Kinetic Theory and Anomalous Transports in the Presence of Nonabelian Phase-Space Berry Curvatures

0.29 MB·

by  Tomoya Hayata

RIKEN-QHP-280, RIKEN-STAMP-33
Kinetic Theory and Anomalous Transports in the Presence of
Nonabelian Phase-Space Berry Curvatures
Tomoya Hayata1 and Yoshimasa Hidaka2
1Department of Physics, Chuo University,
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1-13-27 Kasuga, Bunkyo, Tokyo, 112-8551, Japan
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2Theoretical Research Division, Nishina Center,
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n RIKEN, Wako, Saitama 351-0198, Japan
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(Dated: January 17, 2017)

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Abstract

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l We construct the kinetic theory in (1+2d)-dimensional phase space and time when all abelian
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andnonabelianphase-spaceBerrycurvaturesarenonzero. Thenwecalculateanomaloustransports
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e induced by the Berry curvatures on the basis of the kinetic theory. As an example, we study
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. anomalous charge and spin transports induced by the SU(2) Berry curvatures. We also derive the
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m topological effective theory to reproduce the transports in insulators calculated from the kinetic
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d theory. Such an effective theory is given by the nonabelian phase space Chern-Simons theory.
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PACS numbers: 03.65.Vf,73.43.-f,03.65.Sq,72.10.bg
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Introduction. The wave function in quantum mechanics often acquires the non-trivial
phase under an adiabatic and cyclic process, which is known as the so called Berry phase [1].
The Berry phase and the Berry curvature describe torsion of the wave function in a closed
manifold composed by parameters of the Hamiltonian. They provide us a universal descrip-
tion of anomalous transport phenomena, represented by the seminal works on the quantum
Hall effect [2–7], and the adiabatic charge pumping [8–10].
The effect of the Berry curvature has been incorporated into kinetic theory to study
anomalous transports in metals as well as insulators [11, 12]. An example of this is the
intrinsic contribution to the anomalous Hall effect [13–17]. Later, the modified kinetic
theory has been applied to study anomalous transports in Dirac and Weyl semimetals [18–
22] such as the chiral magnetic effect [23–25]. Interestingly, it has been shown that the Berry
curvature of a Weyl fermion is closely related to the triangle anomaly, and the modified
kinetic theory has been applied to study anomalous transports in high-energy physics under
the name of the chiral kinetic theory [26–29].
The topological nature of the anomalous transports induced by the Berry curvatures can
be understood from the relation with topological field theory. For example, the low-energy
effective theory of the quantum Hall state is given by the (1+2)-dimensional Chern-Simons
theory [30–32], and that of topological insulators [33–39] is given by the so called θ term in
1+3 dimensions [40–42]. Those topological effective theory in real space are generalized to
phase space for describing the electromagnetic responses induced by the phase-space Berry
curvatures [40, 43]. Recently in Ref. [44], we have shown, for the abelian Berry curvatures,
that such a topological effective theory is completely equivalent to the kinetic theory, and
can reproduce the anomalous transports obtained from the modified kinetic theory [45].
In contrast to the abelian case, the anomalous transports in the presence of nonabelian
phase space Berry curvatures await full understanding, which is necessary to study the
nonabeliangeneralizationoftheanomaloustransportssuchastheadiabaticchargepumping,
the chiral magnetic effect, and the anomalous Hall effect. There have been attempts to
derive a kinetic theory in a multi-band case with the nonabelian Berry curvatures from the
derivative expansion of the Wigner function [46–49]. Such a kinetic theory has been applied
to study the intrinsic contribution to the spin Hall effect [50, 51]. However, neither the full
expression of the anomalous transports nor the relation to the Chern-Simons theory has not
been derived unlike the case of the abelian Berry curvature.
