# Chiral anomaly in non-relativistic systems: Berry curvature and chiral
kinetic theory

Lan-Lan Gao  Physics Department and Center for Particle Physics and Field
Theory, Fudan University, Shanghai 200438, China  Xu-Guang Huang
<EMAIL_ADDRESS>Physics Department and Center for Particle Physics
and Field Theory, Fudan University, Shanghai 200438, China  Key Laboratory of
Nuclear Physics and Ion-beam Application (MOE), Fudan University, Shanghai
200433, China

###### Abstract

Chiral anomaly and the novel quantum phenomena it induces have been widely
studied for Dirac and Weyl fermions. In most typical cases, the Lorentz
covariance is assumed and thus the linear dispersion relations are maintained.
However, in realistic materials, such as Dirac and Weyl semimetals, the non-
linear dispersion relations appear naturally. We develop a kinetic framework
to study the chiral anomaly for Weyl fermions with non-linear dispersions by
using the methods of Wigner function and semi-classical equations of motion.
In this framework, the chiral anomaly is sourced by Berry monopoles in
momentum space and could be enhanced or suppressed due to the windings around
the Berry monopoles. Our results can help understand the chiral anomaly-
induced transport phenomena in non-relativistic systems.

###### pacs:

11.30.Rd, 72.10.Bg

##  I Introduction

During the development of condensed matter physics, the concepts and theories
proficient in describing relativistic particles have been often found very
useful. Among them, in recent years, the chiral anomaly which was first
encountered in particle physics is particularly eye-catching in the studies of
three-dimensional topological materials such as Weyl and Dirac semimetals (Yan
and Felser, 2017; Armitage  _et al._ , 2018; Lv  _et al._ , 2021). In Weyl
semimetals, the conduction and valence bands touch at some points (called Weyl
nodes) in the Brillouin zone. The Weyl nodes always appear in pairs with
opposite chirality around which the low-energy excitations have linear
dispersion relations and can be described by Weyl Hamiltonian. In Dirac
semimetals, the pairs of Weyl nodes of opposite chirality overlap to form
Dirac nodes. The Weyl and Dirac semimetals exhibit a number of novel transport
phenomena such as the chiral magnetic effect (CME) Kharzeev  _et al._ (2008);
Fukushima  _et al._ (2008) which is a consequence of the chiral anomaly and
induces special magnetoresistance Li  _et al._ (2016); Huang  _et al._ (2015);
Xiong  _et al._ (2015); Zhang  _et al._ (2016), and novel modification of
Casimir effect Fukushima  _et al._ (2019); Rong  _et al._ (2021).
Interestingly, the CME has also been widely discussed in particle and nuclear
physics and is under intensive search in relativistic heavy-ion collision
experiments Kharzeev  _et al._ (2016); Huang (2016a); Hattori and Huang
(2017); Kharzeev and Liao (2021).

Although the Weyl semimetals are considered to be described by Weyl fermions,
the realistic excitations in solids can be different from the ideal Weyl
Hamiltonian $\propto\bm{\sigma}\cdot{\bm{p}}$ with $\bm{\sigma}$ the Pauli
matrices and ${\bm{p}}$ the momentum away from the Weyl node. Their dispersion
relations typically have tiltings, warpings, and non-linear momentum
dependence, which are absent in the ideal Weyl Hamiltonian. These new features
may be modeled by an extended Weyl Hamiltonian Fang  _et al._ (2012);
Soluyanov  _et al._ (2015); Sun  _et al._ (2015); Huang  _et al._ (2017)

$H=\bm{K}({\bm{p}})\cdot\bm{\sigma}+K_{0}({\bm{p}})$ (1)

with $K_{0}$ and $\bm{K}$ polynomials of ${\bm{p}}$. A number of interesting
phenomena due to the tiltings, warpings, and non-linearity of the Weyl nodes
are discussed including, e.g., the modified anomalous Hall effect Zyuzin and
Tiwari (2016); Nandy  _et al._ (2017) and the enhanced CME Sharma  _et al._
(2017); Yu  _et al._ (2016); Wei  _et al._ (2018).

Moreover, generalizing the ideal Weyl Hamiltonian to include non-linear
momentum dependence is also natural from the view point of low-energy
effective theory. The band structure of a real material is usual very complex.
Even for the usual Weyl semimetals, the linear Weyl Hamiltonian is considered
as a lowest-order effective description valid only near the Weyl nodes. To
describe physics away from the Weyl nodes, terms quadratic or even higher-
power in momentum need to be included.

In this paper, we will first focus on a simple example of the Hamiltonian (1),
given by

$H=\lambda\bm{\sigma}\cdot\bm{p}+\alpha\bm{p}^{2},$ (2)

where $\lambda=\pm$ denotes the chirality of the Weyl node and the parameter
$\alpha$ depends on the band structure of the material which is assumed to be
positive to guarantee the stability of the system. Our purpose is to develop a
chiral kinetic theory (CKT) from quantum field theory to describe the semi-
classical physics of Hamiltonian (2) and to reveal how the chiral anomaly is
self-consistently encoded in such a framework. The underlying methodology is
parallel to its relativistic counterpart for relativistic Dirac or Weyl
fermions that has been intensively discussed over the past few years Gao  _et
al._ (2012); Chen  _et al._ (2013); Huang  _et al._ (2018); Gao  _et al._
(2018); Hidaka  _et al._ (2017); Carignano  _et al._ (2018); Liu  _et al._
(2019); Sheng  _et al._ (2020); Huang  _et al._ (2020); Hattori  _et al._
(2021); Chen and Lin (2021). We then will extend the discussion to the more
general Hamiltonian (1).

The CKT for relativistic systems consists of a set of Lorentz covariant semi-
classical kinetic equations that can well describe the anomalous transport
effects like the CME and chiral vortical effect (CVE) Liu  _et al._ (2019);
Stephanov and Yin (2012); Chen  _et al._ (2014); Huang and Sadofyev (2019). It
also provides a convenient tool to explore the topological characteristics of
the Dirac or Weyl fermions through the effects of Berry curvature in phase
space. For example, the effects of external magnetic or gravitational field
and the transports of Dirac fermions in curved space-time can be well
expressed through the chiral kinetic theory Liu  _et al._ (2019); Chen  _et
al._ (2014); Huang and Sadofyev (2019). The relativistic CKT has also been
applied to condensed matter systems to study Berry-curvature induced transport
phenomena Gorbar  _et al._ (2017); Landsteiner (2014); Gao  _et al._ (2021).

This paper is organized as follows. In section II , We will derive the CKT for
non-relativistic Weyl fermions of type (2) by using the Wigner function
method. We will show that, at the $\hbar$ order (corresponding to one-loop
approximation in field-theory calculation), the divergence of the chiral
current receives a contribution from the singularity in phase space residing
exactly at the Weyl node and gives the chiral anomaly in the same form as
derived from relativistic field theory. In section III, we will directly
derive the semi-classical equations of motion for Weyl fermions of type (1)
and then check the anomaly relation. In section IV, we give comments on our
results and discuss potential future application and extension of our results.
Throughout this paper, we choose the Fermi velocity $v_{F}=1$ and the coupling
constant $e=1$. To simplify the notations, we will use the four-vector
representation like $x^{\mu}=(x_{0}=t,{\bm{x}})$ with index $\mu$ from $0$ to
$3$, $p^{\mu}=(p_{0},{\bm{p}})$, $p\cdot x=p_{0}t-{\bm{p}}\cdot{\bm{x}}$, and
use $\eta^{\mu\nu}=\eta_{\mu\nu}={\rm diag}(1,-1,-1,-1)$ to raise and lower
indices though no Lorentz invariance is assumed.

##  II Wigner function and chiral anomaly

The Wigner function $W(x,p)$ is a generalization of the classical phase-space
distribution function to quantum physics. It is defined as the ensemble
average of Wigner operator given by

$\begin{split}\hat{W}(x,p)=\int d^{4}ye^{-ip\cdot
y/\hbar}{\hat{\psi}}^{\dagger}(x)e^{y\cdot\overleftarrow{D}^{*}/2}\otimes
e^{-y\cdot D/2}\hat{\psi}(x),\end{split}$ (3)

where $\hat{\psi}(x)$ is a Weyl spinor operator and
$[\hat{\psi}^{\dagger}\otimes\hat{\psi}]_{ab}=\hat{\psi}^{\dagger}_{b}\hat{\psi}_{a}$
with $a,b=1$-$2$ being the spinor indices. Note that we have considered the
presence of an external electromagnetic field $A_{\mu}$ so that $D_{\mu}$ is
the covariant derivative acting on the Weyl spinor:

$D_{\mu}\hat{\psi}(x)=\Bigl{(}\partial_{\mu}+iA_{\mu}/\hbar\Bigr{)}\hat{\psi}(x)\,.$
(4)

The Wigner function is given by $W(x,p)=\langle\hat{W}(x,p)\rangle$ where
$\langle\cdots\rangle$ denotes the ensemble average. The equations of motion
(Heisenberg equations) of $\hat{\psi}$ and $\hat{\psi}^{\dagger}$ for
Hamiltonian (2) are given by:

