A way to evaluate directly the so-called consistent form of anomalies in the path integral, which has a form of gauge non-invariance in general, is to use a Pauli-Villars regularization with a bosonic fermion by defining a fermion operator of the form $D =\Dslash (\frac{1-\gamma_{5}}{2}) + \delslash (\frac{1+\gamma_{5}}{2})$ [70]. The Jacobians then completely cancel between the physical fermion and the bosonic Pauli-Villars regulator fields; the Ward-Takahashi identity then contains both the physical fermion and the Pauli-Villars regulator but without anomalies. One reproduces the so-called consistent form of anomalies from the Pauli-Villars mass term in the limit of $m_{PV}\rightarrow \infty$, in which the Pauli-Villars regulator field (except for the regulator mass term) decouples from the physical Hilbert space and only the physical field survives the limit together with anomalies. We illustrate a mechanism similar to the Pauli-Villars regularization in the anomaly evaluation in lattice gauge theory later.

To summarize the analysis in this subsection, we can treat all the known chiral anomalies in the path integral formalism.

§.§ Fermions in lattice gauge theory

Fermions defined on the lattice exhibit some novel properties related to chiral symmetry as was noticed by K. Wilson in his original formulation of the lattice gauge theory [15]. We discuss the novel properties from the point of view of chiral anomalies.

§.§.§ Chiral symmetry and species doubling

We explain the basic aspects of fermions defined on the lattice.
We use the simplest Lagrangian of QED defined on the 4-dimensional hypercubic lattice. We choose the hypercubic lattice since the simplest (Euclidean) hypercubic lattice with the lattice spacing $a$ is considered to be required to restore the Lorentz or $O(4)$ symmetry in the continuum limit.
The action is defined by

\begin{align}\label{lattice QED}
\frac{S_{F}}{a^{4}} =&\sum_{x}\Bigg\{\sum_{\mu}\left(\frac{i}{2a}\right)[\overline{\psi}(x)\gamma^{\mu}U_{\mu}(x)\psi(x+a^{\mu})-\overline{\psi}( x+a^{\mu})\gamma^{\mu}U_{\mu}(x)^{\dagger}\psi(x)] \nonumber\\
+ \sum_{\mu}\left(\frac{r}{2a}\right)[\overline{\psi}(x)U_{\mu}(x)\psi(x+a^{\mu})+\overline{\psi}(x+a^{\mu})U_{\mu}(x)^{\dagger}\psi(x) \nonumber\\
&-2\overline{\psi}(x)\psi(x)] \Bigg\}
\end{align}

\begin{eqnarray}
\end{eqnarray}

This basic structure is valid for a non-Abelian gauge theory also.
The fermions are placed on the lattice point and the gauge field is placed on the link connecting the lattice points.
We sum $x$ over all the lattice points and sum over $\mu=1\sim 4$ with $a^{\mu}=a\hat{\mu}$, where $\hat{\mu}$ stands for the unit vector in the direction of $\mu$.  The action is invariant under the gauge transformations

\begin{eqnarray}
&&U_{\mu}(x)\rightarrow U(\omega(x))U_{\mu}(x)U(\omega(x+a^{\mu}))^{\dagger},\nonumber\\
&&\psi(x)\rightarrow U(\omega(x))\psi(x), \ \ \overline{\psi}(x)\rightarrow \overline{\psi}(x)U(\omega(x))^{\dagger}.
\end{eqnarray}

The first kinetic term in (<ref>) is invariant under the (global) chiral symmetry

\begin{eqnarray}\label{lattice chiral symmetry}
\psi(x)\rightarrow e^{i\gamma_{5}\alpha}\psi(x), \ \ \  \overline{\psi}(x)\rightarrow\overline{\psi}(x) e^{i\gamma_{5}\alpha}
\end{eqnarray}

with a real constant parameter $\alpha$, while the mass term $m_{0}$ and the term with $r/a$ break this chiral symmetry.
In the naive continuum limit $a\rightarrow 0$,
we have

\begin{eqnarray}\label{Wilson fermion near continuum}
S_{F}\simeq \int d^{4}x \left\{ \frac{i}{2}\left[ \overline{\psi}(x)\gamma^{\mu}D_{\mu}\psi(x)-D_{\mu}\overline{\psi}(x)\gamma^{\mu}\psi(x) \right] -m_{0}\overline{\psi}(x)\psi(x) \right\}
\end{eqnarray}

and thus the term with $r/a$, which is called the Wilson term, appears to vanish in the naive continuum limit.

To understand the physical meaning of the Wilson term, we examine the free propagator given by the lattice Lagrangian (<ref>) with $r=0$

\begin{eqnarray}\label{naive lattice propagator}
\Big[ \sum_{\mu}\frac{1}{a}\gamma^{\mu}\sin ak_{\mu} - m_{0} \Big]^{-1}.
\end{eqnarray}

The momenta on the lattice theory are chosen in the fundamental Brillouin zone

\begin{eqnarray}\label{fundamental Brillouin zone}
-\frac{\pi}{2a}\leq k_{\mu}<\frac{3\pi}{2a}.
\end{eqnarray}

If one chooses the momenta near $k_{\mu}\simeq 0$ for each direction in (<ref>), one recovers the continuum propagator

\begin{eqnarray}
[\gamma^{\mu}k_{\mu} - m_{0}]^{-1}
\end{eqnarray}

in the limit $a\rightarrow 0$.
On the other hand, if one chooses
$k_{1}=\pi/a +k_{1}^{\prime}$ with small $k_{1}^{\prime}$, for example, one has in the limit $a\rightarrow 0$

\begin{eqnarray}
\Big[\sum_{\mu=2}^{4}\gamma^{\mu}k_{\mu}
\end{eqnarray}

which after the change of the signature of
$k_{1}^{\prime}$  gives another pole with the mass $m_{0}$. We thus have two particle poles in each momentum direction and in total $2^{4}=16$ fermions. This phenomenon is called the species doubling.

If one chooses $r\neq 0$, which breaks chiral symmetry (strongly),

\begin{eqnarray}
\Big[\sum_{\mu}\frac{1}{a}\gamma^{\mu}\sin ak_{\mu} - m_{0} -\sum_{\mu}\frac{r}{a}(1-\cos ak_{\mu})\Big]^{-1}
\end{eqnarray}

then all the extra fermion poles except for the one at $k_{\mu}\simeq 0$ are eliminated from the physical spectrum in the limit $a\rightarrow0$, since the extra poles have the masses $\sim m_{0} + r/a$.

To understand the species doubling from a symmetry point of view, one may define following Karsten and Smit [16]

\begin{eqnarray}\label{shift operator}
T_{\mu}(x) =\gamma^{\mu}\gamma^{5}\exp[i\pi (x^{\mu}/a)]
\end{eqnarray}

which satisfies the Clifford algebra $T_{\mu}T_{\nu}+T_{\nu}T_{\mu}=2\delta_{\mu\nu}$. One then defines 16 operators

\begin{eqnarray}\label{even operator}
1, T_{1}T_{2}, T_{1}T_{3}, T_{1}T_{4}, T_{2}T_{3}, T_{2}T_{4}, T_{3}T_{4}, T_{1}T_{2}T_{3}T_{4},
\end{eqnarray}

\begin{eqnarray}\label{odd operator}
T_{1}, T_{2}, T_{3}, T_{4}, T_{1}T_{2}T_{3}, T_{2}T_{3}T_{4}, T_{3}T_{4}T_{1}, T_{4}T_{1}T_{2}.
\end{eqnarray}

If one denotes any one of these 16 operators by $T$, one can confirm that the action (<ref>) with $r=0$ (but $m_{0}\neq 0$) is invariant under

\begin{eqnarray}
\psi(x)\rightarrow T(x)\psi(x), \ \ \overline{\psi}(x)\rightarrow \overline{\psi}(x)T^{-1}(x).
\end{eqnarray}

The operator (<ref>) adds a momentum $\pi/a$ in the direction $\mu$, and thus one recognizes that these operators except for $T=1$ generates 15 extra poles starting with the pole at $k_{\mu}=0$. A crucial property  is that all $T$ in (<ref>) commute with $\gamma_{5}$ and all $T$ in (<ref>) anti-commute with $\gamma_{5}$.

If one attempts to define a left-handed massless fermion $[(1-\gamma_{5})/2]\psi(x)$ and $\overline{\psi}(x)[(1+\gamma_{5})/2]$, which means one sets $m_{0}=r=0$ in (<ref>) to ensure the exact chiral symmetry,

\begin{eqnarray}\label{lattice chiral QED}
\frac{S_{0}}{a^{4}} &=&\sum_{x}\sum_{\mu}\left(\frac{i}{2a}\right)[\overline{\psi}(x)\gamma^{\mu}U_{\mu}(x)[(1-\gamma_{5})/2]\psi(x+a^{\mu})\nonumber\\
&&-\overline{\psi}( x+a^{\mu})\gamma^{\mu}U_{\mu}(x)^{\dagger}[(1-\gamma_{5})/2]\psi(x)],
\end{eqnarray}

the species doubling and the symmetry operators in (<ref>) and (<ref>) imply that one inevitably has 8 left-handed fermions and 8 right-handed fermions. (Starting with the right-handed fermion $[(1+\gamma_{5})/2]\psi(x)$ and $\overline{\psi}(x)[(1-\gamma_{5})/2]$, one similarly finds 8 right-handed massless fermions and 8 left-handed massless fermions.)  This statement, which is based on the hypercubic lattice, is generic since the Lorentz invariance in the continuum limit is believed to require the hypercubic lattice. An equivalent statement has been given by Nielsen and Ninomiya which is based on a topological consideration [17]. It is important that this analysis of species doubling itself is not directly based on the analysis of chiral anomalies.

We next analyze the above statement of species doubling from a point of view of  chiral anomalies in lattice gauge theory.  The chiral anomaly on the lattice in the limit $a\rightarrow 0$, which is used as a means to examine to what extent  the lattice theory realizes the continuum theory, has been discussed by many authors by evaluating Feynman diagrams in the past [16]. We here instead discuss the same issue by an analysis of the chiral Jacobian in the path integral.

From the Lagrangian (<ref>), we identity the Euclidean latticized (hermitian) Dirac operator for the simple case with $m_{0}=r=0$ and by setting $\hat{p}_{\mu}=-i\partial_{\mu}$

\begin{eqnarray}
\Dslash \equiv \frac{i}{2a}[\gamma^{\mu}U_{\mu}(x)e^{(-ia\hat{p}_{\mu})}  -e^{(-ia\overleftarrow{\hat{p}}_{\mu})}\gamma^{\mu}U^{\dagger}_{\mu}(x)]
\end{eqnarray}

where $e^{(-ia\hat{p}_{\mu})}$ stands for the difference operator.
The lattice version of the regularized Jacobian (<ref>) for the localized chiral transformation $\psi(x)\rightarrow e^{i\gamma_{5}\alpha(x)}\psi(x)$ and $\overline{\psi}(x)\rightarrow\overline{\psi}(x) e^{i\gamma_{5}\alpha(x)}$, under the global version of which (i.e., global chiral symmetry) the action (<ref>) is invariant,  is given for an infinitesimal $\alpha(x)$

\begin{eqnarray}\label{lattice anomaly1}
\lim_{M\rightarrow \infty}a^{4}\sum_{x} 2i\alpha(x)\sum_{n=1}^{16}tr T^{-1}_{n}\gamma_{5}T_{n} \int_{-\pi/2a\leq k^{\mu}<\pi/2a}\frac{d^{4}k}{(2\pi)^{4}}e^{-ikx}\exp[-(\Dslash)^{2}/M^{2}]e^{ikx}\nonumber\\
\end{eqnarray}

where we used the 16 symmetry operators in (<ref>) and (<ref>) and restricted the momentum range to the physical domain

\begin{eqnarray}
-\pi/2a\leq k^{\mu}<\pi/2a
\end{eqnarray}

starting with the fundamental Brillouin zone (<ref>). Note that $\Dslash^{\dagger}\Dslash=\Dslash^{2}$. If one takes the sum over the 16 operators first, then the Jacobian factor (<ref>) vanishes identically
since $\sum_{n=1}^{16} T^{-1}_{n}\gamma_{5}T_{n}=0$, namely, the lattice massless or massive  theory with
$r=0$ has no chiral anomalies due to 16 species doublers altogether
; anomalies vanish for any $a\neq 0$ and thus smoothly vanish in the limit $a\rightarrow 0$ also. We comment on this fact from a different point of view later.

Coming back to $S_{F}$ in (<ref>), it is important to examine  the contributions to chiral anomaly from the 15 species doublers, which acquire the effective masses of the order $m_{0} + r/a$.
In the small $a$ limit, $a\rightarrow 0$ but $a\neq 0$, the 16 species with the effective masses in each subdomain of the fundamental Brillouin zone

\begin{eqnarray}\label{effective masses}
&&m_{0}\ \ (1,1), \ \ m_{0}+2r/a\ \ (4, -1), \ \ m_{0}+4r/a\ \ (6, 1), \ \ m_{0}+6r/a\ \ (4, -1), \nonumber\\ &&m_{0}+8r/a\ \ (1, 1)
\end{eqnarray}

appear. Here we indicate inside the bracket, (multiplicity, chiral charge); for example, fermions with the effective mass $m_{0}+2r/a$  appear 4 times with the chiral charge $-1$.
Mathematically, it is better defined to consider the $a\rightarrow0$ limit first with $r/a$ kept fixed and later consider the limit $r/a \rightarrow \infty$, for which one can identify the fermion species with the kinetic energy term $\Dslash$ of a continuum fermion with masses of the order $m_{0}+r/a$ in (<ref>). In this limit one obtains the Jacobian (including the effective mass terms for the sake of book-keeping)

\begin{align}\label{lattice anomaly4}
&\lim_{M\rightarrow \infty}\int d^{4}x ~2i\alpha(x)\sum_{n=1}^{16}tr T^{-1}_{n}\gamma_{5}T_{n}\nonumber\\
&\times \lim_{a\rightarrow 0}\int_{-\pi/2a\leq k^{\mu}<\pi/2a}\frac{d^{4}k}{(2\pi)^{4}}\exp[-((i\kslash+\Dslash)^{2} +M^{2}_{n})/M^{2}]
\end{align}

with $M_{n}$ standing for the effective mass of the $n$-th species in (<ref>). This anomaly factor vanishes for $M\rightarrow\infty$ with $M_{n}$ kept fixed; each $M_{n}$ produces the continuum anomaly up to a sign factor of chirality, but the sum $\sum_{n=1}^{16} T^{-1}_{n}\gamma_{5}T_{n}$ gives a vanishing result.

To write the Ward-Takahashi identity for the localized chiral transformation $\psi(x)\rightarrow e^{i\gamma_{5}\alpha(x)}\psi(x)$ and $\overline{\psi}(x)\rightarrow\overline{\psi}(x) e^{i\gamma_{5}\alpha(x)}$, one first evaluates the Jacobian factor, which is given by
(<ref>), that vanishes as we have explained above. Namely, the chiral identity in lattice gauge theory in the present setting is anomaly-free.
To write the Ward-Takahashi identity, we first write the action in the form in the limit $a\rightarrow 0$ but with $r/a$ kept finite (in a symbolic notation),

\begin{eqnarray}
{\cal L}&\simeq& \sum_{n}\int_{V_{n}}\frac{d^{4}k}{(2\pi)^{4}} \frac{d^{4}q}{(2\pi)^{4}}\overline{\psi}(k+q)\Aslash(q)\psi(k) + \sum_{n}\int_{V_{n}}\frac{d^{4}k}{(2\pi)^{4}}\overline{\psi}(k)(-\kslash+M_{n})\psi(k)\nonumber\\
&\simeq&\sum_{n}\int d^{4}x ~\overline{\psi}_{n}(x)[i\Dslash + M_{n}]\psi_{n}(x)
\end{eqnarray}

where we first consider the Fourier transform of the field and then rewrite it by the effective fields representing the species doublers. The symbol $V_{n}$ indicates the $n$-th subdomain in the Brillouin zone.
The Ward-Takahashi identity is then written in the form which is anomaly-free

\begin{eqnarray}\label{Ward-Takahashi for Wilson}
\sum_{n}\partial_{\mu}\langle J_{5n}^{\mu}(x)\rangle
=\sum_{n}2iM_{n}\langle \overline{\psi}_{n}(x)T^{-1}_{n}\gamma_{5}T_{n}\psi_{n}(x)\rangle
\end{eqnarray}

where the right-hand side is the contribution of all the species doublers with the effective masses $M_{n}$ in (<ref>) which break chiral symmetry. In the limit $r/a \rightarrow \infty$, the 15 species doublers with large effective masses decouple from the physical space;  at the same time,
the effective large mass terms of 15 species doublers give rise to the chiral anomalies

\begin{eqnarray}
-[4\times (-1) + 6\times (1) +4\times (-1) +1\times (1)]
\frac{1}{2\pi^{2}}\vec{E}\cdot\vec{B} =\frac{1}{2\pi^{2}}\vec{E}\cdot\vec{B}
\end{eqnarray}

and thus

\begin{eqnarray}
\partial_{\mu}\langle J_{51}^{\mu}(x)\rangle
=2im_{0}\langle \overline{\psi}_{1}(x)\gamma_{5}\psi_{1}(x)\rangle + \frac{1}{2\pi^{2}}\vec{E}\cdot\vec{B}
\end{eqnarray}

which agrees with the perturbative analysis. Readers are referred to the very detailed diagrammatic analysis in [16].
This mechanism is the same as the Pauli-Villars regularization in continuum theory, in which the physical fermion and the (bosonic) Pauli-Villars regulator together give the vanishing Jacobian but the large Pauli-Villars mass term, which breaks chiral symmetry strongly, gives the ordinary chiral anomaly in the large mass limit $m_{PV}\rightarrow\infty$ of the regulator field.

