Introduction Relativistic hydrodynamics with triangle anomaly Effective field theory Kubo formula and effective gravity Microscopic picture and zero modes Superfluidity Discussion Chiral effects in superfluid Sadofyev A. Institute for Theoretical and Experimental Physics June 13, 2012 Sadofyev A. Chiral effects in superfluid
Introduction Relativistic hydrodynamics with triangle anomaly Effective field theory Kubo formula and effective gravity Microscopic picture and zero modes Superfluidity Discussion Plan Introduction 1 Relativistic hydrodynamics with triangle anomaly 2 Equations of motion Entropy current conservation Coefficients Effective field theory 3 Anomaly in the effective field theory Hydrodynamical approximation Kubo formula and effective gravity 4 Model Kinetic coefficients Microscopic picture and zero modes 5 Eigenstates Microscopical current calculation Sadofyev A. Chiral effects in superfluid Superfluidity 6
Introduction Relativistic hydrodynamics with triangle anomaly Effective field theory Kubo formula and effective gravity Microscopic picture and zero modes Superfluidity Discussion Recently, there were intense studies of hydrodynamics of chiral liquids. A crucial novel point is existence of new transport coefficients, overlooked in the text-book approaches. − → j CME = µ 5 → − − → j CVE = µ 5 µ − → , ω , B 2 π 2 π 2 The coefficients are considered to be completely fixed by the coefficient in front of the chiral anomaly. This result could be obtained in a lot of approaches. Sadofyev A. Chiral effects in superfluid
Introduction Relativistic hydrodynamics with triangle anomaly Effective field theory Equations of motion Kubo formula and effective gravity Entropy current conservation Microscopic picture and zero modes Coefficients Superfluidity Discussion Equation of motion for relativistic liquid in external EM field plus anomaly: ∂ µ T µν = F νλ J λ , ∂ µ J µ = CE µ B µ , where T µν = wu µ u ν + Pg µν + τ µν , J µ = nu µ + ν µ For ideal liquid τ µν are ν are absence but in presence of the anomaly situation is not the same! Sadofyev A. Chiral effects in superfluid
Introduction Relativistic hydrodynamics with triangle anomaly Effective field theory Equations of motion Kubo formula and effective gravity Entropy current conservation Microscopic picture and zero modes Coefficients Superfluidity Discussion Entropy non-decreasing for ideal liquid transform to entropy current conservation and in presence of anomaly it takes form ∂ µ s µ = − C µ T E · B . where s µ = su µ . To improve this relation one should introduce another definition of entropy current and turn on ν µ then s µ = su µ − µ T ν µ + D ω µ + D B B µ and ∂ µ s µ ≥ 0 , where ν µ = ξω µ + ξ B B µ ω µ = 1 B µ = 1 2 ǫ µναβ u ν ∂ α u β , 2 ǫ µναβ u ν F αβ Sadofyev A. Chiral effects in superfluid
Introduction Relativistic hydrodynamics with triangle anomaly Effective field theory Equations of motion Kubo formula and effective gravity Entropy current conservation Microscopic picture and zero modes Coefficients Superfluidity Discussion From the modified entropy current conservation ( ∂ µ s µ ) coefficients take following form (arXiv:0906.5044 [hep-th]) µ 3 n ξ = C ( µ 2 − 2 ǫ + p ) 3 µ 2 n ξ B = C ( µ − 1 ǫ + p ) 2 The terms of higher order in chemical potential are not universal and depend on equation of state. It is interesting that in two current model only coefficient of chiral magnetic effect (coefficient in vector current for magnetic field) could be fixed. Sadofyev A. Chiral effects in superfluid
Introduction Relativistic hydrodynamics with triangle anomaly Effective field theory Anomaly in the effective field theory Kubo formula and effective gravity Hydrodynamical approximation Microscopic picture and zero modes Superfluidity Discussion Relation between the chiral anomaly and the coefficients could be studied in effective field theory (arXiv:1012.1958v1 [hep-th]). Lets consider the action for chiral fermions with chemical potential � i ¯ ψγ ρ D ρ ψ + µ ¯ ψγ 0 ψ + µ 5 ¯ ψγ 0 γ 5 ψ � � S eff = dx + S int . One can consider that action after some modification as model of chiral liquid generated by some non-anomalous interaction. To modify action one should bring it to naively Lorentz invariant form � i ¯ ψγ ρ D ρ ψ + µ u ν ¯ ψγ ν ψ + µ 5 u ν ¯ ψγ ν γ 5 ψ � � S eff = dx by introducing of liquid velocity u ν as slowly varying external field. Sadofyev A. Chiral effects in superfluid