$\displaystyle
iD_{t}\hat{\psi}+\hbar\alpha{\bm{D}}^{2}\hat{\psi}+i\lambda{\bm{\sigma}}\cdot{\bm{D}}\hat{\psi}=0,$
(5) $\displaystyle
iD^{*}_{t}\hat{\psi}^{\dagger}-\hbar\alpha{\bm{D}}^{*2}\hat{\psi}^{\dagger}+i\lambda{\bm{D}}^{*}\hat{\psi}^{\dagger}\cdot{\bm{\sigma}}=0.$
(6)

We will consider $\hbar$ as being much smaller than the classical action of
the system and thus make an expansion in $\hbar$. Combining Eqs. (5)-(6) and
the definition (3) for Wigner operator, we derive the equation of motion for
Wigner function accurate at $O(\hbar^{2})$ as

$\begin{split}&\bigg{[}\frac{i\hbar}{2}\Delta_{t}+p_{0}+\lambda\sigma^{i}\left(\frac{i\hbar}{2}\Delta_{i}+p_{i}\right)\\\
&\quad\quad\quad-\alpha\left(\frac{i\hbar}{2}\Delta_{i}+p_{i}\right)\left(\frac{i\hbar}{2}\Delta_{i}+p_{i}\right)\bigg{]}W=0,\end{split}$
(7)

where $\Delta_{\mu}=\partial_{\mu}-F_{\mu\nu}\partial_{p}^{\nu}$ (Here,
$\Delta_{0}=\Delta_{t}$) with
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ the field strength,
$\partial^{\mu}_{p}=\partial/\partial p_{\mu}$, and repeated indices are
summed. In deriving this equation, we have made an expansion in $\hbar$ with
the power counting rule: $p^{\mu}\sim O(1),A_{\mu}\sim O(1)$, and $y^{\mu}\sim
i\hbar\partial^{\mu}_{p}\sim O(\hbar)$, and have used the following equations
which are accurate up to $O(\hbar)$:

$\displaystyle[D_{\mu},e^{-\frac{y\cdot
D}{2}}]=-\frac{iy^{\nu}}{2\hbar}\left(F_{\mu\nu}-\frac{y^{\rho}}{4}\partial_{\rho}F_{\mu\nu}\right)e^{-\frac{y\cdot
D}{2}}+O(\hbar^{2})\,,$ $\displaystyle[\partial_{\mu}^{y},e^{-\frac{y\cdot
D}{2}}]=-\frac{1}{2}e^{-\frac{y\cdot
D}{2}}\left(D_{\mu}-\frac{iy^{\nu}}{4\hbar}F_{\mu\nu}+\frac{iy^{\nu}y^{\rho}}{12\hbar}\partial_{\rho}F_{\mu\nu}\right)$
$\displaystyle\mkern-120.0mu+O(\hbar^{2})\,.$

To reveal the physical content in Eq. (LABEL:eq:Weq-full), we expand the
Wigner function in $su(2)$ algebra,
$W=\frac{1}{2}(\mathcal{V}_{0}+\lambda\sigma_{i}\mathcal{V}_{i})$. Note that
the Wigner function is Hermitian so that the coefficients $\mathcal{V}_{0}$
and $\mathcal{V}_{i}$ are real. The physical meanings of $\mathcal{V}_{0}$ and
$\mathcal{V}_{i}$ become clear if we express them as
$\mathcal{V}_{0}(x,p)={\rm Tr}\,W(x,p)$ and $\bm{\mathcal{V}}(x,p)=\lambda{\rm
Tr}\,[\bm{\sigma}W(x,p)]$. After integration over momentum,
$\mathcal{V}_{0}(x)=\int\frac{d^{4}p}{(2\pi)^{4}}\mathcal{V}_{0}(x,p)=\langle\hat{\psi}^{\dagger}(x)\hat{\psi}(x)\rangle$
111In principle, the momentum-space measure should be
$d^{4}p/(2\pi\hbar)^{4}$, but the $\hbar$ here does not influence any of our
discussions, so we omit it. and
$\bm{\mathcal{V}}(x)=\lambda\langle\hat{\psi}^{\dagger}(x)\bm{\sigma}\hat{\psi}(x)\rangle$.
Thus $\mathcal{V}_{0}(x,p)$ is considered as the phase-space particle number
distribution and $\bm{\mathcal{V}}(x,p)$ can be considered as (part of) the
particle current (See below) or spin distribution in phase space. These
coefficients satisfy the following four equations which can be easily
extracted from Eq. (LABEL:eq:Weq-full):

$\displaystyle\left(p_{0}-\alpha{\bm{p}}^{2}+\alpha\frac{\hbar^{2}}{4}\bm{\Delta}^{2}\right)\mathcal{V}^{i}-p^{i}\mathcal{V}_{0}-\lambda\frac{\hbar}{2}\epsilon^{ijk}\Delta_{j}\mathcal{V}^{k}=0,\;\;\;\;\;\;\;$
(8) $\displaystyle\frac{\hbar}{2}\left(\Delta_{t}\mathcal{V}^{i}-2\alpha
p_{j}\Delta_{j}\mathcal{V}^{i}+\Delta_{i}\mathcal{V}_{0}\right)+\lambda\epsilon^{ijk}p_{j}\mathcal{V}^{k}=0,$
(9)
$\displaystyle\left(p_{0}-\alpha{\bm{p}}^{2}+\alpha\frac{\hbar^{2}}{4}\bm{\Delta}^{2}\right)\mathcal{V}_{0}+p_{i}\mathcal{V}^{i}=0,$
(10) $\displaystyle(\Delta_{t}-2\alpha
p_{j}\Delta_{j})\mathcal{V}_{0}+\Delta_{i}\mathcal{V}^{i}=0.$ (11)

We now solve Eqs. (8)-(11) order by order in $\hbar$. At zeroth order (namely,
the classical level), combining Eqs. (8)-(10), one easily finds that

$\displaystyle\mathcal{V}^{(0)}_{0}$ $\displaystyle=$ $\displaystyle
4\pi\tilde{p}_{0}f^{(0)}(x,p)\delta(\tilde{p}^{2}),$ (12)
$\displaystyle\mathcal{V}^{(0)}_{i}$ $\displaystyle=$ $\displaystyle
4\pi\tilde{p}_{i}f^{(0)}(x,p)\delta(\tilde{p}^{2}),$ (13)

where we have introduced the shorthand notation
$\tilde{p}^{\mu}=(\tilde{p}^{0},\tilde{{\bm{p}}})$ with
$\tilde{p}^{0}=\tilde{p}_{0}=p_{0}-\alpha{\bm{p}}^{2}$,
$\tilde{p}^{i}=-\tilde{p}_{i}=p^{i}$, and
$\tilde{p}^{2}=\tilde{p}^{2}_{0}-\tilde{{\bm{p}}}^{2}$. The function $f^{(0)}$
represents the classical distribution function in phase space Elze  _et al._
(1986); Vasak  _et al._ (1987). Substituting $\mathcal{V}^{(0)}_{0}$ and
$\mathcal{V}^{(0)}_{i}$ into Eq. (11), we obtain the classical (collisionless)
kinetic equation for $f^{(0)}$:

$\delta(\tilde{p}^{2})\left[(p_{0}-\alpha{\bm{p}}^{2})(\Delta_{t}+2\alpha{\bm{p}}\cdot\bm{\Delta})+{\bm{p}}\cdot\bm{\Delta}\right]f^{(0)}=0.$
(14)

To proceed to $O(\hbar)$ order, it is convenient to re-write Eqs. (8)-(11) at
$O(\hbar)$ order in a formally covariant form:

$\displaystyle\tilde{p}_{\mu}\mathcal{V}_{\nu}-\tilde{p}_{\nu}\mathcal{V}_{\mu}+\lambda\frac{\hbar}{2}\epsilon_{\mu\nu\rho\sigma}\tilde{\Delta}^{\rho}\mathcal{V}^{\sigma}=0,$
(15) $\displaystyle\tilde{p}\cdot\mathcal{V}=0,$ (16)
$\displaystyle\tilde{\Delta}\cdot\mathcal{V}=0,$ (17)

where the four-vectors are defined by
$\tilde{\Delta}_{\mu}=(\Delta_{t}-2\alpha p_{j}\Delta_{j},\Delta_{i})$ and
$\mathcal{V}^{\mu}=({\mathcal{V}_{0},\mathcal{V}^{i}})$, and the four-
dimensional Levi-Civita symbol is normalized to
$\epsilon^{0ijk}=-\epsilon_{0ijk}=\epsilon^{ijk}$. The contraction is
performed with metric $\eta_{\mu\nu}$, e.g.,
$\tilde{p}\cdot\mathcal{V}=\eta_{\mu\nu}\tilde{p}^{\mu}\mathcal{V}^{\nu}=\tilde{p}_{0}\mathcal{V}_{0}-\tilde{\bm{p}}\cdot\bm{\mathcal{V}}$.
This set of equations are in the same form as their counterparts in
relativistic CKT and thus can be solved in the same way as stressed in Refs.
Hidaka  _et al._ (2017); Liu  _et al._ (2019); Huang  _et al._ (2018).
Multiplying Eq. (15) by an arbitrary vector $n^{\mu}$ satisfying
$n^{\mu}n_{\mu}=1$ and $n\cdot\tilde{p}\neq 0$ and combining it with the
zeroth order results given in Eqs. (12)-(13) and the constraint (16), we
finally obtain