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In this Letter, we construct the kinetic theory when all abelian and nonabelian phase-
space Berry curvatures in 1+2d dimensions are nonzero. For this purpose we generalize the
kinetic theory in the presence of the nonabelian gauge fields, which has been developed in
the context of quark-gluon-plasma to study the nonequilibrium dynamics of the plasma such
as thermalization with color exchange interactions [52–60]. We generalize it to incorporate
the interactions with the nonabelian gauge fields (Berry curvatures) in momentum space
and study transports induced by the Berry curvatures. As an example, we elaborate the
anomalous transports induced by the SU(2) Berry curvatures. We derive the nonabelian
version of the adiabatic charge pumping, the chiral magnetic effect and the anomalous Hall
effect, and their analogue in the spin transports. Finally, we construct an effective theory to
reproduce the transports obtained from the kinetic theory. We show that such an effective
theory is the nonabelian generalization of the phase space Chern-Simons theory [43, 44].
Kinetic theory and non-abelian Berry connections. We consider the semiclassical dynam-
ics in (1+2d)-dimensional phase space and time. The action has the form:
(cid:90)
(cid:16)
S = dt x˙ ·(p+A)+p˙ ·a
(1)
(cid:17)
+x˙ ·qaˆAaˆ +p˙ ·qaˆaaˆ −ε+A +qaˆAaˆ ,
t t
whereεisanenergyeigenvalueofN-folddegenerateBlochstates|ui(t,ξ ,qaˆ)(cid:105)(i = 1,...,N),
a
which depend on time t and phase space coordinates ξ = (ξ ,...,ξ ) = (x,p) with x =
a 1 2d
(x ,...,x ) and p = (p ,...,p ), and also depend on “color” charges qaˆ (aˆ = 1,...,N2−1).
1 d 1 d
We here introduce the “color” charges as the dynamical degrees of freedom to describe the
dynamics of the “color” currents jaˆ, whose expression in kinetic theory is given later. To
b
keep the gauge invariance of the action (1), qaˆ satisfies
q˙aˆ = −faˆˆbcˆAˆb qcˆ−faˆˆbcˆξ˙ Aˆb qcˆ. (2)
t j j
A , A = (A ,...,A ), and a = (a ,...,a ) are the abelian Berry connections and defined as
t 1 d 1 d
A = tr(cid:104)ui|i∂ |uj(cid:105)/N, A = tr(cid:104)ui|i∇ |uj(cid:105)/N, and a = tr(cid:104)ui|i∇ |uj(cid:105)/N. In the same manner,
t t x p
Aaˆ , Aaˆ = (Aaˆ ,...,Aaˆ ), and aaˆ = (aaˆ ,...,aaˆ ) are the nonabelian Berry connections and
t 1 d 1 d
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
defined as Aaˆ = 2tr taˆ(cid:104)ui|i∂ |uj(cid:105) , Aaˆ = 2tr taˆ(cid:104)ui|i∇ |uj(cid:105) , and aaˆ = 2tr taˆ(cid:104)ui|i∇ |uj(cid:105) ,
t t x p
with taˆ (aˆ = 1,...,N2−1) being the generators of the Lie algebras of SU(N) group normal-
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ized as trtaˆtˆb = δaˆˆb/2. faˆˆbcˆ are the structure constants of the Lie algebras, whose generators
are taˆ ([taˆ,tˆb] = ifaˆˆbcˆtcˆ). In the presence of the external electromagnetic fields, their gauge
potentials are also introduced to the action (1), which can be absorbed into A and A. In the
t
following, we do not distinguish the electromagnetic gauge potentials (field strengths) and
