the axial current.] Chiral Magnetic Wave: relations between all densities ● Anomalous current-density relation ● Let B be along the z-axis, B=(0,0,B): ● Charge conservation: ● Exclude currents: Chiral Magnetic Wave – wave equations ● Relations between densities: Two first-order equations for two variables. ● Diagonalize them and get second order equations: ● ● Sound-like (“gapless”) wave propagation with velocity ● This is Chiral Magnetic Wave Chiral Magnetic Wave – physical picture (1) fluctuation in axial density (2) creates fluctuation in vector current magnetic field B (3) which induces vector density fluctuation at the next point (4) which creates fluctuation of the vector current (5=1) which generates fluctuation of the axial density at the next point (6=2) creates fluctuation in vector current (7=3) etc etc etc. The Chiral Magnetic Wave is a chain-like process. Chiral Magnetic Wave – velocity ● The wave propagates with velocity: At weak magnetic field c is constant. ● At arbitrary field strength: ● Chiral Magnetic Wave – experiment ● Signatures of the Chiral Magnetic Wave in heavy-ion collisions …. Chiral Magnetic Wave – simplest signatures ● ● ● ● Can we see (“feel”) the existence of the Wave just from anomalous relations without making all those steps? Yes, we see the wave just from anomalous transport laws: Signature: linear cross-coupling of chemical potentials and the corresponding currents In this particular example: Vector current – Axial chemical potential Axial current – Vector chemical potential Other waves? ● We just discussed vector and axial currents for chiral fermions in the background of magnetic field: There are also anomalous effects due to rotation! Our plan: ➔ Switch off magnetic field, B = 0. ➔ Rotate the fluid or gas of chiral fermions! ● ● Anomalous currents in a rotating system: Chiral Vortical Wave – feeling of existence ● Non-zero density, zero temperature, no magnetic field, rotation: ● Here we use d explicitly to stress that we have fluctuations. ➔ Coupling between vector/axial currents/densities! ➔ Chiral Vortical Wave! Chiral Vortical Wave – physical picture (1) fluctuation in axial density (2) creates fluctuation in vector current angular velocity W (3) which induces vector density fluctuation at the next point (4) which creates fluctuation of the vector current (5=1) which generates fluctuation of the axial density at the next point (6=2) creates fluctuation in vector current (7=3) etc etc etc. The Chiral Vortical Wave is also a chain-like process. Chiral Vortical Wave – results ● Velocity of wave propagation depends on angular velocity W: ArXiv:1504.03201 Chiral Heat Wave – anomalous transport for energy ● Dissipationless energy transfer flow (energy current): ● Energy flow is defined by the stress-energy tensor: ● Anomalous conductivities: Temperature fluctuations: We must consider thermal fluctuations! Chiral Heat Wave – finding its traces ● Finite temperature, zero density, no magnetic field, rotation: fluctuations in vector (electric) current are negligible fluctuations in axial current depend on temperature fluctuations fluctuations in energy (thermal) current depend on fluctuations of the axial charge ● ● Conclusion: axial charge density and energy density are together, the vector charge should leave does not participate We have a wave of axial charge density and energy density! Chiral Heat Wave ● Coupled axial and energy fluctuations: ● Sound waves of axial charge density and energy density: ● Velocity of propagation: ● This is Chiral Heat Wave (M.Ch., to appear soon) Chiral Heat Wave – physical picture (1) fluctuation in axial density (2) creates fluctuation in energy flow (in energy current) angular velocity W (3) which induces thermal energy (4) which creates fluctuation fluctuation (increase in of the axial current temperature) at the next point (5=1) which generates fluctuation of the axial density at the next point (6=2) creates fluctuation in energy flow (7=3) etc etc etc. The Chiral Heat Wave is again a chain-like process. Summary 1: anomalous transport Chiral fermions form systems which exhibit various dissipationless transport phenomena in equilibrium: A) Chiral Magnetic Effect: generation of the electric current by magnetic field B) Chiral Separation Effect: generation of the axial current by magnetic field C) Chiral Vortical Effects: generation of the vector and axial currents in rotating plasma C) Energy flow in rotating plasma and in magnetic field background Summary 2: chiral waves The same systems possess various massless excitations: A) Chiral Magnetic Wave: coherent sound-like propagation of vector and axial densities along the axis of magnetic field B) Chiral Vortical Wave: vector-axial density propagation along the axis of rotation of the fluid C) Chiral Heat Wave: axial-energy density waves in rotating plasma D) These waves may mix and lead to common vector-axial-energy waves (not discussed in