$\displaystyle\mathcal{V}^{\mu}=4\pi\delta(\tilde{p}^{2})\left[\tilde{p}^{\mu}-\lambda\frac{\hbar}{\tilde{p}^{2}}\tilde{G}^{\mu\nu}\tilde{p}_{\nu}+\lambda\hbar\Sigma^{\mu\nu}\tilde{\Delta}_{\nu}\right]f+O(\hbar^{2}),$

where
$\Sigma_{\mu\nu}=\epsilon_{\mu\nu\rho\sigma}\tilde{p}^{\rho}n^{\sigma}/(2\tilde{p}\cdot
n)$ is called the spin tensor in a frame specified by vector $n^{\mu}$,
$f=f^{(0)}+f^{(1)}$ is the distribution function up to order $O(\hbar)$, and
$\tilde{G}^{\mu\nu}=(1/2)\epsilon^{\mu\nu\rho\sigma}G_{\rho\sigma}$ is the
dual tensor of $G_{\mu\nu}$ which is defined by
$G_{\mu\nu}=-\tilde{\Delta}_{\mu}\tilde{p}_{\nu}$ and related to the field
strength tensor via $G_{0i}=-G_{i0}=F_{0i}-2\alpha
p_{j}F_{ji}=E^{i}+2\alpha\epsilon^{ijk}p^{j}B^{k}$ and
$G_{ij}=F_{ij}=-\epsilon^{ijk}B^{k}$ with $\bm{E},\bm{B}$ the electric and
magnetic fields. Substituting Eq. (II) into Eq. (17), we obtain the
$O(\hbar)$-order kinetic equation for $f$:

$\begin{split}&\delta(\tilde{p}^{2}-\lambda\hbar\Sigma_{\alpha\beta}G^{\alpha\beta})\bigg{[}\tilde{p}\cdot\tilde{\Delta}\\\
&\quad\quad+\lambda\hbar\left(\frac{n_{\sigma}}{\tilde{p}\cdot
n}\tilde{G}^{\sigma\nu}\tilde{\Delta}_{\nu}+\tilde{\Delta}_{\mu}\Sigma^{\mu\nu}\tilde{\Delta}_{\nu}\right)\bigg{]}f(t,{\bm{x}},p_{0},{\bm{p}})=0.\end{split}$
(19)

To derive this equation, we have used the following relations:
$\delta^{\prime}(x)=-\delta(x)/x$,
$\delta^{\prime\prime}(x)=-2\delta^{\prime}(x)/x$,
$\tilde{\Delta}_{\mu}\tilde{G}^{\mu\nu}=0$, and
$\tilde{G}^{\mu\rho}G_{\mu\sigma}\tilde{p}_{\rho}\tilde{p}^{\sigma}=\tilde{G}^{\alpha\beta}G_{\alpha\beta}\tilde{p}^{2}/4$
which can be proven using the Schouten identity
$\epsilon^{[\mu\nu\rho\sigma}\tilde{p}^{\lambda]}=0$ with $[\cdots]$ meaning
the anti-symmetrization operation. The form of Eq. (19) shows clearly
similarity with its relativistic counterpart in, e.g., Ref. Liu  _et al._
(2019).

To see the physical content in Eq. (19) more clearly, it is instructive to
choose $n^{\mu}=(1,0,0,0)$ (the laboratory frame). The Dirac $\delta$ function
in Eq. (19) gives the on-shell condition (dispersion relation) for the
fermions in which the term $-\lambda\hbar\Sigma\cdot G$ expresses a quantum
correction due to the coupling between the magnetic moment and magnetic field:

$\displaystyle
p_{0}=\varepsilon_{p}=\pm|{\bm{p}}|(1\mp\hbar\bm{\Omega}_{\lambda}\cdot\bm{B})+\alpha{\bm{p}}^{2},$
(20)

where the upper (lower) sign corresponds to conduction (valance) band of
Hamiltonian (2). In the following, we focus on the upper sign. The magnetic
moment is determined by a Berry curvature

$\displaystyle\bm{\Omega}_{\lambda}=\lambda\frac{{\bm{p}}}{2|{\bm{p}}|^{3}}$
(21)

of Berry monopole residing at ${\bm{p}}=\bm{0}$ with charge $\lambda$
(chirality):
$\bm{\nabla}_{p}\cdot\bm{\Omega}_{\lambda}=2\pi\lambda\delta^{(3)}({\bm{p}})$.
Our semi-classical scheme is meaningful only when the quantum correction does
not break the band gap $|{\bm{p}}|$; this requires
$\hbar|\bm{\Omega}_{\lambda}\cdot\bm{B}|\ll 1$ leading to
$|{\bm{p}}|\gg\sqrt{\hbar|\bm{B}|}$ (the adiabatic condition) at which our
kinetic description applies. Integrating Eq. (19) over $\tilde{p}_{0}$ from
$0$ to $\infty$, we pickup the contribution of conduction fermions and, after
a tedious calculation, find

$\begin{split}&\Big{\\{}\sqrt{G}\partial_{t}+\left[\bm{v}_{p}+\hbar(\bm{v}_{p}\cdot\bm{\Omega}_{\lambda})\bm{B}+\hbar\tilde{\bm{E}}\times\bm{\Omega}_{\lambda}\right]\cdot\bm{\nabla}_{x}\\\
&\quad+\left[\tilde{\bm{E}}+\bm{v}_{p}\times\bm{B}+\hbar(\tilde{\bm{E}}\cdot\bm{B})\bm{\Omega}_{\lambda}\right]\cdot\bm{\nabla}_{p}\Big{\\}}f(t,{\bm{x}},{\bm{p}})=0,\end{split}$
(22)

where $f(t,{\bm{x}},{\bm{p}})=f(t,{\bm{x}},p_{0}=\varepsilon_{p},{\bm{p}})$,
$\sqrt{G}=1+\hbar\bm{\Omega}_{\lambda}\cdot\bm{B}$ is the quantum corrected
measure of phase space (see below),
$\bm{v}_{p}=\bm{\nabla}_{p}\varepsilon_{p}$ is the single-particle velocity,
and $\tilde{\bm{E}}=\bm{E}-\bm{\nabla}_{x}\varepsilon_{p}$ is a modified
electric field. To derive this equation, we have neglected all the
$O(\hbar^{2})$ terms and used the relations (Note that $\partial_{\mu}$ and
$\partial^{i}_{p}$ on the left-hand and right-hand sides act on different
arguments.):
$\partial_{\mu}f(t,{\bm{x}},{\bm{p}})=\\{[\partial_{\mu}+(\partial_{\mu}\varepsilon_{p})\partial^{0}_{p}]f(t,{\bm{x}},p_{0},{\bm{p}})\\}|_{p_{0}=\varepsilon_{p}}$
and
$\partial^{i}_{p}f(t,{\bm{x}},{\bm{p}})=\\{[\partial^{i}_{p}+(\partial^{i}_{p}\varepsilon_{p})\partial^{0}_{p}]f(t,{\bm{x}},p_{0},{\bm{p}})\\}|_{p_{0}=\varepsilon_{p}}$.
The kinetic equation for valance fermions can be similarly obtained and the
result is in the same form of Eq. (22) but with $\varepsilon_{p}$ understood
as being given by the lower sign in Eq. (20) and $\lambda$ replaced by
$-\lambda$ in all the other terms.

By comparing Eq. (22) with the standard form
$(\partial_{t}+\dot{{\bm{x}}}\cdot\bm{\nabla}_{x}+\dot{{\bm{p}}}\cdot\bm{\nabla}_{p})f=0$
of Boltzmann equation, we extract the single-particle equations of motion:

$\displaystyle\sqrt{G}\dot{{\bm{x}}}$ $\displaystyle=$
$\displaystyle\bm{v}_{p}+\hbar(\bm{v}_{p}\cdot\bm{\Omega}_{\lambda})\bm{B}+\hbar\tilde{\bm{E}}\times\bm{\Omega}_{\lambda},$
(23) $\displaystyle\sqrt{G}\dot{{\bm{p}}}$ $\displaystyle=$
$\displaystyle\tilde{\bm{E}}+\bm{v}_{p}\times\bm{B}+\hbar(\tilde{\bm{E}}\cdot\bm{B})\bm{\Omega}_{\lambda}.$
(24)

Similar set of equations of motion has been derived repeatedly in literature
in different contexts (though some of them are not complete at $\hbar$ order)
for, e.g., band electrons in solids Sundaram and Niu (1999); Xiao  _et al._
(2010, 2005); Son and Yamamoto (2012), trapped cold atoms Huang (2016b),
neutrinos in supernovae Yamamoto (2016), quarks in quark-gluon plasma
Stephanov and Yin (2012); Son and Yamamoto (2013); Chen  _et al._ (2013), and
photons Bliokh and Bliokh (2004); Onoda  _et al._ (2004); Yamamoto (2017);
Huang and Sadofyev (2019). A direct calculation leads to

$\displaystyle\partial_{t}\sqrt{G}+\bm{\nabla}_{x}\cdot(\sqrt{G}\dot{{\bm{x}}})+\bm{\nabla}_{p}\cdot(\sqrt{G}\dot{{\bm{p}}})=2\pi\lambda\hbar\delta^{(3)}({\bm{p}})\bm{E}\cdot\bm{B},$

where we have used the Maxwell equations $\bm{\nabla}_{x}\cdot\bm{B}=0$ and
$\partial_{t}\bm{B}=-\bm{\nabla}_{x}\times\bm{E}$ and have omitted
$O(\hbar^{2})$ terms. This identifies $\sqrt{G}$ as the invariant phase-space
measure except for a singularity at ${\bm{p}}=\bm{0}$ at which a Berry
monopole resides. This Berry monopole contributes anomalous velocity which
could lead to, for example, CME or anomalous Hall effect, and anomalous force,
which could lead to chiral anomaly as we show now.