the real-space abelian Berry connections (curvatures) unless otherwise stated. We employ
the Einstein convention for repeated indices. We define the phase space and time coordi-
nates and the generalized connections as ξ = (ξ ,ξ ) = (t,ξ ), qνˆ = (qˆ0,qˆb) = (1,qˆb), and
µ 0 a a
Aνˆ = (Aνˆ ,Aνˆ ) = (Aνˆ,Aνˆ ) (νˆ = 0,1,...,N2 − 1) with ξ = (x,p), Aˆ0 = −ε + A ,
µ 0 a t a a t t
Aˆ0 = (p + A,a), Aˆb = Aˆb , and Aˆb = (Aˆb,aˆb). The unhatted Greek indices such as
a t t a
µ, and ν run from 0 to 2d. The unhatted Roman indices such as a, and b run from 1 to
2d. The hatted Greek indices such as µˆ, and νˆ run from 0 to N2 −1. The hatted Roman
indices such as aˆ, and ˆb run from 1 to N2 − 1. Following Refs. [43, 44], we absorb the
energy ε and momentum p into the abelian Berry connections. Using these variables, we
can write the action as the topological form: S = (cid:82) qνˆAνˆ dξ , where qˆ0 = 1. As will be seen
µ µ
below, this topological form implies that the anomalous transports are expressed by using
the (1+2d)-dimensional Chern-Simons theory for Aνˆ . The equation of motion reads
µ
qµˆωµˆ ξ˙ = −qνˆωνˆ , (3)
ab b at
where ωµˆ are the generalized Berry curvatures and defined as
νλ
∂Aµˆ ∂Aµˆ
ωµˆ = λ − ν +fµˆαˆβˆAαˆ Aβˆ , (4)
νλ ∂ξ ∂ξ ν λ
ν λ
where fµˆαˆβˆ are the structure constants with fˆ0αˆβˆ = fαˆˆ0βˆ = fαˆβˆˆ0 = 0. It will be helpful to
explicitly show the classical equation of motion in the standard notation in d = 3. To avoid
the lengthy expression, we here neglect the mixed-space Berry curvatures such as Ωµˆ :
xipj
x˙ = ∂ ε−(cid:15) p˙ bµˆ qµˆ −eµˆqµˆ, (5)
i pi ijk j k i
p˙ = −∂ ε+(cid:15) x˙ Bµˆ qµˆ +Eµˆqµˆ, (6)
i xi ijk j k i
where Eµˆ = Ωµˆ , and eµˆ = Ωµˆ (Bµˆ = (cid:15) Ωµˆ /2, and bµˆ = (cid:15) Ωµˆ /2) are the electric
i xit i pit i ijk xjxk i ijk pjpk
and emergent electric fields (magnetic and emergent magnetic fields) in real and momentum
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space, respectively. The definition of Ωµˆ is given by Eq. (4) with replacing Aµˆ by Aµˆ
νλ ν ν
(Those with µˆ = 0 are equal to the U(1) electromagnetic fields and the abelian Berry curva-
tures). The second and third terms in Eq. (6) with µˆ (cid:54)= 0 are the nonabelian generalization
of the electromagnetic Lorentz and Coulomb forces, and the momentum analogue of them
is introduced in Eq. (5). Similarly, in the presence of the mixed-space Berry curvatures, the
mixed-space analogue of the Lorentz force appears, which can be introduced by generalizing
the abelian case as Ω → qµˆΩµˆ in Refs. [12, 44].
νλ νλ
We assume that det(qµˆωµˆ) is nonzero, and then, the equation of motion (3) is written as
ξ˙ = −(cid:2)qµˆωµˆ(cid:3)−1qνˆωνˆ . (7)
a ab bt
(cid:112)
Like the abelian case, the invariant volume element reads ddxddpdN2−1q det(qµˆωµˆ)/(2π)d,
where the integration measure of the color space is chosen so as for the Casimir invariants
of the Lie algebras to be constants of motion with the normalization (cid:82) dN2−1q1 = 1 [60, 61].
