# Adiabatic observables and Berry curvatures in insulators and metals

Raffaele Resta  Istituto Officina dei Materiali IOM-CNR, Strada Costiera 11,
314151 Trieste, Italy  Donostia International Physics Center, 20018 San
Sebastián, Spain

(DRAFT: run through LaTeX on at 3:58)

###### Abstract

A sharp definition of what “adiabatic” means is given; it is then shown that
the time-dependent expectation value of a quantum-mechanical observable in the
adiabatic limit can be expressed—in many cases—by means of the appropriate
Berry curvature. Condensed-matter observables belonging to this class include:
Born effective charges in insulators and in metals, quantized Faraday charges
in electrolytes, and linear dc conductivities (longitudinal and transverse).
Remarkably, the adiabatic limit is well defined even in metals, despite the
absence of a spectral gap therein. For all of the above observables the
explicit Berry-curvature expressions are derived in a general many-body
setting, which also allows for compact and very transparent notations and
formulas. Their conversion into band-structure formulas in the independent-
electron crystalline case is straightforward.

##  I Introduction

The term “adiabatic” intuitively refers to the time evolution of a system
which is driven out of equilibrium by an “infinitely slow” perturbation.
Consider a quantum system in its stationary ground state, whose Hamiltonian
acquires a time dependence: I am addressing here expressions for some given
observables which are exact in the adiabatic limit. Suppose the Hamiltonian is
driven by a parameter $\lambda(t)$: the adiabatic limit is then defined as the
limit $\dot{\lambda}(t)\rightarrow 0$; the adiabatic evolution of a given
observable coincides with the exact one to linear order in $\dot{\lambda}(t)$
and neglects terms of order $\ddot{\lambda}(t)$ and higher. Equivalently, when
the acquired time dependence of the Hamiltonian is harmonic at frequency
$\omega$, the adiabatic evolution of a given observable is exact to linear
order in $\omega$ and differs from the exact evolution by terms of order
$\omega^{2}$ and higher.

Quite often in the electronic-structure literature the adiabaticity issue is
discussed in the framework of the Born-Oppenheimer (also called Born-Huang)
approximation for decoupling the equations of motions of a system of electrons
and nuclei. Throughout this paper, instead, I only address the adiabaticity
issue by itself, by considering a system of $N$ electrons and classical
nuclei, having ideally infinite masses. The expressions derived in the present
paper are exact in the adiabatic limit, i.e. to linear order in the
adiabaticity parameter $\omega$; whether they can be applied as such to a
given molecule, liquid, or solid and to a given phenomenon depends of course
on the accuracy (or otherwise) of the Born-Oppenheimer approximation in the
case study.

Adiabaticity is deeply rooted in electron transport theory. This is clearly
demonstrated by a well known apparent paradox: continuity equation is
obviously violated if both the charge density and the current density are
evaluated as the expectation values of the corresponding operators on the
instantaneous adiabatic ground state. In the simple case of a time-reversal
(T) invariant system the current so evaluated vanishes at all times, while the
charge density is clearly time-dependent. The missing term is provided by
adiabatic-transport theory, which owes to the work of ThoulessThouless83  and
BerryBerry84  in the 1980s. The focus of the present paper is on interacting
many-electron systems; in this case the theory owes to a classical 1984
paperNiu84  by Niu and Thouless. Adiabatic transport theory is rooted in
quantum geometry by means of quantities which today are commonly called Berry
curvatures.rap_a20 ; Xiao10 ; Vanderbilt

This work is devoted to addressing geometry-wise a number of condensed-matter
observables: Born effective charges in solids and liquids (insulating and
metallic), Faraday’s transported charges in electrolytes, and linear dc
conductivities (both longitudinal and transverse). All these observables are
adiabatic (according to the above definition), and all will be cast in terms
of the appropriate Berry curvature. The Berry-curvature nature of the above
observables is nowadays obvious in one case (transverse dc conductivity;Xiao10
; Nagaosa10 ) little known in other cases (Born charges and Faraday’s
chargesPendry84 ; rap_a29 ; Grasselli19 ); probably unexpected in one case
(longitudinal dc conductivity). Addressing adiabatic responses in metals may
look surprising, given the absence of a spectral gap therein (in the
thermodynamic limit); nonetheless I am going to show that it is legitimate
within Kohn’s conceptual approach to dc conductivity,Kohn64 ; rap157 ; rap165
adopted here.