The $U(1)$ Noether current $j^{\mu}=(j^{0},j^{i})$ associated with Hamiltonian
(2) is easily expressed by $\mathcal{V}^{\mu}$,

$\displaystyle j^{0}(x)$ $\displaystyle=$
$\displaystyle\langle\hat{\psi}^{\dagger}(x)\hat{\psi}(x)\rangle=\int\frac{d^{4}p}{(2\pi)^{4}}\mathcal{V}^{0}(x,p),$
(26) $\displaystyle j^{i}(x)$ $\displaystyle=$
$\displaystyle\lambda\langle\hat{\psi}^{\dagger}(x)\sigma^{i}\hat{\psi}(x)\rangle-i\hbar\alpha\langle\hat{\psi}^{\dagger}(x)(D_{i}-\overleftarrow{D}^{*}_{i})\hat{\psi}(x)\rangle$
(27) $\displaystyle=$
$\displaystyle\int\frac{d^{4}p}{(2\pi)^{4}}\left[\mathcal{V}^{i}(x,p)+2\alpha
p^{i}\mathcal{V}^{0}(x,p)\right].$

Substituting Eq. (II) and after a lengthy but straightforward calculation, we
obtain 222Here, we only present the contributions from conduction fermions. If
the contributions of the valance fermions are taken into account, additional
terms should be added to $j^{0}$ and $j^{i}$ with the same forms as Eqs. (28)
and (II) but with a minus sign for the magnetization current.

$\displaystyle j^{0}(x)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}{\bm{p}}}{(2\pi)^{3}}\sqrt{G}f(t,{\bm{x}},{\bm{p}}),$
(28) $\displaystyle\bm{j}(x)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}{\bm{p}}}{(2\pi)^{3}}\sqrt{G}\left[\dot{{\bm{x}}}-\hbar|{\bm{p}}|\bm{\Omega}_{\lambda}\times\bm{\nabla}_{x}\right]f(t,{\bm{x}},{\bm{p}}).$

Note that the second term in Eq. (II) can be written as
$\bm{\nabla}_{x}\times\bm{M}$ with the magnetization
$\bm{M}=\hbar\int\sqrt{G}|{\bm{p}}|\bm{\Omega}_{\lambda}fd^{3}{\bm{p}}/(2\pi)^{3}$
and thus represents a magnetization current. Now, using Eqs. (22) and (II),
one can finds that the $U(1)$ current is not conserved at $O(\hbar)$:

$\partial_{\mu}j^{\mu}=\lambda\frac{\hbar}{4\pi^{2}}\bm{E}\cdot\bm{B}f(t,{\bm{x}},{\bm{p}}=\bm{0}).$
(30)

If the Fermi surface encloses the point ${\bm{p}}=\bm{0}$ so that
$f({\bm{p}}=\bm{0})=1$ at $T=0$, the above relation recover exactly the chiral
anomaly relation. Therefore, in the kinetic framework, the chiral anomaly is
sourced by the Berry monopole at the Weyl node. Further more, although the
presence of $\alpha{\bm{p}}^{2}$ terms in Hamiltonian (2) modifies the single-
particle dispersion relation and thus the kinetic equation, it does not change
the form of chiral anomaly relation as long as the Fermi surface encloses the
Berry monopole.

Before closing this section, we comment on the Wigner-function method in
comparison with the methods in Sundaram and Niu (1999); Xiao  _et al._ (2010,
2005); Son and Yamamoto (2012); Huang (2016b) based on the wave-packet
dynamics of band electrons. Unlike the wave-packet method which can be
considered as a bottom-up treatment starting from the single-particle quantum
mechanics of the band electrons, the starting point of the Wigner-function
method is the underlying quantum field theory. It allows a top-down treatment
of the quantum dynamics in a manner of an expansion in $\hbar$. The kinetic
equation and the corresponding single-particle equations of motion can be
systemcatically obtained to an arbitrary order in $\hbar$ expansion. Moreover,
the Wigner-function method can be easily extended to the study of bosons.

##  III Semi-classical equations of motion and Chiral anomaly

The above discusses focus on Hamiltonian (2), we now consider the more general
Hamiltonian (1). We first derive the semi-classical equations of motion of
single particle for Hamiltonian (1). This can be achieved by multiple methods.
Here, we use the following well established result Stiepan and Teufel (2013):
Suppose $H({\bm{x}},{\bm{k}})=H_{0}({\bm{x}},{\bm{k}})+\hbar
H_{1}({\bm{x}},{\bm{k}})+O(\hbar^{2})$ with $({\bm{x}},{\bm{k}})$ the
canonical pairs of mechanical variables. For each isolated eigenvalue
$h_{0}({\bm{x}},{\bm{k}})$ of $H_{0}({\bm{x}},{\bm{k}})$, there is associated
a classical system with an $\hbar$-dependent Hamilton function
$h({\bm{x}},{\bm{k}})=h_{0}({\bm{x}},{\bm{k}})+\hbar
h_{1}({\bm{x}},{\bm{k}})+O(\hbar^{2})$ and a modified symplectic form
$G({\bm{x}},{\bm{k}})=G_{0}+\hbar\Omega({\bm{x}},{\bm{k}})+O(\hbar^{2})$
($\Omega$ is a generalized Berry curvature), where

$\displaystyle h_{1}$ $\displaystyle=$ $\displaystyle{\rm
tr}\left(H_{1}\pi_{0}\right)-\frac{i}{2}{\rm
tr}\left(\pi_{0}\\{\pi_{0},H_{0}-h_{0}\\}\right),$ (31)
$\displaystyle\Omega_{\alpha\beta}$ $\displaystyle=$ $\displaystyle-i{\rm
tr}\left(\pi_{0}[\partial_{z^{\alpha}}\pi_{0},\partial_{z^{\beta}}\pi_{0}]\right).$
(32)

Here, the trace is over inner space (for our case, the spin) and ${\bm{x}}$
and ${\bm{k}}$ are collectively denoted by $z^{\alpha}$ ($\alpha=1,\cdots,6$).
$\pi_{0}$ is spectral projection to the eigenenergy $h_{0}$, i.e.,
$H_{0}\pi_{0}=h_{0}\pi_{0}$ and $\pi_{0}^{2}=\pi_{0}$. The zeroth order
symplectic form is given by $G_{0}=i\sigma_{y}\otimes{\bf 1}_{3}$. The Poisson
bracket is defined by
$\\{A,B\\}=\sum_{i=1}^{3}(\partial_{k^{i}}A\partial_{x^{i}}B-\partial_{x^{i}}A\partial_{k^{i}}B)$.
The commutator is defined by $[A,B]=AB-BA$. If the Hamiltonian depends on time
explicitly, $H=H(t,{\bm{x}},{\bm{k}})$, we just add the canonical pair $(t,E)$
and apply the previous procedure to a new Hamiltonian
$\tilde{H}(t,{\bm{x}},E,{\bm{k}})=E+H(t,{\bm{x}},{\bm{k}})$. Its spectral
projections $\pi_{0}(t,{\bm{x}},{\bm{k}})$ are independent of $E$ and the
classical Hamilton function is
$h(t,{\bm{x}},E,{\bm{k}})=E+h_{0}(t,{\bm{x}},{\bm{k}})+\hbar
h_{1}(t,{\bm{x}},{\bm{k}})+O(\hbar^{2})$ with symplectic form
$G=G_{0}+\hbar\Omega(t,{\bm{x}},{\bm{p}})$, where $h$ and $\Omega$ are
computed from the instantaneous Hamilton function $H(t,{\bm{x}},{\bm{p}})$ as
before. Note that with symplectic form $G$ the semi-classical equations of
motion are

$\displaystyle G_{\alpha\beta}\dot{z}^{\beta}=-\frac{\partial
h}{\partial{z^{\alpha}}},$ (33)

and the invariant phase-space measure is changed to

$\displaystyle\sqrt{G}\equiv\sqrt{{\rm
det}\;(G_{\alpha\beta})}=1+\frac{\hbar}{2}(\Omega_{x^{i}k^{i}}-\Omega_{k^{i}x^{i}})+O(\hbar^{2}).$

For Hamiltonian (1), $H_{1}=0$, and ${\bm{p}}={\bm{k}}-\bm{A}(t,{\bm{x}})$ is
the kinetic momentum if external electromagnetic field is applied. The
eigenenergy and the corresponding projection are

