A path integral derivation of the equations of anomalous Hall effect

Kazuo Fujikawa 1 and Koichiro Umetsu 2

1 Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS),

RIKEN, Wako 351-0198, Japan

2 Laboratory of Physics, College of Science and Technology, and Junior
College,

Funabashi Campus, Nihon University, Funabashi, Chiba 274-8501, Japan

###### Abstract

A path integral (Lagrangian formalism) is used to derive the effective
equations of motion of the anomalous Hall effect with Berry’s phase on the
basis of the adiabatic condition $|E_{n\pm 1}-E_{n}|\gg 2\pi\hbar/T$, where
$T$ is the typical time scale of the slower system and $E_{n}$ is the energy
level of the fast system. In the conventional definition of the adiabatic
condition with $T\rightarrow{\rm large}$ and fixed energy eigenvalues, no
commutation relations are defined for slower variables by the Bjorken-Johnson-
Low prescription except for the starting canonical commutators. On the other
hand, in a singular limit $|E_{n\pm 1}-E_{n}|\rightarrow\infty$ with specific
$E_{n}$ kept fixed for which any motions of the slower variables $X_{k}$ can
be treated to be adiabatic, the non-canonical dynamical system with deformed
commutators and the Nernst effect appear. In the Born-Oppenheimer
approximation based on the canonical commutation relations, the equations of
motion of the anomalous Hall effect is obtained if one uses an auxiliary
variable $X_{k}^{(n)}=X_{k}+{\cal A}^{(n)}_{k}$ with Berry’s connection ${\cal
A}^{(n)}_{k}$ in the absence of the electromagnetic vector potential
$eA_{k}(X)$ and thus without the Nernst effect. It is shown that the gauge
symmetries associated with Berry’s connection and the electromagnetic vector
potential $eA_{k}(X)$ are incompatible in the canonical Hamiltonian formalism.
The appearance of the non-canonical dynamical system with the Nernst effect is
a consequence of the deformation of the quantum principle to incorporate the
two incompatible gauge symmetries.

##  1 Introduction

Historically, the geometric phase or Berry’s phase [1] has been recognized as
a forgotten phase of the state vector in the naive applications of the
adiabatic theorem. The topological aspects of Berry’s phase such as a Dirac
monopole-like structure in the level crossing problem in the precise adiabatic
limit are well-known [2]. The use of Berry’s phase in the analyses of the
anomalous Hall effect [3, 4] and the anomalous spin Hall effect [5] has,
however, revealed the more general aspects of Berry’s phase, namely, a
dynamical system which is not quantized in the conventional canonical
formalism of quantum mechanics [6] leading to deformed commutators. The
related non-commutative geometry has been known for many years if one chooses
suitable variables in the Born-Oppenheimer approximation which is by itself
quantized by the canonical commutation relations [7, 8].

If one understands Berry’s phase as a characteristic property of the quantum
mechanical system in the adiabatic limit [9, 10], it is puzzling why Berry’s
phase leads to a non-canonical system which deforms the canonical commutation
relations dictated by the principle of quantum mechanics 111In the present
article, Berry’s phase is considered only in the context of quantum
mechanics.. The purpose of the present article is to clarify the puzzling
aspect by presenting a systematic derivation of the equations of motion of the
anomalous Hall effect by the path integral (Lagrangian) formalism, starting
with the quantum system defined by the standard canonical commutation
relations. We use the second quantized formulation of Berry’s phase which
makes it transparent that Berry’s phase appears when one throws away some
terms causing the level crossing in the adiabatic limit [11], rather than the
common view that Berry’s phase is an additional term in the adiabatic limit.

To be explicit, the effective equations of motion of the anomalous Hall
effect, which incorporate Berry’s phase near the level crossing point, are
customarily adopted as (see, for example, [6])

$\displaystyle\dot{x}_{k}=-\Omega_{kl}(\vec{p})\dot{p}_{l}+\frac{\partial\epsilon_{n}(\vec{p})}{\partial
p_{k}},\ \ \dot{p}_{k}=-eF_{kl}(\vec{x})\dot{x}_{l}+e\frac{\partial}{\partial
x_{k}}\phi(\vec{x})$ (1.1)

by adding the adiabatic Berry’s phase $\Omega_{kl}\dot{p}_{l}$ to the
equations of motion as an extra induced term. Here $\epsilon_{n}(\vec{p})$
essentially stands for the n-th energy level of the fast system. The magnetic
flux $\Omega_{kl}(\vec{p})$ of Berry’s phase, which is assumed to be a genuine
Dirac monopole form [12] for the moment, and the electromagnetic tensor
$F_{kl}$ are defined by

$\displaystyle\Omega_{kl}=\frac{\partial}{\partial p_{k}}{\cal
A}_{l}-\frac{\partial}{\partial p_{l}}{\cal A}_{k},\ \ \
F_{kl}=\frac{\partial}{\partial x_{k}}A_{l}-\frac{\partial}{\partial
x_{l}}A_{k},$ (1.2)

respectively. The monopole is located at the origin of momentum space, i.e.,
at the level crossing point in the adiabatic level-crossing problem, but it is
now assumed to be a genuine particle in the momentum space. Here we defined
$p_{l}=\hbar k_{l}$ to write all the equations in terms of $p_{l}$ compared to
the notation in [6], to keep track of the $\hbar$ factor in a transparent way.

It is known that the equations in (1.1) are derived without using any
commutation relations from [13]

$\displaystyle S=\int dt[p_{k}\dot{x}_{k}-eA_{k}(\vec{x},t)\dot{x}_{k}+{\cal
A}_{k}(\vec{p})\dot{p}_{k}-\epsilon_{n}(\vec{p})+e\phi(\vec{x})],$ (1.3)

A simplified version of this action was used in [6]. It is usually assumed
that the action (1.3) with Berry’s phase included is a fundamental classical
action, and it has been shown [13] that the action (1.3), which is known to
belong to a non-canonical system [6], is treated by a modified canonical
formalism inverting a symplectic matrix defined by the action in the extended
phase space formalism [14]. They then derived the deformed Poisson brackets
induced by the genuine monopole curvature
$\Omega_{kl}=\epsilon^{klm}\Omega_{m}$,

$\displaystyle\\{x_{k},x_{l}\\}=\frac{\epsilon^{klm}\Omega_{m}}{1+e\vec{B}\cdot\vec{\Omega}},\
\ \
\\{p_{k},x_{l}\\}=-\frac{\delta_{kl}+e\Omega_{k}B_{l}}{1+e\vec{B}\cdot\vec{\Omega}},$
$\displaystyle\\{p_{k},p_{l}\\}=-\frac{\epsilon^{klm}eB_{m}}{1+e\vec{B}\cdot\vec{\Omega}},$
(1.4)

where the factors containing $\Omega_{k}$ are anomalous [13]. The background
field method applied to (1.3) in the path integral formalism using the
Bjorken-Johnson-Low (BJL) prescription [15], which is briefly summarized in
Appendix, gives the commutation relations [16]

$\displaystyle[x_{k},x_{l}]=i\hbar\frac{\epsilon^{klm}\Omega_{m}}{1+e\vec{B}\cdot\vec{\Omega}},\
\ \
[p_{k},x_{l}]=-i\hbar\frac{\delta_{kl}+e\Omega_{k}B_{l}}{1+e\vec{B}\cdot\vec{\Omega}},$
$\displaystyle[p_{k},p_{l}]=-i\hbar\frac{\epsilon^{klm}eB_{m}}{1+e\vec{B}\cdot\vec{\Omega}}.$
(1.5)

which are exact for either $B_{m}=0$ or $\Omega_{m}=0$. These anomalous
commutation relations are the main features of the action (1.3), in contrast
to the canonical commutation relations used in the Born-Oppenheimer
approximation.

We summarize briefly our main assertion in this paper: We start with a master
Hamiltonian formalism and derive the action (1.3) by a second quantized path
integral formulation [11] using the adiabatic condition $|E_{n\pm 1}-E_{n}|\gg
2\pi\hbar/T$, where $T$ is the typical time scale of the slower system and
$E_{n}$ is the energy level of the fast system. In the conventional adiabatic
limit, $T\rightarrow$ large with $E_{n}$ fixed, the extension of the action
(1.3) to the domain of variables covering the full phase space including rapid
movement is not allowed, and thus the equal-time commutation relations are not
determined by the BJL prescription 222 The basic idea of the BJL prescription,
which is more general than the canonical quantization in the sense that it
works for the system with quantum anomalies also, is that the equal-time
commutation relation of two operators $A(t)$ and $B(t)$ is determined by the
analysis of the short-time limit of the time ordered product $\langle
TA(t_{1})B(t_{2})\rangle$ with $t_{1}\rightarrow t_{2}$, which in turn means
that the infinitely large frequencies of the Fourier transform of $\langle
TA(t_{1})B(t_{2})\rangle$ determine the equal-time commutation relation.. One
may recognize the starting canonical commutation relations by going back to
the starting Lagrangian. In contrast, a singular limit of the adiabatic
condition with $|E_{n\pm 1}-E_{n}|\rightarrow\infty$ allows any motion of the
slower variables to be regarded as adiabatic motion; we regard that this limit
corresponds to the customary treatment of (1.3). In this limit, however, one
recognizes that one cannot quantize the action (1.3) in a canonical manner
[13], although the action (1.3) is derived starting with a formalism defined
in the conventional canonical commutation relations. It is customary to regard
(1.3) as a classical action and applies a second-time quantization by applying
an extended phase space formalism [14], for example, and one obtains the
deformed Poisson brackets (1) or the deformed commutators (1) by a background
field method [16]. The deformed phase space volume associated with the
deformed Poisson brackets leads to a modified density of states $\propto
1+e\vec{B}\cdot\vec{\Omega}$ and the Nernst effect in thermal functions [6,
17]. We emphasize that (1.3) is formally invariant under both gauge
transformations of the electromagnetic field and of Berry’s phase.