A large part of the literature on related topics is formulated at the
independent-electron level, in a band structure framework: this is the case of
Born charges (in insulatorsBaroni01 ; Vanderbilt  and in metalsDreyer22 ) and
of transverse dc conductivity.Nagaosa10  Here instead I address an interacting
many-electron system, possibly noncrystalline; the present results generalize
some of the established independent-electron ones, but also allow for very
compact notations and formulas and very transparent derivations. A simple
prescription is given for converting our compact formulas into their band-
structure counterparts, i.e. into Fermi-volume integrals.

This paper is organized as follows. Sec. II and Sec. III are formulated in a
general quantum-mechanical setting. In the former a generic Berry curvature is
defined and its main properties are outlined; in the latter it is shown how
the Berry curvature yields the time-evolution of an observable in the
adiabatic limit. Sec. IV sets the notations for the Hamiltonian and the
eigenstates of an interacting $N$-electron system. In Sec. V it is proved that
the charge density and the current density—both evaluated in the adiabatic
limit—conserve continuity equation; this applies on the same ground to
molecules and to condensed matter. From this point onwards the paper
specializes to solids and liquids, whose many-body boundary conditions are
specified in Sec. VI. Sec. VII addresses Kohn’s approachKohn64  to macroscopic
charge current and its adiabatic evolution, in the case of both metals and
insulators. Sec. VIII shows that the many-body Berry curvature provides a very
compact expression for the Born effective charges in solids and liquids; the
following Sec. IX applies the same concepts to the Faraday laws of
electrolysis and proves the topological quantization of the transported
charges. Sec X provides a very transparent derivation of the famous Kohn’s
formula for longitudinal dc conductivity,Kohn64  showing that even this
property can be recast in terms of a Berry curvature. Sec. XI addresses the
recentDreyer22  issue of the Born effective charges in metals and their
relationship to dc conductivity; it is shown that the two properties are
related by the same kind of Berry curvature. Sec. XII shows that the present
general setting allows for a one-line proof of the expression for the
anomalous Hall conductivity. Its quantization—in the special case of a two-
dimensional insulator—is also proved. Some conclusions are drawn in Sec. XIII.

Apppendix A proves continuity equation for an interacting $N$-electron system
in the exact case when both potentials (scalar and vector) depend on time. The
following Appendices convert the many-body formulas into their independent-
electron counterparts: in B for a generic case, in C for a crystalline system
of Bloch electrons.

##  II The Berry curvature: fundamentals

Let $\hat{H}$ be a time-independent Hamiltonian and let $|\Psi_{n}\rangle$ be
its eigenstates with eigenvalues $E_{n}$; $\hat{H}$ depends on a couple of
generic real parameters $(\kappa,\lambda)$: the eigenstates and eigenvalues
are parameter-dependent as well. The ground state is assumed to be
nondegenerate for all $(\kappa,\lambda)$. The Berry curvature is by
definitionrap_a20 ; Xiao10 ; Vanderbilt

$\displaystyle\Omega(\kappa,\lambda)$ $\displaystyle=$ $\displaystyle
i(\langle\partial_{\kappa}\Psi_{0}|\partial_{\lambda}\Psi_{0}\rangle-\langle\partial_{\lambda}\Psi_{0}|\partial_{\kappa}\Psi_{0}\rangle)$
(1) $\displaystyle=$ $\displaystyle-2\,\mbox{Im
}\langle\partial_{\kappa}\Psi_{0}|\partial_{\lambda}\Psi_{0}\rangle;$

it is a quasi-static quantity, in the sense that its definition requires
solely the instantaneous ground eigenstate; yet it encodes the lowest-order
effect of the excited states on the adiabatic evolution.

The parameters may have various physical interpretations and different
dimensions; the curvature has the inverse dimensions of the product
$\kappa\lambda$. For a macroscopic homogeneous system $\Omega(\kappa,\lambda)$
is an extensive quantity.

The Berry curvature admits a sum-over-states (i.e. Kubo) formula, first
displayed in the original Berry’s paper:Berry84

$\Omega(\kappa,\lambda)=-2\mbox{Im}\sum_{n\neq
0}\frac{\langle\Psi_{0}|\partial_{\kappa}\hat{H}|\Psi_{n}\rangle\langle\Psi_{n}|\partial_{\lambda}\hat{H}|\Psi_{0}\rangle}{(E_{0}-E_{n})^{2}}.$
(2)

The formula perspicuously shows that the Berry curvature encodes—to lowest
order—the effect of the excited states on the ground state when the quantum
system is transported in the parameter space. Eq. (2) has also the virtue of
showing that the curvature becomes ill defined whenever the ground state is
degenerate with the first excited state.