We generalize color space to include qˆ0 (qˆ0 itself is a constant of motion, namely, qˆ0 = 1),
and write the integration measure as dN2q. Since ω is a skew symmetric matrix, whose
determinant is written as det(qµˆωµˆ) = Pf(qµˆωµˆ)2 with the Pfaffian
1
Pf(qµˆωµˆ) = (cid:15) qµˆ1ωµˆ1 ···qµˆdωµˆd , (8)
2dd! a1···a2d a1a2 a2d−1a2d
where(cid:15) isthetotallyanti-symmetrictensor((cid:15) = 1). Wenotethatthedeterminant
a1···a2d 12···2d
of a real skew matrix is always nonnegative. We find that the inverse matrix is given as [44]
(cid:15)
(cid:2)qµˆωµˆ(cid:3)−1 = baa1···a2d−2 qµˆ1ωµˆ1 ···qµˆd−1ωµˆd−1 . (9)
ab 2d−1(d−1)!Pf(qνˆωνˆ) a1a2 a2d−3a2d−2
In our kinetic theory, the current can be calculated by averaging the velocity of particles,
ξ˙ , over phase and color space (ξ ,qµˆ) with the invariant volume element. We consider the
a a
distribution function n(t,ξ ), which is independent of the color charges such as the thermal
a
equilibrium distribution. Then by using Eqs. (7) and (9), the current density in phase space
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jµˆ (t,ξ ) is given as, by integrating over color space,
a a
jµˆ (t,ξ ) = (cid:90) dN2q (cid:112)det(qνˆωνˆ)qµˆξ˙ n(t,ξ)
a a (2π)d a
(−1)νc (cid:15)
= µˆµˆ1...µˆd aba1···a2d−2ωµˆ1 ···ωµˆd−1 ωµˆdn(t,ξ), (10)
(2π)d2d−1(d−1)! a1a2 a2d−1a2d−2 bt
˙
where ξ is the solution of the equation of motion (7), and we introduced the sign of the
a
(cid:112)
Pfaffian (−1)ν = Pf(qνˆωνˆ)/ det(qνˆωνˆ) (In our kinetic regime, the sign is negative [44]).
c = (cid:82) dN2qqµˆqµˆ1···qµˆd is a symmetric tensor and invariant under the adjoint action
µˆµˆ1...µˆd
of the Lie group. Since c = c , whether c becomes nonzero or not is
ˆ0µˆ1...µˆm µˆ1...µˆm µˆµˆ1...µˆd
determined by c = (cid:82) dN2−1qqaˆ1···qaˆm (m = 1,...,d + 1), which are written only by
aˆ1...aˆm
using the Casimir invariants of the Lie algebras [61]. Integrating Eq. (10) with respect to
p , we obtain the current in real space. Also the local charge density is given as jµˆ =
i 0
(cid:82) dN2q(cid:112)det(qνˆωνˆ)qµˆn(t,ξ)/(2π)d. For a band insulator (n(t,ξ ) = 1), we have
a
(−1)ν
jµˆ (ξ) = c (cid:15) ωµˆ1 ···ωµˆd . (11)
0 2dd! µˆµˆ1...µˆd a1···a2d a1a2 a2d−1a2d
Anomalous charge and spin transports. The general expression of anomalous transports
isgivenbyEq.(10). WehereelaboratetheonesinducedbytheinterplayoftheSU(2)phase-
space Berry curvatures and the U(1) electromagnetic fields. The SU(2) Berry curvatures
arise e.g., in the Luttinger model [62]: H = (cid:80)5 d Γ , where Γ are the generators of the
a=1 a a a
SO(5) Clifford algebras. The standard Berry curvatures in momentum space are obtained
by the pull back from those in the five-dimensional d space to momentum space [49–51].
The SU(2) Berry curvatures in phase space and time are also obtained by the pull back to
mixed space when d depends on x and t, e.g., by applying strain spatial gradient and lasers.
First we consider the adiabatic charge pumping in the presence of time-periodic pertur-
bation. By integrating over the period of perturbation T with the Fermi-Dirac distribution
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n(p) in the vanishing external fields, in d = 3, it is given as
(cid:90) dtd3p
P (x )= −e jˆ0(t,x,p)
i i (2π)3 i
(cid:90) dtd3p(cid:104)1 (cid:16) (cid:17)
= e δ Ωaˆ −Ωaˆ eaˆ
(2π)3 2 ij xkpk pixj j
1
− (cid:15) (cid:15) Ωaˆ Ωaˆ ∂ ε
4 ikl jm¯n¯ pm¯xk pn¯xl pj (12)
1 1
+ Baˆbaˆ ·∇ ε+ eB eaˆ ·baˆ
2 i p 2 i
1 (cid:16) (cid:17)(cid:105)
+ (cid:15) baˆ Eaˆ −Ωaˆ baˆ eE n(p),
2 ijk k j pk¯xk k¯ j
where e > 0 is the electric charge. The first and second lines are the nonabelian correction to
the Thouless pumping in the presence of the mixed-space Berry curvatures. The nonabelian
generalizationoftheThoulesspumping[8]contributesonlytothespinpumping(Seebelow).