In the Born-Oppenheimer approximation that is based on canonical commutation
relations, one first observes that the gauge symmetry of Berry’s connection is
a measure of the validity of Born-Oppenheimer approximation where the total
system is written as $\Psi=\sum_{l}\varphi_{l}(P)\phi_{l}(x,P)$. The gauge
symmetry of Berry’s connection implies that the subsystem
$\varphi_{l}(P)\phi_{l}(x,P)$, which consists of a specific energy level of
the fast system $\phi_{l}(x,P)$ and the slower system $\varphi_{l}(P)$ with
Berry’s connection ${\cal A}^{(l)}_{k}(P)$, is treated independently of the
total system $\Psi$. This independence of each subsystem is not compatible
with the electromagnetic gauge symmetry of the vector potential $eA_{k}$,
which acts on all the subsystems universally in
$\Psi=\sum_{l}\varphi_{l}(P)\phi_{l}(x,P)$. Thus a subsystem which satisfies
the gauge symmetry of Berry’s connection cannot incorporate consistently the
electromagnetic potential $eA_{k}$, as we demonstrate explicitly. Namely, the
electromagnetic gauge symmetry and the gauge symmetry of Berry’s connection
are not compatible in the Born-Oppenheimer approximation which is defined by
canonical commutation relations. We also discuss an algebraic manifestation of
this incompatibility. We argue that this incompatibility in the Born-
Oppenheimer approximation is a restatement of the failure of the canonical
quantization of the action (1.3) in the presence of the electromagnetic vector
potential $eA_{k}$ and Berry’s phase. This is because if the canonical
quantization of the action (1.3) should be defined, it would imply that the
Born-Oppenheimer approximation, which is defined by canonical commutation
relations, satisfies the gauge symmetry of the electromagnetism and the gauge
symmetry of Berry’s connection simultaneously. We explicitly demonstrate that
the Born-Oppenheimer approximation, when converted to the Lagrangian (path
integral) formalism, can incorporate the vector potential $eA_{k}(X)$ in a
gauge invariant manner, but the resulting action is not amenable to the
conventional canonical formulation of quantum mechanics. We conclude that the
equations of motion of the anomalous Hall effect (1.1) are generally valid in
the precise adiabatic limit, but the appearance of the non-canonical dynamical
system and the Nernst effect is based on the deformation of the quantum
principle to incorporate the two incompatible gauge symmetries.

##  2 Path integral derivation

We use the path integral formulation of Berry’s phase discussed in [11]. We
adopt the Hamiltonian of the form $H=H_{0}+H_{1}$ assuming that the slower
particle is charged with $q=-e$ ($e>0$) and the fast particle is neutral for
simplicity,

$\displaystyle H_{0}(X,P+eA(X))$ $\displaystyle=$
$\displaystyle\frac{1}{2M}(P_{k}+eA_{k}(X))^{2}-e\phi(X),$ $\displaystyle
H_{1}(x,p;P+eA(X))$ $\displaystyle=$ $\displaystyle
H_{1}(x_{k},p_{k};P_{k}+eA_{k}(X))$ (2.1)

with the canonical quantization of fast variables

$\displaystyle[p_{k},x_{l}]=\frac{\hbar}{i}\delta_{kl},\ \ [p_{k},p_{l}]=0,\ \
[x_{k},x_{l}]=0$ (2.2)

and the canonical quantization of slower variables

$\displaystyle[P_{k},X_{l}]=\frac{\hbar}{i}\delta_{kl},\ \ [P_{k},P_{l}]=0,\ \
[X_{k},X_{l}]=0$ (2.3)

by treating $H_{0}(X,P)$ as the slower system; from now on, we use the capital
characters $X_{k}$ and $P_{k}$ for slower variables without stated otherwise (
in contrast, the lower case letters are sometimes used for slower variables
also to respect the past usages). The use of covariant derivative,
$P_{k}+eA_{k}(X)$, ensures the electromagnetic gauge invariance. We discuss
the case of a single fast particle and a single slow particle, for simplicity,
although our use of the second quantization can cover a slightly more general
case 333The general proof of the adiabatic theorem [10] includes the case of
degenerate states. We can incorporate the case with the degeneracy of
$n$-states of the fast system using an internal $U(n)$ symmetry for a single
particle. See also [18].. We start with the fundamental path integral

$\displaystyle Z$ $\displaystyle=$ $\displaystyle\int{\cal
D}\overline{P}_{k}{\cal
D}X_{k}\exp\\{\frac{i}{\hbar}\int_{0}^{T}dt[(\overline{P}_{k}(t)-eA_{k}(X(t)))\dot{X}_{k}(t)-H_{0}(X(t),\overline{P}(t))]\\}$
$\displaystyle\times$ $\displaystyle\int{\cal D}\psi^{\star}{\cal
D}\psi\exp\\{\frac{i}{\hbar}\int_{0}^{T}dt\int
d^{3}x[\psi(t,\vec{x})^{\star}i\hbar\partial_{t}\psi(t,\vec{x})-\psi(t,\vec{x})^{\star}H_{1}(\frac{\hbar}{i}\vec{\nabla},\vec{x};\overline{P}(t))\psi(t,\vec{x})]\\}$

with

$\displaystyle\overline{P}_{k}(t)=P_{k}(t)+eA_{k}(X(t))$ (2.5)

where we used the invariance of the path integral measure ${\cal D}P_{k}{\cal
D}X_{k}={\cal D}\overline{P}_{k}{\cal D}X_{k}$. We use the second quantization
for the fast system, which is convenient to analyze the level crossing and
Berry’s phase [11]. It is confirmed that the BJL prescription reproduces the
canonical commutation relations (2.3) and

$\displaystyle[\psi(t,\vec{x}),\psi^{\dagger}(t,\vec{y})]=\delta(\vec{x}-\vec{y})$
(2.6)

in the path integral formula of (2). We expand the field variable
$\psi(t,\vec{x})$ into a complete set of orthonormal bases
$\\{\phi_{k}(\vec{x};\overline{P}(t))\\}$ defined by

$\displaystyle
H_{1}(\frac{\hbar}{i}\vec{\nabla},\vec{x};\overline{P}(t))\phi_{n}(\vec{x};\overline{P}(t))=E_{n}(\overline{P}(t))\phi_{n}(\vec{x};\overline{P}(t)),$
$\displaystyle\psi(t,\vec{x})=\sum_{n}a_{n}(t)\phi_{n}(\vec{x};\overline{P}(t)).$
(2.7)

We then have

$\displaystyle Z$ $\displaystyle=$ $\displaystyle\int{\cal
D}\overline{P}_{k}{\cal
D}X_{k}\exp\\{\frac{i}{\hbar}\int_{0}^{T}dt[(\overline{P}_{k}(t)-eA_{k}(X(t)))\dot{X}_{k}(t)-H_{0}(X(t),\overline{P}(t))]\\}$
$\displaystyle\times$ $\displaystyle\int\Pi_{n}{\cal D}a^{\star}_{n}{\cal
D}a_{n}\exp\\{\frac{i}{\hbar}\int_{0}^{T}dt\sum_{n}[a_{n}^{\star}(t)i\hbar\partial_{t}a_{n}(t)-E_{n}(\overline{P}(t))a_{n}^{\star}(t)a_{n}(t)]\\}$
$\displaystyle\hskip
56.9055pt\times\exp\\{\frac{i}{\hbar}\int_{0}^{T}dt\sum_{n,l}\langle
n,t|i\hbar\partial_{t}|l,t\rangle a_{n}^{\star}(t)a_{l}(t)\\}$
$\displaystyle\simeq$ $\displaystyle\int{\cal D}\overline{P}_{k}{\cal
D}X_{k}\exp\\{\frac{i}{\hbar}\int_{0}^{T}dt[(\overline{P}_{k}-eA_{k}(X))\dot{X}_{k}-H_{0}(X,\overline{P})]\\}$
$\displaystyle\times$ $\displaystyle\int\Pi_{n}{\cal D}a^{\star}_{n}{\cal
D}a_{n}\exp\\{\frac{i}{\hbar}\int_{0}^{T}dt\sum_{n}[a_{n}^{\star}(t)i\hbar\partial_{t}a_{n}(t)-\left(E_{n}(\overline{P})-{\cal
A}^{(n)}_{k}(\overline{P})\dot{\overline{P}^{k}}\right)a_{n}^{\star}(t)a_{n}(t)]\\}$

where we defined

$\displaystyle\langle n,t|i\hbar\partial_{t}|l,t\rangle$ $\displaystyle=$
$\displaystyle\int
d^{3}x\phi^{\dagger}_{n}(\vec{x};\overline{P}(t))i\hbar\partial_{t}\phi_{l}(\vec{x};\overline{P}(t)),$
$\displaystyle{\cal A}^{(n)}_{k}(\overline{P})\dot{\overline{P}^{k}}(t)$
$\displaystyle\equiv$ $\displaystyle\langle n,t|i\hbar\partial_{t}|n,t\rangle$
(2.9) $\displaystyle=$ $\displaystyle\int
d^{3}x\phi^{\dagger}_{n}(\vec{x};\overline{P}(t))i\hbar\frac{\partial}{\partial\overline{P}^{k}(t)}\phi_{n}(\vec{x};\overline{P}(t))\dot{\overline{P}^{k}}(t).$

The diagonal element ${\cal A}^{(n)}_{k}(\overline{P})\dot{\overline{P}^{k}}$
is commonly called Berry’s phase. The assumption of the diagonal dominance,
which was used in the last step of the above path integral formula,
corresponds to the adiabatic approximation and it is valid only for

$\displaystyle|E_{n\pm
1}(\overline{P})-E_{n}(\overline{P})|\gg\hbar\frac{2\pi}{T_{s}}$ (2.10)

where $T_{s}$ stands for the typical time scale of the slower system; usually
$T_{s}$ is estimated by the period of the slowly varying variable
$\overline{P}_{k}(t)$, $\overline{P}_{k}(0)=\overline{P}_{k}(T_{s})$. We
understand that $\hbar(2\pi/T_{s})$ stands for the typical energy scale
contained in $\overline{P}_{k}(t)$ 444We use $T_{s}$ for the period of the
slower dynamical system to distinguish it from the T-ordering operation,
whenever necessary. It is convenient to choose the upper-bound of the time
integral in the path integral (2) to agree with this period $T_{s}$.. This
condition means that the movement (or energy) of the slower system is much
slower (or smaller) than the level spacing of the fast system. If this
adiabaticity condition is not satisfied, one needs to retain the non-diagonal
transition elements $\langle n,t|i\hbar\partial_{t}|l,t\rangle$ in (2), and
the path integral is reduced to the starting expression (2) after summing over
$n$ and $l$. This shows that one would recover the starting universal
canonical commutation relations for the slower variables (2.3) if one
evaluates the fast system exactly without using the adiabatic approximation
555For the moment, we understand the condition (2.10) in the manner of the
standard adiabatic condition $T\rightarrow{\rm large}$ with energy eigenvalues
fixed. in the present setting.