$\displaystyle h_{0}$ $\displaystyle=$
$\displaystyle\pm|\bm{K}(t,{\bm{x}},{\bm{k}})|+K_{0}(t,{\bm{x}},{\bm{k}})+A_{0}(t,{\bm{x}}),$
(35) $\displaystyle\pi_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(1\pm\hat{\bm{K}}\cdot\bm{\sigma}),$ (36)

where $\hat{\bm{K}}=\bm{K}/|\bm{K}|$. Substituting them into Eqs. (31) and
(32), we find

$\displaystyle h_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{\partial\hat{\bm{K}}}{\partial
k^{i}}\times\frac{\partial\hat{\bm{K}}}{\partial x^{i}}\cdot\bm{K},$ (37)
$\displaystyle\Omega_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\pm\frac{1}{2}\frac{\partial\hat{\bm{K}}}{\partial
z^{\alpha}}\times\frac{\partial\hat{\bm{K}}}{\partial
z^{\beta}}\cdot\hat{\bm{K}},$ (38)

and thus the semi-classical equations of motion read

$\displaystyle\dot{x}^{i}$ $\displaystyle=$
$\displaystyle\left(\delta^{ij}+\hbar\Omega_{k^{i}x^{j}}\right)\frac{\partial
h}{\partial k^{j}}-\hbar\Omega_{k^{i}k^{j}}\frac{\partial h}{\partial
x^{j}}+\hbar\Omega_{k^{i}t},$ (39) $\displaystyle\dot{k}^{i}$ $\displaystyle=$
$\displaystyle\left(-\delta^{ij}+\hbar\Omega_{x^{i}k^{j}}\right)\frac{\partial
h}{\partial x^{j}}-\hbar\Omega_{x^{i}x^{j}}\frac{\partial h}{\partial
k^{j}}-\hbar\Omega_{x^{i}t}.$ (40)

Using ${\bm{x}}$ and ${\bm{p}}$ (the kinetic momentum) as independent variable
is more convenient. In this case, the equations of motion (33) become very
simple and take the same form as Eqs. (23) and (24),

$\displaystyle\sqrt{G}\dot{{\bm{x}}}$ $\displaystyle=$
$\displaystyle\bm{v}_{p}+\hbar(\bm{v}_{p}\cdot\bm{\Omega})\bm{B}+\hbar\tilde{\bm{E}}\times\bm{\Omega},$
(41) $\displaystyle\sqrt{G}\dot{{\bm{p}}}$ $\displaystyle=$
$\displaystyle\tilde{\bm{E}}+\bm{v}_{p}\times\bm{B}+\hbar(\tilde{\bm{E}}\cdot\bm{B})\bm{\Omega}.$
(42)

where $\tilde{\bm{E}}=\bm{E}+\bm{\nabla}_{x}\varepsilon_{p}$ with
$\varepsilon_{p}=\pm|\bm{K}({\bm{p}})|+K_{0}({\bm{p}})+h_{1}({\bm{x}},{\bm{p}})$,
${\bm{v}_{p}}=\bm{\nabla}_{p}\varepsilon_{p}$, and
$\Omega^{i}=\frac{1}{2}\epsilon^{ijk}\Omega_{p^{j}p^{k}}$. With these
equations of motion, the collisionless kinetic equation can be written as
$(\partial_{t}+\dot{{\bm{x}}}\cdot\bm{\nabla}_{x}+\dot{{\bm{p}}}\cdot\bm{\nabla}_{p})f=0$
whose explicit form is same as Eq. (22):

$\begin{split}&\Big{\\{}\sqrt{G}\partial_{t}+\left[\bm{v}_{p}+\hbar(\bm{v}_{p}\cdot\bm{\Omega})\bm{B}+\hbar\tilde{\bm{E}}\times\bm{\Omega}\right]\cdot\bm{\nabla}_{x}\\\
&\quad+\left[\tilde{\bm{E}}+\bm{v}_{p}\times\bm{B}+\hbar(\tilde{\bm{E}}\cdot\bm{B})\bm{\Omega}\right]\cdot\bm{\nabla}_{p}\Big{\\}}f(t,{\bm{x}},{\bm{p}})=0.\end{split}$
(43)

Let us seek for an action in the form $S=\int
dt[U^{i}(t,{\bm{x}},{\bm{p}})\dot{x}^{i}+W^{i}(t,{\bm{x}},{\bm{p}})\dot{p}^{i}-\xi(t,{\bm{x}},{\bm{p}})]$
for Eqs. (41) and (42). Using the least action principle, we determine
$U^{i},W^{i}$, and $\xi$ as $U^{i}=p^{i}+A^{i}(t,{\bm{x}})$, $W^{i}=-\hbar
a^{i}({\bm{p}})$, and $\xi=\varepsilon_{p}+A_{0}(t,{\bm{x}})$:

$\displaystyle S=\int
dt\left[({\bm{p}}+\bm{A}(t,{\bm{x}}))\dot{{\bm{x}}}-\hbar\bm{a}({\bm{p}})\cdot{{\bm{p}}}-\varepsilon_{p}-A_{0}(t,{\bm{x}})\right],$

where $\bm{a}({\bm{p}})$ is determined by the condition
$\bm{\nabla}_{p}\times\bm{a}=\bm{\Omega}$ so that $\bm{a}$ is the Berry
connection. Using $S$ we obtain the number density and current as

$\displaystyle j^{\mu}(x)$ $\displaystyle=$
$\displaystyle-\int\frac{d^{3}\bm{y}d^{3}{\bm{p}}}{(2\pi)^{3}}\sqrt{G}\frac{\delta
S(\bm{y},{\bm{p}})}{\delta A_{\mu}(t,{\bm{x}})}f(t,{\bm{x}},{\bm{p}}).$ (45)

The results are

$\displaystyle j^{0}(x)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}{\bm{p}}}{(2\pi)^{3}}\sqrt{G}f(t,{\bm{x}},{\bm{p}}),$
(46) $\displaystyle\bm{j}(x)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}{\bm{p}}}{(2\pi)^{3}}\sqrt{G}\left[\dot{{\bm{x}}}\mp\hbar|\bm{K}({\bm{p}})|\bm{\Omega}\times\bm{\nabla}_{x}\right]f(t,{\bm{x}},{\bm{p}}),$

where the second term in $\bm{j}$ is because $h_{1}({\bm{x}},{\bm{p}})$
depends on $\bm{A}$. Note the similarity with equations (28) and (II). The
divergence of $j^{\mu}$ reads (omitting $O(\hbar^{2})$ terms)

$\displaystyle\partial_{\mu}j^{\mu}$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}{\bm{p}}}{(2\pi)^{3}}\left[\partial_{t}\sqrt{G}+\bm{\nabla}_{x}\cdot(\sqrt{G}\dot{{\bm{x}}})+\bm{\nabla}_{p}\cdot(\sqrt{G}\dot{{\bm{p}}})\right]f$
(48) $\displaystyle=$
$\displaystyle\hbar\bm{E}\cdot\bm{B}\int\frac{d^{3}{\bm{p}}}{(2\pi)^{3}}\bm{\nabla}_{p}\cdot\bm{\Omega}({\bm{p}})f(t,{\bm{x}},{\bm{p}}),$

where we have used 333In terms of the generalized Berry curvature
$\Omega_{\alpha\beta}$, it can be expressed by
$\frac{\partial\sqrt{G}}{\partial
t}+\frac{\partial\sqrt{G}\dot{x}^{i}}{\partial
x^{i}}+\frac{\partial\sqrt{G}\dot{p}^{i}}{\partial
p^{i}}=\Theta_{p^{i}x^{i}t}+\Theta_{p^{j}x^{j}x^{i}}\partial_{p^{i}}\varepsilon_{p}+\Theta_{x^{j}p^{j}p^{i}}\partial_{x^{i}}\varepsilon_{p}$
where
$\Theta_{\alpha\beta\gamma}=\partial_{\alpha}\Omega_{\beta\gamma}+\partial_{\beta}\Omega_{\gamma\alpha}+\partial_{\gamma}\Omega_{\beta\alpha}$
is the Berry monopole charge function which is the exteriror derivative of
Berry curvature Gao  _et al._ (2021).