Introduction Relativistic hydrodynamics with triangle anomaly Effective field theory Anomaly in the effective field theory Kubo formula and effective gravity Hydrodynamical approximation Microscopic picture and zero modes Superfluidity Discussion After calculating of anomaly in the effective filed theory we get ∂ µ J 5 µ = − 1 4 π 2 ǫ µναβ ( ∂ µ ( A ν + µ u ν ) ∂ α ( A β + µ u β )+ ∂ µ ( µ 5 u ν ) ∂ α ( µ 5 u β )) ∂ µ J µ = − 1 2 π 2 ǫ µναβ ∂ µ ( A ν + µ u ν ) ∂ α ( µ 5 u β ) It should be noted that there are terms of higher orders in chemical potentials. However that contributions are not anomalous and moreover depend on details of IR cutoff µ 3 δξ ∼ , ǫ IR ∼ ( ǫ + p ) / n ǫ IR Sadofyev A. Chiral effects in superfluid
Introduction Relativistic hydrodynamics with triangle anomaly Effective field theory Anomaly in the effective field theory Kubo formula and effective gravity Hydrodynamical approximation Microscopic picture and zero modes Superfluidity Discussion One can take naive hydrodynamical limit by substituting < J µ > = nu µ , < J 5 ,µ > = n 5 u µ and hydrodynamical currents could be redefined as 1 1 J 5 ,µ = n 5 u µ + 2 π 2 ( µ 2 + µ 2 5 ) ω µ + 2 π 2 µ B µ J µ = nu µ + 1 1 π 2 ( µµ 5 ) ω µ + 2 π 2 µ 5 B µ This result coincides with the answer obtained through pure hydrodynamical consideration. Sadofyev A. Chiral effects in superfluid
Introduction Relativistic hydrodynamics with triangle anomaly Effective field theory Model Kubo formula and effective gravity Kinetic coefficients Microscopic picture and zero modes Superfluidity Discussion The other way to obtain CME and CVE is consideration of effective gravity. One can consider slowly moving chiral liquid as in the rest frame by performing following coordinate transformation ds = − dt 2 + 2 v i dtdx i + dx 2 In that frame Dirac action should be modified � � � dx i ¯ ψγ a e ρ i ∂ ρ − ω ab S = ρ + A ρ ψ, a is spin-connection and e ρ where ω ab a is vierbien. One readily finds ρ that in the low velocity limit this effective theory coincides with the previous one. Sadofyev A. Chiral effects in superfluid
Introduction Relativistic hydrodynamics with triangle anomaly Effective field theory Model Kubo formula and effective gravity Kinetic coefficients Microscopic picture and zero modes Superfluidity Discussion The chiral kinetic coefficients could be obtained in the theory with effective gravity as a linear response. Lets consider only CVE in the axial current here then one should calculate the following correlators: − ik c 2 k 2 � J a T 0 b �| ω =0 σ = lim k c → 0 ǫ abc and after some calculation (arXiv:1103.5006v2 [hep-ph]): σ = µ 2 + µ 2 + T 2 5 4 π 2 12 It was shown that two terms are proportional to the chiral and gravitational anomalies respectively. Sadofyev A. Chiral effects in superfluid
Introduction Relativistic hydrodynamics with triangle anomaly Effective field theory Eigenstates Kubo formula and effective gravity Microscopical current calculation Microscopic picture and zero modes Superfluidity Discussion Despite of evaluation methods variety there is non-answered question of the chiral effects microscopic realization. To clarify that one could consider Dirac operator eigenstates. The Dirac Hamiltonian in external constant magnetic field uniform in the third direction is H = − i ( ∂ i − ieA i ) γ 0 γ i + m γ 0 , where we can remove chemical potential as an energy shift and Dirac equation takes form − H R ψ L + m ψ R = E ψ L , H R ψ R + m ψ L = E ψ R , where H R = ( − i ∂ i + eA i ) σ i and H 2 R + m 2 � ψ R = E 2 ψ R � Sadofyev A. Chiral effects in superfluid
Introduction Relativistic hydrodynamics with triangle anomaly Effective field theory Eigenstates Kubo formula and effective gravity Microscopical current calculation Microscopic picture and zero modes Superfluidity Discussion We can go to momentum eigenstates − i ∂ 3 ψ R = p 3 ψ R and then H R = p 3 σ 3 + H ⊥ , where H ⊥ = ( − i ∂ a + eA a ) σ a , a = 1 , 2. The eigenstates of H R can be expressed in terms of eigenstates of H ⊥ ([ H ⊥ , H 2 R ] = 0). So each eigenstates of H ⊥ generates two eigenstate of H R and the zero modes of H ⊥ are simultaneously eigenstates of H R with eigenvalue ǫ = p 3 σ 3 . Sadofyev A. Chiral effects in superfluid
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