The third line is the nonabelian chiral magnetic effect. There are two contributions: One is
the nonabelian generalization of the standard chiral magnetic effect, which survives even in
the absence of the time-dependent perturbation. The other is induced by external AC fields,
and understood as the spectral flow in real space. According to the axial anomaly equation,
inthepresenceofnonzeroE·B orEaˆ·Baˆ, theelectricchargeispumpedinmomentumspace
by the magnitude of (E ·B)b or (Eaˆ ·Baˆ)b, which is the conventional spectral flow. Since
our kinetic theory is completely symmetric in phase space, in the presence of nonzero e·b or
eaˆ·baˆ, the spectral flow arises in real space, and the electric charge is pumped in real space
by the magnitude of (e·b)B or (eaˆ·baˆ)B. The last line gives the nonabelian anomalous Hall
effect, and the correction in the presence of the mixed-space Berry curvatures. We remark
here that there also exists the term, which comes from the anomalous velocity due to the
Zeeman energy shift in the nonabelian magnetic field, jˆ0(t) = (cid:82) d3p/(2π)3maˆ·Baˆ(t)∂ n(p),
i pi
where maˆ is the nonabelian magnetic moment. This is the nonabelian generalization of the
gyrotropic magnetic effect [63]. However, the gyrotropic magnetic effect induces only the
AC current, so that it vanishes in Eq. (12).
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Next the adiabatic spin pumping is given as
(cid:90) dtd3p
Paˆ(x ) =2λ jaˆ(t,x,p)
i i (2π)3 i
(cid:90) dtd3p(cid:104) 1 1 (cid:16) (cid:17)
=2λ − eaˆ + δ Ωaˆ −Ωaˆ ∂ ε
(2π)3 2 i 2 ij xkpk pixj pj
3 (cid:16) (cid:17)
− (cid:15) (cid:15) Ωaˆ Ωˆb eˆb +Ωˆb Ωaˆ eˆb +Ωˆb Ωˆb eaˆ
40 ikl jm¯n¯ pm¯xk pn¯xl j pm¯xk pn¯xl j pm¯xk pn¯xl j
1 3 (cid:16) (cid:17)
+ eB baˆ ·∇ ε+ Baˆeˆb ·bˆb +Bˆb eaˆ ·bˆb +Bˆb eˆb ·baˆ
2 i p 20 i i i
1 3 (cid:16) (cid:17)(cid:105)
+ (cid:15) baˆ eE − (cid:15) Ωaˆ bˆb Eˆb +Ωˆb baˆ Eˆb +Ωˆb bˆb Eaˆ n(p), (13)
2 ijk k j 20 ijk pk¯xk k¯ j pk¯xk k¯ j pk¯xk k¯ j
where λ is the total angular momentum of the degenerate Bloch states [50, 51]. The first
and second lines are the nonabelian Thouless pumping and the correction in the presence of
the mixed-space Berry curvatures. The third line is the spin analogue of the chiral magnetic
effect. Only the first term persists in the absence of the external AC fields. The last line
gives the spin Hall effect, and the correction by the mixed-space Berry curvatures. In fact
the first term reproduces the semiclassical result obtained in Refs. [50, 51] in the absence of
the time-dependent perturbation.
Topological effective field theory. We here show that the abelian phase-space Chern-
SimonstheoryderivedinRefs.[43,44]canbegeneralizedtothenonabelianBerrycurvatures.
WeintroducetheChern-SimonsLagrangiandensityL asthe(1+2d)-formdefinedthrough
CS
(−1)ν
dL = c ωνˆ1 ∧···∧ωνˆd+1, (14)
CS (2π)d(d+1)! νˆ1...νˆd+1
whereωνˆ = ωνˆ dξ ∧dξ /2 = dAνˆ+fνˆµˆλˆAµˆ∧Aλˆ/2withAνˆ = Aνˆ dξ . Wenotethatc
αβ α β α α νˆ1...νˆd+1
issymmetricandinvariantundertheadjointactionoftheLiegroup. ThentheChern-Simons
(cid:82)
action is given as S = L , where M is the entire manifold of phase space and time.