As for the fermion number transformation $\psi(x)\rightarrow e^{i\alpha}\psi(x)$ and $\overline{\psi}(x)\rightarrow\overline{\psi}(x) e^{-i\alpha}$, which is an exact symmetry of (<ref>), it is confirmed that one obtains the (trivially) vanishing Jacobian and thus the fermion number is conserved in the vector-like QED.

The absence of the chiral anomaly in the limit $a\rightarrow 0$ but $a\neq 0$ is indicated by the explicit form of the above Jacobian factor (<ref>) and supported by the short-distance idea of anomaly by K. Wilson, namely, the anomaly does not appear when one cuts-off all the short distances.  On the one hand, the inevitable appearance of the chiral anomaly for a single fermion defined in the continuum is known, as in the conventional analysis. These two facts combined imply that any chiral symmetric theory once defined on the lattice automatically re-arranges itself to have the spectrum with species doublers, which is overall anomaly-free to ensure the  smooth limit $a\rightarrow 0$. This view of the inevitable appearance of 16 species doublers (in the $O(4)$ symmetric Euclidean theory) is physically attractive.
We emphasize that the phenomenon of  species doublers arises in the present scheme from the  chiral symmetry generated by $\gamma_{5}$ in the fundamental Lagrangian defined on the lattice.

§.§.§ Spectral flow in a Dirac sea with a finite depth

We would like to comment further on the issue of the  species doublers and spectral flow.  A precise Lagrangian formulation of species doubling, which appears inevitably when one imposes the $\gamma_{5}$ symmetry,  has been discussed in $d=4$ space-time already. This is symbolically stated as  the absence of a (well-defined left-handed) massless “neutrino” on the lattice [17], and the related inevitable appearance of species doublers seems to be often assumed in condensed matter physics also [52] [ The mechanism of the appearance of species doublers in condensed matter physics does not have a solid physical basis.].

The notion of species doublers means that a single “local” fermion and thus a single “local” current defined on the lattice actually describes the multiple species of fermions, i.e., doublers  in momentum space in the limit $a\rightarrow 0$, if the chiral symmetry $\gamma_{5}$ is imposed on the underlying lattice theory.  (One may recall that the Wilson fermion (<ref>) without chiral symmetry does not generate species doublers.)
For example, if one attempts to define a left-handed massless fermion

\begin{eqnarray}
\psi_{L}(x)=[(1-\gamma_{5})/2]\psi(x)
\end{eqnarray}

on the lattice, the species doubling implies that one inevitably obtains (at least) both a left-handed fermion and a right-handed fermion in momentum space in the limit $a\rightarrow 0$.  Similarly, $\psi_{R}(x)=[(1+\gamma_{5})/2]\psi(x)$ implemented on the lattice inevitably induces (at least) both  a right-handed fermion and a left-handed fermion in momentum space in the limit $a\rightarrow 0$.  As another aspect of the species doubling,
the chiral fermion such as $\psi_{L}(x)$, which would contain chiral gauge anomaly  (<ref>) when coupled to a gauge field  in continuum theory and thus inconsistent by itself, contains no chiral gauge anomaly when placed on the lattice which cuts-off the short distances and thereby avoiding the inconsistency induced by the anomaly. Instead, we are forced to have species doublers for $a\neq0$ (and as a result, you may say that it becomes anomaly-free).  A massless Dirac fermion in continuum consists of  $\psi_{L}$ and $\psi_{R}$ and thus anomaly-free in (vector-like) gauge symmetry, but placed on the lattice both $\psi_{L}$ and $\psi_{R}$ would require their species doublers; for $a\rightarrow 0$, one would then have multiple species of vector-like Dirac  fermions (and thus, for example, the massless “electron” and  “muon” appear, although one originally intended to define only the “electron”), and in this sense the construction is inconsistent.  In any case, the notion of species doubling on the lattice implies that one would obtain at least twice as many fermions in momentum space  in the limit $a\rightarrow 0$ than originally intended, if the chiral symmetry defined by $\gamma_{5}$ is imposed on the lattice Lagrangian.

On the other hand, a  recent progress of the Ginsparg-Wilson fermion, which arises from the idea of block transformation on the lattice, allows a definition of  a single Dirac fermion without doublers on the lattice [72, 73]. See [57] for further references. The fermion field in such a theory is exponentially local  [74, 75] (stated intuitively, the construction does not appear to be manifestly local but actually becomes local in the limit $a\rightarrow 0$)  without any species doublers. In contrast, the original construction of the lattice fermion by K. Wilson is called ultra-local for which species doublers appear when $\gamma_{5}$ symmetry is imposed.  In the formulation of Ginsparg and Wilson, the exact chiral symmetry is realized by a new effective chiral operator $\Gamma_{5}$ that is regarded as a deformation of $\gamma_{5}$ for finite $a$. One then obtains a non-trivial index and a Jacobian as a symmetry breaking factor under chiral transformation using $\Gamma_{5}$ in the path integral formulation of chiral identity [76, 77]. One thus obtains a correct anomaly in a natural manner in the continuum limit [78] and the path integral formulation of chiral anomalies become logically more consistent than in the case of the continuum theory (in the sense that the Jacobian is well-defined for a finite theory). A more detailed account of the Ginsparg-Wilson fermion shall be given later.

To understand what is going on in the species doubling intuitively, it is instructive to consider a $d=1+1$ dimensional lattice fermion.  We thus consider a Hamiltonian (i.e., continuous time) formulation of a free Dirac fermion based on the simplest discretization without the electromagnetic field for the moment

\begin{eqnarray}\label{two-dimension}
H=\sigma_{3}\frac{\sin ap}{a}+\sigma_{1}m_{0}+\sigma_{1}\frac{r}{a}(1-\cos ap)
\end{eqnarray}

with a constant $r$ which is called the Wilson parameter. See also Ambjorn et al. [64] and Nielsen and Ninomiya [49] for a related analysis; we here comment briefly on this problem from our point of view.  Note that $H(p)$ in (<ref>) has a period $2\pi/a$ in $p$. If one recalls that chiral $\gamma_{5}$ is given by $\sigma_{3}$ in this notation,
the first term is chiral invariant but the second and third terms do not commute with $\gamma_{5}$ and thus break chiral symmetry. The chiral symmetric Hamiltonian is thus given by setting $m_{0}=r=0$,

\begin{eqnarray}
H_{0}=\sigma_{3}\frac{\sin ap}{a}.
\end{eqnarray}

We have the energy spectrum of $H_{0}(p)$ as

\begin{eqnarray}
\epsilon^{(0)}_{\pm}(p)=\pm \frac{\sin ap}{a}
\end{eqnarray}

where $\epsilon^{(0)}_{\pm}(p)$ correspond to chirality $\gamma_{5}=\pm 1$, respectively.  Note that $\epsilon^{(0)}_{\pm}(p+2\pi/a)=\epsilon^{(0)}_{\pm}(p)$. See Fig.<ref>. In terms of Weyl fermions, this exhibits the spectrum of  Weyl fermions (in $d=1+1$) for small $|\epsilon^{(0)}_{\pm}(p)|$, and $\epsilon^{(0)}_{\pm}(p)=0$ at $p=0$ and $p=\pi/a$ in the Brillouin zone. Moreover, $\epsilon^{(0)}_{+}(p)$ with $\gamma_{5}=1$ near $p=\pi/a$ has the same  structure as $\epsilon^{(0)}_{-}(p)$ with $\gamma_{5}=-1$ near $p=0$, and similarly, $\epsilon^{(0)}_{-}(p)$ near $p=\pi/a$ has the same  structure as $\epsilon^{(0)}_{+}(p)$ near $p=0$.

The energy spectrum of a $d=2$ massless Dirac fermion. $\epsilon^{(0)}_{\pm}(p)$ corresponds to chirality $\gamma_{5}=\pm 1$, respectively.

Namely, we have species doubling of the Weyl fermion for chiral invariant theory $H_{0 L}=-\frac{\sin ap}{a}$, for example, in the limit $a\rightarrow 0$. (In the conventional continuum theory, a single massless Dirac fermion in $d=2$ will contain one right-handed Weyl fermion and one left-handed Weyl fermion both at $p=0$.)

We may write the solution corresponding to $\epsilon^{(0)}_{-}(p)$ in the form

\begin{eqnarray}\label{doublers}
\psi_{L}(x)&=&\int_{-\pi/2a}^{\pi/2a}\frac{dp}{(2\pi)}e^{-ipx}\psi_{L}(p)+\int_{\pi/2a}^{3\pi/2a}\frac{dp}{(2\pi)}e^{-ipx}\psi_{L}(p)\nonumber\\
&+&e^{i\pi x^{1}/a}\int_{-\pi/2a}^{\pi/2a}\frac{dp}{(2\pi)}e^{-i\epsilon^{(0)}_{+}(p)t+ipx^{1}}\psi_{L}(p+\pi/a)\nonumber\\
&\equiv&e_{L}(x)+e^{i\pi x^{1}/a}\sigma_{1}\sigma_{3}\mu_{R}(x)
\end{eqnarray}

by choosing the Brillouin zone $-\pi/2a\leq p<3\pi/a$. We defined formally two fields $e_{L}(x)$ and $\mu_{R}(x)$ although they are actually part of a single field $\psi_{L}(x)$; when one discusses short distance properties  such as the chiral anomaly of $\psi_{L}(x)$, which imply the maximum extension in momentum space by uncertainty principle,  one cannot separate $e_{L}(x)$ and $\mu_{R}(x)$ for finite $a\neq 0$. We emphasize that $e_{L}(x)$ and $\mu_{R}(x)$ separately cannot define  well-defined local fields for $a\neq 0$ since half of the Brillouin zone is missing in them. Nevertheless, this notation is useful to understand the following discussions.

We note that the chiral current and the fermion number current are anomaly-free
in the lattice regularized

\begin{eqnarray}
{\cal L}=\overline{\psi}_{L}(x)\frac{i}{2a}[\gamma^{\mu}U_{\mu}(x)e^{(-ia\hat{p}_{\mu})}  -e^{(-ia\overleftarrow{\hat{p}}_{\mu})}\gamma^{\mu}U^{\dagger}_{\mu}(x)] \psi_{L}(x)
\end{eqnarray}

in $d=2$ Euclidean space-time for $a\neq 0$. (In fact, one can show that each species doubler, which contains two fermion poles, is anomaly free for $a\neq 0$ by evaluating the Jacobian factor for each species doubler separately in momentum space; this is analyzed in the next subsection.) This is also consistent with the idea of chiral anomaly as a short-distance effect by K. Wilson [56]; the lattice theory which cuts-off the short distance effect does not contain the anomaly. The present lattice model  is thus completely anomaly-free, and all the symmetries specified by $\gamma_{5}$ of the Lagrangian or Hamiltonian hold in the naive form.

One may introduce an infinitesimal external gauge potential $A_{1}(t)$ which is spatially constant in the gauge $A_{0}=0$. One may then examine the semi-classical movement of the electron under the external uniform electric field $eE=e\partial_{t}A_{1}$ starting with the configuration $\epsilon^{(0)}_{-}(p)$ with all the negative energy states $0\leq p \leq \pi/a$ being filled initially as in Fig.<ref>, in analogy with the motion of the electron in condensed matter physics.

A schematic picture of the spectral flow in $d=1+1$ dimensions for the state with the energy spectrum $\epsilon^{(0)}_{-}(p)$. A particle is produced near $p=0$ and a hole is produced near $p=\pi/a$.

A particle creation at the momentum close to $p=0$, driven by the Lorentz force $eE$ of the external weak field, implies a hole creation close to $p=\pi/a$ as in Fig.<ref>, namely, we have a pair production

\begin{eqnarray}\label{d=2-pairproduction}
\psi_{L}+ \bar{\psi}_{L}\ \ \ {\rm or}\ \ \ e_{L}+ \bar{\mu}_{R}.
\end{eqnarray}

with the fermion number balance

\begin{eqnarray}\label{fermion-number balance}
\Delta N&=&eE\Delta t/(2\pi/L) - eE\Delta t/(2\pi/L)\nonumber\\
&=&\frac{e}{2\pi}\int_{0}^{L} dx\int_{0}^{\Delta t}dt \partial_{t}A_{1} -
\frac{e}{2\pi}\int_{0}^{L} dx\int_{0}^{\Delta t}dt \partial_{t}A_{1}=0
\end{eqnarray}

where the momentum increase $eE\Delta t$  by the exerted Lorentz force was divided by the level spacing $2\pi/L$ in momentum space assuming the size of the box to be $L$ to count the fermion number. We have no net fermion number production.

\begin{eqnarray}
\frac{e}{2\pi}\partial_{t}A_{1} =\frac{e}{4\pi}\epsilon_{\mu\nu}F^{\mu\nu}
\end{eqnarray}

in the $A_{0}=0$ gauge agrees with the chiral anomaly in the present $d=1+1$ case, one might be tempted to interpret the above formula (<ref>) by saying that each species doubler produces the standard chiral anomaly but these anomalies are canceled among the species doublers, although the present lattice model is anomaly-free and thus all the considerations above are normal and naive ones; besides, no local field on the lattice with $a\neq 0$ is defined for each species doubler separately. It is shown that the above naive  expectation is not realized; if one wants to understand the anomaly of the species doubler in $-\pi/2a <p\leq \pi/2a $ in the present manner, one needs to consider the configuration in Fig.<ref>.
If one considers a particle production near the Fermi level $\epsilon_{-}=0$, one may consider this time that a hole is generated in the deep inside the Dirac sea at $p=\pi/2a$ by the spectral flow.

A schematic picture of a spectral flow in $d=1+1$ dimensions with a particle production near $p=0$ and a hole production near $p=\pi/2a$.

Thus a particle-hole pair creation $e_{L} + \bar{e}_{L}$ with

\begin{eqnarray}
\Delta N
&=&\frac{e}{2\pi}\int_{0}^{L} dx\int_{0}^{\Delta t}dt \partial_{t}A_{1} -
\frac{e}{2\pi}\int_{0}^{L} dx\int_{0}^{\Delta t}dt \partial_{t}A_{1}=0
\end{eqnarray}

takes place driven by the Lorentz force of the external electric field $eE$ in $d=2$ rather than a net particle production which would be induced by the anomaly.  If one considers the other species doubler defined in $\pi/2a<p\leq 3\pi/2a$ together, the spectral flow fills the hole at $p=\pi/2a$ in Fig.<ref> and produces a hole near $p = 3\pi/2a$, and  obtains $\Delta N=0$ again. Combined together one recovers the result in Fig.<ref>, which is a well-defined and naive relation. In essence, one has basically a single fermion from a point of view of anomalies in the presence of species doubling for $a\neq 0$, although formally one has two fermions.

To have a possible net particle production associated with the chiral anomaly of a Weyl fermion defined in $-\pi/2a<p\leq \pi/2a$, one has to generate the infinitely deep Dirac sea by setting $a\rightarrow 0$ first in Fig.<ref> to have an infinite number of degrees of freedom and then a potential net particle production $\Delta N\neq 0$ would be allowed in the picture of spectral flow. Our view is that the spectral flow in a finite lattice theory cannot realize the chiral anomaly. Besides, to analyze the actual particle production, one would need to ensure the gauge anomaly cancellation by postulating some extra flavors.

Incidentally, the customary assumption of the inevitable  appearance of a pair of "Weyl fermions" (species doublers) in the band diagram in condensed matter physics, which are based on the pseudo-chiral symmetry (implied by the two neighboring bands), is not a prediction of the chiral symmetry of any basic theory such as a fundamental  Hamiltonian (i.e., a multi-electron Schrödinger Hamiltonian) [49]. It is a consequence of simulating the spectrum in condensed matter physics by a chiral gauge theory on the lattice.