The slower system standing on the specific n-th level

$\displaystyle a_{n}^{\dagger}(0)|0\rangle$ (2.11)

of the fast system, namely, the fast system being constrained to the n-th
level, is described by the path integral derived from (2) 666The conversion of
the path integral to the time-evolution operator for the variables
$\\{a_{n},a_{n}^{\dagger}\\}$ is  $\displaystyle\int\Pi_{n}{\cal
D}a^{\star}_{n}{\cal
D}a_{n}\exp\\{\frac{i}{\hbar}\int_{0}^{T}dt\sum_{n}[a_{n}(t)^{\star}i\hbar\partial_{t}a_{n}(t)$
$\displaystyle\hskip 113.81102pt-\left(E_{n}(\overline{P}(t))-{\cal
A}^{(n)}_{k}(\overline{P}(t))\dot{\overline{P}^{k}}(t)\right)a_{n}^{\star}(t)a_{n}(t)]\\}$
$\displaystyle\rightarrow\exp\\{\sum_{n}\frac{-i}{\hbar}\int_{0}^{T}dt\left(E_{n}(\overline{P}(t))-{\cal
A}^{(n)}_{k}(\overline{P}(t))\dot{\overline{P}^{k}}(t)\right)a^{\dagger}_{n}(0)a_{n}(0)\\}$
by noting $a^{\dagger}_{n}(t)a_{n}(t)=a^{\dagger}_{n}(0)a_{n}(0)$.

$\displaystyle Z_{n}$ $\displaystyle=$ $\displaystyle\int{\cal
D}\overline{P}_{k}{\cal D}X_{k}$ $\displaystyle\times$
$\displaystyle\exp\\{\frac{i}{\hbar}\int_{0}^{T}dt[(\overline{P}_{k}-eA_{k}(X))\dot{X}_{k}-H_{0}(X,\overline{P})-(E_{n}(\overline{P})-{\cal
A}^{(n)}_{k}(\overline{P})\dot{\overline{P}_{k}})]\\}$

with $H_{0}(X,\overline{P})=\frac{1}{2M}\overline{P}_{k}^{2}-e\phi(X)$. This
formula is valid under the crucial constraint (2.10). The Lagrangian appearing
in this path integral

$\displaystyle L_{n}=(\overline{P}_{k}-eA_{k}(X))\dot{X}_{k}+{\cal
A}^{(n)}_{k}(\overline{P})\dot{\overline{P}_{k}}-(\frac{1}{2M}\overline{P}_{k}^{2}+E_{n}(\overline{P}))+e\phi(X)$
(2.13)

precisely agrees with the common Lagrangian in condensed matter physics (1.3)
if one identifies

$\displaystyle\overline{P}_{k}\rightarrow p_{k},\ \ X_{k}\rightarrow x_{k},\ \
{\rm and}\ \
\epsilon_{n}(p)=\frac{1}{2M}\overline{P}_{k}^{2}+E_{n}(\overline{P}),$ (2.14)

respectively, and if one uses the adiabatic Berry’s phase ${\cal A}^{(n)}_{k}$
associated with the specific $n$-th level of the fast system. Our path
integral formulation thus naturally reproduces the common formulas (1.3) in
the precise adiabatic limit but with the adiabatic Berry’s phase ${\cal
A}^{(n)}_{k}$ instead of a genuine Dirac monopole. The action appearing in (2)
is invariant under a gauge transformation of Berry’s phase

$\displaystyle{\cal A}^{(n)}_{k}(\overline{P})\rightarrow{\cal
A}^{(n)}_{k}(\overline{P})+\frac{\partial}{\partial\overline{P}_{k}}\omega^{(n)}(\overline{P})$
(2.15)

with different $\omega^{(n)}(\overline{P})$ for each $n$ if one chooses the
periodic boundary condition $\overline{P}_{k}(0)=\overline{P}_{k}(T)$. The
induced Berry’s phase term is invariant by itself

$\displaystyle\int_{0}^{T}dt\\{{\cal
A}^{(n)}_{k}(\overline{P})+\frac{\partial}{\partial\overline{P}_{k}}\omega^{(n)}(\overline{P})\\}\dot{\overline{P}_{k}}$
$\displaystyle=$ $\displaystyle\int_{0}^{T}dt{\cal
A}^{(n)}_{k}(\overline{P})\dot{\overline{P}_{k}}+\int_{0}^{T}dt\frac{d}{dt}\omega^{(n)}(\overline{P}(t))$
(2.16) $\displaystyle=$ $\displaystyle\int_{0}^{T}dt{\cal
A}^{(n)}_{k}(\overline{P})\dot{\overline{P}_{k}}.$

This gauge invariance is a direct consequence of the adiabatic (diagonal)
approximation in (2), which is based on the crucial condition (2.10). This
gauge invariance is thus a measure of the validity of the adiabatic
approximation.

###  2.1 Equations of motion without using commutation relations

It is important that the path integral (2) with the Lagrangian (2.13) is an
approximation in the following two senses: Firstly, it is an approximation
since all the states other than $n$ and their mixings with the state $n$ have
been neglected. Secondly, to justify the above truncation, the valid energy
domain of the path integral formula (for slower variables) is limited to the
very slow motions of the slower variables. If one goes outside this energy
domain, the above path integral formula with the given $L_{n}$ is not
accurate.

We recall that the equations of motion are defined in the path integral
without using canonical commutation relations: We illustrate it by starting
with the identity

$\displaystyle\int{\cal D}\overline{P}_{k}{\cal
D}X_{k}\exp[\frac{i}{\hbar}S(\overline{P},X)]$ (2.17) $\displaystyle=$
$\displaystyle\int{\cal D}(\overline{P}_{k}+\alpha_{k}){\cal
D}(X_{k}+\beta_{k})\exp[\frac{i}{\hbar}S(\overline{P}+\alpha,X+\beta)]$
$\displaystyle=$ $\displaystyle\int{\cal D}\overline{P}_{k}{\cal
D}X_{k}\exp[\frac{i}{\hbar}S(\overline{P}+\alpha,X+\beta)]$

where $S=\int dtL$ and $\alpha_{k}(t)$ and $\beta_{k}(t)$ are infinitesimal
arbitrary functions of $t$ but independent of $\overline{P}_{k}$ and $X_{k}$.
The first equality in (2.17) is an identity since the path integral in terms
of $\\{\overline{P}_{k},\ X_{k}\\}$ and the path integral in terms of
$\\{\overline{P}_{k}+\alpha_{k},\ X_{k}+\beta_{k}\\}$ are the same, namely,
the change of the naming of path integral variables does not change the path
integral itself. The basic postulate of the path integral is that

$\displaystyle{\cal D}(\overline{P}_{k}+\alpha_{k}){\cal
D}(X_{k}+\beta_{k})={\cal D}\overline{P}_{k}{\cal D}X_{k}$ (2.18)

namely, no nontrivial Jacobian for this form of the change of variables, and
thus we obtain the last line of (2.17). After expanding the action in the
linear order of the infinitesimal $\alpha_{k}$ and $\beta_{k}$ in the last
expression of (2.17), one obtains

$\displaystyle\langle T^{\star}\int
dt^{\prime}\alpha_{k}(t^{\prime})\frac{\delta
S}{\delta\overline{P}_{k}(t^{\prime})}\rangle=0,$ $\displaystyle\langle
T^{\star}\int dt^{\prime}\beta_{k}(t^{\prime})\frac{\delta S}{\delta
X_{k}(t^{\prime})}\rangle=0.$ (2.19)

By taking the functional derivative of these relations with respect to
$\alpha_{k}(t)$ and $\beta_{k}(t)$, respectively, one obtains (1.1)

$\displaystyle\langle
T^{\star}\\{\dot{X}_{k}(t)+\Omega^{(n)}_{kl}(\overline{P})\dot{\overline{P}}_{l}(t)-\frac{\partial[\frac{1}{2M}\overline{P}_{k}^{2}+E_{n}(\overline{P})](t)}{\partial\overline{P}_{k}}\\}\rangle=0,$
$\displaystyle\langle
T^{\star}\\{\dot{\overline{P}}_{k}(t)+eF_{kl}(X)\dot{X}_{l}(t)-e\frac{\partial}{\partial
X_{k}}\phi(X(t))\\}\rangle=0,$ (2.20)

with the Berry’s curvature defined in the $n$-th level of the fast system

$\displaystyle\Omega^{(n)}_{kl}(\overline{P})=\frac{\partial{\cal
A}^{(n)}_{l}(\overline{P})}{\partial\overline{P}_{k}}-\frac{\partial{\cal
A}^{(n)}_{k}(\overline{P})}{\partial\overline{P}_{l}}.$ (2.21)

In the present simple case, the difference between the covariant
$T^{\star}$-product and the ordinary $T$-product does not appear, and thus one
may replace $T^{\star}$ by $T$ in the above formula. It is important that the
quantum equations of motion are derived without using the commutation
relations in the path integral (Lagrangian) formalism.

###  2.2 Commutation relations

The fundamental canonical commutation relations are always valid in the
precise treatment of the fast system without adiabatic approximations. This
observation is natural and consistent with our basic path integral formulation
(2); if one sums over all the levels $n$ and $l$ of the fast system, one comes
back to the starting fundamental formula (2) for which one obviously recovers
the basic canonical commutation relations by the BJL procedure for the slower
variables. The basic canonical commutation relations are valid exactly in this
case, which covers all the allowed energy range of the fast and slower
systems.

It is important how one incorporates the above general observation into the
understanding of the path integral formula for a slower system

$\displaystyle Z_{n}$ $\displaystyle=$ $\displaystyle\int{\cal
D}\overline{P}_{k}{\cal D}X_{k}\exp\\{\frac{i}{\hbar}\int_{0}^{T}dtL_{n}\\}$
(2.22)

with the Lagrangian (2.13) which agrees with the common Lagrangian in
condensed matter physics (1.3).