$\displaystyle\partial_{t}\sqrt{G}+\bm{\nabla}_{x}\cdot(\sqrt{G}\dot{{\bm{x}}})+\bm{\nabla}_{p}\cdot(\sqrt{G}\dot{{\bm{p}}})$
$\displaystyle=$
$\displaystyle\hbar\bm{\nabla}_{p}\cdot\bm{\Omega}({\bm{p}})\tilde{\bm{E}}\cdot\bm{B},$

which is a direct consequence of Eqs. (41) and (42). The quantity
$\bm{\nabla}_{p}\cdot\bm{\Omega}$ vanishes except for points where
$\bm{\Omega}$ is singular. In fact, substituting the expression for
$\bm{\Omega}$, we have

$\displaystyle\bm{\nabla}_{p}\cdot\bm{\Omega}({\bm{p}})$ $\displaystyle=$
$\displaystyle{\rm det}\left(\frac{\partial K^{a}}{\partial
p^{i}}\right)\bm{\nabla}_{K}\cdot\bm{\Omega}_{K}$ (50) $\displaystyle=$
$\displaystyle\pm 2\pi\,{\rm det}\left(\frac{\partial K^{a}}{\partial
p^{i}}\right)\delta^{(3)}(\bm{K}),$

where the repeated indices all summed over $1-3$ and
$\bm{\Omega}_{K}=\pm\hat{\bm{K}}/(2\bm{K}^{2})$ is the Berry curvature in
$\bm{K}$-space. To derive this relation, we have used the Schouten identity
$\epsilon^{ijk}K^{[a}_{,j}K^{b}_{,k}K^{d}_{,i}\Omega_{K}^{c]}=0$ with
$K^{a}_{,j}=\partial K^{a}/\partial p^{j}$. The determinant in the right-hand
side of Eq. (50) represents the Berry monopole charge in $\bm{K}$-space and is
mathematically a Jacobian for the map from $\bm{p}$-space to $\bm{K}$-space
which makes the integral of $\bm{\nabla}_{p}\cdot\bm{\Omega}$ have clear
topological meaning:

$\displaystyle\frac{1}{2\pi}\int
d^{3}{\bm{p}}\bm{\nabla}_{p}\cdot\bm{\Omega}({\bm{p}})$ $\displaystyle=$
$\displaystyle\pm\sum_{\bm{p}^{*}}N_{{\bm{p}}^{*}},$ (51)

where ${\bm{p}}^{*}$ is the points in ${\bm{p}}$-space where
$\bm{K}({\bm{p}}^{*})=0$ (i.e., the location of Berry monopole in
${\bm{p}}$-space) and $N_{{\bm{p}}^{*}}$ is the winding number of map
${\bm{p}}\rightarrow\bm{K}=\bm{K}({\bm{p}})$ around ${\bm{p}}^{*}$. Collecting
the above results, we obtain the following relation expressing the chiral
anomaly:

$\displaystyle\partial_{\mu}j^{\mu}$ $\displaystyle=$
$\displaystyle\pm\frac{\hbar}{4\pi^{2}}\bm{E}\cdot\bm{B}\sum_{\bm{p}^{*}}N_{{\bm{p}}^{*}}f(t,{\bm{x}},{\bm{p}}^{*}).$
(52)

This extends Eq. (30) to a more general case described by Hamiltonian (1). It
shows that the term $K_{0}$ in Hamiltonian (1) does not change the form of the
anomaly relation and contributes to the chiral anomaly only through the
distribution function $f$. We emphasize that the relation (52) can also be
understood through the index theorem Yee and Yi (2020).

To end this section, we give the expressions for CME of fermions of type (1)
for an equilibrium distribution function which depends only on the energy
$\varepsilon_{p}$, $f=f(\varepsilon_{p})$,

$\displaystyle\bm{j}_{\rm CME}$ $\displaystyle=$
$\displaystyle\mp\frac{\hbar}{4\pi^{2}}\bm{B}\sum_{\bm{p}^{*}}N_{{\bm{p}}^{*}}F(\varepsilon_{{\bm{p}}^{*}}),$
(53)

where $F({\varepsilon_{p}})$ satiesfies
$F^{\prime}(\varepsilon_{p})=f(\varepsilon_{p})$ and the boundary condition
$F(\varepsilon_{p})\rightarrow 0$ for ${\bm{p}}\rightarrow\infty$.

##  IV Summary and outlook

In summary, we have thoroughly examined chiral anomaly for fermions described
by Hamiltonians (1) and (2) via the kinetic approach. For Hamiltonian (2), we
use the Wigner function method and derive the corresponding chiral kinetic
equation up to $O(\hbar)$ order. Comparing to the relativistic case, the only
change is the on-shell condition which constrains also the single-particle
equations of motion and the distribution function. The chiral anomaly is seen
to be sourced by the Berry monopole at ${\bm{p}}=\bm{0}$ and is formally
unchanged comparing to the relativistic case. We then extend the analysis to
the more general Hamiltonian (1) by directly applying the semi-classical
expansion to it. This amounts to map the $O(\hbar)$ order behavior of the
quantum dynamics of a spinful particle to the classical dynamics of a spinless
particle equipped with a modified $\hbar$-dependent Hamilton function and
sympletic form. The Berry monopole charge is given by the winding number of
the map from ${\bm{p}}$-space to $\bm{K}$-space around each Berry monopole.
Due to this, the chiral anomaly relation is correspondly changed [see Eq.
(52)].

From the above analysis, the source of chiral anomaly from the point of view
of kinetic theory is clearly displayed, showing that the chiral anomaly has an
infrared origin stemming from the singularities in phase space (the Berry
monopole). This could be helpful for understanding the related topological
transport phenomena in non-relativistic systems. It would be also interesting
to make a field-theoretical analysis for Hamiltonians (1) and (2) to reveal
the connection of chiral anomaly with ultraviolet divergence in field theory.
Such an analysis has been performed for Hamiltonian (1) without $K_{0}$ term
Yee and Yi (2020) and will be presented for the case with $K_{0}$ in a future
work.

###### Acknowledgements.

The authors thank Sahal Kaushik, Dmitri Kharzeev, Yu-Chen Liu, Satoshi Nawata,
Evan Philip, Xin-Li Sheng, and Yong-Shi Wu for helpful discussions. This work
is supported by NSFC under Grant No. 12075061 and Shanghai NSF under Grant No.
20ZR1404100.

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Special Research Centre for the Subatomic Structure of Matter (CSSM),
Department of Physics, University of Adelaide, Adelaide, South Australia 5005,
Australia School of Physical Science and Technology, Lanzhou University,
Lanzhou 730000, China Research Centre for Hadron and CSR Physics, Lanzhou
University and Institute of Modern Physics of CAS, Lanzhou 730000, China
Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics
of Gansu Province, Key Laboratory of Quantum Theory and Applications of MoE,
and Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou
730000, China Jülich Supercomputing Centre, Forschungszentrum Jülich, D-52428
Jülich, Germany School of Physical Sciences, University of Chinese Academy of
Sciences (UCAS), Beijing 100049, China Southern Center for Nuclear-Science
Theory (SCNT), Institute of Modern Physics, Chinese Academy of Sciences,
Huizhou 516000, Guangdong Province, China

# Understanding the nature of baryon resonances

Derek B. Leinwebercssm  Curtis D. Abellcssm  Liam C. Hockleycssm
Waseem Kamlehcssm  Zhan-Wei Liulslhlt  Finn M. Stokescssmju
Anthony W. Thomascssm  Jia-Jun Wuucsc cssmcssmlslslhlhltltjujuucucscsc

###### Abstract

This presentation opens with a brief review of lattice QCD calculations
showing the $2s$ radial excitation of the nucleon sits at approximately 2 GeV,
well above the Roper resonance position. We then proceed to reconcile this
observation with experimental scattering data. While the idea of dressing
quark-model states in a coupled-channel analysis to describe scattering data
has been around for decades, it’s now possible to bring these descriptions to
the finite-volume of lattice QCD for confrontation with lattice-QCD
calculations. This combination of lattice QCD and experiment demands that we
reconsider our preconceived notions about the quark-model and its excitation
spectrum. We close with a discussion of an unanticipated resolution to the
missing baryon resonances problem.

##  1 Introduction

Perhaps the most surprising and unanticipated discovery in low-lying baryon
spectroscopy is the discovery that the $2s$ radial excitation of the nucleon
lies at approximately 2 GeV [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], far from
the Roper resonance of the nucleon at 1.44 GeV [13]. So ingrained is the idea
that the Roper resonance is associated with the $2s$ excitation of the simple
quark model, the literature even refers to the $2s$ quark-model excitation as
the Roper state. We now know this is not the case.

Remarkably the idea that the $\Lambda(1405)$ resonance is not a quark model
state [14, 15] is widely accepted. Here even the two-pole structure associated
with attractive coupled-channel effects in the $\pi\Sigma$ and $\bar{K}N$
channels is understood [16, 17, 18].

However, this is not necessarily the case for the Roper resonance.
Nevertheless, the Roper resonance should now be understood to be generated by
rescattering in the coupled channels of $\pi N$, $\sigma N$, and $\pi\Delta$.
Here the $\sigma N$, and $\pi\Delta$ channels are the resonant contributions
from the three-body $\pi\pi N$ channel.

In the following section, we briefly review the lattice QCD calculations
revealing that the radial excitation of the nucleon sits at 2 GeV. Attempts to
find more exotic descriptions of the Roper resonance in lattice QCD are also
reviewed in Sec. 2.

With the radial excitation entrenched at 2 GeV, analysis turns to the $\pi N$
phase shift data, exploring whether one can describe this data without a low-
lying radial excitation near the Roper resonance. Here Hamiltonian Effective
Field Theory is used to bring the scattering data of experiment to the finite
volume of lattice QCD to confront the results of contemporary lattice QCD
calculations. This research is briefly reviewed in Sec. 3.

These discoveries provide a novel and unanticipated resolution of the long-
standing missing baryon resonances problem and this solution is presented in
the closing section.