§.§.§ No well-defined anomaly for each species doubler for $a\neq 0$

In view of the anomaly calculation as a Jacobian and the analysis of simple examples of the spectral flow for $a\neq 0$ on the lattice in the preceding subsection  combined with the observation of K. Wilson as to anomalies as the  short-distance effects, we argue that the anomaly for each species doubler separately is not well-defined (or absent) for $a\neq 0$.
We  re-examine the Jacobian calculation in (<ref>).
The lattice version of the regularized Jacobian (<ref>) for the localized chiral transformation $\psi(x)\rightarrow e^{i\gamma_{5}\alpha(x)}\psi(x)$ and $\overline{\psi}(x)\rightarrow\overline{\psi}(x) e^{i\gamma_{5}\alpha(x)}$ is given by

\begin{eqnarray}\label{lattice anomaly-x}
\lim_{M\rightarrow \infty}a^{4}\sum_{x} 2i\alpha(x)\sum_{n=1}^{16}tr T^{-1}_{n}\gamma_{5}T_{n} \int_{-\pi/2a\leq k^{\mu}<\pi/2a}\frac{d^{4}k}{(2\pi)^{4}}e^{-ikx}\exp[-(\Dslash)^{2}/M^{2}]e^{ikx}\nonumber\\
\end{eqnarray}

where $
\Dslash = \frac{i}{2a}[\gamma^{\mu}U_{\mu}(x)e^{(-ia\hat{p}_{\mu})}  -e^{(-ia\overleftarrow{\hat{p}}_{\mu})}\gamma^{\mu}U^{\dagger}_{\mu}(x)] $. In this expression,
we used the 16 symmetry operators in (<ref>) and (<ref>) and restricted the momentum range to the physical domain
$-\pi/2a\leq k^{\mu}<\pi/2a$
starting with the fundamental Brillouin zone (<ref>).
If one adopts a view that the continuum limit defines a  physical theory, one would say that each species doubler gives the conventional chiral anomaly in the limit $a\rightarrow0$ (and $M\rightarrow \infty$ later), but because of the species doubling these anomalies cancel completely. Thus the chiral symmetric theory is anomaly free because of the species doubling on the lattice (actually, the species-wise cancellation of anomalies takes place even for $a\neq0$ in the present case due to the symmetry of the hypercubic lattice). This is the common interpretation of the absence of chiral anomaly on the lattice.

On the other hand, from a view point of adopting the theory with finite $a\neq 0$ to be physical, one may first examine the contribution of the physical domain specified by $T_{1}$  for fixed $a\neq 0$ in the above anomaly formula. For fixed $a$, the operator $\Dslash$ is a bounded operator,
$||\Dslash|| = O(1/a)$. Thus the exponential factor goes away in the limit $M\rightarrow\infty$. In this limit, the volume of the momentum space is finite $(\pi/a)^{4}/(2\pi)^{4}$ for $a\neq0$, but the trace operation with $\gamma_{5}$
vanishes ${\rm tr}\gamma_{5}=0$. Namely, the contribution of each species doubler separately vanishes.  This is consistent with the picture of K. Wilson [56], since the finite $a$ cuts off the short distance contributions  completely and thus no anomaly appears.

As long as the notion of species doublers plays a central role, it implies that the spectrum is continuously connected beyond the boundaries of species doublers and the height of the boundaries of species doublers is finite. Thus the notions of species doublers, which imply $a\neq0$, and the well-defined chiral anomaly for each species doubler separately are not compatible. (To our knowledge, no explicit evaluation of the chiral anomaly for each species doubler separately has been performed in lattice gauge theory with fixed $a$ at $a\neq 0$.)  It is a safe statement that the chiral anomaly is not well-defined for each species doubler separately, which is defined in a part of the Brillouin zone and thus not a local field in space-time for $a\neq 0$.

§.§ Modified chiral symmetry on the lattice: Ginsparg-Wilson fermion

A definition of chiral symmetric fermion on the lattice using the idea of a block transformation has been proposed by Ginsparg and Wilson [18, 72, 73, 74, 75]. This scheme utilizes the chiral symmetry generated by a new $\Gamma_{5}$ which is different from the conventional $\gamma_{5}$. This construction may
be useful, not only conceptually but also practically, in considering an effective fermion such as a massless Dirac fermion (and possibly a Weyl fermion) without species doublers in condensed matter and related fields.
Although the construction of chiral fermions in this scheme is still not completed, some aspects of this construction are interesting. We would like to explain the basic aspects of this construction in this subsection. See also D. Kaplan [79] for  more general schemes of chiral fermions and anomalies on the lattice.

§.§.§ Dirac fermion theory without species doublers

The  Ginsparg-Wilson fermion is defined by

\begin{eqnarray}\label{GWfermion}
\end{eqnarray}

on the Euclidean $d=4$ hypercubic lattice, for example; $D$ is a lattice generalization of the covariant Dirac derivative
${\not \!\! D}$.
This fermion operator $D$ satisfies the Ginsparg-Wilson relation which is suggested by the block transformation [18]

\begin{eqnarray}\label{GWrelation}
\gamma_{5}D + D\gamma_{5} = aD\gamma_{5}D
\end{eqnarray}

where $a$ stands for the lattice spacing.
We also impose the hermiticity condition common in Euclidean theory

\begin{eqnarray}
\end{eqnarray}

It is known that one can construct the operator $D$ that satisfies the Ginsparg-Wilson relation with the hermiticity condition  and  contains a single species in the fundamental Brillouin zone, which is mainly concentrated  in the physical  sub-domain

\begin{eqnarray}
-\pi/2a\leq p_{\mu}<\pi/2a
\end{eqnarray}

without any species doublers. See Neuberger in [72, 73] for an explicit construction of the operator $D$ and the locality analysis [74, 75]; see also  [76, 77] and [78] for the detailed analysis of the construction in connection with the anomaly evaluation. The following discussions are, however, understood without having the explicit form of $D$, as the analysis is mostly based on the formal relation (<ref>) and the fact that the species doubling is absent. Intuitively, $D$ contains the Dirac operator appearing in the Wilson action (<ref>) inside it (and thus no doubling with $2\pi/a$ periodic in momentum space) and one uses a modified chiral symmetry operator to define the exact chiral symmetry for $D$, since the chiral symmetry generated by $\gamma_{5}$ is broken in (<ref>).

One can then define the massless Dirac fermion for the vector-like gauge theory such as in QED and QCD (quantum chromodynamics) by a symbolic notation on the lattice

\begin{eqnarray}\label{vector-like Dirac fermion}
\int {\cal D}\overline{\psi}{\cal D}\psi \exp\left[ \sum_{x,y}\bar{\psi}(x)D(x,y)\psi(y) \right]=
\int {\cal D}\overline{\psi}{\cal D}\psi \exp\left[ \int \overline{\psi}D\psi \right].
\end{eqnarray}

One may define the operator

\begin{eqnarray}
\hat{\gamma}_{5}=\gamma_{5}(1-aD), \ \ \ \hat{\gamma}_{5}^{2}=1
\end{eqnarray}

where we used the Ginsparg-Wilson relation (<ref>).
One may then define a (modified) chiral transformation

\begin{eqnarray}\label{chiral-transformation}
\psi\rightarrow e^{i\alpha \hat{\gamma}_{5}}\psi, \ \ \bar{\psi}\rightarrow \bar{\psi}e^{i\alpha \gamma_{5}}
\end{eqnarray}

under which the action is invariant since

\begin{eqnarray}
\gamma_{5}D+D\hat{\gamma}_{5}=0
\end{eqnarray}

due to the Ginsparg-Wilson relation.
Under the transformation (<ref>), we have a Jacobian of the form (this natural appearance of the Jacobian is an advantage of the Ginsparg-Wilson construction, compared to the continuum case discussed in the proceeding subsection)

\begin{eqnarray}\label{lattice-jacobian}
\exp \left[ -2i{\rm Tr}\alpha\frac{1}{2}(\gamma_{5}+\hat{\gamma}_{5}) \right]=\exp\left[-2i{\rm Tr}\alpha\Gamma_{5} \right],
\end{eqnarray}

\begin{eqnarray}\label{new gamma5}
\Gamma_{5}\equiv \gamma_{5}\left( 1-\frac{1}{2}aD \right).
\end{eqnarray}

The above chiral symmetry (<ref>) may formally be replaced by [77]

\begin{eqnarray}\label{chiral-transformation2}
\psi\rightarrow e^{i\alpha \Gamma_{5}}\psi, \ \ \bar{\psi}\rightarrow \bar{\psi}e^{i\alpha
(1-\frac{1}{2}aD)\gamma_{5} }
\end{eqnarray}

using the relation

\begin{eqnarray}
\end{eqnarray}

arising from the Ginsparg-Wilson relation and resulting in the same Jacobian factor (<ref>).

To evaluate the Jacobian (<ref>), one  may classify all the normalizable eigenstates
of the hermitian operator on a finite lattice [80]

\begin{eqnarray}
H\equiv \gamma_{5}D,
\end{eqnarray}

\begin{eqnarray}
\end{eqnarray}

\begin{eqnarray}
(\phi_{n},\phi_{m})=\phi^{\dagger}_{n}\phi_{m}\equiv \sum_{x}\phi^{\dagger}_{n}(x)\phi_{m}(x)=\delta_{nm}.
\end{eqnarray}

All the states on the lattice with a finite spacing $a$ are categorized into the following 3 classes
using the basic relation derived from (<ref>), which is an analogue of $\gamma_{5}\Dslash+\Dslash\gamma_{5}=0$ in continuum theory,

\begin{eqnarray}\label{chiral-algebra}
\Gamma_{5}H + H\Gamma_{5}=0
\end{eqnarray}

with $\Gamma_{5}=\gamma_{5}-\frac{1}{2}aH$ in (<ref>):

(i) Zero modes ($n_{\pm}$ states),

\begin{eqnarray}
H\phi_{n} = 0,\  \gamma_{5}\phi_{n} = \pm \phi_{n},
\end{eqnarray}

since such $\phi_{n} $ can be chosen as the eigenstate of $\Gamma_{5}$, $\Gamma_{5}\phi_{n} = \pm \phi_{n}$.

(ii) Highest states ($N_{\pm}$ states) with $\Gamma_{5}\phi_{n}=0$,

\begin{eqnarray}\label{higheststates}
H\phi_{n} = \pm\frac{2}{a}\phi_{n},\ \gamma_{5}\phi_{n} = \pm \phi_{n},
\end{eqnarray}

(iii) Remaining paired states with $0 < |\lambda_{n}|< 2/a$,

\begin{eqnarray}\label{paired states}
H\phi_{n}=\lambda_{n}\phi_{n},\   H(\Gamma_{5}\phi_{n})=-\lambda_{n}(\Gamma_{5}\phi_{n}),
\end{eqnarray}

with $\Gamma_{5}\phi_{n}\neq 0$
and  $(\phi_{n}, \Gamma_{5}\phi_{n})=0$. Note that $\lambda_{n}$ and $-\lambda_{n}$ appear in pairs. Sandwiching the Ginsparg-Wilson relation (<ref>) by $\phi^{\dagger}_{n}\gamma_{5}$
and $\phi_{n}$, one obtains $(\phi_{n}\gamma_{5},\phi_{n})=\frac{a}{2}\lambda_{n}$ for $\lambda_{n}\neq0$, and thus $2/a \geq |\lambda_{n}|$, namely, the cases analyzed above cover all the possible eigenvalues of $H$.

The sum rule

\begin{eqnarray}
\end{eqnarray}

holds, which  is a result of

\begin{eqnarray}\label{sumrule}
{\rm Tr}\gamma_{5}=\sum_{n} (\phi_{n}, \gamma_{5}\phi_{n})=n_{+}+N_{+}-(n_{-}+N_{-})=0
\end{eqnarray}

that holds even for non-Abelian Yang-Mills fields in the finite dimensional functional space; this relation shows that no chiral anomaly for the symmetry generated by the fundamental $\gamma_{5}$ in the form of the Jacobian. Note that we used

\begin{eqnarray}
\sum_{n} (\phi_{n}, \gamma_{5}\phi_{n})=\sum_{n} (\phi_{n}, \Gamma_{5}\phi_{n}) + \sum_{n} (\phi_{n}, \frac{1}{2}H\phi_{n})=0
\end{eqnarray}

for the states in (<ref>).
The quantum breaking of effective chiral symmetry (non-trivial Jacobian) in (<ref>) is described by the Atiyah-Singer-type index [80]

\begin{eqnarray}\label{index}
{\rm Tr}\Gamma_{5}=\sum_{n} (\phi_{n}, \Gamma_{5}\phi_{n})=n_{+}-n_{-},
\end{eqnarray}

which implies the chiral asymmetry for $n_{+}-n_{-}\neq 0$.
The difference of $\Gamma_{5}$ and $\gamma_{5}$ is very important; the effective chiral symmetry $\Gamma_{5}$ projects out the highest states $N_{\pm}$ in (<ref>) on a finite lattice and realizes the chiral asymmetry (<ref>).  It may be natural to define the Hilbert space for $a\neq 0$ in the present scheme using $\Gamma_{5}$, which is a good symmetry of the action, then those  $N_{\pm}$ states are projected out from the evaluation of the index as in (<ref>) (and also in the smooth limit $a\rightarrow 0$).

The index is also written with a regulator as

\begin{eqnarray}\label{index2}
\sum_{n} (\phi_{n}, \Gamma_{5}e^{-H^{2}/M^{2}}\phi_{n})=n_{+}-n_{-},
\end{eqnarray}

which is independent of the values of $M$. If one first considers $a\rightarrow 0$ and then $M\rightarrow \infty$,
the index $n_{+}-n_{-}$ is given by the Chern-Pontryagin number, $\sim\int d^{4}x F\tilde{F}$ in the continuum limit for non-Abelian gauge theory, which is an analogue of the integral of $\frac{1}{4\pi^{2}}\vec{E}\cdot\vec{B}$ in Abelian theory.
The index vanishes for Abelian theory, but we assume the formal index relation since a local version of ${\rm Tr}\Gamma_{5}$ gives the correct anomaly in the continuum limit; we define a local version of (<ref>) using the plane waves and the trace with respect to Dirac indices (and Yang-Mills  indices  in the case of non-Abelian theory) [78],

\begin{align}\label{local-index}
&\lim_{M\rightarrow \infty}\lim_{a\rightarrow 0}\int_{B} \frac{d^{4}p}{(2\pi)^{4}}{\rm tr} \left\{e^{-ipx}\Gamma_{5}\exp\left[ -\frac{(\gamma_{5}D)^{2}}{M^{2}} \right]e^{ipx} \right\}\nonumber\\
\end{align}

where the momentum integral is over the fundamental Brillouin zone; the eigenvalues of $H=\gamma_{5}D$ blow up outside the physical domain $-\pi/2a\leq p_{\mu}<\pi/2a$ in the limit $a\rightarrow 0$ first and this information relies on the specific construction of $D$ in addition to the Ginsparg-Wilson relation. In the physical domain, $\gamma_{5}D\rightarrow \gamma_{5}\Dslash$ in the limit $a\rightarrow 0$. Also, $\Gamma_{5}\sim \gamma_{5} -\frac{1}{2}aM \rightarrow \gamma_{5}$ inside the expression (<ref>) in the limit $a\rightarrow 0$ with $M$ kept finite, and one thus recovers the continuum formula of the Jacobian evaluation for  $\gamma_{5}$,

\begin{eqnarray}\label{anomaly3}
\lim_{M\rightarrow \infty}{\rm Tr}\{ 2i\alpha(x)\gamma_{5}\exp[-\Dslash^{2}/M^{2}]\}
=\int d^{4}x~\alpha(x)\frac{1}{4\pi^{2}}\vec{E}\cdot\vec{B}.
\end{eqnarray}

The (formally integrated) relation, which is an analogue of the Atiyah-Singer index theorem,

\begin{eqnarray}
n_{+}-n_{-}=\int d^{4}x~\frac{1}{4\pi^{2}}\vec{E}\cdot\vec{B}
\end{eqnarray}

is concluded from (<ref>).

As for the relation
$ {\rm Tr}\gamma_{5}=\sum_{n} (\phi_{n}, \gamma_{5}\phi_{n})=n_{+}+N_{+}-(n_{-}+N_{-})=0$, this is a specific relation for a lattice with $a\neq 0$. But it contains information on an anomaly relation as above in the limit $a\rightarrow 0$ with a suitable regularization, which replaces $N_{+}-N_{-}$ by the anomaly factor in the limit $a\rightarrow 0$. In fact, if one starts with ${\rm Tr}\Gamma_{5}$,

\begin{align}
{\rm Tr}\Gamma_{5}&={\rm Tr} \left(\gamma_{5}-\frac{a}{2}H \right)\nonumber\\
&={\rm Tr} \left( -\frac{a}{2}H \right)\nonumber\\
&=\left( -\frac{1}{2} \right) \int d^{4}x ~tr \gamma_{5} \int_{B} \frac{d^{4}k}{(2\pi)^{4}}e^{-ikx}aDe^{ikx}
\end{align}

and using the explicit formula for the Ginsparg-Wilson operator $D$, one can show that

\begin{eqnarray}
-(N_{+}-N_{-})=\lim_{a\rightarrow 0} \left( -\frac{1}{2} \right) tr \gamma_{5} \int_{B} \frac{d^{4}k}{(2\pi)^{4}}e^{-ikx}aDe^{ikx}= -\int d^{4}x ~\frac{1}{4\pi^{2}}\vec{E}\cdot\vec{B}.
\end{eqnarray}

We have thus demonstrated that the massless Dirac fermion for vector-like gauge theory such as QCD and QED can be formulated using the Ginsparg-Wilson fermion without any species doublers by maintaining the exact chiral symmetry defined by $\gamma_{5}$ and $\hat{\gamma}_{5}$ (or essentially $\Gamma_{5}$), and the chiral anomaly is evaluated as a Jacobian factor  as in continuum theory and in fact in a more logically consistent manner.
This is a major achievement of the Ginsparg-Wilson fermion.