When one discusses the commutation relations, one may consider two main
options:
(i) The above effective Lagrangian is defined in the (conventional) adiabatic
limit $T\rightarrow$ large and thus it is not used to quantize the slower
variables by the BJL prescription, for which infinitely large frequencies are
crucial. A natural extension of the above Lagrangian to the non-adiabatic
domain, where one defines the commutation relations by the BJL procedure and
thus one can estimate the plausible commutators, is to go back to the starting
well-defined path integral formula (2) which gives rise to the normal
canonical commutation relations

$\displaystyle[P_{k},X_{l}]=\frac{\hbar}{i}\delta_{kl},\ \ [P_{k},P_{l}]=0,\ \
[X_{k},X_{l}]=0$ (2.23)

with $\overline{P}_{k}(t)=P_{k}(t)+eA_{k}(X(t))$. This is based on the
assumption that the starting Lagrangian is valid in the non-adiabatic domain
as well as the adiabatic domain where one recognizes Berry’s phase. This
appears to be logically consistent and we call this interpretation as a
standard adiabatic approximation in this article; the procedure described
above shall be illustrated later in a concrete example of a Weyl fermion
appearing in the two-level crossing in condensed matter physics. In this
option, the path integral (2.22) is a formal object that may be used to derive
the equations of motion of the anomalous Hall effect (1.1), which is derived
without using commutation relations. The formula (1.1) is then regarded as a
useful but approximate formula valid only in the standard adiabatic condition
(2.10) 777 In this view point, our understanding of the adiabatic Berry’s
phase in the anomalous Hall effect is analogous to that of Schwinger’s
anomalous magnetic moment in QED, which modifies the low-energy effective
equations of motion of spin by an order $\hbar$ correction but does not modify
the canonical commutation relations of the electron field. . In this
understanding, we do not encounter the Nernst effect.
(ii) The second option may be to adopt a singular limit of the adiabatic
condition (2.10) defined by

$\displaystyle|\epsilon_{n\pm 1}-\epsilon_{n}|\rightarrow\infty$ (2.24)

with fixed
$\epsilon_{n}=\frac{1}{2M}\overline{P}_{k}^{2}+E_{n}(\overline{P})$, for which
any ordinary motions of slower variables $X_{k}(t)$ and $P_{k}(t)$ are
regarded as adiabatic. One may then treat the above effective adiabatic
Lagrangian (2.13) to be a classical one since the commutation relations are
generally modified in this singular limit. The Lagrangian is then extended to
the non-adiabatic domain as it is and then quantized by a suitable means.
Another important property is that Berry’s phase
$\Omega^{(n)}_{kl}(\overline{P})$ becomes generally an exact Dirac monopole in
this singular limit. This is the scheme one applied to (1.3) in the past. One
then applies a renewed (second-time) quantization by BJL or other methods to
the effective Lagrangian (1.3) or equivalently to (2.13) and recognizes the
appearance of the non-canonical system with anomalous Poisson brackets (1) or
anomalous commutation relations (1) in the background field method.

As for the appearance of non-canonical property (i.e., not quantized in a
conventional manner) of the Lagrangian (1.3) or (2.13) for $eA_{k}\neq 0$
888The canonical quantization is possible for $eA_{k}=0$, if one uses
$X^{(l)}_{k}=X_{k}-{\cal A}^{(l)}_{k}$ as an auxiliary variable., we attribute
it to the truncation of the well-defined quantum system (2) to sub-systems in
which both the gauge symmetry of the electromagnetic vector potential and the
newly introduced gauge symmetry of the adiabatic Berry’s phase (2.15) are
formally preserved in the singular limit (2.24), but they are incompatible in
the Hamiltonian formalism. The algebraic compatibility shall be analyzed in
subsection 3.2 later in the context of the Born-Oppenheimer approximation.

This second option singles out the slower system constrained to the
neighborhood of a specific $n$-th energy level $E_{n}$ of the fast system by a
singular limit of the adiabatic condition (2.24). If one is interested in a
specific $n$-th energy band and its intra-band physics such as the Nernst
effect in condensed matter physics, this second option may be physically a
sensible one.

##  3 Born-Oppenheimer approximation and constrained dynamics

###  3.1 A manageable model

An analysis of a manageable model in the Born-Oppenheimer approximation, which
is readily compared to the scheme with Berry’s phase in Section 2, shall be
given in this section to illustrate that the Born-Oppenheimer approximation,
which operates within the scheme of the canonical Hamiltonian formalism, does
not deform the principle of quantum mechanics in the slower system, although
the auxiliary variables which satisfy the deformed commutation relations are
commonly used to describe the anomalous Hall effect [7, 8].

In the formulation of the Born-Oppenheimer approximation, one starts with the
time-independent master Schrödinger equation (using a simplified model
described by $x^{k}$ and $X^{k}$, as an example),

$\displaystyle[H_{0}(X,P)+H_{1}(x,p;P)]\Psi(x,P)=E\Psi(x,P)$ (3.1)

which implies that we adopt the representation of $x_{k}$ and $P_{k}$
diagonal, although the canonical formalism is basically symmetric with respect
to coordinates and momenta; the possible inclusion of the vector potential
$eA_{k}(X)$ and the associated complications will be discussed later. The
canonical quantization of the fast variables is

$\displaystyle[p_{k},x_{l}]=\frac{\hbar}{i}\delta_{kl},\ \ [p_{k},p_{l}]=0,\ \
[x_{k},x_{l}]=0$ (3.2)

and the canonical quantization of the slower variables is given by

$\displaystyle[P_{k},X_{l}]=\frac{\hbar}{i}\delta_{kl},\ \ [P_{k},P_{l}]=0,\ \
[X_{k},X_{l}]=0.$ (3.3)

The present analysis goes through for any finite $N$ number of fast
coordinates $x_{k}$ without any significant modification. We however consider
a single freedom for each of the fast and slower systems, for simplicity.

As a first step, we confirm that the typical quantum mechanical solutions of
$H_{0}(X,P)$ generate slow motions compared to the expected motion in $H_{1}$;
we thus treat $P_{k}$ as slower variables. We then expand the total wave
function

$\displaystyle\Psi(x,P)=\sum_{n}\varphi_{n}(P)\phi_{n}(x,P)$ (3.4)

by solving the equation

$\displaystyle H_{1}(x,p;P)\phi_{n}(x,P)=E_{n}(P)\phi_{n}(x,P)$ (3.5)

with $P_{k}$ treated as background variables; the states $\\{\phi_{n}(x,P)\\}$
are assumed to form a complete orthonormal basis set of the fast system. By
inserting (3.4) into the equation (3.1) and multiplying by
$\phi^{\star}_{l}(x,P)$ and integrating over $x_{k}$, one obtains

$\displaystyle\sum_{n}\\{\int
d^{3}x\phi_{l}(x,P)^{\star}H_{0}(X,P)\phi_{n}(x,P)+E_{n}(P)\delta_{l,n}\\}\varphi_{n}(P)=E\varphi_{l}(P).$
(3.6)

We tentatively adopt in this subsection for simplicity

$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2M}P_{k}^{2}+\frac{M\omega_{0}^{2}}{2}X_{k}^{2}$ (3.7)

which makes the analysis transparent without extra technical complications. We
give later a non-trivial example. Using the completeness relation
$\sum_{l^{\prime}}\phi^{\star}_{l^{\prime}}(x,P)\phi_{l^{\prime}}(y,P)=\delta^{3}(x^{k}-y^{k})$,
we have

$\displaystyle\sum_{n}\sum_{l^{\prime}}\\{\frac{M\omega_{0}^{2}}{2}(\delta^{ll^{\prime}}\frac{-\hbar}{i}\nabla_{k}+{\cal
A}^{ll^{\prime}}_{k}(P))(\delta^{l^{\prime}n}\frac{-\hbar}{i}\nabla_{k}+{\cal
A}^{l^{\prime}n}_{k}(P))+\frac{1}{2M}P_{k}^{2}\delta_{ln}$
$\displaystyle\hskip
42.67912pt+E_{n}(P)\delta_{l,n}\\}\varphi_{n}(P)=E\varphi_{l}(P)$ (3.8)

where $X_{k}=\frac{-\hbar}{i}\frac{\partial}{\partial
P_{k}}=\frac{-\hbar}{i}\nabla_{k}$ and

$\displaystyle{\cal A}^{ll^{\prime}}_{k}(P)=\int
d^{3}x\phi_{l}(x,P)^{\star}\frac{-\hbar}{i}\frac{\partial}{\partial
P_{k}}\phi_{l^{\prime}}(x,P).$ (3.9)

If one assumes the diagonal dominance (adiabatic approximation), namely, if
one assumes that the slower variables $P_{k}$ do not cause a sizable mixing of
fast systems described by $\phi_{l}(x,P)$ (see (2.10)), or assuming that the
properties of the slower system $\varphi_{l}(P)$ are well described by
ignoring the effects of all the states $\phi_{l^{\prime}}(x,P)$ with
$l^{\prime}\neq l$, one obtains

$\displaystyle\\{\frac{M\omega_{0}^{2}}{2}(X_{k}+{\cal
A}^{(l)}_{k}(P))(X_{k}+{\cal
A}^{(l)}_{k}(P))+\frac{1}{2M}P_{k}^{2}+E_{l}(P)\\}\varphi_{l}(P)=E\varphi_{l}(P)$
(3.10)

where we defined

$\displaystyle{\cal A}^{(l)}_{k}(P)\equiv{\cal A}^{ll}_{k}(P)=\int
d^{3}x\phi_{l}(x,P)^{\star}\frac{-\hbar}{i}\frac{\partial}{\partial
P_{k}}\phi_{l}(x,P),$ (3.11)

which is often called Berry’s connection for the specific level $l$; ${\cal
A}^{(l)}_{k}(P)$ is an order $\hbar$ quantity. This is the standard formula of
the Born-Oppenheimer approximation.

It is thus obvious that the slower variables $X^{k}$ are quantized in the
standard manner only once as in (3.3), and the deformation of commutation
relations $[X^{k}(t),X^{l}(t)]\neq 0$, is not induced by Berry’s connection.
The system of the $l$-th subsector (3.10) has an interesting gauge symmetry
defined by the simultaneous transformations

$\displaystyle{\cal A}^{(l)}_{k}(P)\rightarrow{\cal
A}^{(l)}_{k}(P)+\hbar\partial_{k}\alpha^{(l)}(P),$
$\displaystyle\varphi_{l}(P)\rightarrow e^{i\alpha^{(l)}(P)}\varphi_{l}(P)$
(3.12)

which keeps $\Psi_{l}(x,P)=\varphi_{l}(P)\phi_{l}(x,P)$ of each $l$-th
subsector invariant for a choice of $\alpha^{(l)}(P)$ for each $l$
independently; the gauge variation of ${\cal A}^{(l)}_{k}(P)$, which is
induced by the phase change of $\phi_{l}(x,P)$, keeps $\Psi_{l}(x,P)$
invariant if one compensates it by the phase change of $\varphi_{l}(P)$. This
gauge symmetry is thus a measure of the independence of the specific
$\Psi_{l}(x,P)$ from other sectors with $l^{\prime}\neq l$, which is a measure
of the validity of the adiabatic approximation. This gauge symmetry is
manifest in the equation (3.10) if one recalls that the combination

$\displaystyle X^{(l)}_{k}=X_{k}+{\cal A}^{(l)}_{k}(P)$ (3.13)

defines a covariant derivative with respect to (3.1) [7, 21], which satisfies

$\displaystyle[X^{(l)}_{k},X^{(l)}_{m}]=-\frac{\hbar}{i}[\partial_{k}{\cal
A}^{(l)}_{m}(P)-\partial_{m}{\cal A}^{(l)}_{k}(P)].$ (3.14)

In this derivation, we used the momentum representation where the momentum is
diagonal, $X_{k}=-\frac{\hbar}{i}\frac{\partial}{\partial P_{k}}$, which
satisfies the standard canonical commutation relations (3.3).