##  2 Lattice QCD

The first hint that there was a problem with the excitation energy of the $2s$
radial excitation of the nucleon in full $2+1$ flavour lattice QCD [19] was
reported in the preprint of Ref. [1]. By using superpositions of Gaussian-
shaped smeared quark sources, this work presents the first attempt to excite
the $2s$ radial excitation of the nucleon in full QCD.

By combining narrow and wide Gaussian sources with opposite signs, one can
create a node in the wave function of the quark distributions within the
nucleon. The well established generalised eigenvalue solution [20, 21] is
employed to calculate the manner in which the Gaussian-smeared quark sources
are superposed to create the $2s$ radial excitation on the lattice.

Figure 3 of Ref. [1] illustrates the invariance of the $2s$ excitation energy
obtained. It was impossible to generate a low-lying $2s$ excitation. These
results were subsequently confirmed by the HSC collaboration at heavy quark
masses [22] and others [2, 3]. A consensus began to emerge in 2015 [4]
particularly with the advent of the Athens Model Independent Analysis Scheme
[3]. The wave functions of both the $2s$ and $3s$ excitations were directly
calculated in lattice QCD [5, 6] and their profiles compare well with the
expectations of a constituent quark model [5].

More exotic five-quark descriptions for the Roper resonance were pursued in
Refs. [7, 8]. However, no new low-lying states were observed. Similarly the
consideration of hybrid baryons [9] did not reveal any new low-lying states in
the Roper channel.

Next generation calculations appeared in 2016 with a high-statistics
calculation coming from the CSSM [10] and the very first insights of the role
of two-particle scattering states from Lang et al. [11]. Here the three-quark
dominated state continued to be observed at $\sim 2$ GeV, confirming earlier
observations [1, 2, 3, 4, 5, 6, 7, 8], even when lower-lying scattering states
are included in the correlation matrix.

The role of chiral symmetry in lattice fermion actions was explored
quantitatively in Ref. [23] to see if the explicit breaking of chiral symmetry
in Wilson-clover fermion actions was responsible for the large $2s$ excitation
energy. A direct analysis of lattice correlation functions from Wilson-clover
and overlap fermion actions – the latter providing a lattice implementation of
chiral symmetry – revealed no differences in the spectrum.

Since then, parity-expanded variational analysis (PEVA) techniques [24] have
been developed to explore the electromagnetic form factors of both the even
and odd-parity excitations of the nucleon [12] and their electromagnetic
transitions to the ground state [25]. With regard to the $2s$ excitation, the
charge radius of the excited proton is larger than the ground state, in accord
with expectations. Moreover, the magnetic moments of the excited $2s$ proton
and neutron calculated in lattice QCD agree with the ground-state magnetic
moments, again consistent with the expectations of a $2s$ excitation.

The most recent calculations of the nucleon spectrum are focused on the
lowest-lying scattering states [18, 26, 27, 28]. As described in detail in the
next section, these energy levels are consistent with $\pi N$ scattering data
from experiment, providing confidence in their world-leading computationally-
challenging analysis methods.

In summary, there is overwhelming evidence that the $2s$ radial excitation of
the nucleon has been observed in lattice QCD. On a lattice volume of 3 fm on a
side, the state sits at 1.9(1) GeV. One then naturally asks, is this result
even consistent with experiment? It is possible to describe $\pi N$ scattering
data in the absence of a low-lying single-particle quark-model state. If so,
then what is the Roper resonance?

##  3 Hamiltonian effective field theory

###  3.1 Infinite volume

The idea of dressing quark-model states in a coupled-channel analysis to
describe scattering data has been around for decades [29]. However, recent
advances in understanding the nature of baryon excitations are flowing from
the novel ability to bring these coupled-channel descriptions to the finite-
volume of lattice QCD for confrontation with lattice-QCD calculations. This
combination of lattice QCD and experiment demands that we reconsider our
preconceived notions about the quark-model and its excitation spectrum.

Herein, the infinite volume world of experiment and the finite-volume world of
lattice-QCD are bridged by Hamiltonian effective field theory (HEFT), a
nonperturbative extension of effective field theory incorporating the Lüscher
formalism [30, 31]. HEFT calculations typically commence in infinite volume
with an aim to describe experimental scattering data [10, 32, 33, 34, 35, 36].
Single and non-interacting two-particle states mix via one-to-two and two-to-
two vertices as depicted in Fig. 1.

$\pi(-\vectorsym{k})$$N(\vectorsym{k})$$N_{0}(\vectorsym{0})$$\pi(-\vectorsym{k^{\prime}})$$\Delta(\vectorsym{k^{\prime}})$$\pi(-\vectorsym{k})$$N(\vectorsym{k})$$\pi(-\vectorsym{k^{\prime}})$$\Delta(\vectorsym{k^{\prime}})$

Figure 1: Examples of one-to-two (left) and two-to-two (right) vertices
contributing to $\pi N$ scattering in the rest frame. Here $N_{0}$ denotes the
single-particle $2s$ radial excitation of the nucleon.

The functional form of the one-to-two vertices are governed by heavy-baryon
chiral perturbation theory [31, 34] and the two-to-two vertices are described
by separable potentials which facilitate a closed form solution of the
standard coupled-channel equations [35]. Here the potentials become
phenomenological, adopting a momentum dependence as demanded by the scattering
data [32]. Remarkably, the Lüscher formalism embedded within HEFT ensures the
low-lying finite-volume spectrum is independent of the phenomenology used to
describe the data, provided the data is described accurately [34].

Of course HEFT can be used in the more traditional manner where lattice QCD
results constrain the Hamiltonian and infinite-volume scattering observables
are predicted [37, 38]. In the baryon sector, there is a lack of precision
results showing the subtle shifts of the non-interacting spectra due to the
finite volume of the lattice. It is better to bring the precision experimental
scattering data to the finite volume of lattice QCD.

As one moves up in the spectrum, more coupled channels come into effect.
Reference to the partial decay widths of the resonances under investigation
can inform the essential channels to be included in the calculation. For
example, the $\pi N$, $\sigma N$ and $\pi\Delta$ channels are all observed in
the partial decay widths of the Roper resonance making their inclusion
essential to understanding the structure of the Roper resonance. In the
absence of experimental analyses of the $\pi N$ to $\pi\Delta$ and $\sigma N$
scattering amplitudes, there is insufficient information to uniquely constrain
the Hamiltonian from experimental data alone.

As a result, fits to the experimental data provide a variety of Hamiltonians,
all describing the experimental $\pi N$ phase shift and inelasticity but
describing the composition of the spectrum in a different manner. It is here,
that the finite-volume predictions of the various coupled-channel fits can be
confronted with lattice QCD results.

###  3.2 Finite volume

In going from infinite volume to finite volume, the continuum of momentum
states becomes quantised to the momenta available on a finite periodic volume.
Scattering potentials constrained by experiment pick up finite-volume factors
and form the elements of a matrix Hamiltonian. One then solves the Schrödinger
equation

$\langle\,i\,|\,H\,|\,j\,\rangle\>\langle\,j\,|\,E_{\alpha}\,\rangle=E_{\alpha}\,\langle\,i\,|\,E_{\alpha}\,\rangle\,,$
(1)

where $|\,i\,\rangle$ and $|\,j\,\rangle$ are the non-interacting basis states
taking the discrete momenta available on the periodic volume (e.g.
$|\,\pi(\vectorsym{k})\,N(-\vectorsym{k})\,\rangle\,$), $E_{\alpha}$ is the
energy eigenvalue for the finite-volume energy eigenstate labelled by
$\alpha$, and $\langle\,i\,|\,E_{\alpha}\,\rangle$ is the eigenvector of the
Hamiltonian matrix $\langle\,i\,|\,H\,|\,j\,\rangle$ describing the
composition of the finite-volume energy eigenstate $\alpha$ in terms of the
non-interacting basis states $|\,i\,\rangle$.

The ability of HEFT to describe the pion mass dependence of the finite-volume
energy eigenstates and provide a description of their composition is key to
the advances being made. Of particular importance is the contribution of the
single-particle basis state to the energy eigenstates. As analyses of the
nucleon spectrum in the excitation regime of the Roper are performed with
local three-quark interpolating fields, the overlap of the single-particle
basis state $|\,N_{0}\,\rangle$ with the energy eigenstate
$|\,\alpha\,\rangle$ provides information of the energy eigenstates most
likely to be excited in lattice QCD calculations.

More quantitatively, Bär et al. [39] provided a $\chi$PT estimate of the
coupling between a smeared nucleon interpolating field and a non-interacting
pion-nucleon basis state

$\frac{3}{16}\,\frac{1}{(f_{\pi}\,L)^{2}\,E_{\pi}\,L}\left(\frac{E_{N}-m_{N}}{E_{N}}\right)\approx
10^{-3}\,,$ (2)

where $E_{\pi}$ and $E_{N}$ are on-shell pion and nucleon energies. The
numerical estimate is based on a 3 fm lattice and the lowest nontrivial
momentum contribution where the coupling is largest. Here the $1/L^{3}$
dependence of the coupling is manifest as the non-interacting two-particle
momentum state is spread uniformly throughout the lattice volume.