This final result may be compared with the formulation with the Wilson action using $\gamma_{5}$ in (<ref>); the same anomaly relation is obtained by the chiral symmetry defined by $\gamma_{5}$ in the continuum limit $a\rightarrow 0$, which is however strongly broken by the large effective masses  of the order $r/a$  of species doublers for any finite $a\neq 0$ in the Wilson action.  The major issue is the compatibility of the presence of exact chiral symmetry and the absence of species doublers.

§.§.§ Lattice Chiral fermions without species doublers

It is interesting to examine if one can incorporate the chiral fermions (Weyl fermions) in the scheme of Ginsparg and Wilson.
We define two projection operators,

\begin{eqnarray}\label{projectors}
P_{\pm}=\frac{1}{2}(1\pm \gamma_{5}),\ \ \ \hat{P}_{\pm}=\frac{1}{2}(1\pm \hat{\gamma}_{5})
\end{eqnarray}

and define the chiral components by

\begin{eqnarray}\label{chiral-components}
\psi_{R,L}=\hat{P}_{\pm}\psi, \ \ \bar{\psi}_{L,R}=\bar{\psi}P_{\pm}
\end{eqnarray}

which satisfy (in a simplified notation)

\begin{eqnarray}\label{GWfermion2}
S=\int \bar{\psi}_{L}D\psi_{L}+\int \bar{\psi}_{R}D\psi_{R}
\end{eqnarray}

using  the identity arising from the Ginsparg-Wilson relation

\begin{eqnarray}\label{decomposition of Ginsparg-Wilson operator}
\end{eqnarray}

We thus define the left-handed massless fermion on the lattice by

\begin{eqnarray}\label{chiral fermion}
\int {\cal D}\overline{\psi}_{L}{\cal D}\psi_{L} \exp \left[ \int \overline{\psi}_{L}D\psi_{L} \right]
\end{eqnarray}

which satisfies the exact chiral symmetry generated by $P_{+}$ and $\hat{P}_{-}$ and no species doublers appear.
The present definition of the chiral fermion slightly differs from the conventional definition in continuum theory, in that we do not define the generator of chiral gauge symmetry in the present construction. We instead project the vector-like theory by  gauge field dependent chiral projection operators to define a chiral fermion theory.

In the interacting case with gauge fields included in $D(U)$, one may tentatively assume

\begin{eqnarray}\label{chiral fermion2}
\int {\cal D}U{\cal D}\overline{\psi}_{L}{\cal D}\psi_{L} \exp\left[\int \overline{\psi}_{L}D\psi_{L}\right].
\end{eqnarray}

As for the fermion number anomaly associated with

\begin{eqnarray}
\psi_{L}\rightarrow e^{i\alpha(x)}\psi_{L}, \ \ \  \overline{\psi}_{L}\rightarrow \overline{\psi}_{L}e^{-i\alpha(x)}
\end{eqnarray}

one has the Jacobian

\begin{eqnarray}
J=\exp[-{\rm Tr}i\alpha(x)\hat{P}_{-} +{\rm Tr} i\alpha(x) P_{+}]=\exp[i {\rm Tr} \alpha(x) \Gamma_{5}].
\end{eqnarray}

This anomaly may be evaluated in the continuum limit just as (<ref>). In the continuum limit $a\rightarrow 0$ for a smooth gauge field configurations, it is confirmed that one recovers the continuum path integral formula with the correct fermion number anomaly without encountering the species doublers. In this sense, the present scheme has (partially) achieved an important task to define a lattice chiral fermion without species doublers with a desired properties of a chiral fermion in the continuum limit.

However, the truly satisfactory definition of the path integral for a chiral fermion is still missing. The remaining issue is to define the path integral measure to analyze the anomalies associated with gauge symmetry. Simply stated, it is not easy to generate the consistent form of gauge anomalies in the path integral defined on the lattice. Since the lattice theory is supposed to define everything finite and thus consistently, the final formulation needs to produce the consistent form of gauge anomaly by a Jacobian factor.
This task has been achieved for an Abelian gauge theory   [81, 82], but the formulation for non-Abelian theory is still missing. In continuum theory, one usually defines the so-called covariant form of anomaly which is sufficient to analyze the anomaly cancellation in the Standard Model [71, 67]; in this case, one can treat the Jacobian for the gauge anomaly by imitating the analysis of Atiyah-Singer-type index, namely, as in the manner we have done so far in this article. There is a well-defined prescription to convert the covariant form of anomaly, which is easy to evaluate, to the consistent form of anomaly in continuum theory [57]; this fact shows that the anomaly cancellation condition is the same for both forms of anomalies in continuum theory.

Technically, the Ginsparg-Wilson operator is exponentially local [74, 75] while the conventional lattice Lagrangian is ultra-local; this difference may be another reason why one evades the appearance of species doubling in the Ginsparg-Wilson scheme.

§.§.§ Comment on Ginsparg-Wilson fermion and related issues

It is interesting to examine if one can use the Ginsparg-Wilson fermion [18, 72, 74, 75], which contains no species doubling, to simulate an effective fermion  in the context of practical applications [The idea of species doublers asserts that even a massless Dirac fermion (not a chiral fermion) such as in massless QED cannot avoid the appearance of species doublers in the conventional lattice formulation.].
This is compared with the original proposal of Wilson (<ref>) which gives the correct chiral anomaly but the chiral symmetry defined in terms of $\gamma_{5}$ is broken  strongly in the Lagrangian.
The massless Dirac fermion, which is defined without technical complications, is given by (<ref>)

\begin{eqnarray}\label{vector-like Dirac fermion-2}
\int {\cal D}\overline{\psi}{\cal D}\psi \exp\left[ \int \overline{\psi}D\psi \right].
\end{eqnarray}

This construction, when one should suitably rotate back to the Minkowski theory,  may be useful in the applications to the condensed matter physics, for example. The conventional chiral $U(1)$ anomaly and consistent chiral anomaly, if necessary, are evaluated.
One may simulate a massless Dirac fermion (such as Dirac semi-metal) coupled to the electromagnetic field without worrying about the species doubling.

The Ginsparg-Wilson fermion gave an impressive progress in the lattice gauge theory, but not complete yet. In this respect, one may still examine other constructions [79], and also the staggered fermions [83, 84] and its generalizations, for example. The situation in the staggered fermion is not rosy, as was forcefully criticized by M. Creutz [85] among others. But the issue of constructing satisfactory lattice fermions is still alive and remains interesting.

§ SOME EXAMPLES IN NUCLEAR PHYSICS AND RELATED FIELDS

§.§ Species doubling and chiral anomaly

We first comment on  the general aspects of the proposal of Nielsen and Ninomiya [49] on the possible implications of chiral anomalies in condensed matter physics. This paper contains many stimulating ideas and  it has been very influential in the recent developments, although this paper does not mention Berry's phase explicitly. The main theme of this paper is the use of a chiral fermion in lattice gauge theory with the notion of species doublers to simulate the Weyl fermion in condensed matter physics.

The authors also made important contributions to the subject of species doubling [17]. They argued that the species doubling is an inevitable consequence of the definition of fermions on a general class of lattice. They make use of
the torus structure of the momentum space description of lattice fermions, for example,

\begin{eqnarray}
-\frac{\pi}{2a}\leq p_{\mu}<\frac{3\pi}{2a},
\end{eqnarray}

in the case of the hypercubic lattice (in their $1+3$ lattice setting [83], they use the 3-dimensional torus) and the important assumption of the existence of the well-defined chiral charge or charge density specified by $\gamma_{5}$. The existence of the well-defined chiral charge implies the absence of chiral anomaly, as is mentioned below.
They also analyze an interesting construction of chiral fermions starting with the two-level crossing in  lattice theory, which is similar to our derivation of Berry's phase in the two-level crossing problem in Section 2.
Interested readers are referred to the original work [17].

Stated simply for an explicit example, the notion of species doublers means that the chiral fermion which is invariant under the chiral symmetry generated by $\gamma_{5}$

\begin{eqnarray}\label{lattice chiral fermionS5}
S=\int d^{4}x ~\overline{\psi}(x) \left[ \gamma^{\mu}i(\partial_{\mu} -ieA_{\mu}) \right]\left( \frac{1-\gamma_{5}}{2} \right) \psi(x),
\end{eqnarray}

inevitably contains multiple species of chiral fermions when transcribed to a theory defined on the lattice by preserving $\gamma_{5}$ symmetry, which is a generalization of the analysis of K. Wilson[15]. In the present case defined on a 4-dimensional hypercubic lattice, which is Lorentz invariant in the continuum limit, one in fact obtains 8 left-handed fermions and 8 right-handed fermions  as has been confirmed by an elementary analysis in Section 4. The crucial assumption is the existence of chiral symmetry generated by $\gamma_{5}$. We have already seen in Section 4 that the original lattice model of K. Wilson [15], which does not have the chiral symmetry, does not contain species doublers  for $a\rightarrow0$.

The action (<ref>) transcribed to a theory on a lattice satisfies the naive chiral identity, namely,

\begin{eqnarray}\label{No anomalyS5}
\partial_{\mu}J^{\mu}_{L}=0
\end{eqnarray}

for a chiral current $J^{\mu}_{L}$ defined on the lattice. The theory defined on the lattice is well-regularized, and if chiral symmetry is preserved, leads to an anomaly-free Ward-Takahashi identities when combined with ${\rm Tr}\gamma_{5}=0$, which ensures the vanishing Jacobian. The above conservation implies that both the charge conservation and the fermion number conservation are satisfied in a naive form.

Coming back to the analysis of possible implications of anomalies in condensed matter physics in [49], their basic assumption is the realization of chiral anomalies in terms of spectral flow. This is partly because it is not simple to evaluate chiral anomalies directly in their scheme of Weyl fermions on the lattice. As we emphasized in Section 4, a local field in space-time is not defined for each species doubler
separately, since the species doubler is defined only in a part of the Brillouin zone in momentum space; if one should be able to define a local field in space-time using only a part of the Brillouin zone, one would define such a field from the beginning instead of using the entire fundamental Brillouin zone.  This loss of locality of each species doubler separately suggests that the chiral anomaly, which is a short-distance effect [56], is not well-defined for
each species doubler separately [Very intuitively, a left-handed chiral fermion on the lattice in the scheme of species doubling is left-handed in a half of the Brillouin zone and right-handed in the other half of the Brillouin zone.]. In fact, we illustrated this fact for a simple $1+1$ dimensional model in Fig.4.2 and Fig.4.3 in Section 4. A semi-classical analysis with the assumed classical equations of motion in momentum space works to explain some aspects of chiral anomaly such as the particle creation of a left-handed fermion in the case of continuum theory, for which one can confirm the existence of the chiral anomaly by the conventional method. But it does not work in lattice theory with $a\neq 0$ where the chiral charge is assumed to be well-defined with the appearance of species doublers; the well-defined chiral charge implies the absence of chiral anomaly.  The spectral flow instead implies the pair production in a general sense in the case of a  smooth spectrum covering the neighboring species doublers in lattice theory; in a theory with well-defined chiral charges and thus with no chiral anomalies (<ref>), it is natural that the spectral flow does not generate chiral anomalies. A statement equivalent to this fact is also found in [49].

We thus understand that the analysis in [49] suggests  that the analogue of chiral anomaly in condensed matter physics is the pair production in a general sense in lattice gauge theory. In fact, they state in Abstract of their paper [49], “
For such materials, in the presence of parallel electric and strong magnetic
fields, there exists an effect similar to the ABJ  (Adler-Bell-Jackiw) anomaly that is the movement of the electrons in the energy-momentum
space from the neighborhood of one degeneracy point to another one”, which is a very accurate and interesting observation, to be consistent with (<ref>). This effect may be termed properly as phenomenon similar to the chiral anomaly, although  in their lattice model one has no chiral anomaly for a local current due to the appearance of species doublers.

In the case of the Ginsparg-Wilson fermion [18, 72, 74, 75] discussed in Section  4, we have shown that one can construct a suitable latticized vector-like QCD (or QED) Lagrangian with chiral symmetry but  without species doublers, which evades the no-go argument of Nielsen and Ninomiya, but with modified chiral operator $\Gamma_{5}$ and the action is defined with a more general notion of locality. As for the chiral fermion, one can construct a single Weyl fermion without species doublers for the Abelian gauge theory using the Ginsparg-Wilson fermion. But one then encounters the chiral anomaly even for the lattice fermion arising from ${\rm Tr}\Gamma_{5}\neq 0$ and has to add other “flavors” to cancel the gauge anomaly, just as in the Standard Model in continuum  where the anomaly cancellation among leptons and quarks is essential. Thus the doubling of Weyl fermions in a  different context is required if one asks the absence of chiral gauge anomalies in the framework of  the Ginsparg-Wilson fermion in lattice theory. The difference between the Ginsparg-Wilson fermion and the conventional lattice fermion with species doublers is that we have a well-defined anomaly for the chiral fermion in the Ginsparg-Wilson fermion and thus it should be canceled by other flavors, while in the construction with species doublers with $\gamma_{5}$,  the absence of anomaly is ensured by the appearance of species doublers which are not local fermion fields separately for $a\neq 0$.

§.§ Chiral magnetic effect

We comment on  the “chiral magnetic effect” first suggested by Nielsen and Ninomiya in condensed matter physics [49], although the term “chiral magnetic effect” was not used. The early use of the term “chiral magnetic effect” appears in  Kharzeev, McLerran and Warringa [86] with an emphasis on the effect of the sphaleron. The notion of the sphaleron is crucial and central to the chiral magnetic effect in nuclear physics, while no notion of the sphaleron appears in the condensed matter contents. In this sense, these two phenomena should better be distinguished, although we use the term “chiral magnetic effect” for both phenomena for simplicity.

We  first mention the chiral magnetic effect in condensed matter physics, as one of the concrete physical suggestions [49]. This effect is generically written in the form

\begin{eqnarray}
J_{k}= \frac{e^{2}}{4\pi^{2}}(\mu_{R}-\mu_{L}) B_{k}
\end{eqnarray}

where $J_{k}$ stands for the electromagnetic current induced by a strong external magnetic field $B_{k}$ and  the difference $(\mu_{1}-\mu_{2})$ of the chemical potentials of two states with different chirality in the thermal equilibrium [49].  The crucial aspect of this relation is that the induced current is proportional to the direction of the external magnetic field. The current thus satisfies $\vec{\nabla}\cdot \vec{J}=0$ naively.
The  chiral magnetic effect in the quark-gluon plasma is written in a similar form under a suitable assumption and can be tested by
experiments [50].
Our interest in the context of the present review is whether this relation is regarded as a consequence of  quantum anomalies and possibly Berry's phase.

§.§.§ Chiral magnetic effect in condensed matter physics

We examine the analysis of Nielsen and Ninomiya [49]. As we have already emphasized, their setting of a lattice model does not contain any chiral anomalies in the conventional sense. They assume that the idea of chiral invariant species doublers in lattice gauge theory is applicable to simulate the condensed matter physics defined in the Brillouin zone, and thus the species doublers ensure the absence of chiral anomalies. Intuitively speaking, their analysis corresponds to the chiral Lagrangian

\begin{eqnarray}\label{chiral fermion on the latticeS5}
S=\int dt d^{3}x ~\overline{\psi}(x) \left[\gamma^{0}i(\partial_{t} -ieA_{0}) + \gamma^{k}i(\partial_{k} -ieA_{k}) \right] \left(\frac{1-\gamma_{5}}{2} \right)\psi(x)
\end{eqnarray}

placed on a lattice. This Lagrangian is invariant under the chiral transformation defined by $\gamma_{5}$ and that all the naive operations are well-defined in a theory defined on the discrete lattice, and thus the chiral identities hold without any anomalies if one uses ${\rm Tr}\gamma_{5}=0$.
Moreover, the local chiral fermion number current and the local chiral electromagnetic current are proportional to each other. One thus obtains the single relation on the lattice

\begin{eqnarray}\label{chiral current conservationS5}
\partial_{\mu}J^{\mu}_{L}(x)=0
\end{eqnarray}

as already mentioned in (<ref>). One naively encounters the  left- and right-handed fermions in momentum space  because of the assumed species doubling on the lattice, but these species doublers are not local fields in the real space-time for $a\neq0$. One thus has  no anomaly relations, and only the conserved local current as in (<ref>).
This is different from the massless Dirac fermion in continuum where one encounters two Weyl fermions

\begin{align}\label{continuum massless DiracS5}
S=&\int d^{4}x \overline{\psi}(x) \left[\gamma^{0}i(\partial_{t} -ieA_{0}) + \gamma^{k}i(\partial_{k} -ieA_{k}) \right] \left(\frac{1-\gamma_{5}}{2} \right)\psi(x) \nonumber\\
&+\int d^{4}x ~\overline{\psi}(x) \left[\gamma^{0}i(\partial_{t} -ieA_{0}) + \gamma^{k}i(\partial_{k} -ieA_{k}) \right] \left(\frac{1+\gamma_{5}}{2}\right)\psi(x)
\end{align}

for which one can define two local currents with an anomaly [46, 47]

\begin{eqnarray}
\partial_{\mu}J^{\mu}(x)=0, \ \ \ \partial_{\mu}J^{\mu}_{5}(x)= \frac{e^{2}}{2\pi^{2}}\vec{E}\cdot\vec{B}.
\end{eqnarray}

From the point of view of chiral anomalies, these two theories are very different. Besides, one  encounters more than two Weyl-type fermions in the momentum space in (<ref>) and two Weyl fermions in (<ref>).

