Chiral kinetic theory

Transcription

1 Chiral kinetic theory. 1/12 Chiral kinetic theory M. Stehanov U. of Illinois at Chicago with Yi Yin

2 Chiral kinetic theory. 2/12 Motivation Interesting alications of chiral magnetic/vortical effect involve highly non-equilibrium conditions such as heavy-ion collisions. Derivations of chiral magnetic effect are usually done in equilibrium. Can we generalize this descrition to non-equilibrium conditions? Kinetic descrition is a solution. There are limitations: Classical descrition Weakly couled But it would be an imortant ste for our understanding of the anomalous effects. Condensed matter literature: Sundaram-Niu (1999), Wong-Tserkovnyak (2011), Son-Yamamoto, Son-Sivak (2012). In the context of CME, CVE - Loganayagam-Surowka.

3 Aroach Kinetic equation describes the motion of articles in the regime where collisions are rare enough that motion between collisions is classical. In terms of the distribution function f(t, x, ) the equation is df dt f t + f xẋ + f ṗ = C[f]. We think of a cloud of articles each of which follows classical trajectory x(t), (t). As a result the distribution evolves with time. But if you follow a local volume along the trajectory, the number of articles in it can only be changed by collisions. Ignore collisions for now. Clearly, the number of articles in the hase sace cannot change. The hase-sace density obeys conservation equation. How can a kinetic equation account for anomalous article creation? Also, how can classical equation account for quantum anomaly? Two words: Chiral kinetic theory. 3/12

4 Aroach Kinetic equation describes the motion of articles in the regime where collisions are rare enough that motion between collisions is classical. In terms of the distribution function f(t, x, ) the equation is df dt f t + f xẋ + f ṗ = C[f]. We think of a cloud of articles each of which follows classical trajectory x(t), (t). As a result the distribution evolves with time. But if you follow a local volume along the trajectory, the number of articles in it can only be changed by collisions. Ignore collisions for now. Clearly, the number of articles in the hase sace cannot change. The hase-sace density obeys conservation equation. How can a kinetic equation account for anomalous article creation? Also, how can classical equation account for quantum anomaly? Two words: Berry monoole Chiral kinetic theory. 3/12

5 Chiral kinetic theory. 4/12 Path integral Consider a Weyl article: H = σ For each momentum it is a two-state system. With ga 2. Transition amlitude. Insert comlete sets at every time slice: f e ih(t f t i ) i =... 1,x 1 X λ 1,s 1... x 2, s 2 1, λ 1 1, λ 1 e ih t x 1, s 1 x 1, s 1... =» DxD P ex j i tf ff ( ẋ σ )dt t i fi is ath-ordered roduct of ex{ iσ t} over each hase sace ath x(t), (t).

6 Chiral kinetic theory. 5/12 Classical limit To take the classical limit we diagonalize V σ V = σ 3 at each oint:...v 2 V 2 ex{ iσ 2 t}v 2 V 2 V 1 V 1 ex{ iσ 1 t}v 1 V 1... =... V 2 ex{ i 2 σ 3 t} V 2 V 1 extra rotation ex{ i 1 σ 3 t}v 1... V 2 V 1 ex( iâ ) where â = iv V. If we did not insist on diagonalizing σ, we could choose arbitrary U(2) rotation, say V V U. This results in a gauge transformation of the action such that σ 3 U σ 3 U and â U â U + iu U Corresonds to the free choice of the hase and sin quantization direction for the momentum states:, s U, s. We chose helicity basis at each to diagonalize σ.

7 Chiral kinetic theory. 6/12 Abelian rojection and the monoole f e ih(t f t i ) i =»V f DxD P ex j tf ff i ( ẋ σ 3 â ṗ)dt V i t i fi To describe classical motion of a article of a given helicity we need to be able to neglect off-diagonal transitions, which are caused by the off-diagonal comonents of â. I.e., â ṗ 2. Forces (ṗ) should not be too strong ( B 2 ). However, we cannot neglect the diagonal comonents Berry hases! Although non-abelian â is ure gauge, abelian comonent [â ] 11 a is non-trivial: the field of a monoole at = 0 (as in t Hooft, Polyakov): b a = ˆ 2 2. Singularity is due to level crossing at = 0, where classical descrition breaks down.

8 Chiral kinetic theory. 7/12 Classical action and equations In the external ordinary magnetic field B = A the clasical action is then I = tf t i ( ẋ + A ẋ a ṗ)dt The equations of motion can be obtained by variations: ẋ ˆ + b ṗ = 0; ṗ + B ẋ = 0 Solving for ẋ and ṗ we can then write the desired equation: f t + f xẋ + f ṗ = 0. NB: the invariant measure on the hase sace is G d3 xd 3 (2π) 3, where G = (1 + b B) 2 is the det of the 6x6 matrix G AB = where ξ A = (x, ). No effect from b without B (ṗ = 0). 2 L ξ A ξ B 2 L ξ B ξ A,

9 Chiral kinetic theory. 8/12 The current density can be calculated as were G ẋ = ˆ + B(ˆ b) and Thus Current and CME j = j = f ˆ regular current Gfẋ + B d 3 (2π) 3. f(ˆ b) CME Using b = ˆ 2 2, E : j CME = B 1 4π 2 0 f(e, ˆ)dE (cf. Loganayagam-Surowka in isotroic case) and is = 1 µb for FD distribution. 4π2

10 EM anomaly Turn on electric field: Solve for ẋ and ṗ: ẋ = ˆ + ṗ b; ṗ = E + ẋ B. G ẋ = ˆ + E b + B(ˆ b). Gṗ = E + ˆ B + b(e B). Integrate kinetic equation t f + ẋ x f + ṗ f = 0 over R G: n t + j = (E B) f b = 1 4π 2 E B f 0. where b = 2πδ 3 () and f 0 f =0. j = Gfẋ = f ˆ + E regular current anom. Hall current =0 in equilibrium fb + B f(ˆ b) CME Chiral kinetic theory. 9/12

11 Strictly seaking we cannot treat region near = 0 classically. Define a hase sace current density with 6+1 comonents: (ρ, ρẋ, ρṗ), where ρ = Gf. It obeys continuity equaton: ρ t + (ρẋ) x + (ρṗ) = 0. Exclude non-classical <, where B, define (n, j ) = R ρ(1, ẋ). Then u to O( / )2 > n t + j = ds ρṗ (2π) 3. Anomaly is matched by the net flow into the classical region of hase sace classical region J anom = ρṗ (2π) = E B 3 4π 2 f ˆ 4π 2. quantum region through the cutoff surface S. Chiral kinetic theory. 10/12

12 Chiral kinetic theory. 11/12 CVE Relace Lorentz force with Coriolis force: ṗ = 2 ω ẋ I.e., B 2 ω. The non-equilibrium CVE is then j CVE = ω 2 f(ˆ b) which is j CVE = ω 1 4π 2 0 f(e, ˆ) 2EdE (cf. Loganayagam-Surowka in isotroic case) and is = 1 4π 2 µ2 ω for FD distribution. NB: B is not µω.

13 Conclusions and discussion Kinetic descrition of CME, CVE and anomaly: f t + f xẋ + f ṗ = C[f] with ẋ = ˆ + ṗ b; ṗ = E + ẋ B and the Berry monoole b = 2 3. Out-of-equilibrium CME, CVE: j CME = B f(ˆ b); j CVE = 2ω f(ˆ b). The descrition is limited by > B. The anomaly works inside <. In the classical region, without collisions, articles cannot be created or destroyed. They can enter or exit through the boundary at =. The net flux is 1 4π 2 E B f 0 Chiral kinetic theory. 12/12