We now discuss the corresponding equations of motion of slower variables to
compare the final formula with the one given by Berry’s phase. The effective
Hamiltonian for the slower system constructed on the $l$-th level of the fast
system is given by (3.10)

$\displaystyle
H_{l}(P)=\frac{1}{2M}P_{k}^{2}+\frac{M\omega_{0}^{2}}{2}(X_{k}+{\cal
A}^{(l)}_{k}(P))(X_{k}+{\cal A}^{(l)}_{k}(P))+E_{l}(P)$ (3.15)

with $E_{l}(P)$ arising from the $l$-th level of the fast system. We examine
the quantum mechanical equations of motion of slower variables generated by
the effective Hamiltonian, which is constrained to the $l$-th level of the
fast system, using the canonical commutation relations (in the Heisenberg
picture)

$\displaystyle\dot{P}_{m}$ $\displaystyle=$
$\displaystyle\frac{i}{\hbar}[H_{l},P_{m}]=-M\omega_{0}^{2}(X_{m}+{\cal
A}^{(l)}_{m}(P)),$ $\displaystyle\dot{X}_{m}$ $\displaystyle=$
$\displaystyle\frac{i}{\hbar}[H_{l},X_{m}]$ (3.16) $\displaystyle=$
$\displaystyle\frac{M\omega_{0}^{2}}{2}[(X_{k}+{\cal
A}^{(l)}_{k}(P))\frac{\partial}{\partial P_{m}}{\cal
A}^{(l)}_{k}(P)+\frac{\partial}{\partial P_{m}}{\cal
A}^{(l)}_{k}(P)(X_{k}+{\cal A}^{(l)}_{k}(P))]$
$\displaystyle+\frac{\partial}{\partial P_{m}}[\frac{P^{2}_{k}}{2M}+E(P)]$
$\displaystyle=$ $\displaystyle-\frac{\partial}{\partial P_{m}}{\cal
A}^{(l)}_{k}(P)\dot{P}_{k}+\frac{\partial}{\partial
P_{m}}[\frac{P^{2}_{k}}{2M}+E_{l}(P)]$

where the last expression is valid when one ignores the possible operator
ordering problem 999The issue of operator ordering appears when one replaces
$[(X_{k}+{\cal A}^{(l)}_{k}(P))\frac{\partial}{\partial P_{m}}{\cal
A}^{(l)}_{k}(P)+\frac{\partial}{\partial P_{m}}{\cal
A}^{(l)}_{k}(P)(X_{k}+{\cal A}^{(l)}_{k}(P))]$ by $2\frac{\partial}{\partial
P_{m}}{\cal A}^{(l)}_{k}(P)(X_{k}+{\cal A}^{(l)}_{k}(P))$. This kind of
ordering problem is common in the operator formulation in quantum mechanics.
In the present case, ${\cal A}^{(l)}_{k}(P)$ is of order $O(\hbar)$ and we are
interested in the order $O(\hbar)$ corrections to the equations of motion. We
can thus ignore the operator ordering issue which would give $O(\hbar^{2})$
terms.. If one uses the auxiliary variables $X^{(l)}_{m}\equiv X_{m}+{\cal
A}^{(l)}_{m}(P)$ in (3.13) specific to the $l$-th level, one has an equivalent
set of equations of motion (again by ignoring the possible operator ordering
problem) [7]

$\displaystyle\dot{P}_{m}$ $\displaystyle=$
$\displaystyle-M\omega_{0}^{2}X^{(l)}_{m},$ $\displaystyle\dot{X}^{(l)}_{m}$
$\displaystyle=$
$\displaystyle-\Omega^{(l)}_{mk}(P)\dot{P}_{k}+\partial_{m}[\frac{P^{2}_{k}}{2M}+E_{l}(P)]$
(3.17)

with $\Omega^{(l)}_{mk}(P)=\frac{\partial}{\partial P_{m}}{\cal
A}^{(l)}_{k}(P)-\frac{\partial}{\partial P_{k}}{\cal A}^{(l)}_{m}(P).$

If one uses the electromagnetic scalar potential $e\phi(X)$ instead of the
harmonic potential, (3.1) becomes

$\displaystyle\dot{P}_{m}$ $\displaystyle=$ $\displaystyle
e\frac{\partial}{\partial X^{(l)}_{m}}\phi(X^{(l)}),$
$\displaystyle\dot{X}^{(l)}_{m}$ $\displaystyle=$
$\displaystyle-\Omega^{(l)}_{mk}(P)\dot{P}_{k}+\partial_{m}[\frac{P^{2}_{k}}{2M}+E_{l}(P)]$
(3.18)

The set of equations (3.1) correspond to the common equations of the anomalous
Hall effect in condensed matter physics (1.1) [7].

One may notice that the auxiliary variables $X^{(l)}_{m}$ (3.13) give rise to
the non-commutative geometry

$\displaystyle[P_{k},P_{l}]=0,\ \
[P_{k},X^{(l)}_{m}]=\frac{\hbar}{i}\delta_{km},\ \
[X^{(l)}_{k},X^{(l)}_{m}]=-\frac{\hbar}{i}[\partial_{k}{\cal
A}^{(l)}_{m}(P)-\partial_{m}{\cal A}^{(l)}_{k}(P)],$ (3.19)

specific to the $l$-th level of the fast system, in contrast to the starting
canonical commutation relations (3.3). But this is not the deformation of the
principle of quantum mechanics, since these commutation relations of
$X^{(l)}_{m}=X_{m}+{\cal A}^{(l)}_{m}(P)$ are dictated by the canonical
commutation relations (3.3) of the variables $X_{m}$ and $P_{k}$; every
quantity expressed in terms of $X_{m}$ and $P_{k}$ are described by the
conventional canonical commutation relations, and thus no Nernst effect in
(3.1). The appearance of noncommutative geometry in the sense of (3.19) is an
artifact of the choice of the auxiliary variables $X^{(l)}_{m}$ specific to
the subsystem defined by $\Psi_{l}(x,P)=\varphi_{l}(P)\phi_{l}(x,P)$. In the
present example, $X_{m}(=X^{(l)}_{m}-{\cal A}^{(l)}_{m}(P))$ stand for the
proper coordinates of the total system of slower particles and exhibit no non-
commutative geometry.

We next examine the electromagnetic current. The measurement of the transverse
velocity as an anomalous Hall effect implies that one adopts the measured
electric current

$\displaystyle j_{m}\equiv-e\dot{X}^{(l)}_{m}=-e\dot{X}_{m}-e\dot{{\cal
A}}^{(l)}_{m}=-e\dot{X}_{m}-e\left(\frac{\partial}{\partial P_{k}}{\cal
A}^{(l)}_{m}\right)\dot{P}_{k}.$ (3.20)

On the other hand, if one evaluates the current in the starting Hamiltonian
(3.1) by gauging the vector potential one has

$\displaystyle J_{k}$ $\displaystyle=$ $\displaystyle-\frac{\delta}{\delta
A_{k}(X)}[H_{0}(X,P+eA_{k}(X))+H_{1}(x,p;P+eA_{k}(X))]|_{A_{k}=0}$ (3.21)
$\displaystyle=$ $\displaystyle-e\frac{\delta}{\delta
P_{k}}[H_{0}(X,P+eA_{k}(X))+H_{1}(x,p;P+eA_{k}(X))]|_{A_{k}=0}$
$\displaystyle=$ $\displaystyle-e\frac{\delta}{\delta
P_{k}}[H_{0}(X,P)+H_{1}(x,p;P)]$ $\displaystyle=$ $\displaystyle-e\dot{X}_{k}$

using the equations of motion for the starting exact Hamiltonian. Thus the
exact current before one makes any approximation agrees with the time
derivative of the slower coordinates. If one recalls that the total wave
function is given by
$\Psi(x,P)=\sum_{l}\Psi_{l}(x,P)=\sum_{l}\varphi_{l}(P)\phi_{l}(x,P)$ and one
is analyzing the anomalous Hall effect in the subspace
$\Psi_{l}(x,P)=\varphi_{l}(P)\phi_{l}(x,P)$, one may expect the relation

$\displaystyle\frac{d}{dt}\sum_{l}{\cal A}^{(l)}_{m}(P)=0$ (3.22)

in (3.20). This fundamental property (3.22) is ensured by (2) in the
Lagrangian formalism, which states that the exact evaluation of the fast
system does not generate Berry’s phase, combined with the adiabatic condition
(2.10) which implies the diagonal dominance. In the Hamiltonian formalism one
can argue the same conclusion directly from (3.11) after a suitable
regularization 101010Intuitively, if one defines the completeness relation of
$\phi_{l}(x,P)$ by
$\sum_{l}\phi_{l}(x,P)\phi^{\dagger}_{l}(y,P^{\prime})=\delta(x-y)\delta_{P,P^{\prime}}$,
one has $\frac{d}{dt}\sum_{l}A_{l}(P)=\frac{d}{dt}\int dx\lim_{y\rightarrow
x}\delta(x-y)(1/\Delta P)[\delta_{P+\Delta P,P}-\delta_{P,P}]=0$.. The
relation (3.22) implies that the electric current conservation is satisfied
when one considers all the bands together, although the current conservation
is generally modified for each band separately in the Born-Oppenheimer
approximation based on the canonical commutators, if one adopts the current
(3.20). This observation implies that the introduction of the electromagnetic
vector potential $eA_{k}(X)$, which couples to the current, is not
straightforward in the Hamiltonian of the Born-Oppenheimer approximation. This
issue is analyzed in the next subsection.

