Download Chiral kinetic theory

Chiral kinetic theory
M. Stephanov
U. of Illinois at Chicago
with Yi Yin
Chiral kinetic theory – p. 1/12
Motivation
Interesting applications 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 description to non-equilibrium conditions?
Kinetic description is a solution.
There are limitations:
Classical description
Weakly coupled
But it would be an important step for our understanding of the anomalous effects.
Condensed matter literature:
Sundaram-Niu (1999), Wong-Tserkovnyak (2011), Son-Yamamoto,
Son-Spivak (2012). In the context of CME, CVE - Loganayagam-Surowka.
Chiral kinetic theory – p. 2/12
Approach
Kinetic equation describes the motion of particles in the regime where collisions
are rare enough that motion between collisions is classical.
In terms of the distribution function f (t, x, p) the equation is
∂f
∂f
∂f
df
≡
+
ẋ +
ṗ = C[f ].
dt
∂t
∂x
∂p
We think of a “cloud” of particles each of which follows classical trajectory x(t),
p(t). As a result the distribution evolves with time. But if you follow a local volume
along the trajectory, the number of particles in it can only be changed by collisions.
Ignore collisions for now.
Clearly, the number of particles in the phase space cannot change. The
phase-space density obeys conservation equation.
How can a kinetic equation account for anomalous particle creation?
Also, how can classical equation account for quantum anomaly?
Two words:
Chiral kinetic theory – p. 3/12
Approach
Kinetic equation describes the motion of particles in the regime where collisions
are rare enough that motion between collisions is classical.
In terms of the distribution function f (t, x, p) the equation is
∂f
∂f
∂f
df
≡
+
ẋ +
ṗ = C[f ].
dt
∂t
∂x
∂p
We think of a “cloud” of particles each of which follows classical trajectory x(t),
p(t). As a result the distribution evolves with time. But if you follow a local volume
along the trajectory, the number of particles in it can only be changed by collisions.
Ignore collisions for now.
Clearly, the number of particles in the phase space cannot change. The
phase-space density obeys conservation equation.
How can a kinetic equation account for anomalous particle creation?
Also, how can classical equation account for quantum anomaly?
Two words: Berry monopole
Chiral kinetic theory – p. 3/12
Path integral
Consider a Weyl particle:
H =σ·p
For each momentum p it is a two-state system. With gap 2|p|.
Transition amplitude. Insert complete sets at every time slice:
−iH(tf −ti )
hf |e
|ii = . . .
=
»Z
Z
X
p 1 ,x 1 λ ,s
1 1
. . . hx2 , s2 |p1 , λ1 ihp1 , λ1 |e−iH∆t |x1 , s1 ihx1 , s1 | . . .
 Z
DxDp P exp i
tf
ti
ff–
(p · ẋ − σ · p)dt
fi
is path-ordered product of exp{−iσ · p∆t} over each phase space path x(t), p(t).
Chiral kinetic theory – p. 4/12
Classical limit
To take the classical limit we diagonalize Vp† σ · pVp = |p|σ3 at each point:
. . .Vp 2 Vp†2 exp{−iσ · p2 ∆t}Vp 2 Vp†2 Vp 1 Vp†1 exp{−iσ · p1 ∆t}Vp 1 Vp†1 . . .
= . . . Vp 2 exp{−i|p2 |σ3 ∆t} Vp†2 Vp 1 exp{−i|p1 |σ3 ∆t}Vp†1 . . .
| {z }
extra rotation
Vp†2 Vp 1 ≈ exp(−iâp · ∆p)
where
âp = −iVp† ∇p Vp .
If we did not insist on diagonalizing σ · p, we could choose arbitrary U (2)
rotation, say Vp → Vp Up .
This results in a “gauge transformation” of the “action” such that
−|p|σ3 → −|p|Up† σ3 Up
and âp → Up† âp Up + iUp† ∇p Up
Corresponds to the free choice of the phase and spin quantization direction for
the momentum states: |p, si → Up |p, si.
We chose helicity basis at each p to diagonalize σ · p.
Chiral kinetic theory – p. 5/12
Abelian projection and the monopole
»
hf |e−iH(tf −ti ) |ii = Vp f
Z
 Z
DxDp P exp i
tf
ti
ff
–
(p · ẋ − |p|σ3 − âp · ṗ)dt Vp†i
fi
To describe classical motion of a particle of a given helicity we need to be able
to neglect off-diagonal transitions, which are caused by the off-diagonal
components of âp . I.e., |âp · ṗ| ≪ 2|p|.
Forces (ṗ) should not be too strong ( B ≪ |p|2 ).
However, we cannot neglect the diagonal components – Berry phases!
Although non-abelian âp is pure gauge, abelian component [âp ]11 ≡ ap is
non-trivial: the field of a “monopole” at |p| = 0 (as in ’t Hooft, Polyakov):
b ≡ ∇ p × ap =
p̂
.