the talk) Summary 3: experiment and applications Some of these effects are already seen in: 1) High energy physics: heavy-Ion collisions (RHIC/LHC) 2) Solid state physics: Dirac semimetals 3) Our pocket: probably, consumer electronics in N years. Intrinsic chiral magnetic effect in Dirac semimetals M. A. Zubkov Institute of Theoretical and Experimental Physics, Moscow, and Centre national de la recherche scientifique (CNRS) University of Tours, France M. N. Chernodub CNRS, University of Tours, France Arxiv:1508.03114 Previous Summary 3: experiment and applications Some of the anomalous transport effects were seen in: 1) High energy physics: heavy-Ion collisions (RHIC/LHC) 2) Solid state physics: Weyl and Dirac semimetals Soon: probably, consumer electronics in N years. Very important! What are Weyl and Dirac Semimetals? Simplest ways to think about these materials: 1) A three-dimensional crystal which possesses fermion quasiparticles in the electronic spectrum. 2) There are right-handed and left-handed quasiparticles. 3) The quasiparticles are electrically charged. 4) A three-dimensional version of graphene. 5) A very distant similarity to quark-gluon plasma (light quarks are nearly massless particles compared to their energies). Semimetals: real crystal materials ArXiv:1503.01304 ArXiv:1503.09188 1mm – 1 cm ● Good crystals ● Chiral fermionic quasiparticles Energy spectrum of a Weyl semimetal Energy ArXiv:1503.09188 TaAs Momentum ArXiv:1503.01304 Dirac semimetal ZrTe 5 ArXiv:1412.6543 Illustration: Chiral Magnetic Effect in a semimetal 1 Following ArXiv:1412.6543: Illustration: Chiral Magnetic Effect in a semimetal 2 Consider a gas of free chiral fermions. The chiral density is chiral chemical potential temperature usual chemical potential velocity of fermions (“speed of light” for the fermion excitations) = Fermi velocity (v = c/300) Evolution of chiral density (anomaly, the ideal case): coming from the standard expression: Following ArXiv:1412.6543 Illustration: Chiral Magnetic Effect in a semimetal 3 Evolution of chiral density (the real case): axial anomaly Equilibrium: decay of chiral charge through chirality-changing transitions (expressed via the relaxation time) equilibrium chiral density: the decay rate and production rate are the same: Following ArXiv:1412.6543 Illustration: Chiral Magnetic Effect in a semimetal 4 Equilibrium chiral density: from and small Chiral Magnetic Effect: Following ArXiv:1412.6543 Illustration: Chiral Magnetic Effect in a semimetal 5 Contribution to the current from anomaly and from the Chiral Magnetic Effect: Contribution to total conductivity depends on direction and strength of magnetic field: Experimentally measured, confirmed! Following ArXiv:1412.6543 Dislocation in semimetals The crystals of semimetals are not perfect, they may have defects which are called “dislocations” screw dislocation edge dislocation Burgers vector b: gives shifts of the atomic layers at the dislocation; it is a constant vector for each dislocation Intrinsic magnetic flux at a dislocation Fermionic quasiparticles feel the dislocations as: 1) gravity (small for typical dislocations, ignore it) 2) magnetic field (large, we take it into account) Total flux of the effective magnetic field: total flux Burgers vector (characterizes the strength of the dislocation) distance between Dirac nodes in momentum space internal, crystal-specific parameter direction of the dislocation Fermion zero mode at the dislocation Each dislocation carries a fermion zero mode which is 1) bound to the dislocation 2) propagates along the dislocation 3) has zero mass 4) appears due to magnetic flux induced by the dislocation Similar to a superconducting cosmic string. Intrinsic Chiral Magnetic Effect (idea) Each dislocation carries the magnetic flux. So, we do not need an external magnetic flux for 1) anomaly external (voltage) intrinsic (internal, due to dislocation) 2) Chiral Magnetic Effect: Thus, the external electric field generates anomalous current at the dislocation: “Intrinsic Chiral Magnetic Effect” Intrinsic Chiral Magnetic Effect (estimations) Generation of the current due to anomalous processes at the dislocation in the presence of external electric field: E anomalous contribution due to dislocation (absent, if the electric field is perpendicular to the dislocation) usual Ohm conductivity Estimations: Conclusions 1. Dislocations in the crystal structures of Dirac semimetals carry effective magnetic field. 2. Dislocations behave as strings with magnetic flux. 3. In external electric field the effective magnetic field leads to the axial anomaly, and generates a chiral charge density around the dislocation. 4. Then, an intrinsic magnetic field of the dislocation leads to an intrinsic Chiral Magnetic Effect which generates an electric current along the dislocation. 5. The Intrinsic Chiral Magnetic Effect is proposed to be experimentally measured in Dirac semimetals.