Noting the small magnitude of the overlap between the local interpolating
field and the two-particle basis states, one concludes that the state excited
by the local interpolating field is the only local state in the Hamiltonian
basis, the single-particle basis state, $\ket{N_{0}}$. Thus, we associate the
three-quark nucleon interpolating field $\bar{\chi}$ acting on the QCD vacuum,
$\ket{\Omega}$, with the bare $2s$-nucleon basis state of HEFT, via
$\bar{\chi}(0)\ket{\Omega}=\ket{N_{0}}$. The most likely energy eigenstate to
be observed in lattice QCD is the energy eigenstate having the largest overlap
with $\ket{N_{0}}$ as indicated by a survey of
$\langle\,N_{0}\,|\,E_{\alpha}\,\rangle$ over $\alpha$.

Remarkably, there is sufficient information in the eigenvectors of HEFT to
actually simulate what lattice QCD correlation functions of three-quark
operators look like

$G_{N_{0}}(t)=\sum_{i}\left|\braket{N_{0}}{E_{i}}\right|^{2}e^{-E_{i}t}\,.$
(3)

This was examined in Ref. [35] and the favourable comparison with lattice QCD
results is noteworthy.

###  3.3 Confrontation with Lattice QCD

Noting that experimental $\pi N$ scattering data alone is not enough to
uniquely constrain the Hamiltonian, we bring the coupled-channel predictions
to the finite volume of lattice QCD to confront the predictions with lattice
QCD determinations of the mass spectrum.

Of critical importance is where the states excited by local three-quark
interpolating fields lie in the spectrum. HEFT not only predicts the positions
of the energy eigenstates but also their composition. A careful comparison of
the composition in HEFT with the interpolating fields used to excite the
states in lattice QCD is very powerful in discriminating between various
descriptions of the experimental scattering data. To be clear, all coupled-
channel analyses considered herein describe the experimental $\pi N$
scattering data well and generate poles in the complex plane in agreement with
the particle data group [40].

To differentiate between the many possible descriptions of experimental data
alone, we label two fits of interest by the bare mass, $m_{0}$, of the single-
particle state associated with the $2s$ radial excitation of the quark model,
namely $m_{0}=1.7$ GeV and $m_{0}=2.0$ GeV. The former value leads to the
familiar scenario where the quark-model state is dressed by meson-baryon
states to lie in the regime of the Roper resonance.

The excellent description of the $\pi N$ scattering data for both of these
scenarios is presented in Fig. 1 of Ref. [33] where there is very little to
differentiate between the two fits. The differences become apparent in the
finite volume of the lattice. These HEFT predictions of the eigenstate
energies are presented in Fig. 2.

Figure 2: The finite volume spectra predicted by HEFT [33] (solid lines) for a
lattice QCD volume of (3 fm)3 are confronted with results from lattice QCD
(bullets and squares). The left plot is the familiar scenario with $m_{0}=1.7$
GeV and the right-hand plot has $m_{0}=2.0$ GeV. Line colours illustrate the
energy eigenstates with the largest quark-model single-particle components in
the order red, blue, green, and orange. These are the most likely states to be
excited with local three-quark interpolating fields in lattice QCD. Bullets
indicate the lattice results from Lang et al. [11] and squares are from the
CSSM [10]. Open symbols denote energy eigenstates dominated by two-particle
momentum projected interpolating fields and full symbols denote energy
eigenstates dominated by local three-quark interpolating fields.

The left-hand plot of Fig. 2 with $m_{0}=1.7$ GeV illustrates the prediction
of a low-lying state dominated by a single-particle quark-model like state in
the Roper regime, as indicated by the red curve from HEFT. The problem with
this fit is that this single-particle dominated state in the regime of the
Roper resonance is not seen in lattice QCD. While there is a lattice QCD point
on the red curve at $\sim 1.5$ GeV, this state was excited by a momentum
projected $\pi N$ interpolating field [11]. It is not dominated by a three-
quark operator.

The right-hand plot of Fig. 2 with $m_{0}=2.0$ GeV is in accord with lattice
QCD results. Here the states predominantly excited by three-quark
interpolating fields are associated with HEFT eigenstates with large single-
particle components indicated by colour on the eigenstate energy lines.

Moreover, the composition of the scattering states predicted in HEFT with
$m_{0}=2.0$ GeV match the interpolating fields used to excite the states in
lattice QCD. In both calculations, the lowest excitation at $\sim 1.35$ GeV is
dominated by $\sigma N$ at zero back-to-back momentum and the next excitation
at $\sim 1.5$ GeV is dominated by $\pi N$ with the lowest nontrivial back-to-
back momentum. Table 1 presents a scorecard to evaluate the quality of these
two HEFT descriptions.

Table 1:  Scorecard for the agreement between lattice QCD calculations and two HEFT descriptions of experimental $\pi N$ scattering phase shifts and inelasticities. The $m_{0}=2.0$ GeV description characterises the Roper resonance as arising from coupled channel rescattering.  Criteria |  $m_{0}=1.7$ GeV |  $m_{0}=2.0$ GeV
---|---|---
Describes experimental data well and produces poles in accord with PDG. | ✔ | ✔
1st lattice scattering state created via $\sigma N$ interpolator has dominant $\langle\,\sigma N\,|\,E_{1}\,\rangle$ in HEFT. | ✔ | ✔
2nd lattice scattering state created via $\pi N$ interpolator has dominant $\langle\,\pi N\,|\,E_{2}\,\rangle$ in HEFT. | ✗ | ✔
Lattice-QCD states excited with 3-quark interpolators are associated with HEFT states with large $\langle\,{N_{0}}\,|\,E_{\alpha}\,\rangle$. | ✗ | ✔
HEFT predicts three-quark states existing in lattice QCD. | ✗ | ✔
Table 2: Quark-model predictions [41] for the energies of the radial excitations of the nucleon and the $\Delta$ in units of GeV are compared with contemporary lattice QCD calculations [6, 10, 42].  State | Quark Model | Lattice QCD | State | Quark Model | Lattice QCD
---|---|---|---|---|---
$N1/2^{+}\ 2s$ | $1.54$ | $1.90(6)$ | $\Delta 3/2^{+}\ 1s$ | $1.23$ | $1.26(2)$
$N1/2^{+}\ 3s$ | $1.77$ | $2.60(7)$ | $\Delta 3/2^{+}\ 2s$ | $1.80$ | $2.14(5)$
$N1/2^{+}\ 4s$ | $1.88$ | $3.60(5)$ | $\Delta 3/2^{+}\ 3s$ | $1.92$ | $3.10(17)$
$N1/2^{+}\ 5s$ | $1.98$ |  | $\Delta 3/2^{+}\ 4s$ | $1.99$ |

In the valid HEFT description where $m_{0}=2.0$ GeV, the $\pi\Delta$ channel
takes on an enhanced role [33] in describing the experimental $\pi N$ phase
shift and inelasticity. In this case the coupling is large and combines with
$\pi N$ and $\sigma N$ channels to generate a pole in the complex plane. The
Roper resonance arises from coupled channel rescattering in the $\pi N$,
$\sigma N$ and $\pi\Delta$ channels, the latter two being the resonant
contributions of the three-body $\pi\pi N$ contribution. This is the nature of
the Roper resonance.

##  4 Resolution of the missing baryon resonances problem

We close by discussing the impact of the $2s$ radial excitation of the nucleon
being observed at 1.9 GeV in lattice QCD calculations near the physical point
[6, 10, 11] as opposed to 1.44 GeV as anticipated historically [43, 41].

As the single-particle quark model state is mixed with nearby two-particle
states to form resonances, we anticipate the $2s$ radial excitation is largely
associated with the $N1/2^{+}(1880)$ resonance observed in photoproduction and
to a smaller extent the $N1/2^{+}(1710)$ as it is only 170 MeV away. This is
supported by the right-hand plot of Fig. 2 where the large single-particle
basis state contributions are illustrated in colour.

In this light, it seems erroneous to tune the parameters of the quark model to
place the radial excitation of the nucleon at 1440 MeV. The proliferation of
radial excitations below 2 GeV is a direct consequence of this
misidentification. Table 2 reports the predictions of the quark model [41]
tuned to place the $2s$ excitation at 1540 MeV, a little higher than the Roper
resonance in anticipation of some meson-baryon dressings. These predictions
for the nucleon and $\Delta$ contrast the reality of lattice QCD calculations.

Had the quark model been tuned to place the $2s$ excitation at 1.9 GeV, the
next excitations would be well above 2 GeV. In this regard, the missing
resonance problem may be regarded as further evidence that the $2s$ radial
excitation of the nucleon is not associated with the Roper resonance.

###### Acknowledgements.

This research was undertaken with the assistance of resources from the
National Computational Infrastructure (NCI), and by the Phoenix HPC service at
the University of Adelaide. This research was supported by the Australian
Research Council through ARC Discovery Project Grants Nos. DP190102215 and
DP210103706 (D.B.L.). J.-J. Wu was supported by the National Natural Science
Foundation of China under Grant Nos. 12175239 and 12221005, and by the
National Key R&D Program of China under Contract No. 2020YFA0406400. Z.-W. Liu
was supported by the National Natural Science Foundation of China under Grant
Nos. 12175091, 11965016, 12047501, and 12247101, and the 111 Project under
Grant No. B20063.

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