They then introduce the idea of the spectral flow in the setting of (<ref>) [49]. The spectral flow can describe the essence of the chiral anomaly appearing in a theory which gives rise to chiral anomalies in the conventional evaluation, by analyzing the local properties in momentum space, namely, on the basis of the behavior near the origin of the momentum space.  But the spectral flow does not generate the chiral anomaly for a theory which has no anomalies in the conventional sense. The spectral flow is based on the behavior near the origin, while the chiral anomaly is sensitive to the global behavior of the spectrum, in particular, the ultraviolet behavior of the spectrum.

In their analysis, the spectral flow thus induces the pair production in a general sense, as we already emphasized (and also mentioned in their paper [49]); the species doubler separately, which is not a local field for $a\neq 0$, does not give well-defined anomalies in the conventional sense.  The analysis in [49] is thus regarded as a semi-classical analysis in a theory which is intrinsically anomaly-free.
Thus technical details of their analysis aside, their results do not contain the chiral anomaly as we understand in field theory (except for the limit $a\rightarrow0$ for which species doublers become dis-connected chiral fermions, but then their interpretation in the context of condensed matter physics is not obvious.). But the pair production in a general sense gives rise to interesting effects such as the chiral magnetic effect.

One thus starts with the conservation condition of the chiral current in (<ref>).
This conservation means two facts. Firstly, the possible gauge anomaly does not appear, and one can thus ensure the charge conservation condition. Secondly, the  chiral fermion number current has no anomaly, namely, no net particle production. As we have emphasized, each species doubler is not a local field in space-time and thus the notion of anomaly for each species doubler separately is ill-defined for $a\neq 0$.  Nevertheless they apply a classical spectral flow to each species doubler separately and infer the particle creation rate which is the same as the chiral anomaly naively assumed for each species doubler separately, namely, the particle creation rate under the parallel $E$ and $B$

\begin{eqnarray}
\pm \frac{e^{2}}{4\pi^{2}}EB
\end{eqnarray}

for the right-handed and left-handed species doubler, respectively; this value is sensible in the limit $a\rightarrow 0$, and may be that it is a reasonable guess without any other ways to estimate it. Assuming that the external electric force $E$ pushes up the Fermi surface of the right-handed doubler  $\mu_{R}$ and pushes down the left handed doubler $\mu_{L}$, they obtain the energy balance condition for the generated electromagnetic current $J_{A}$

\begin{eqnarray}
EJ_{A}=\frac{e^{2}}{4\pi^{2}} EB (\mu_{R} - \mu_{L}),
\end{eqnarray}

\begin{eqnarray}\label{chiral magnetic effect1}
J_{A}=\frac{e^{2}}{4\pi^{2}} B (\mu_{R} - \mu_{L})
\end{eqnarray}

which is a generic form of the proposed chiral magnetic effect in a theory of two species doublers of Weyl-type fermions.
They also define the relaxation time $\tau_{1}$ for the transfer from the right-handed Weyl cone into the left-handed  Weyl cone, which are described by species doublers,  by the external electric field $E$

\begin{eqnarray}\label{chemical potentialS5}
(\mu_{R} - \mu_{L})=evE\tau_{1}
\end{eqnarray}

with the effective velocity $v$ of the Weyl fermion appearing at the level crossing point

\begin{eqnarray}
\epsilon(p)=\pm v\vec{p}\cdot\vec{\sigma}.
\end{eqnarray}

Readers are referred to the original paper for more details [49].

If one combines (<ref>) with (<ref>), the magnitude of anomalous vector current $J_{A}$ is formally proportional to $EB$ and in this sense  analogous to the anomalous particle creation itself (or more precisely, the pair production rate in their model) but now dealing with an electromagnetic  current.

The derivation of (<ref>) is based on a semi-classical analysis in an anomaly-free (lattice) theory by assuming a specific form of the particle creation rate for each species doubler separately implied by the idea of spectral flow. The creation rate is reasonable for a semi-classical flow of the particle, i.e., a pair production rate in a general sense induced by  the applied $EB$. The notion of quantum anomaly we know in field theory does not appear,  but nevertheless an analogue of the chiral anomaly appears and leads to a  suggestion of the stimulating phenomenon (<ref>).

§.§.§ Chiral magnetic effect in quark-gluon plasma

We next discuss the derivation of the chiral magnetic effect in the quark-gluon plasma. This effect has been studied in connection with the effect of the sphaleron such as in [86]. More recent analyses of the effect in the context of nuclear physics are found in
[87, 88, 89], for example.
We comment on the derivation of the chiral magnetic effect in the quark-gluon plasma following Fukushima, Kharzeev and Warringa (FKW) [50].  They assume that the chiral chemical potential $\mu_{5}$, which gives rise  to the left-right asymmetry, is generated by some mechanism (such as the spharelon) in the basic QCD; their scheme is thus logically more transparent and differs in an important way from the chiral magnetic effect we discussed in the context of condensed matter physics. They then examine the mechanism of the chiral magnetic effect induced by the external strong magnetic field $B$ in great detail; they present several different ways to derive the formula of the proposed chiral magnetic effect and show that all of them lead essentially to the same conclusion.

They apply first the idea of the (justified) spectral flow to the Dirac fermion in (<ref>). They then infer the right-hand and left-handed particle creation  at

\begin{eqnarray}\label{spectral flowS5}
\pm \frac{p_{F}}{2\pi}\cdot \frac{eB}{2\pi} =\pm \frac{e^{2}}{4\pi^{2}}\vec{E}\cdot\vec{B}t
\end{eqnarray}

using the transverse Landau level density $eB/2\pi$ and the longitudinal momentum change near the Fermi surface by the applied electric field $p_{F}=eEt$.  The rate of the change of the chiral particle number $N_{5}=N_{R}-N_{L}$ is then given by

\begin{eqnarray}\label{spectral flow2S5}
\frac{d^{4}N_{5}}{dtd^{3}x}=\frac{e^{2}}{2\pi^{2}}\vec{E}\cdot\vec{B}
\end{eqnarray}

which agrees with the rate given by the chiral anomaly for the Dirac fermion (<ref>) in continuum.  When one writes the chiral chemical potential $2\mu_{5}=\mu_{R}-\mu_{L}$, the energy balance of the particle creation is given by

\begin{eqnarray}\label{energy balanceS5}
\int d^{3}x ~\vec{E}\cdot\vec{j}=\mu_{5}\frac{dN_{5}}{dt}=\frac{e^{2}\mu_{5}}{2\pi^{2}}\int d^{3}x ~\vec{E}\cdot\vec{B}
\end{eqnarray}

where $\vec{j}$ is a current which is supposed to be generated by $\mu_{5}$ and the chiral anomaly. By choosing $\vec{E}$ constant and parallel to $\vec{B}$, one obtains the formula of the chiral magnetic effect

\begin{eqnarray}\label{integrated chiral magnetic effect}
\vec{J}=\int d^{3}x ~\vec{j}= \frac{e^{2}\mu_{5}}{2\pi^{2}}\int d^{3}x ~\vec{B}.
\end{eqnarray}

They then set $\vec{E}=0$ in the final formula.

This derivation is close to that of Nielsen and Ninomiya [49], but there are important differences; namely, the chiral anomaly for two Weyl fermions separately exists in the present case, which one may evaluate in the conventional manner also, and thus one may confirm that the estimate by the spectral flow agrees with the chiral anomaly. Secondly, they set $\vec{E}=0$ in the end since $\mu_{5}$ is assumed to be generated by QCD; this makes the chiral magnetic effect  induced purely by the magnetic field. This derivation may show that the chiral magnetic effect is the consequence of the chiral anomaly for the Dirac fermion, provided that $\mu_{5}$ is generated by QCD separately.

In this derivation, two technical questions come to our mind: Firstly, if the effect arises from the genuine chiral anomaly, which is valid for any space-time dependence of $A_{\mu}(t,\vec{x})$ allowed in the model,  the general (very rapid) time dependence of $A_{\mu}(t,\vec{x})$ might influence the final result. Secondly, the presence of $\mu_{5}$ in the action may influence the chiral anomaly evaluation itself. As for the first question, one may choose $B$ as $F_{12}(t,\vec{x})$ and $E$ as $F_{03}(t,\vec{x})$, and then $\vec{E}$ and $\vec{B}$ are chosen almost independently. One may thus conclude the local relation

\begin{eqnarray}\label{local version of chiral magnetic effect}
\vec{j}(t,\vec{x})= \frac{e^{2}\mu_{5}}{2\pi^{2}} \vec{B}(t,\vec{x})
\end{eqnarray}

from (<ref>) instead of the integrated version (<ref>), and one may conclude

\begin{eqnarray}
\vec{\nabla}\cdot\vec{j}(t,\vec{x})= 0
\end{eqnarray}

which is a strong constraint. This may imply that the relation (<ref>), which is not manifestly Lorentz invariant, is valid for the static cases; this is natural at (near) thermal equilibrium.

As for the second question of the possible effect of the chemical potential on the anomaly evaluation, FKW analyze the effective action

\begin{eqnarray}\label{effective action with chemical potential}
S=\int d^{4}x ~\overline{\psi}(x)[i\Dslash -m]\psi(x)
\end{eqnarray}

\begin{eqnarray}
\Dslash =\gamma^{\mu}(\partial_{\mu} -ieA_{\mu} -ie A^{5}_{\mu}\gamma_{5}).
\end{eqnarray}

FKW then adopt the anomaly relation for the vector current $j^{\mu}(x)$,

\begin{eqnarray}\label{V-A anomaly}
\partial_{\mu}j^{\mu}(x)= -\frac{e^{3}}{4\pi^{2}}F^{A}_{\mu\nu}
\tilde{F}^{V, \mu\nu}
\end{eqnarray}

with $F^{A}_{\mu\nu} =\partial_{\mu}A^{5}_{\nu}-\partial_{\nu}A^{5}_{\mu}$ and $F^{V}_{\mu\nu} =\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. The evaluation of this relation proceeds as follows; one may  start with the evaluation of the divergence of the gauge invariant vector current, namely, the gauge invariant fermion number current $J^{\mu}(x)=\overline{\psi(x)}\gamma^{\mu}\psi(x)$.
This is evaluated in perturbation theory in the path integral, for example, by

\begin{align}
\frac{e^{2}}{2!}\left(\frac{i}{\hbar} \right)^{2}\int d^{4}yd^{4}z\left[\langle T^{\star}J^{\mu}(x)\hat{j}^{\alpha}(y)j_{5}^{\beta}(z)\rangle
+\langle T^{\star}J^{\mu}(x)j_{5}^{\beta}(z)\hat{j}^{\alpha}(y)\rangle\right] A_{\alpha}(y)A^{5}_{\beta}(z)
\end{align}

where $\hat{j}^{\alpha}(x)$ has formally the same form as $J^{\mu}(x)$ and $j_{5}^{\beta}(x)$ is the corresponding axial current. By applying the divergence $\partial^{x}_{\mu}$, one obtains

\begin{eqnarray}\label{anomaly1-S5}
\partial_{\mu}J^{\mu}(x)= -\frac{e^{2}}{4\pi^{2}}F^{A}_{\mu\nu}
\tilde{F}^{V, \mu\nu}
\end{eqnarray}

which is obtained from (the twice of ) a triangle diagram[Note that the standard chiral anomaly (<ref>) is written in the form $-\frac{e^{2}}{8\pi^{2}}F^{V}_{\mu\nu}
\tilde{F}^{V, \mu\nu}$.]

\begin{eqnarray}
\langle T^{\star}J^{\mu}(x)\hat{j}^{\nu}(y)j_{5}^{\sigma}(z)\rangle
\end{eqnarray}

with the conservation conditions $\partial_{\nu}\hat{j}^{\nu}=0$ and $\partial_{\sigma}j_{5}^{\sigma}=0$. The anomaly (<ref>) agrees with the anomaly (<ref>) used by FKW, and this shows that the vector current used in their analysis is the gauge invariant (specific) fermion number current times the charge $e$ (instead of the source current of the electromagnetic field). We thus set

\begin{eqnarray}
j^{\mu}(x)= eJ^{\mu}(x).
\end{eqnarray}

From the relation (<ref>), FKW  infer using an argument of the derivative expansion

\begin{eqnarray}\label{induced currentS5}
j^{\mu}(x)= -\frac{e^{3}}{4\pi^{2}}\epsilon^{\mu\nu\sigma\rho}A^{5}_{\nu}
\end{eqnarray}

which is the leading term for the very slowly varying $A^{5}_{\mu}$. If one sets $eA^{5}_{0}=\mu_{5}$, which is assumed to be constant, and $eA^{5}_{k}=0$, one finally obtains

\begin{eqnarray}\label{derivative expansion}
\vec{j} =\frac{e^{2}\mu_{5}}{2\pi^{2}}\vec{B}
\end{eqnarray}

which agrees with (<ref>) in the static limit with constant $\vec{B}$ in the spirit of the derivative expansion.

This anomaly interpretation (<ref>) of the chiral magnetic effect is mathematically transparent, and it is important that they use the gauge invariant (specific) fermion number current instead of the source current of the electromagnetic current, which differs from the fermion number current by a Bose symmetrization factor of $1/2$ in the present model. The anomaly in the electromagnetic source current spoils the gauge invariance and thus needs to be canceled by some means [One may recall that gauge anomalies are canceled among leptons and quarks in the Standard Model, while the gauge invariant lepton and quark number currents have physical meanings satisfying $B-L=0$.].  The anomaly in the gauge invariant fermion number current is consistent with the spectral flow argument used in (<ref>). How to measure the fermion number current directly is however an interesting issue. See [55] for an analysis of  the related  problem together with an extensive list of references.

FKW also gives other derivations of their suggested relation, which are indirectly related to the anomaly consideration. As a result. they present a general formula of the chiral magnetic effect, which is applicable to QCD plasma,

\begin{eqnarray}
J=N_{c}\sum_{f} q_{f}\left[\frac{q_{f}\Phi}{2\pi}\right]_{F}\frac{L_{z}\mu_{5}}{\pi}
\end{eqnarray}

where  $N_{c}=3$ is the color factor, $q_{f}= \frac{2e}{3}, \frac{-e}{3},\ {\rm and} \ \frac{-e}{3}$ are the charges of quarks. The flux $\Phi$ stands for the magnetic flux and $L_{z}$ is the size of the system in the direction of the magnetic field. $\left[x\right]_{F}$ is the floor function which
is the largest integer smaller than x.

This last formula, which contains the (indirect) effect  of the chiral anomaly in the presence of $\mu_{5}$,  provides a definite prediction and can be tested readily by experiments.  The presence of $\mu_{5}$ provided by QCD, which makes the derivation of the chiral magnetic effect transparent, is different from the other scheme of the chiral magnetic effect already mentioned [49]; the  chiral magnetic effect thus becomes completely independent of $\vec{E}$.

As for the derivation of $\mu_{5}$ from QCD, which is discussed in detail in [50, 86, 87, 88, 89], it is beyond the scope of the present review. See also [90].

§.§ Chiral anomaly from Berry's phase

A similarity of Berry's phase in some physical situations with chiral anomalies has been recognized by various people in the past [11, 12, 91, 92]. As a well-known example among early suggestions, Stone mentioned the similarity of the monopole-type term induced by Berry's phase with the Wess-Zumino term in quantum anomalies [11]. This problem was analyzed in detail in [9]; the essence
is summarized as follows.  The chiral anomaly appears in the shoulder of the functional determinant of the fermion operator

\begin{eqnarray}\label{functional determinant}
{\rm det}\Dslash \rightarrow e^{i\alpha(A)}{\rm det}\Dslash
\end{eqnarray}

with the chiral anomaly factor $\alpha(A)$ such as the Wess-Zumino term. On the other hand, Berry's phase is defined level-wise, namely, a different form of Berry's phase for each energy level of the Dirac operator $\Dslash$ separately. Thus two notions are very distinct in this sense.

Recently, a direct derivation of chiral gauge anomaly
from Berry's phase via the anomalous Hall effect has been suggested in [13, 14].
Their analysis is partly regarded as a variant of Stone's analysis, and it is associated with each energy level of the system instead of the total determinant in the context mentioned above (<ref>), as is shown later. Their derivations themselves are interesting as a derivation of an object similar to chiral anomaly (but the final result does not quite agree). In the following we briefly sketch the essence of the derivation in the paper by Son and Yamamoto [13] as a representative one among related attempts to derive the chiral  anomaly from the anomalous Hall effect, and we add critical comments from our point of view.

Their derivation [13] contains many logical steps which are finally combined in the derivation of $U(1)$ gauge anomaly for a theory defined in the Brillouin zone, although the final result does not agree with what one expects. The crucial ingredient in their derivation is the classical anomalous Poisson brackets induced by the anomalous Hall effect and the simulation of band crossing in condensed matter physics by a chiral lattice gauge theory with species doublers, which are used to ensure the anomaly cancellation at the end.  In the analysis of the anomaly associated with species doublers in lattice theory, we already emphasized that the chiral gauge anomaly of each species doubler by itself is ill-defined for a finite lattice spacing $a\neq0$ in Section 4, since each species doubler is defined in a part of the fundamental Brillouin zone and thus gives rise to a non-local field in space-time.