###  3.2 Gauge symmetries of electromagnetism and Berry’s phase

The introduction of the electromagnetic vector potential $A_{k}(X)$ into the
adiabatic Hamiltonian with Berry’s connection does not proceed in a simple
manner in the Born-Oppenheimer approximation, which is defined by canonical
commutation relations. This complication is related to the appearance of the
non-canonical system in the common description of the anomalous Hall effect
(1.3) with Berry’s phase, since the Born-Oppenheimer approximation operates in
a canonical Hamiltonian formalism and a natural formulation with both
$A_{k}(X)$ and ${\cal A}^{(l)}_{k}(P)$ would imply a canonical system instead
of the non-canonical system in (1.3).

We comment on the origin of this complication from a point of view of
algebraic consistency in the framework of the Born-Oppenheimer approximation.
We start with the canonical universal coordinates of the slower system

$\displaystyle[X_{k},X_{l}]=0,\ \ \ [X_{k},P_{l}]=i\hbar\delta_{kl},\ \ \
[P_{k},P_{l}]=0.$ (3.23)

We define the covariant derivative $P_{k}+eA_{k}(X)$ after gauging the
electromagnetic interaction and we define the covariant derivative
$X_{k}+{\cal A}^{(n)}_{k}(P)$ after inducing Berry’s connection in the
Hamiltonian formalism. The gauge symmetry of Berry’s connection is a measure
of the validity of the adiabatic approximation by constraining the coordinates
of the slower system to $\varphi_{n}(P)$. We have

$\displaystyle[P_{k}+eA_{k}(X),P_{l}+eA_{l}(X)]=-i\hbar
e(\partial_{k}A_{l}(X)-\partial_{l}A_{k}(X)),$ $\displaystyle[X_{k}+{\cal
A}^{(n)}_{k}(P),X_{l}+{\cal A}^{(n)}_{l}(P)]=i\hbar(\partial_{k}{\cal
A}^{(n)}_{l}(P)-\partial_{l}{\cal A}^{(n)}_{k}(P)],$
$\displaystyle[P_{k}+eA_{k}(X),X_{l}+{\cal
A}^{(n)}_{l}(P)]=-i\hbar\delta_{kl}-e[{\cal A}^{(n)}_{k}(P),A_{l}(X)]$ (3.24)

where the last relation complicates the analysis, although all the commutation
relations are controlled by the original canonical commutation relations
(3.23). The last relation in (3.2) shows that the curvature on the right-hand
side is non-vanishing, which implies that the two gauge symmetries are not
commuting 111111 Besides, it contains the term $[{\cal
A}^{(n)}_{k}(P),eA_{l}(X)]$ which is non-local and described by a Moyal
product.. This algebraic complication is the cause of the failure of the Born-
Oppenheimer approximation, which operates in the framework of the canonical
Hamiltonian formalism, in the presence of the electromagnetic potential
$eA_{k}(X)$. In the Hamiltonian formalism in (3.15), if one gauges the
electromagnetic vector potential $eA_{k}(X)$ by the minimal gauge principle
$P_{k}\rightarrow P_{k}+eA_{k}(X)$, one obtains

$\displaystyle H_{l}=\frac{1}{2M}(P_{k}+eA_{k}(X))^{2}$ (3.25)
$\displaystyle+\frac{M\omega_{0}^{2}}{2}(X_{k}+{\cal
A}^{(l)}_{k}(P_{k}+eA_{k}(X)))(X_{k}+{\cal
A}^{(l)}_{k}(P_{k}+eA_{k}(X)))+E_{l}(P_{k}+eA_{k}(X))$

but this is not consistent since $X_{k}$ in $eA_{k}(X)$ needs to be replaced
by the covariant derivative $X_{k}+{\cal A}^{(l)}_{k}(P)$ in the Hamiltonian
formalism, ad infinitum.

We now show that the path integral (Lagrangian formalism) is flexible and show
how it realizes the electric gauge invariance by sacrificing the canonical
quantization. One may start with

$\displaystyle L$ $\displaystyle=$ $\displaystyle P_{k}\dot{X}_{k}-H_{l}(X,P)$
(3.26) $\displaystyle=$ $\displaystyle
P_{k}\dot{X}_{k}-[\frac{1}{2M}P_{k}^{2}+\frac{M\omega_{0}^{2}}{2}(X_{k}+{\cal
A}^{(l)}_{k}(P))(X_{k}+{\cal A}^{(l)}_{k}(P))+E_{l}(P)]$

where $H_{l}(X,P)$ is the Born-Oppenheimer Hamiltonian in (3.15). One may then
define the path integral

$\displaystyle\int{\cal D}P{\cal D}X\exp\\{\frac{i}{\hbar}\int
dt[P_{k}\dot{X}_{k}-H_{l}(X,P)]\\}$ (3.27)

which gives rise to the canonical commutation relations if the BJL
prescription is applied by assuming that $H_{l}(X,P)$ is valid for any motion
of $P_{k}$ and $X_{k}$. Cf. (2.24). One may next rewrite this path integral as

$\displaystyle\int{\cal D}P{\cal D}X^{(l)}\exp\\{\frac{i}{\hbar}\int
dt[P_{k}\dot{X}^{(l)}_{k}+{\cal
A}^{(l)}_{k}(P)\dot{P}_{k}-\left(\frac{1}{2M}P_{k}^{2}+\frac{M\omega_{0}^{2}}{2}(X^{(l)}_{k})^{2}+E_{l}(P)\right)]\\}$
(3.28)

where $X^{(l)}_{k}=X_{k}+{\cal A}^{(l)}_{k}(P)$ and we used ${\cal D}P{\cal
D}X={\cal D}P{\cal D}X^{(l)}$. The BJL prescription when applied to this path
integral gives rise to canonical $[X_{k},X_{l}]=0$, $[P_{k},P_{l}]=0$ and
$[X_{k},P_{l}]=i\hbar\delta_{kl}$, although with non-commutative geometry
$[X^{(l)}_{k},X^{(l)}_{n}]=-\frac{\hbar}{i}\Omega^{(l)}_{kn}$ for the
auxiliary variable $X^{(l)}_{k}$. One may add an electromagnetic gauge
invariant term $-\int dteA_{k}(X^{(l)})\dot{X}^{(l)}_{k}$ to the action by
hand, then one obtains

$\displaystyle\int{\cal D}P{\cal D}X^{(l)}\exp\\{\frac{i}{\hbar}\int
dt[P_{k}\dot{X}^{(l)}_{k}-eA_{k}(X^{(l)})\dot{X}^{(l)}_{k}+{\cal
A}^{(l)}_{k}(P)\dot{P}_{k}$ $\displaystyle\hskip
99.58464pt-\left(\frac{1}{2M}P_{k}^{2}+\frac{M\omega_{0}^{2}}{2}(X^{(l)}_{k})^{2}+E_{l}(P)\right)]\\}.$
(3.29)

The above extra electromagnetic coupling would be induced from the minimally
coupled Hamiltonian

$\left(\frac{1}{2M}(P_{k}+eA_{k}(X^{(l)}))^{2}+\frac{M\omega_{0}^{2}}{2}(X^{(l)}_{k})^{2}+E_{l}(P_{k}+eA_{k}(X^{(l)}))\right)$

by a change of the variable $P_{k}+eA_{k}(X^{(l)})\rightarrow P_{k}$ if the
term ${\cal A}^{(l)}_{k}(P)\dot{P}_{k}$ should be absent, but in reality
because of ${\cal A}^{(l)}_{k}(P)\dot{P}_{k}$ the manipulation in Hamiltonian
formalism does not work. Note that the current to which the vector potential
$eA_{k}(X^{(l)})$ at the point $X^{(l)}$ couples is chosen as the current
$\dot{X}^{(l)}_{k}$. Cf. (3.20). The term $-\int
dteA_{k}(X^{(l)})\dot{X}^{(l)}_{k}$ is analogous to the Wess-Zumino term by
satisfying the gauge invariance by itself in the Lagrangian formalism. See
also (2.16). The Lagrangian in (3.2), which is confirmed not to be quantized
in a canonical manner, agrees with (1.3) or (2.13) if one changes the dummy
path integral variable $X^{(l)}_{k}\rightarrow X_{k}$ and replaces the
harmonic potential by the scalar potential
$\frac{M\omega_{0}^{2}}{2}(X_{k})^{2}\rightarrow-e\phi(X_{k})$. This analysis
shows how the Lagrangian formalism maintains the electromagnetic gauge
invariance in the presence of $eA_{k}$ by sacrificing the canonical formalism
of quantum mechanics.

The Nernst effect is attributed to the non-canonical behavior and a
modification of the density of states $D=(1+e\vec{B}\cdot\vec{\Omega})$ [6],
which is derived from the deformed commutators (1) [17, 8]; the same result is
obtained by a more general treatment of the action (1.3) [13] in a singular
adiabatic limit (2.24), namely, for any ordinary movements of slower variables
in (1.3). From a point of view of gauge symmetries, the Nernst effect is
regarded as a manifestation of the stricture of the electromagnetic gauge
symmetry when the slower freedom is constrained to a specific $n$-th state
$\varphi_{n}(P)$ in (3.4) by imposing the gauge invariance of Berry’s
connection, as we have seen in the framework of the Born-Oppenheimer
approximation defined by canonical commutators121212In the formulation in
Section 2, the singular limit of the adiabatic condition (2.24) effectively
constrains the system to a specific state with $E_{n}$.. Note that the
breaking of the gauge symmetry of Berry’s connection would imply that the
adiabatic approximation is not accurate, while the breaking of the
electromagnetic gauge symmetry would imply the inconsistency of the theory.

##  4 Discussions and conclusion

We attempted to understand why Berry’s phase defined in the adiabatic limit
leads to a non-canonical dynamical system with deformed commutators that have
been opened up by the applications of Berry’s phase to the anomalous Hall
effect. We start with the derivation of the action (1.3) or (2.13) in the path
integral (Lagrangian) formalism based on the generic adiabatic condition

$\displaystyle|E_{n\pm 1}-E_{n}|\gg\hbar\frac{2\pi}{T}$ (4.1)

where $T$ stands for the typical time scale of the slower system $X_{k}$, and
$E_{n}$ is the $n$-th energy level of the fast system. Regardless of the
specific interpretations of the condition (4.1), one can derive the equations
of motion of the anomalous Hall effect (1.1). In the conventional
understanding of the adiabatic condition by letting $T\rightarrow$ large with
fixed energy eigenvalues $E_{n}$ of the fast system, the BJL prescription is
not applicable. But the conventional canonical commutation relations of slower
variables are reproduced by applying the BJL prescription to the original
exact Lagrangian, which works for the adiabatic as well as non-adiabatic
motions. The same conclusion is attained using an exactly solvable model of
Berry’s phase [19] which covers from the adiabatic domain to the non-adiabatic
domain. In this understanding of Berry’s phase, no Nernst effect arises.