2|p|2
Singularity is due to level crossing at p = 0, where classical description breaks
down.
Chiral kinetic theory – p. 6/12
Classical action and equations
In the external ordinary magnetic field B = ∇ × A the clasical action is then
I=
Z
tf
ti
(p · ẋ + A · ẋ − |p| − ap · ṗ)dt
The equations of motion can be obtained by variations:
ẋ − p̂ + b × ṗ = 0;
ṗ + B × ẋ = 0
No effect from b without B (ṗ = 0).
Solving for ẋ and ṗ we can then write the desired equation:
∂f
∂f
∂f
+
ẋ +
ṗ = 0.
∂t
∂x
∂p
√
d3 xd3 p
NB: the invariant measure on the phase space is G
,
(2π)3
∂2L
∂2L
2
−
,
where G = (1 + b · B) is the det of the 6x6 matrix GAB =
˙
∂ξA ∂ ξB
∂ξB ∂ ξ̇A
where ξA = (x, p).
Chiral kinetic theory – p. 7/12
Current and CME
The current density can be calculated as
j=
Z √
Gf ẋ
p
were
√
G ẋ = p̂ + B(p̂ · b)
Thus
j=
and
Z
Z
p
≡
Z
f p̂ + B
p
| {z }
|
regular current
Using b =
p̂
, E ≡ |p|:
2|p|2
jCME
d3 p
.
(2π)3
1
=B 2
4π
Z
Z
f (p̂ · b)
p
{z
CME
}
∞
f (E, p̂)dE
0
(cf. Loganayagam-Surowka in isotropic case) and is =
1
µB for FD distribution.
4π 2
Chiral kinetic theory – p. 8/12
EM anomaly
Turn on electric field:
ẋ = p̂ + ṗ × b;
ṗ = E + ẋ × B.
Solve for ẋ and ṗ:
√
G ẋ = p̂ + E × b + B(p̂ · b).
√
Gṗ = E + p̂ × B + b(E · B).
Integrate kinetic equation ∂t f + ẋ · ∇x f + ṗ · ∇p f = 0 over
∂n
+ ∇ · j = (E · B)
∂t
Z
f ∇p b =
p
R √
p
G:
1
E · B f0 .
4π 2
where ∇p b = 2πδ 3 (p) and f0 ≡ f |p=0 .
j=
Z √
Z
Gf ẋ =
f p̂ + E ×
p
p
| {z }
regular current
|
|
Z
{z
fb + B
p
}
anom. Hall current
{z
=0 in equilibrium
}
|
Z
f (p̂ · b)
p
{z
CME
}
Chiral kinetic theory – p. 9/12
Strictly speaking ...
... we cannot treat region near p = 0 classically.
Define a phase space current density with 6+1 components: (ρ, ρẋ, ρṗ), where
√
ρ = Gf . It obeys continuity equaton:
∂(ρẋ)
∂(ρṗ)
∂ρ
+
+
= 0.
∂t
∂x
∂p
√
Exclude non-classical |p| < ∆, where ∆ ∼ B, define
R
(n∆ , j∆ ) = |p|>∆ ρ(1, ẋ). Then up to O(∆/|p|)2
∂n∆
+ ∇ · j∆ =
∂t
Z
dS∆ ·
Anomaly is matched by the net flow into the
classical region of phase space
Janom =
ρṗ
E·B
p̂
=
f
.
3
2
2
(2π)
4π
4π|p|
ρṗ
.
(2π)3
classical region
quantum
region
through the cutoff surface S∆ .
Chiral kinetic theory – p. 10/12
CVE
Replace Lorentz force with Coriolis force:
ṗ = 2|p|ω × ẋ
I.e., B → 2|p|ω .
The non-equilibrium CVE is then
jCVE = ω
which is
jCVE
1
=ω 2
4π
Z
2|p|f (p̂ · b)
p
Z
∞
f (E, p̂) 2EdE
0
(cf. Loganayagam-Surowka in isotropic case) and is =
1 2
µ ω for FD distribution.
4π 2
NB: B is not “∼” µω.
Chiral kinetic theory – p. 11/12
Conclusions and discussion
Kinetic description of CME, CVE and anomaly:
∂f
∂f
∂f
+
ẋ +
ṗ = C[f ]
∂t
∂x
∂p
with
ẋ = p̂ + ṗ × b;
ṗ = E + ẋ × B
and the Berry monopole
b=
Out-of-equilibrium CME, CVE:
Z
jCME = B
f (p̂ · b);
p
.
2|p|3
jCVE = 2ω
p
The description is limited by |p| > ∆ ≫
√
Z
f (p̂ · b)|p|.
p
B.
The anomaly “works” inside |p| < ∆. In the classical
region, without collisions, particles cannot be created or
destroyed. They can enter or exit through the boundary at
|p| = ∆. The net flux is 4π1 2 E · B f0
Chiral kinetic theory – p. 12/12