§.§.§ Various forms of chiral $U(1)$ anomaly

In the analysis of $U(1)$-type gauge anomalies, the distinction of the covariant form of gauge anomaly and the consistent form of gauge anomaly is important, both in the technical sense and in physical implications.
We thus explain the difference between the gauge invariant fermion number current and the gauge non-invariant electromagnetic source current in chiral Abelian gauge theory defined in  continuum. It becomes  transparent if one uses the perturbation approach.
One may start with the chiral fermion in the presence of the background gauge field

\begin{eqnarray}\label{chiral Abelian gauge theoryS5}
S=\int d^{4}x \left\{ \overline{\psi}_{L}(x)i\gamma^{\mu}[\partial_{\mu}- iA_{\mu}(x)]\psi_{L}(x) - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} \right\}.
\end{eqnarray}

One may define the fermion number current by the variational derivative with $\psi(x)\rightarrow e^{i\alpha(x)}\psi(x)$ and $\overline{\psi_{L}}(x)\rightarrow \overline{\psi_{L}}(x)e^{-i\alpha(x)}$, namely,

\begin{eqnarray}\label{chiral fermion-number current}
\end{eqnarray}

which contains an anomaly

\begin{eqnarray}\label{chiral fermion-number anomaly}
\partial_{\mu}J_{L}^{\mu}(x)=-\frac{1}{4\pi^{2}}\vec{E}\cdot \vec{B}.
\end{eqnarray}

Thus the gauge invariant fermion number current in a chiral Abelian gauge theory  is not conserved.
This is evaluated in perturbation theory in the path integral, for example, by

\begin{eqnarray}
\frac{1}{2!}(\frac{i}{\hbar})^{2}\int d^{4}yd^{4}z\langle T^{\star}J_{L}^{\mu}(x) j_{L}^{\alpha}(y) j_{L}^{\beta}(z)\rangle A_{\alpha}(y)A_{\beta}(z)
\end{eqnarray}

and applying the divergence $\partial^{x}_{\mu}$. To define $J_{L}^{\mu}(x)$ to be gauge invariant is to define the chiral currents $j_{L}^{\alpha}(y)$  and $j_{L}^{\beta}(z)$ coupled to gauge fields, which formally have the same form as (<ref>),  to be invariant under the gauge transformations of $A_{\alpha}(y)$ and $A_{\beta}(z)$, respectively. One thus evaluates the triangle diagram

\begin{eqnarray}\label{triangle diagram}
\langle T^{\star}J_{L}^{\mu}(x) j_{L}^{\alpha}(y) j_{L}^{\beta}(z)\rangle
\end{eqnarray}

by imposing the conservation (gauge invariance) conditions on $j_{L}^{\alpha}(y)$  and $j_{L}^{\beta}(z)$,

\begin{eqnarray}\label{current conservation condition}
\partial_{\alpha}j_{L}^{\alpha}(y)=\partial_{\beta}j_{L}^{\beta}(z)=0.
\end{eqnarray}

One then finds the anomalous relation for $J_{L}^{\mu}(x)$ in (<ref>) [46, 47].

To define the gauge anomaly, one examines a vertex of three gauge fields and thus  needs to treat all the currents in (<ref>) on an equal-footing (to ensure the Bose symmetry), namely,

\begin{eqnarray}\label{consistent anomalyS5}
\langle T^{\star}\hat{j}_{L}^{\mu}(x)\hat{j}_{L}^{\alpha}(y) \hat{j}_{L}^{\beta}(z)\rangle
&=& \frac{1}{3}[\langle T^{\star}J_{L}^{\mu}(x) j_{L}^{\alpha}(y) j_{L}^{\beta}(z)\rangle\nonumber\\
&+&\langle T^{\star} j_{L}^{\mu}(x) j_{L}^{\alpha}(y)J_{L}^{\beta}(z)\rangle+\langle T^{\star} j_{L}^{\mu}(x)J_{L}^{\alpha}(y) j_{L}^{\beta}(z)\rangle]
\end{eqnarray}

where $\hat{j}_{L}^{\mu}(x)$ is the source current of the electromagnetic field appearing at a vertex of three gauge fields.
Note that the three currents $\hat{j}_{L}^{\mu}(x)$, $J_{L}^{\mu}(x)$ and $ j_{L}^{\mu}(x)$ are identical if there should be no anomaly, but actually very different due to the anomaly. The differences of these three currents are specified  operationally when one evaluates the triangle diagram; if one imposes the Bose symmetry on three currents, one obtains $\hat{j}_{L}^{\mu}(x)$ on the left-hand side of (<ref>). If one treats two of the three currents in a Bose symmetric manner together with the gauge invariance condition as in (<ref>), the currents are specified to be  $ j_{L}^{\mu}(x)$. As for $J_{L}^{\mu}(x)$, no specification other than the Lorentz and chiral properties in the present context.

One then finds from (<ref>) by noting (<ref>)

\begin{eqnarray}
\partial^{x}_{\mu}\langle T^{\star}\hat{j}_{L}^{\mu}(x)\hat{j}_{L}^{\alpha}(y) \hat{j}_{L}^{\beta}(z)\rangle =
\frac{1}{3}\partial^{x}_{\mu}[\langle T^{\star}J_{L}^{\mu}(x) j_{L}^{\alpha}(y) j_{L}^{\beta}(z)\rangle]
\end{eqnarray}

which means in terms of the operator notation using (<ref>)

\begin{eqnarray}\label{chiral gauge anomaly}
\partial_{\mu}\hat{j}_{L}^{\mu}(x) =-\frac{1}{3}\frac{1}{4\pi^{2}}\vec{E}\cdot \vec{B}.
\end{eqnarray}

This is the chiral gauge anomaly (<ref>) we discussed in Section 4

\begin{eqnarray}\label{gauss2}
\partial_{t}G(t,\vec{x})=\frac{i}{\hbar}[H, G(t,\vec{x})]=-\frac{1}{3}\frac{1}{4\pi^{2}}\vec{E}\cdot \vec{B}.
\end{eqnarray}

Note that the anomaly discussed  by Nielsen and Ninomiya [49] using the argument of fermion creation corresponds to the chiral fermion number anomaly (<ref>) in the context of continuum theory [In the lattice theory with species doublers, the gauge anomaly does not appear for $a\neq 0$. Thus the electric charge conservation is ensured, and the particle production takes place under this constraint. Thus only the pair production in a general context is allowed for $a\neq 0$.], while the method to derive the anomaly discussed by Son and Yamamoto corresponds to the calculational scheme of the (consistent) gauge anomaly (<ref>) (but the result gives the covariant  anomaly), as we explain below. Thus two anomalies are quite different even in the context of continuum theory; the gauge invariant anomaly (<ref>) has a physical meaning, while the gauge non-invariant anomaly (<ref>) which needs to be canceled  has no physical meaning by itself.

§.§.§ Derivation of chiral anomaly from anomalous Hall effect

Coming back to  the possible derivation of chiral gauge anomaly in [13] which contains many logical steps,  we mention the aspects which are related to Berry's phase and the characteristic features of chiral anomalies in the context of the present review. The effective equations of the anomalous Hall effect, which
incorporate Berry's phase  near the level crossing point [35, 28], play a central role in their analysis. The equations of motion (<ref>) are commonly summarized by
the effective action (<ref>) [38]

\begin{eqnarray}\label{action-1S5}
S=\int dt[p_{k}\dot{x}_{k} - eA_{k}(\vec{x},t)\dot{x}_{k} + {\cal A}_{k}(\vec{p})\dot{p}_{k}-\epsilon_{n}(\vec{p})+e\phi(\vec{x},t)],
\end{eqnarray}

and the equations of motion are  derived from the (classical) action for which one  assumes a variational principle.  Without the electromagnetic vector potential $eA_{k}=0$, the action is reduced to the canonical action (<ref>) and thus no anomalous quantum commutation relations and no Nernst effect are induced.  Without $eA_{k}$, one still has the anomalous Hall effect as in the Born-Oppenheimer approximation (<ref>), but no quantum anomaly is  induced. Namely, within the framework of quantum mechanics, the anomalous Hall effect does not induce the quantum anomaly.

If one adds a non-vanishing electromagnetic vector potential $eA_{k}\neq 0$, the system (<ref>) becomes non-canonical and thus classical due to the incompatibility of Berry's phase and the electromagnetic vector potential  $eA_{k}$, as was emphasized in Section 3.  When one reduces the quantum system to a classical system by adding  $eA_{k}$, one induces the Poisson brackets  if  one assumes
an extended phase space formalism[40],

\begin{eqnarray}\label{Poisson bracketS5}
&&\{x_{k},x_{l}\}=\frac{\epsilon^{klm}\Omega_{m}}{1+e\vec{B}\cdot\vec{\Omega}} , \ \ \ \{p_{k},x_{l}\}=-\frac{\delta_{kl}+e\Omega_{k}B_{l}}{1+e\vec{B}\cdot\vec{\Omega}},\nonumber\\
&&\{p_{k},p_{l}\}=- \frac{\epsilon^{klm}eB_{m}}{1+e\vec{B}\cdot\vec{\Omega}},
\end{eqnarray}

for an assumed genuine Dirac monopole $\Omega_{kl}=\epsilon^{klm}\Omega_{m}$.

Son and Yamamoto [13] operate precisely in this classical scheme with Berry's phase and the electromagnetic vector potential. In the second paper of theirs [93] a more abstract argument is given, but we follow the detailed presentation in their original paper [13].  To be specific, they start with the action

\begin{eqnarray}\label{action-2S5}
S=\int dt[p_{k}\dot{x}_{k} - A_{k}(\vec{x},t)\dot{x}_{k} + {\cal A}_{k}(\vec{p})\dot{p}_{k}-H(\vec{x},\vec{p})],
\end{eqnarray}

with $H(\vec{x},\vec{p})$ unspecified and assuming that ${\cal A}_{k}(\vec{p})\dot{p}_{k}$ is given by the genuine Dirac monopole and that the same Poisson brackets in (<ref>) hold (the electric charge $e$ is suppressed).
They then evaluate the classical commutation relations of the density operator $n(\vec{x})$ of the (chiral) fermion number

\begin{eqnarray}\label{anomalous commutatorS5}
[n(\vec{x}), n(\vec{y})] = -i\left( \vec{\nabla}\times \vec{\sigma} +\frac{k}{4\pi^{2}}\vec{B}\right)\cdot \vec{\nabla}\delta(\vec{x}-\vec{y})
\end{eqnarray}

using the above anomalous Poisson bracket with

\begin{eqnarray}
\sigma_{i}(\vec{x})= -\int \frac{d^{3}p}{(2\pi)^{3}}p_{i}\Omega_{k}\frac{\partial n_{\vec{p}}(\vec{x})}{\partial p_{k}}
\end{eqnarray}

where $n_{\vec{p}}(\vec{x})$ is the fermion number density in the Fermi liquid description, and

\begin{eqnarray}\label{flux of Berry's phase}
k=\frac{1}{2\pi}\int d\vec{S}\cdot \vec{\Omega}.
\end{eqnarray}

Eq.(<ref>) may be compared with (<ref>) in the related continuum theory.

They then consider a conserved energy operator $H$ with the static  electromagnetic potential $A_{k}$ (which does not generate the electric field)  but without the scalar potential, and evaluate

\begin{eqnarray}
\dot{n}=i[H, n] =-\vec{\nabla}\cdot \vec{j}
\end{eqnarray}

\begin{eqnarray}\label{Berry phase induced current}
j_{k}=\int \frac{d^{3}p}{(2\pi)^{3}}\left[ -\epsilon_{\vec{p}}\frac{\partial n_{\vec{p}}}{\partial p_{k}} -(\Omega_{i}\frac{\partial n_{\vec{p}}}{\partial p_{i}})\epsilon_{\vec{p}}B_{k} -\epsilon_{\vec{p}}\epsilon^{klm}\Omega_{l} \frac{\partial n_{\vec{p}}}{\partial p_{m}} \right].
\end{eqnarray}

They next turn on the electric field by adding a term to the Hamiltonian

\begin{eqnarray}\label{total HamitonianS5}
H^{\prime} = H + \int d^{3}x \phi(\vec{x})n(\vec{x})
\end{eqnarray}

\begin{eqnarray}\label{derivation of anomaly relationS5}
\dot{n}&=&i[H^{\prime}, n] \nonumber\\
&=&-\vec{\nabla}\cdot \vec{j} - \left(\vec{\nabla}\times\vec{\sigma} +\frac{k}{4\pi^{2}}\vec{B}\right)\cdot \vec{\nabla}\phi(\vec{x})
\end{eqnarray}

which is rewritten as an anomalous conservation law

\begin{eqnarray}\label{anomalyS5}
\partial_{t}n + \vec{\nabla}\cdot \vec{j}^{\prime} = \frac{k}{4\pi^{2}}\vec{E}\cdot \vec{B},
\end{eqnarray}

\begin{eqnarray}
\vec{j}^{\prime}=\vec{j} +\vec{E}\times \vec{\sigma}.
\end{eqnarray}

This is a sketch of the derivation of the relation (<ref>) from the Poisson brackets associated with the anomalous Hall effect. The readers are referred to the original paper for further details [13].

This is an interesting derivation of the chiral relation from the anomalous Hall effect. We now want to add comments on their derivation from our point of view. To be precise, their derivation of (<ref>) is based on the Poisson brackets and thus classical.
Their derivation is based on the Hamiltonian (<ref>) and the Poisson brackets (<ref>). The Hamiltonian (<ref>)
shows that the operator $n(\vec{x})$ is the source of the electromagnetic field $\phi(\vec{x})$, while the value of their anomalous term (<ref>) shows a specific form of the anomalous term for the gauge invariant fermion number density (<ref>).  To be more explicit, the equal time Poisson brackets used in  (<ref>)

\begin{eqnarray}\label{commuttorS5}
\dot{n}(\vec{x})=i[\int d^{3}y \phi(\vec{y})n(\vec{y}), n(\vec{x})] + ...
\end{eqnarray}

shows that it is natural to identify the operator $n(\vec{x})$ appearing on the left-hand side being equal to the operator $n(\vec{x})$ coupled to the gauge field
$\phi(\vec{y})$ in the Hamiltonian. Namely, the evaluation method is the anomaly contained in the source current of the electromagnetic field, i.e., corresponds to (<ref>) in the continuum theory. Unlike the Feynman diagrams, there is no means to distinguish two $n(\vec{x})$ appearing in the commutation relations (<ref>); the relation  (<ref>) in continuum theory, for example, is based on this consideration. Mathematically, this fact shows that the anomalous term (<ref>) should be the Bose symmetric gauge anomaly in (<ref>), namely, the correct result implied by their derivation should be

\begin{eqnarray}\label{anomaly2S5}
\partial_{t}n + \vec{\nabla}\cdot \vec{j}^{\prime} = \frac{1}{3}\frac{k}{4\pi^{2}}\vec{E}\cdot \vec{B},
\end{eqnarray}

but, nevertheless, their anomaly (<ref>) corresponds to the covariant anomaly.  The relation (<ref>) looks like a chiral anomaly but the derivation method and the result do not correspond to any known anomaly.

Physically, this implies that  both quantities $n(\vec{x})$ and $\vec{j}^{\prime}$, which are the source currents of the electromagnetic field, (and $H^{\prime}$ also) are gauge non-invariant (i.e., not conserved) and thus unphysical quantities, just as $G$ and $H$ in (<ref>). If the anomaly (<ref>) is canceled by another species doubler, the Hamiltonian $H^{\prime}$ would become physical, as far as the  quantum anomaly is concerned.

In the derivation of possible chiral anomaly in [13], their basic assumption is that the classical anomalous Poisson brackets (<ref>)  contain all the information of chiral anomalies and all the remaining operations can be done naively. But this assumption is not warranted to be valid when one has  anomalies. Note that all the currents in (<ref>), $\hat{j}_{L}^{\mu}(x)$, $J_{L}^{\mu}(x)$ and $ j_{L}^{\mu}(x)$, are identical without quantum anomalies (and also classically)  but different due to quantum anomalies in continuum; the identification and the distinction of current operators are subtle in the analysis of quantum anomalies.   The crucial factor $1/3$ is obtained most simply by a Bose symmetric Feynman diagram evaluation as in (<ref>).

The present  analysis implies that  the specification of the  anomalous Poisson brackets of the semi-classical density $n(\vec{x})$ (<ref>) alone is not sufficient to analyze the chiral anomaly. To refute the criticism with regard to (<ref>), it would be necessary to specify how to distinguish the covariant current from the consistent form of current based on (<ref>).  So far we treated the anomaly evaluation as the one in continuum theory.
If one should assume the species doubling in the Brillouin zone to cancel the anomaly (<ref>) in [13], one would also need to analyze how to define anomaly for each non-local species doubler separately for $a\neq0$; we, however, emphasized that no anomaly appears  for each non-local species doubler separately.