On the other hand, if one considers a singular limit of the adiabatic
condition

$\displaystyle|E_{n\pm 1}-E_{n}|\rightarrow\infty$ (4.2)

with $E_{n}$ fixed, the effective adiabatic Lagrangian is generally converted
to be a classical one (since the commutation relations are generally modified
in this limit ) and any ordinary motion of the slower variables satisfies the
adiabatic condition (4.1). This singular limit changes the adiabatic
Lagrangian, such as (1.3) and (2.13), to a dynamical system which is not
quantized in a canonical manner in the presence of the vector potential
$eA_{k}(X)$. One thus arrives at the deformed commutation relations such as
(1) and (1), which in turn leads to the Nernst effect. In the absence of the
vector potential $eA_{k}(X)=0$, one still encounters the non-commutative
geometry but dynamically one has a canonical system and thus no Nernst effect.

In the Born-Oppenheimer approximation, which is defined in a Hamiltonian
formalism with canonical commutation relations, one can derive the equations
of motion of the anomalous Hall effect (1.1) in the absence of the vector
potential $eA_{k}(X)=0$ if one introduces the auxiliary variables

$\displaystyle X^{(n)}_{k}=X_{k}+{\cal A}^{(n)}_{k}(P),$ (4.3)

but no Nernst effect. In the Hamiltonian formulation of the Born-Oppenheimer
approximation, one cannot incorporate the vector potential $eA_{k}(X)$ in a
consistent manner. We attribute this failure of the Born-Oppenheimer
approximation in the presence of $eA_{k}(X)$ to the incompatibility of the
gauge symmetry of Berry’s connection, which projects the total system of
$\Psi(x,P)=\sum_{n}\varphi_{n}(P)\phi_{n}(x,P)$ to a specific subsystem
$\varphi_{n}(P)\phi_{n}(x,P)$, and the electromagnetic gauge symmetry which
acts universally over all the subsystems of $\Psi(x,P)$. Note that the
variable (4.3) is a covariant derivative for the subsystem
$\varphi_{n}(P)\phi_{n}(x,P)$ of this gauge symmetry of Berry’s connection. We
explicitly demonstrated that the Born-Oppenheimer approximation, when
converted to the Lagrangian (path integral) formalism, can incorporate the
vector potential $eA_{k}(X)$ in a gauge invariant manner, but the resulting
action is not amenable to the conventional canonical formulation of quantum
mechanics.

We thus conclude that the equations of motion of the anomalous Hall effect
(1.1) are generally valid in the adiabatic limit, but the interpretation of
the dynamical system represented by the action (1.3) is more involved. The
appearance of the non-canonical dynamical system and the Nernst effect is
based on the deformation of the quantum principle to incorporate the two
incompatible gauge symmetries of electromagnetic vector potential and Berry’s
connection.

From a point of view of practical applications, if one is interested in a
specific energy band and its intra-band physics in condensed matter physics,
the choice of the singular limit of the adiabatic condition (4.2) may be a
sensible one. The constrained dynamics of the slower system to the specific
$n$-th state imposed by a singular limit of the adiabatic condition combined
with the vector potential $eA_{k}(X)$ is then responsible for the deformation
of the canonical formalism which leads to the deformed commutators and the
Nernst effect in the adiabatic action of the anomalous Hall effect.

## Acknowledgements

We thank Naoto Nagaosa, Shinichi Deguchi and Kenji Fukushima for helpful
discussions and suggestions. The present work is supported in part by JSPS
KAKENHI (Grant No.18K03633).

##  Appendix A Two-band model and commutation relations

We illustrate that the general analysis in Section 2 works in the simple Weyl
model with an explicit form of Berry phase. The model is defined by

$\displaystyle H=-\mu\vec{p}(t)\cdot\vec{\sigma}$ (A.1)

where $\vec{\sigma}$ stands for the pseudo spin that describes the upper and
lower crossing bands.

We thus start with the Schrödinger equation
$i\hbar\partial_{t}\psi(t)=H\psi(t)$ or the Lagrangian (in the spirit of the
second quantization) given by [20]

$\displaystyle L=\psi^{\dagger}(t)[i\hbar\partial_{t}-H]\psi(t)$ (A.2)

where the two-component spinor $\psi(t)$ specifies the movement of upper and
lower levels which appear in the band-crossing problem. In the present
context, the fast variables are given by the spin freedom $\psi(t)$
characterized by the energy scale $\mu|\vec{p}(t)|$, and the slower variables
are given by the angular freedom of $\vec{p}(t)$. For simplicity, we assume
the magnitude $|\vec{p}(t)|$ to be time independent. We then perform a time-
dependent unitary transformation $\psi(t)=U(\vec{p}(t))\psi^{\prime}(t),\ \
\psi^{\dagger}(t)={\psi^{\prime}}^{\dagger}(t)U^{\dagger}(\vec{p}(t))$ with
$U(\vec{p}(t))^{\dagger}\mu\vec{p}(t)\cdot\vec{\sigma}U(\vec{p}(t))=\mu|\vec{p}|\sigma_{3}$.
This unitary transformation is explicitly given by a $2\times 2$ matrix
$U(\vec{p}(t))=\left(v_{+}(\vec{p})\ v_{-}(\vec{p})\right)$, where

$\displaystyle
v_{+}(\vec{p})=\left(\begin{array}[]{c}\cos\frac{\theta}{2}e^{-i\varphi}\\\
\sin\frac{\theta}{2}\end{array}\right),\ \ \
v_{-}(\vec{p})=\left(\begin{array}[]{c}\sin\frac{\theta}{2}e^{-i\varphi}\\\
-\cos\frac{\theta}{2}\end{array}\right)$ (A.7)

which correspond to the use of instantaneous eigenfunctions of the operator
$\mu\vec{p}(t)\cdot\vec{\sigma}$, namely,
$\mu\vec{p}(t)\cdot\vec{\sigma}v_{\pm}(\vec{p})=\pm\mu|\vec{p}|v_{\pm}(\vec{p})$,
where
$\vec{p}(t)=|\vec{p}|(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta)$
with time dependent $\theta(t)$ and $\varphi(t)$.

Based on this transformation, the equivalence of two Lagrangians is derived,
namely, $L$ in (A.2) and

$\displaystyle
L^{\prime}={\psi^{\prime}}^{\dagger}[i\hbar\partial_{t}+\mu|\vec{p}|\sigma_{3}+U(\vec{p}(t))^{\dagger}i\hbar\partial_{t}U(\vec{p}(t))]\psi^{\prime}.$
(A.8)

The starting Hamiltonian (A.1) is thus replaced by [20]

$\displaystyle H^{\prime}(t)$ $\displaystyle=$
$\displaystyle-\mu|\vec{p}|\sigma_{3}-U(\vec{p}(t))^{\dagger}i\hbar\partial_{t}U(\vec{p}(t))$
(A.11) $\displaystyle=$
$\displaystyle-\mu|\vec{p}|\sigma_{3}-\hbar\left(\begin{array}[]{cc}\frac{(1+\cos\theta)\dot{\varphi}}{2}&\frac{\dot{\varphi}\sin\theta+i\dot{\theta}}{2}\\\
\frac{\dot{\varphi}\sin\theta-i\dot{\theta}}{2}&\frac{(1-\cos\theta)\dot{\varphi}}{2}\end{array}\right).$

We note that $2\mu|\vec{p}|=|\epsilon_{1}-\epsilon_{2}|$ in the notation of
(2.10) in Section 2. The adiabatic condition (2.10) is thus written

$\displaystyle 2\mu|\vec{p}|\gg 2\pi\hbar/T,$ (A.12)

where $T$ is the typical time scale of the slower dynamical variable and
customarily taken as the period of the slower dynamical variable (i.e.,
angular freedom) of $\vec{p}(t)$, and we estimate $\dot{\varphi}\sim 2\pi/T$.
We thus have from (A.11)

$\displaystyle
H^{\prime}_{ad}\simeq-\mu|\vec{p}|\sigma_{3}-\hbar\left(\begin{array}[]{cc}\frac{(1+\cos\theta)\dot{\varphi}}{2}&0\\\
0&\frac{(1-\cos\theta)\dot{\varphi}}{2}\end{array}\right).$ (A.15)

Berry’s phase is given by the second term and one recognizes the familiar
monopole-like expression in the diagonal 131313Berry’s phase in a realistic
two-band model in condensed matter physics in the adiabatic limit has been
analyzed in detail by Nagaosa [21]..

The BJL prescription, whose essence is illustrated in Appendix B, requires the
condition

$\displaystyle\mu|\vec{p}|\ll 2\pi\hbar/T=\hbar\omega$ (A.16)

for the fixed $\mu|\vec{p}|$ to analyze the equal-time commutation relations
of slower variables. To see the implications of the non-adiabatic condition
(A.16) explicitly, it is convenient to perform a further unitary
transformation of the fermionic variable
$\psi^{\prime}(t)=U(\theta(t))\psi^{\prime\prime}(t),\ \ \
{\psi^{\prime}(t)}^{\dagger}={\psi^{\prime\prime}}^{\dagger}(t)U^{\dagger}(\theta(t))$
with [11]

$\displaystyle
U(\theta(t))=\left(\begin{array}[]{cc}\cos\frac{\theta}{2}&-\sin\frac{\theta}{2}\\\
\sin\frac{\theta}{2}&\cos\frac{\theta}{2}\end{array}\right)$ (A.19)

in addition to (A.8), which diagonalizes the dominant Berry’s phase term. The
Hamiltonian (A.11) then becomes

$\displaystyle H^{\prime\prime}(t)$ $\displaystyle=$
$\displaystyle-\mu|\vec{p}|U(\theta(t))^{\dagger}\sigma_{3}U(\theta(t))+(U(\vec{p}(t))U(\theta(t)))^{\dagger}\frac{\hbar}{i}\partial_{t}(U(\vec{p}(t))U(\theta(t)))$
(A.24) $\displaystyle=$
$\displaystyle-\mu|\vec{p}|\left(\begin{array}[]{cc}\cos\theta&-\sin\theta\\\
-\sin\theta&-\cos\theta\end{array}\right)-\hbar\left(\begin{array}[]{cc}\dot{\varphi}&0\\\
0&0\end{array}\right).$

Note that the first term is bounded by $\mu|\vec{p}|$ and the second term is
dominant for the non-adiabatic case $\mu|\vec{p}|\ll 2\pi\hbar/T$. We
emphasize that both (A.11) and (A.24) are exact expressions.