CS M CS
Although the Chern-Simons action itself has a complicated form for the nonabelian Berry
curvatures in higher dimensions, the Chern-Simon current (i.e., the equation of motion)
obtained from the variation of S with respect to Aνˆ has a simple form (See e.g., Ref [64]):
CS µ
∂S
jνˆ = CS
µ ∂Aνˆ
µ
(−1)ν
= c (cid:15) ωνˆ1 ···ωνˆd , (15)
(2π)d2dd! νˆνˆ1...νˆd µµ1...µ2d µ1µ2 µ2d−1µ2d
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which recovers Eqs. (10) with n(t,ξ ) = 1 and (11). We find that the transports obtained
a
from the kinetic theory can generally be expressed by the Chern-Simons current (15). We
here comment on the semiclassical approximation with respect to qaˆ. We treat qaˆ as the
commutable variables, by considering the (cid:126) → 0 limit with qaˆ ∼= (cid:126)taˆ fixed in [(cid:126)taˆ,(cid:126)tˆb] =
i(cid:126)faˆˆbcˆ(cid:126)tcˆ, where (cid:126) is the Planck constant. Because of the classical limit (commutable qaˆ ∼=
(cid:126)taˆ), the numerical factors of the Chern-Simons current (15) might not be correct except
for the one in which c is the constant of motion, and corresponds to the Casimir
νˆ1...νˆd+1
invariant up to the normalization factor. From the comparison with the effective theory
of the higher dimensional quantum Hall effect [40], we infer that the full quantum results
are recovered only by the modification of c in Eqs. (14) and (15) as c →
νˆ1...νˆd+1 νˆ1...νˆd+1
(cid:80) tr(cid:2)tνˆ1···tνˆd+1(cid:3)/(d+1)!. This discrepancy between the classical and quantum results
perm
wouldberesolvedbythecalculationbasedontheWignerfunctionformalismwithaccurately
treating the star products of qaˆ.
Concluding remarks. We have constructed the kinetic theory when all abelian and non-
abelian phase-space Berry curvatures in 1 + 2d dimensions are nonzero, by generalizing
the one in the presence of the nonabelian gauge fields in real space to incorporate the in-
teractions with the nonabelian gauge fields (Berry curvatures) in phase space. Then we
have calculated anomalous transports induced by the Berry curvatures. To show its utility,
we have studied the anomalous transports induced by the SU(2) Berry curvatures such as
the nonabelian generalization of the adiabatic charge pumping, the chiral magnetic effect
and the anomalous Hall effect, and the spin analogue of them. The transports given in
Eqs. (12) and (13) would arise in materials whose low-energy dynamics is described by e.g.,
the Luttinger model, which has the nonzero SU(2) Berry curvatures.
We have also derived the effective theory to reproduce the transports in insulators calcu-
lated from the kinetic theory. We have shown that such an effective theory is the nonabelian
generalization of the phase space Chern-Simons theory given in Ref. [43, 44]. We have rev-
eled the strong connection between the kinetic theory and the Chern-Simons theory, which
is independent of spatial dimensions or the details of Berry curvatures. Our kinetic theory
providestheframeworktostudyhowthetopologicalelectromagneticresponsesininsulators,
which are usually analyzed by using the topological field theory, are modified in metals.
There are several future applications of our work. One is a generalization to the nonequi-
librium distribution. We can calculate the anomalous transports in a steady state or in
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a non-adiabatic process by using the relaxation time approximation. It is also interesting
to consider the color-dependent distribution. Another is to estimate the magnitude of the
anomalous transports by using some model such as the Luttinger model. Although the ad-
vantage of our analysis is to predict the existence of the anomalous transports in a model
independent way, it is also important to predict the magnitude of them for the implication
to experiments. Those analysis are left for future studies.
ThisworkwassupportedbyJSPSGrant-in-AidforScientificResearch(No: JP16J02240).
This work was also partially supported by JSPS KAKENHI Grants Numbers 15H03652,
16K17716 and the RIKEN interdisciplinary Theoretical Science (iTHES) project.
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