§.§.§ Remarks on the anomalous Hall effect and chiral anomalies

The local gauge symmetry of Berry's connection enforces each $\varphi_{n}(P)\phi_{n}(x,P)$ in the total  wave function $\Psi(x,P)=\sum_{n}\varphi_{n}(P)\phi_{n}(x,P)$ to be independent in the Born-Oppenheimer approximation. In contrast, the electromagnetic gauge symmetry associated with the vector potential $eA_{k}$ acts universally on all the states in $\Psi(x,P)$. These two independent gauge symmetries cannot be compatible, and they render the action of  the anomalous Hall effect non-canonical and classical if one adds $eA_{k}$. The conventional applications of Berry's phase to the anomalous Hall effect with anomalous Poisson brackets are thus classical and not quantum [35]. In contrast, the quantum mechanical anomalous Hall effect as described by the Born-Oppenheimer approximation, which cannot incorporate $eA_{k}$,  does not produce any quantum anomaly nor the Nernst effect (<ref>).

The distinction of the covariant form of anomaly and the consistent form of anomaly is crucial in the analysis of anomalies. It would be nice if one would analyze this issue and specify what kind of anomaly one is obtaining by the classical Poisson bracket approach to possible anomalies. In their presentation [13], the mathematical formulation
corresponds to the consistent form of anomaly but the actual value of the anomaly is the covariant anomaly; their formula  as it stands does not correspond to any known anomaly.

The use of the modified Poisson brackets associated with the anomalous Hall effect implies that one is analyzing physics in the specific $n$-th level, which may be natural in condensed matter physics, namely,  each possible (classical) anomalous Poisson bracket is associated with each level  $n$ of the fast system. The analysis in
[13] is thus consistent with the past analysis of possible anomalies by Stone [11], but
it differs from the conventional
understanding of the chiral anomaly defined universally for the field, which includes all the levels such as ${\rm det}\Dslash$ in (<ref>) [9] [This aspect is supported by the detailed evaluations of various chiral anomalies [95].]. See also [96].
If one evaluates the fast system exactly without adiabatic approximations, no anomalous Poisson brackets appear as was emphasized in Section 3. The derivation of the anomaly from anomalous Poisson brackets (<ref>) thus differs from the common belief of chiral anomalies as the exact property of  field theory. In other words, the anomalous Poisson brackets  (and the possible anomaly in [13]) arise when one discards most terms linear in time-derivative in the exact expression such as in (<ref>).

In other attempts to derive chiral anomalies from Berry's phase such as in Stephanov and Yin [14], the notion of chiral kinetic theory is fundamental. This notion, which is beyond the scope of the present review, is widely used in nuclear physics. The fact that they attempt to derive the chiral anomaly from a specific band $n$ on the basis of classical Poisson brackets is still the  features common with other approaches to the derivation of chiral anomalies from Berry's phase.

Our conclusion of the subject of the possible derivation of chiral anomalies from Berry's phase is that it is interesting to recognize phenomena “similar” to the chiral anomaly on the basis of  Berry's phase, which are however not quite the same as the chiral anomaly.

§.§.§ Generic Berry's phase and  commutation relations

It was argued in [36] that Berry's phase as an adiabatic quantum effect  does not introduce the anomalous commutation relations containing a monopole. The analysis is based on the adiabatic properties of Berry's phase, namely, the adiabatic Berry's phase is defined in the adiabatic approximation of level crossing without specifying any interaction terms and not a soliton-like localized object away from the precise adiabatic limit. In addition, the commutation relations are the short-time effect as is clearly shown by the BJL prescription which is explained in Appendix A; this is also the case in the canonical quantization which uses the symplectic  forms retaining all terms with time derivatives. The adiabatic  Berry's phase is thus “resolved” in the short-time limit. This fact is intuitively illustrated by (<ref>) and (<ref>) in the Weyl model in subsection 3.3 ; the exact solution of Berry's phase discussed in Appendix B  covers  these two (snapshot) solutions. The exact solution thus interpolates from the adiabatic Berry's phase represented by a monopole (<ref>) to the non-adiabatic trivial solution (<ref>) which is crucially important in the analysis of commutation relations.

We thus argued in [36]  that the quantum effect of adiabatic Berry's phase, which is defined in the framework of quantum mechanics, does not deform canonical commutation relations by the   monopole.
But this analysis of the exact Berry's phase is not required by the main example chosen in the present review, namely, the common analysis of the anomalous Hall effect based on the classical anomalous Poisson bracket  with an assumed point-like monopole.
As long as one analyzes the anomalous Hall effects approximately in terms of  the classical Poisson brackets with an assumed point-like monopole as in the literature, a different aspect of the problem  involving Berry's phase is emphasized; we followed this practice in the main part of this review.

The above fact is the main reason why we did not discuss much the contents related to Appendix B, namely, Berry's phase as a non-local object implied by the quantum mechanical adiabatic approximation, although we believe it to be fundamental and presented it in Appendix B in detail.  The vanishing of Berry's phase terms in all the Poisson brackets such as (<ref>), which is the view in [36],  corresponds to the point of view (i) in subsection 3.2.3.

§ DISCUSSIONS AND CONCLUSION

We have discussed two main themes, Berry's phase and chiral anomalies, in this review. We emphasized the basic aspects of these notions rather than their practical applications.  We here add a comment on the topological aspects of Berry's phase and then summarize the essential aspects of these two notions.

§.§ Berry's phase and topology

Historically, the topological properties of Berry's phase were first recognized by the observation of the Longuet-Higgins' phase change rule [2], namely, the phase change of a particle when it encircles the level crossing point. The topological aspects of Berry's phase themselves were not emphasized much in the present review. This phase change rule is important when one discusses the issue if the phase change rule implies the level crossing in general on the basis of topological considerations. The analysis based on an exactly solvable model in Appendix B clearly shows that the (potential) level crossing in the adiabatic limit generally implies the appearance of both topological phase and the consequential phase change rule.
But one cannot argue the inverse, namely, that the topology inevitably enforces the level crossing in a general situation; one cannot argue that the topology of Berry's phase  enforces the breakdown of the adiabatic theorem and thus leading to level crossing.
On the contrary, it is natural to consider that the topology arises from the suppressed level crossing combined with the precise adiabatic assumption.
It is an interesting issue to understand the basic mechanism why the topological phase appears in the level crossing phenomenon in the adiabatic limit. One might understand that it  is related to the adiabatic theorem which states that no level crossing takes place in the ideal adiabatic limit; this may imply an appearance of some kind of  obstruction to the level crossing, which may be identified with the monopole singularity in the precise adiabatic limit, that deflects the approaching “particle” from any direction away to avoid the collision at the crossing point. Further illustrations on the Longuet-Higgins phase change rule and topology change are found  in Appendix B.5, which are  based on an exactly solvable model.

§.§ Berry's phase and its universal property

Berry's phase is a product of the adiabatic approximation, which does not specify the interaction term, for example. It depends only on the notion of adiabaticity. For example, Berry's phase in a quantum mechanical  system and Berry's connection in the Born-Oppenheimer approximation could be very different in practice. But it is remarkable that they share an important  property in common; Berry's phase in an anomalous Hall effect cannot co-exist with the electromagnetic vector potential in a canonical Hamiltonian formalism. We clarified the qualitative reason why it happens to be so in the Born-Oppenheimer approximation, and we utilized  this specific universal property in the present analysis.

Berry's phase itself assumes various forms when one changes the parameters which characterize Berry's phase even in the two-dimensional model. See Appendix B. The topology of a specific two-dimensional model is characterized by a Dirac monopole-like configuration in the adiabatic limit, but when one analyzes the commutation properties of Berry's phase, the completely different (non-adiabatic) parameter domains are important. In the present review, however,  we mainly utilized the above universal property in the anomalous Hall effect.

§.§ Species doubling and absence of anomalies on the lattice

As for the chiral gauge theory defined on the lattice, which is used to simulate the “Weyl fermion” in condensed matter physics, one uses a chiral $\gamma_{5}$ invariant  action (in a continuum notation) such as

\begin{eqnarray}
S =\int dtd^{3}x \left\{ \overline{\psi}(t,\vec{x})\left[ i\gamma^{\mu}\partial_{\mu}-ieA_{\mu}(t,\vec{x}) \right] \left(\frac{1-\gamma_{5}}{2} \right)\psi(t,\vec{x}) \right\}
\end{eqnarray}

defined on a heypercubic lattice, for simplicity. One then encounters the so-called species doublers. The species doublers are similar to conventional chiral fermions but different in some important aspects.
On the lattice, the momentum is limited in the Brillouin zone which one may choose $-\frac{\pi}{2a} \leq p_{\mu}\leq \frac{3\pi}{2a}$ for finite $a\neq0$. One of the 16 species doublers in the present context  is defined in, for example,

\begin{eqnarray}
\frac{\pi}{2a} \leq p_{\mu}\leq \frac{3\pi}{2a}
\end{eqnarray}

which clearly shows that each species doubler is defined in a part of the Brillouin zone and thus it is not a local field in space-time.
According to the idea of K. Wilson, the chiral anomaly is a short distance phenomenon. Generally, the chiral anomaly for the chirally symmetric fermion defined by $\gamma_{5}$ on the lattice vanishes, since the short-distance of space-time is completely cut-off for $a\neq0$.
Thus  the chiral anomaly is not defined for the lattice fermion in general and for each non-local species doubler either.
In fact, to our knowledge, no explicit evaluation of chiral anomaly for each species doubler separately in a $\gamma_{5}$ invariant theory has been performed with fixed $a\neq0$.
One may use the idea of the spectral flow to estimate the chiral anomaly for each species doubler separately, but it is not well-defined as stated above.

To realize the chiral anomaly on the lattice, one needs to modify the definition of the chiral generator to $\Gamma_{5}$ (and the notion of locality) such as in the Ginsparg-Wilson fermion. In this modified scheme, one can show that a non-vanishing Jacobian is produced by the chiral transformation generated by $\Gamma_{5}$, which correctly gives rise to the conventional chiral anomaly ${\rm Tr}\Gamma_{5}\neq 0$ explicitly for
the limit $a\rightarrow 0$.  We have sketched how to define a massless Dirac fermion without species doublers on the lattice in the framework of the Ginsparg-Wilson fermion.

In conclusion, it is hoped that the analyses of the basic aspects of Berry's phase and chiral anomalies  in this review will be useful in the future applications of these two important notions.

§ ACKNOWLEDGEMENTS

We thank Kenji Fukushima for suggesting us to write the present review.  One of us (KF) thanks Naoto Nagaosa for helpful comments on the basic aspects of the anomalous Hall effect in condensed matter physics. KF also thanks Shinichi Deguchi for the collaborations on Berry's phase which provided a part of the basic materials of this review, and Yoshio Kikukawa and Hiroshi Suzuki for  helpful comments on chiral fermions on the lattice. We also thank Masaru Hongo for useful comments on the presentation of the manuscript.
The present work is supported in part by JSPS KAKENHI (Grant No.18K03633).

§ BJORKEN-JOHNSON-LOW  PRESCRIPTION

In this Appendix, we briefly summarize the basic rules of the Bjorken-Johnson-Low (BJL) prescription [41]
which becomes useful in the present discussions. The BJL prescription was introduced to understand the equal-time commutation relations of two current operators (current algebras) from a point of view of the short-time limit of the product of two operators. It works for the analysis of
current operators appearing in the anomalous Ward-Takahashi identities for which the conventional canonical formulation does not work in general. The early use of the BJL prescription was given by Jackiw and Johnson [61] and Adler [62]. The prescription presented here is arranged so that it applies to  systems where the canonical quantization may not be well-defined as well as to conventional systems.

The canonical quantization of the Hamiltonian

\begin{eqnarray}\label{Lagrangian-1}
H=\frac{p_{k}^{2}}{2m} +V(x_{k})
\end{eqnarray}

or an equivalent Lagrangian

\begin{eqnarray}\label{Lagrangian-2}
L=p_{k}\dot{x}_{k} - \left[\frac{p_{k}^{2}}{2m} +V(x_{k}) \right]
\end{eqnarray}

are defined by the equal-time commutation relations  [We use the same notation $x_{k}(t)$, for example, for both classical and quantum variables, but our notational convention will not cause any confusions in the present paper.]

\begin{eqnarray}\label{canonical commutators}
[x^{k}(t), p^{l}(t)]= i\hbar \delta_{kl}, \ \ [x^{k}(t), x^{l}(t)]= 0, \ \ [p^{k}(t), p^{l}(t)]= 0
\end{eqnarray}

supplemented by the equations of motion

\begin{eqnarray}
\dot{x_{k}}=\frac{p_{k}}{m}, \ \ \dot{p}_{k} = -\frac{\delta}{\delta x_{k}} V(x_{l}).
\end{eqnarray}

These commutation relations
are not modified by the size or shape of the potential $V(x_{k})$ and the mass parameter (including higher order corrections in $\hbar$), as long as they are not singular. The canonical commutation relations are very universal and only the equations of motion are case-dependent. At first sight, it might look physically  strange that even a very large potential does not modify the canonical commutation relations.  Our analysis of the no-deformation of the canonical commutation relations even by the appearance of adiabatic Berry's phase is related to this universality, although the low energy effective equations of motion may be modified.

Our understanding of the above aspect of the canonical commutation relations is based on the fact that they are defined by the motion with very large “frequencies” of phase space variables, of which meaning is explained below. Thus any large but finite non-singular potential or even the kinetic energy term does not modify the canonical commutation relations.
A definition of canonical commutation relations which explicitly incorporates this fact is convenient when one analyzes
general commutation relations  in quantum mechanics and quantum field theory, in particular, in the  cases where the conventional canonical formulation is not applicable. The BJL prescription  is convenient in these respects.

We first illustrate this prescription for the free Lagrangian without $V(x)$ and $\frac{p^{2}}{2m}$. We thus start with

\begin{eqnarray}
\end{eqnarray}

and the phase space path integral of the form [ The path integral is originally derived by using the canonical quantization, but once the path integral is defined one may use it for more general cases. The time-ordering product is easily evaluated by the path integral. The time-ordering product defined by the path integral is denoted by the $T^{\star}$ product, which agrees with the T-product in the canonical operator formalism in many cases but differs in some important cases, in particular, in the cases associated with the equations of motion.  One can define the path integral  (<ref>) better by adding a suitable kinetic energy term and taking an appropriate limit later, but we forgo an analysis of the refinement here; the $i\epsilon$ prescription in (<ref>) is defined by this procedure. ]

\begin{eqnarray}\label{path integral-0}
\langle T^{\star}p^{k}(t)x^{l}(t^{\prime})\rangle =\int {\cal D}x^{k}{\cal D}p^{k}\{p^{k}(t)x^{l}(t^{\prime})\} \exp \left[\frac{i}{\hbar}\int dt  ~p^{k}(t)\dot{x^{k}}(t) \right].
\end{eqnarray}

A careful evaluation by adding source functions gives rise to the correlations functions

\begin{eqnarray}\label{free propagator}
&&\int dt e^{i\omega(t-t^{\prime})}\langle T^{\star}p^{k}(t)x^{l}(t^{\prime})\rangle=\frac{\hbar\omega}{\omega^{2} +i\epsilon}\delta_{kl}, \nonumber\\
&&\langle T^{\star}x^{k}(t)x^{l}(t^{\prime})\rangle=0,\nonumber\\
&&\langle T^{\star}p^{k}(t)p^{l}(t^{\prime})\rangle=0
\end{eqnarray}

where Feynman's $i\epsilon$ prescription is generalized by $\epsilon=\epsilon_{1} + i\mu^{2}$ using two small positive parameters $\epsilon_{1}$ and $\mu^{2}$ to define the correlation function precisely.

The BJL prescription is based on two basic rules:

(i) The $T^{\star}$ product is replaced by the ordinary $T$ product which is well-defined for the equal-time limit $t=t^{\prime}$ if

\begin{eqnarray}\label{BJL-1}
\lim_{\omega\rightarrow \infty}\int dt e^{i\omega(t-t^{\prime})}\langle T^{\star}p^{k}(t)x^{l}(t^{\prime})\rangle=\lim_{\omega\rightarrow \infty}\frac{\hbar\omega}{\omega^{2} +i\epsilon}\delta_{kl}=0
\end{eqnarray}

that is an analogue of the Riemann-Lebesgue lemma, which states that the Fourier transform of a smooth function vanishes at large frequencies.