The relation (A.24) shows that Berry’s phase becomes trivial for the non-
adiabatic limit (A.16) with fixed $\mu|\vec{p}|$ and thus no modification of
commutation relations by Berry’s phase 141414This property has been argued to
be the case in [19] using an exactly solvable model of Berry’s model [22]..

On the other hand, if one considers a singular limit
$2\mu|\vec{p}|=|\epsilon_{2}-\epsilon_{1}|\rightarrow\infty$ in the adiabatic
criterion $2\mu|\vec{p}|\gg 2\pi\hbar/T$ in (A.12), the adiabatic formula
(A.15) becomes exact together with an exact Dirac monopole for any movement of
slower angular variables. This condition with the present two-band model leads
to the action (1.3) with $A_{k}=0$ and $\phi=0$. One would thus obtain the
non-commutative geometry (1)

$\displaystyle[X_{k},X_{l}]=i\hbar\Omega_{kl}.$ (A.25)

We next show the absence of anomalous commutation relations induced by Berry’s
phase in the present two-level crossing model if one analyzes the exact
effective Hamiltonian without recourse to adiabatic approximations. We first
rewrite the exact formula (A.11) as

$\displaystyle
H^{\prime}(t)=-\mu|\vec{p}|\sigma_{3}-\hbar\left(\begin{array}[]{cc}\frac{(1+\cos\theta)}{2}&\frac{\sin\theta}{2}\\\
\frac{\sin\theta}{2}&\frac{(1-\cos\theta)}{2}\end{array}\right)\dot{\varphi}-\hbar\left(\begin{array}[]{cc}0&\frac{i}{2}\\\
\frac{-i}{2}&0\end{array}\right)\dot{\theta},$ (A.30)

which shows that $\dot{\theta}$ has no non-trivial conjugate variable in the
phase space. It vanishes when one integrates $\int dtH^{\prime}(t)$ to define
the action. The variable $\dot{\varphi}$ appears to have non-trivial conjugate
variables, which are Berry’s phases and their off-diagonal partners, and thus
could contribute to the modification of commutation relations. But one can
confirm that the matrix multiplying $\dot{\varphi}$ has a vanishing
determinant and a unit trace,

$\displaystyle{\rm
det}\left(\begin{array}[]{cc}\frac{(1+\cos\theta)}{2}&\frac{\sin\theta}{2}\\\
\frac{\sin\theta}{2}&\frac{(1-\cos\theta)}{2}\end{array}\right)=0,\ \ \ {\rm
Tr}\left(\begin{array}[]{cc}\frac{(1+\cos\theta)}{2}&\frac{\sin\theta}{2}\\\
\frac{\sin\theta}{2}&\frac{(1-\cos\theta)}{2}\end{array}\right)=1$ (A.35)

independently of $\theta$, which imply the eigenvalues $1$ and $0$ and thus
dynamically trivial. To see this fact explicitly, we refer to the unitary
equivalent (A.24). We emphasize that both (A.30) and (A.24) are exact
expressions in the scheme of [20]; (A.24) is accurate in the non-adiabatic
limit with the time derivative term being diagonal.

The unitary equivalent exact (A.24) shows that $\dot{\varphi}$ which
multiplies would-be Berry’s phases and their off-diagonal partners, does not
have any non-trivial conjugate variable, and thus does not contribute to the
modification of commutation relations. The first term in (A.24) is essentially
the same as the starting expression (A.1) if one considers the azimuthally
symmetric configurations. Thus would-be Berry’s phases and their off-diagonal
partners in a precise treatment generate no extra non-trivial time-derivative
terms in the action, and thus gives no anomalous commutation relations.

This fact agrees with the observation made in connection with (2) in Section
2. The second quantized part in (2) except for the time derivative terms
$a_{n}^{\star}(t)i\hbar\partial_{t}a_{n}(t)$ corresponds to the present
(A.24). If one sums over $n$ and $l$ in (2), the path integral is reduced to
the original path integral (2) without any extra time derivative terms added,
for which the canonical commutation relations of slower variables are
recovered by the BJL prescription. This analysis shows that the anomalous
commutation relations of slower variables do not appear if no adiabatic
approximations are made in the sector of fast variables 151515 To be precise,
there is a difference between (A.30) and (2) in Section 2. In (2), the fast
system and slower system are clearly separated, while in the Weyl model (A.30)
this separation is not perfect as is seen in (A.24). Thus we have no Berry’s
phase for the exact evaluation of the fast system in (2), but we have a non-
trivial Berry’s phase for an exactly solvable model [22, 23] which is close to
the Weyl model. The behavior of Berry’s phase at the non-adiabatic limit is,
however, similar in either case and becomes trivial as in (A.24) which is
relevant when one analyzes the possible deformation of commutation relations
by the BJL prescription..

##  Appendix B BJL analysis of commutation relations

The basic idea of the BJL prescription [15] is that the equal-time commutator
of operators $A(t)$ and $B(t)$ is determined by the short-time limit of the
correlation function $\langle TA(t_{1})B(t_{2})\rangle$ with $t_{1}\rightarrow
t_{2}$, which implies the large frequency limit of the Fourier transform of
$\langle TA(t_{1})B(t_{2})\rangle$. It is illustrated how the BJL prescription
works in the present context for the simple case with $eA_{k}(X(t))=0$. We
start with the path integral (2) but with $eA_{k}(X(t))=0$, for which the path
integral is well-defined quantum mechanically. We assume that the adiabatic
Lagrangian is extended to the non-adiabatic domain of slower variables (by
considering a singular limit of the adiabatic condition (4.2))

$\displaystyle Z_{n}$ $\displaystyle=$ $\displaystyle\int{\cal
D}\overline{P}_{k}{\cal D}X_{k}$ $\displaystyle\times$
$\displaystyle\exp\\{\frac{i}{\hbar}\int_{0}^{T}dt[\overline{P}_{k}(t)\dot{X}_{k}(t)+{\cal
A}_{k}(\overline{P}(t))\dot{\overline{P}_{k}}(t)-(\frac{1}{2M}\overline{P}_{k}^{2}+E_{n}(\overline{P}))+e\phi(X)]\\}.$

To sketch a BJL prescription, we analyze a simplified system instead of (B),

$\displaystyle L=\overline{P}_{k}\dot{X}_{k}+{\cal
A}_{k}(\overline{P})\dot{\overline{P}_{k}}$ (B.2)

by ignoring the terms which do not modify the commutation relations. We start
with the free part $L_{0}=\overline{P}_{k}\dot{X}_{k}$ which gives (2.3); the
starting correlation functions are defined by 161616Feynman’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.

$\displaystyle\langle
TX^{k}(t)\overline{P}^{l}(t^{\prime})\rangle=\int\frac{d\omega}{2\pi}e^{i\omega(t-t^{\prime})}\frac{\hbar\omega}{\omega^{2}+i\epsilon}\delta_{kl},$
$\displaystyle\langle TX^{k}(t)X^{l}(t^{\prime})\rangle=0,\ \ \langle
T\overline{P}^{k}(t)\overline{P}^{l}(t^{\prime})\rangle=0.$ (B.3)

We use the conventional $T$ product instead of the covariant $T^{\star}$
product since they agree in the present simple cases. By multiplying the
Fourier transform of the first relation $\int
dte^{-i\omega(t-t^{\prime})}\langle
TX^{k}(t)\overline{P}^{l}(t^{\prime})\rangle=\frac{\hbar\omega}{\omega^{2}+i\epsilon}\delta_{kl}$
by $\omega$ and taking the limit $\omega\rightarrow\infty$, one obtains the
standard commutation relations
$[X^{k}(t),\overline{P}^{l}(t)]=i\hbar\delta_{kl}$. Similarly,
$[X_{k},X_{l}]=[\overline{P}^{k},\overline{P}^{l}]=0$ by the BJL prescription.

We next use the term containing Berry’s phase $L^{\prime}={\cal
A}_{k}(\overline{P})\dot{\overline{P}_{k}}$ as a perturbation. One then
obtains the correction term by the perturbation

$\displaystyle\langle TX^{k}(t)X^{l}(t^{\prime})\rangle_{(1)}=\int{\cal
D}\overline{P}_{k}{\cal D}X_{k}\\{X^{k}(t)X^{l}(t^{\prime})\\}$ (B.4)
$\displaystyle\hskip
85.35826pt\times\exp\\{\frac{i}{\hbar}\int_{0}^{T}dt[\overline{P}_{k}(t)\dot{X}_{k}(t)+{\cal
A}_{k}(\overline{P})\dot{\overline{P}_{k}}]\\}$ $\displaystyle=$
$\displaystyle\int dt_{1}dt_{2}\langle
TX^{k}(t)\overline{P}^{k^{\prime}}(t_{1})\rangle\\{\frac{\delta}{\delta\overline{P}^{k^{\prime}}(t_{1})}\frac{\delta}{\delta\overline{P}^{l^{\prime}}(t_{2})}(\frac{i}{\hbar})\int
dsL^{\prime}(s)\\}\langle
T\overline{P}^{l^{\prime}}(t_{2})X^{l}(t^{\prime})\rangle$ $\displaystyle=$
$\displaystyle\int ds\langle
TX^{k}(t)\overline{P}^{k^{\prime}}(s)\rangle(\frac{i}{\hbar})\Omega_{k^{\prime}l^{\prime}}(s)\langle
T\dot{\overline{P}}^{l^{\prime}}(s)X^{l}(t^{\prime})\rangle$

where the curvature induced by Berry’s phase is written in the form
$\Omega_{k^{\prime}l^{\prime}}(s)$. Using (B), we thus obtain

$\displaystyle\int dte^{-i\omega(t-t^{\prime})}\langle
TX^{k}(t)X^{l}(t^{\prime})\rangle_{(1)}=\frac{\hbar\omega}{\omega^{2}+i\epsilon}(\frac{i}{\hbar})\Omega_{kl}(\overline{P}(t^{\prime}))(-i\hbar).$
(B.5)

Taking the limit $\omega\rightarrow\infty$ after multiplying the both hand
sides by $\omega$, one obtains the non-commutative geometry [6, 13]

$\displaystyle[X^{k}(t),X^{l}(t)]=-\frac{\hbar}{i}\Omega_{kl}(\overline{P}(t)).$
(B.6)

It has been argued in [19] that (B.6) is reduced to an ordinary canonical
commutation relation, $[X^{k}(t),X^{l}(t)]=0$, if one uses an exactly solvable
model of Berry’s phase [22, 23] in the evaluation (B.5).

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