(ii)We define the equal-time commutation relations by the procedure

\begin{align}\label{BJL-2}
&\omega\int dt ~e^{i\omega(t-t^{\prime})}\langle T p^{k}(t)x^{l}(t^{\prime})\rangle
\nonumber\\
=&\int dt ~e^{i\omega(t-t^{\prime})}i\frac{d}{dt}\langle T p^{k}(t)x^{l}(t^{\prime})\rangle\nonumber\\
=&\int dt ~e^{i\omega(t-t^{\prime})}\left\{i\delta(t-t^{\prime})[p^{k}(t),x^{l}(t)]+\langle Ti\frac{d}{dt}p^{k}(t)x^{l}(t^{\prime})\rangle \right\}\nonumber\\
=&\omega\frac{\hbar\omega}{\omega^{2} +i\epsilon}\delta_{kl}
\end{align}

where we used $T p^{k}(t)x^{l}(t^{\prime})=\theta(t-t^{\prime}) p^{k}(t)x^{l}(t^{\prime}) + \theta(t^{\prime}-t) x^{l}(t^{\prime})p^{k}(t)$.
By considering the limit $\omega\rightarrow \infty$ in (<ref>), we conclude

\begin{eqnarray}\label{BJL-3}
[p^{k}(t),x^{l}(t)] &=&-i\hbar\delta_{kl}, \nonumber\\
\int dt e^{i\omega(t-t^{\prime})}\langle Ti\frac{d}{dt}p^{k}(t)x^{l}(t^{\prime})\rangle&=&0
\end{eqnarray}

where we used the fact (i) that the Fourier transform of the ordinary
T-product $\langle Ti\frac{d}{dt}p^{k}(t)x^{l}(t^{\prime})\rangle$ vanishes for large $\omega$.
The second expression of (<ref>) is consistent with the equations of motion $\frac{d}{dt}p^{k}(t)=0$ given by $L_{0}$.  We thus obtain the operator relations

\begin{eqnarray}
[p^{k}(t),x^{l}(t)] = -i\hbar\delta_{kl}, \ \ \frac{d}{dt}p^{k}(t)=0.
\end{eqnarray}

From the rest of the correlation functions in (<ref>), we obtain similarly

\begin{eqnarray}
&&[x^{k}(t), x^{l}(t)] = 0, \ \ \frac{d}{dt}x^{k}(t)=0, \nonumber\\
&&[p^{k}(t), p^{l}(t)] = 0,  \ \ \frac{d}{dt}p^{k}(t)=0.
\end{eqnarray}

We thus recover the canonical commutation relations and the quantum equations of motion implied by

We next illustrate the effect of the potential term using

\begin{eqnarray}
L_{(1)}=p^{k}(t)\dot{x}^{k}(t) -V(x)
\end{eqnarray}

by treating $L^{\prime}= -V(x)$ as a perturbation. The starting correlation functions are the same as in (<ref>) except for the  correlation function which incorporates $-V(x)$ is given by

\begin{align}
&\langle T^{\star}p^{k}(t)p^{l}(t^{\prime})\rangle
=\int {\cal D}x^{k}{\cal D}p^{k}\{p^{k}(t)p^{l}(t^{\prime})\}\exp \left\{\frac{i}{\hbar}\int dt \left[ p^{k}(t)\dot{x^{k}}(t)-V(x) \right] \right\} \nonumber\\
=&  \int dt_{1}dt_{2} \langle Tp^{k}(t)x^{l^{\prime}}(t_{1}) \rangle \left\{\frac{-i}{\hbar}\int ds\frac{\delta}{\delta x^{l^{\prime}}(t_{1})}\frac{\delta}{\delta x^{k^{\prime}}(t_{2})}V(x(s)) \right\} \langle Tx^{k^{\prime}}(t_{2})p^{l}(t^{\prime})\rangle \nonumber\\
=&\int ds\int\frac{d\omega}{2\pi}\frac{d\omega^{\prime}}{2
\pi}e^{-i\omega(t-s)}\frac{\hbar\omega}{\omega^{2} +i\epsilon}\left( \frac{-i}{\hbar} \right) \partial_{k}\partial_{l}V(x(s))e^{-i\omega^{\prime}(s-t^{\prime})}
\frac{-\hbar\omega^{\prime}}{(\omega^{\prime})^{2} +i\epsilon}.
\end{align}

Using the BJL procedure  for the Fourier transform of $\langle T^{\star}p^{k}(t)p^{l}(t^{\prime})\rangle$ by multiplying it by $\omega$ and then taking the limit $\omega \rightarrow \infty$,
one obtains (after some re-arrangements)

\begin{eqnarray}
&&[p^{k}(t), p^{l}(t)]=0,\nonumber\\
&&\langle T[\dot{p}^{k}(t)+ \partial_{k}V(x(t))]p^{l}(t^{\prime})\rangle
\end{eqnarray}

One thus recovers the universal canonical commutation relations and the equations of motion $\dot{p}^{k}(t)+\partial_{k}V(x(t))=0$ implied by $L_{(1)}=p^{k}(t)\dot{x}^{k}(t) -V(x)$ when combined with $[x^{k}(t), x^{l}(t)] = 0, \ \ \dot{x}^{k}(t)=0$ and $[p^{k}(t),x^{l}(t)] = -i\hbar\delta_{kl}$.

The general strategy is that one uses the equations of motion when one encounters with $\dot{x}^{k}(t)$ or $\dot{p}^{k}(t)$ in the BJL prescription, and thus only the universal canonical commutation relations are obtained.
So far, the procedure is essentially a re-phrasing of the conventional canonical formulation of quantum mechanics in terms of the path integral.  Our purpose of formulating the commutation relations in the present manner is that one may be able to deal with the situation, where the equations of motion defined by the path integral are modified by an induced term at low energies but still the canonical commutation relations, which are defined in the high frequency domain, may be maintained without modifications.

Another useful example is the particle moving in electromagnetic potentials

\begin{eqnarray}\label{Lagrangian-3}
&=&P_{k}(t)\dot{x^{k}}(t) - \frac{1}{2m}(P_{k} +eA_{k}(x))^{2} +  e\phi(x).
\end{eqnarray}

We then obtain the canonical commutation relations from the first term of (<ref>)

\begin{eqnarray}\label{commutator-2}
[x^{k}(t), P^{l}(t)]= i\hbar \delta_{kl}, \ \ [x^{k}(t), x^{l}(t)]= 0, \ \ [P^{k}(t), P^{l}(t)]= 0
\end{eqnarray}

which are universally valid.
If one defines  an auxiliary variable

\begin{eqnarray}\label{appendix covariant derivative}
\end{eqnarray}

the Lagrangian is modified to

\begin{eqnarray}\label{Lagrangian-3-2}
&=&p_{k}(t)\dot{x^{k}}(t) - eA_{k}(x)\dot{x^{k}}(t)- \frac{1}{2m} p_{k}^{2} +  e\phi(x).
\end{eqnarray}

and the commutation relations  are re-written as

\begin{eqnarray}\label{commutator-10}
&&[x^{k}(t), p^{l}(t)]= i\hbar \delta_{kl}, \ \ [x^{k}(t), x^{l}(t)]= 0,\nonumber\\
&& [p^{k}(t), p^{l}(t)]= -i\hbar e \left(\frac{\partial}{\partial x_{k}(t)}A_{l}(x(t))-\frac{\partial}{\partial x_{l}(t)}A_{k}(x(t))\right).
\end{eqnarray}

The derivative coupling (namely, the time-derivative $\dot{x^{k}}(t)$ in $eA_{k}(x)\dot{x^{k}}(t)$), as is well-known, modifies the commutation relations written in the form (<ref>) depending on the functional form of $A_{k}$. But the important fact is that the universally valid canonical commutation relations (<ref>) are always valid without any modifications by higher order effects in $\hbar$; the commutators in (<ref>) are obtained form (<ref>) by the replacement (<ref>).

Another comment is  that the potential which generates a constant magnetic field $B$,

\begin{eqnarray}
A^{k}(x(t))=\left(-\frac{1}{2}Bx^{2}(t), \frac{1}{2}Bx^{1}(t), 0\right),
\end{eqnarray}

for example, contains an arbitrary large frequencies in the sense that this potential is assumed to be valid for very slow movement as well as for very fast movement of $x^{k}(t)$, by definition. In the conventional formulation of quantum mechanics, we do not write the time dependence explicitly (Schrödinger picture) in the premise that we can generate the time dependence by the operation $e^{i\hat{H}t} A^{k}(x(0))e^{-i\hat{H}t}=A^{k}(x(t))$ (in the Heisenberg picture), which contains the frequencies covering all the spectra of $\hat{H}$ [In the BJL prescription, we use infinitely large $\omega$ to ensure that the canonical commutation relations are valid for any sensible form of the Hamiltonian.]. It may happen that a specific potential does not survive this operation

\begin{eqnarray}
e^{i\hat{H}t} {\cal A}^{k}(x(0))e^{-i\hat{H}t}\neq {\cal A}^{k}(x(t))
\end{eqnarray}

since an assumed specific functional form of the potential such as ${\cal A}^{k}(x(t))$ is valid only for a specific very small range of frequencies contained in $x^{k}(t)$; what we have in mind is the adiabatic Berry's phase which assumes a genuine monopole form at the limit of  adiabatic slow motion but changes its functional form at higher frequencies in non-adiabatic domain.  In such a case, the conventional canonical formulation based on the time independent variables is not reliable in general and an alternative scheme such as the BJL prescription becomes useful.

§ WHAT IS THE MONOPOLE IN BERRY'S PHASE?

The exact characterization of the monopole appearing in Berry's phase and its possible topology change to a dipole are important in the applications of Berry's phase. This is a major issue in the classical analysis of the topological aspects of Berry's phase. We present an analysis of this issue using an exactly solvable version of Berry's original model of a spin placed in the rotating magnetic field. This exact solution reproduces the results of a generic model of Berry's phase in both adiabatic and non-adiabatic limits and provides a smooth interpolation between these two limiting cases. The topology change is visualized as a lensing effect of the monopole in the parameter space.

§.§ Exactly solvable model of Berry's phase

Berry's phase is defined in association with the adiabatic theorem, and  the Dirac monopole-like configuration is found in the generic two-level crossing problem in the precise adiabatic limit.  It has been shown that the monopole disappears in the nonadiabatic limit in Section 2. It is important to understand precisely how the Dirac monopole, which appears in the adiabatic limit, disappears when one goes to the parameter domain of nonadiabatic motion.
This problem is also related to the basic issue of the topology change of the (apparently) topological phase.  To analyze those issues in a reliable manner, we consider an exactly solvable model of Berry's phase, in which the fast moving variables are given by the spin degrees of freedom and the slow variables are given by the background C-number rotating magnetic field $\vec{B}(t)$.
This model is the same as the model discussed by Berry in his original paper [3] but the time dependence of parameters which define the model are more restrictive. To be specific, we consider  the Schrödinger equation [3]

\begin{eqnarray}\label{starting equation}
\end{eqnarray}

for the Hamiltonian
describing the motion of a magnetic moment $\mu\hbar\vec{\sigma}$ placed in a rotating magnetic field

\begin{eqnarray}\label{solvable model}
\vec{B}(t)=B(\sin\theta\cos\varphi(t),
\sin\theta\sin\varphi(t),\cos\theta )
\end{eqnarray}

with $\vec{\sigma}$ standing for Pauli matrices. The naively expected level crossing takes place at the vanishing external field $B=0$.
It has been noticed that the equation (<ref>) is exactly solved if one restricts the movement of the magnetic field to the form
$\varphi(t)=\omega t$ with constant $\omega$, and constant $B$ and $\theta$ [97]. It will be explained later that this particular parametrization (<ref>) maintains the essence of Berry's phase.
The exact solution is written as

\begin{align}\label{eq-exactamplitude1}
\psi_{\pm}(t)
{\cal A}_{\pm}(\vec{B})\cdot\frac{d\vec{B}}{d t}dt\right]
\end{align}

\begin{eqnarray}\label{exact eigenfuntion}
\cos\frac{1}{2}(\theta-\alpha) e^{-i\varphi(t)}\\
\sin\frac{1}{2}(\theta-\alpha)
\end{array}\right), \ \ \
\sin\frac{1}{2}(\theta-\alpha) e^{-i\varphi(t)}\\
\end{array}\right).
\end{eqnarray}

It is important that these solutions differ from the so-called instantaneous solutions used in the adiabatic approximation, which are given by  setting $\alpha=0$; the following analysis of topology change is not feasible using the instantaneous solutions.   The parameter $\alpha(\theta,\eta)$ is defined by $\mu\hbar B\sin\alpha= (\hbar\omega/2)\sin(\theta-\alpha)$ or equivalently [97]

\begin{eqnarray}\label{cotangent}
\cot\alpha(\theta,\eta)=\frac{\eta+\cos\theta}{\sin\theta}
\end{eqnarray}

with $\eta=2\mu\hbar B/\hbar\omega$ for $0\leq \theta\leq\pi$, which specifies the branch of the cotangent function.

The second term in the exponential of the exact solution (<ref>)
is customarily called Berry's phase and defined by a potential-like object (or connection) [We fix the sign convention of Berry's phase by the expression (<ref>) in this section.]

\begin{eqnarray}\label{ES Berry's phase}
{\cal A}_{\pm}(\vec{B})\equiv w_{\pm}^{\dagger}(t)\left(-i\hbar\frac{\partial}{\partial \vec{B}} \right)w_{\pm}(t).
\end{eqnarray}

This potential describes an azimuthally symmetric static magnetic monopole-like object in the present case.

The solution (<ref>) to the Schrödinger equation is confirmed by evaluating

\begin{align}
=&\{ i\hbar\partial_{t}w_{\pm}(t)+w_{\pm}(t)[w_{\pm}^{\dagger}(t)\big(\hat{H}
=&\{ i\hbar\partial_{t}w_{\pm}(t)+w_{\pm}(t)[w_{\pm}^{\dagger}(t)\big(\hat{H}
\end{align}

where we used
$w_{\mp}^{\dagger}\big(\hat{H}-i\hbar\partial_{t}\big)w_{\pm}=0$ by noting (<ref>), and the completeness relation $w_{+}w_{+}^{\dagger}+w_{-}w_{-}^{\dagger}=1$.

The parameter
$\eta\geq 0$ in (<ref>) is written as

\begin{eqnarray}\label{parameter2}
\eta=\frac{2\mu\hbar B}{\hbar\omega}=\frac{\mu BT}{\pi}
\end{eqnarray}

when one defines the period $T=2\pi/\omega$. The parameter $\eta$ is a ratio of the two different energy scales appearing in the model, namely, the static energy $2\mu\hbar B$ of the dipole moment in an external magnetic field  and the kinetic energy (rotation energy) $\hbar\omega$: $\eta\gg 1$  (for example, $T\rightarrow \infty$ for any finite $B$) corresponds to the adiabatic limit, and $\eta\ll 1$ (for example, $T\rightarrow 0$ for finite $B$) corresponds to the non-adiabatic limit. In a mathematical treatment of the adiabatic theorem, the precise adiabaticity is defined by $T\rightarrow \infty$ with fixed $B$ [4].

The parameter $\alpha(\theta,\eta)$ in (<ref>) is normalized as $\alpha(0,\eta)=0$
by definition.
The analysis of the transition from $\eta>1$ to
$\eta<1$ through the critical value $\eta=1$ is important.
In Fig.B.1, we show the relation between $\alpha(\theta,\eta)$ and $\theta$ with emphasis on the transition region near $\eta=1$ given by (<ref>) [98, 99].

The topology change at the parameter value $\eta=1$ determined by Eq. (<ref>). Note the limiting values of $\alpha(\theta,\eta)$, $\alpha(\theta,0)=\theta$ and $\alpha(\theta,\infty)=0$.

For the parameters
with a small positive $\epsilon$, the value $\alpha(\theta,\eta)$ departs from the common value $\frac{1}{2}\theta$ assumed at around $\theta=0$ and splits into two branches for the values of the parameter $\theta$ close to  $\theta=\pi$. We have $\alpha(\pi,\eta)=0$ for $\eta=1+\epsilon$ and  $\alpha(\pi,\eta)=\pi$ for $\eta=1-\epsilon$, respectively,  with the slopes

\begin{eqnarray}\label{singularity}
\frac{d\alpha(\theta,\eta)}{d\theta}|_{\theta=\pi} =\mp \frac{1}{\epsilon}
\end{eqnarray}

for $\eta=1\pm \epsilon$, respectively, using (<ref>). We thus observe the singular jump characteristic to the topology change in terms of $\alpha(\theta,\eta)$ at $\eta=1$. It will be shown  later that the topology of the monopole-like object is specified by the value

\begin{eqnarray}
\lim_{\theta\rightarrow\pi} \alpha(\theta, \eta) = 0, \frac{1}{2}\pi, \pi,
\end{eqnarray}

for $\eta>1$, $\eta=1$ and $\eta<1$, respectively.

The extra phase factor for one period of motion is written explicitly as,

\begin{align}\label{Berry's phase1}
\exp\left[-\frac{i}{\hbar}\oint
{\cal A}_{\pm}(\vec{B})\cdot\frac{d\vec{B}}{d t}dt\right]=&\exp \left\{-i\oint \frac{-1\mp\cos(\theta-\alpha(\theta,\eta))}{2}d\varphi \right\}\nonumber\\
=&\exp \left\{-i\oint\frac{1\mp\cos(\theta-\alpha(\theta,\eta))}{2}d\varphi+2\pi i\right\}\nonumber\\
=&\exp\left\{-\frac{i}{\hbar}\Omega_{\pm} \right\},
\end{align}

with the monopole-like integrated flux [98]

\begin{eqnarray}\label{solid-angle}
\Omega_{\pm}
&=&\hbar\oint \frac{[1\mp\cos(\theta-\alpha(\theta,\eta))]}{2B\sin\theta}B\sin\theta d\varphi .
\end{eqnarray}