The curvature is a geometrical quantity defined by the evolution of the
ground-state projector

${\hat{P}}=|\Psi_{0}\rangle\langle\Psi_{0}|$ (3)

in the parameter space, and can equivalently be expressed directly in terms of
${\hat{P}}$:

$\Omega(\kappa,\lambda)=-i\,\mbox{Tr
}\\{{\hat{P}}\,[\,\partial_{\kappa}{\hat{P}},\partial_{\lambda}{\hat{P}}\,]\\},$
(4)

where “Tr” indicates the trace over the Hilbert space. The form of Eq. (4)
explicitly displays the gauge-invariance of $\Omega$: a gauge-transformation
modifies $|\Psi_{0}\rangle$ by a phase factor, ergo leaves ${\hat{P}}$ and the
curvature unchanged.

##  III Adiabatic transport

Let us suppose here that $\hat{H_{t}}$ is instead a time-dependent
Hamiltonian, and $|\Psi\rangle$ one of its solutions:

$\hat{H}_{t}|\Psi\rangle=i\hbar|\dot{\Psi}\rangle;$ (5)

the time-dependent energy is

$E(t)=\langle\Psi|\,\hat{H}_{t}\,|\Psi\rangle.$ (6)

Let us also assume that a generic observable $\hat{O}$ can be written as the
derivative of the Hamiltonian with respect to some parameter $\kappa$:

$\hat{O}=\partial_{\kappa}\hat{H}_{t},\qquad\mbox{time independent}.$ (7)

The $\kappa$-derivative of Eq. (6) yields

$\partial_{\kappa}E(t)=\langle\Psi|\,\hat{O}\,|\Psi\rangle+i\hbar\,(\,\langle\partial_{\kappa}\Psi|\dot{\Psi}\rangle-\langle\dot{\Psi}|\partial_{\kappa}\Psi\rangle\,).$
(8)

The time evolution of the observable’s expectation value is therefore:

$\displaystyle O(t)$ $\displaystyle\equiv$
$\displaystyle\langle\Psi|\,\hat{O}\,|\Psi\rangle$ (9) $\displaystyle=$
$\displaystyle\partial_{\kappa}E(t)-i\hbar\,(\,\langle\partial_{\kappa}\Psi|\dot{\Psi}\rangle-\langle\dot{\Psi}_{\kappa}|\partial_{\kappa}\Psi\rangle\,).$

Next suppose that the time-dependence of $\hat{H_{t}}$ owes to the time
dependence of a parameter $\lambda$ entering it. Upon replacing the exact
time-dependent energy and state vector in Eq. (9) with the instantaneous
ground-state eigenvalue and eigenvector one gets the adiabatic evolution of
the expectation value in terms of the Berry curvature as

$O(t)=\partial_{\kappa}E_{0}-\hbar\,\Omega(\kappa,\lambda)\,\dot{\lambda}(t),$
(10)

where $|\Psi_{0}\rangle$, $E_{0}$, and the curvature depend implicitly on
time. To the best of the author’s knowledge, a similar formulation first
appeared in the Niu-Thouless 1984 paper,Niu84 ; Xiao10  where it is also
explicitly shown that the exact time evolution of $O(t)$ differs from Eq. (10)
by terms proportional to $\ddot{\lambda}(t)$ and higher.

For a time-independent $\lambda$ Eq. (10) coincides with the Hellmann-Feynman
theorem: therefore the equation can be regarded as the adiabatic
generalization of it. Owing to Eq. (7), one may also rewrite Eq. (10) as

$O(t)=\langle\Psi_{0}|\,\hat{O}\,|\Psi_{0}\rangle-\hbar\,\Omega(\kappa,\lambda)\,\dot{\lambda}(t).$
(11)

Notice that—in the most general case—both terms in Eq. (11) contribute; the
first term vanishes e.g. when the unperturbed Hamiltonian is T-invariant and
the operator $\hat{O}$ is T-odd (e.g. a current or an angular momentum).

One may also consider the more general case where the time-independent
operator $\hat{O}$ cannot be expressed as an Hamiltonian derivative: then the
adiabatic evolution of ${O(t)}$ obtains by replacing the curvature in Eq. (11)
with the corresponding Kubo formula, Eq. (2):

$\displaystyle{O(t)}$ $\displaystyle=$
$\displaystyle\langle\Psi_{0}|\,\hat{O}\,|\Psi_{0}\rangle$ (12)
$\displaystyle+$ $\displaystyle 2\hbar\dot{\lambda}(t)\,\mbox{Im}\sum_{n\neq
0}\frac{\langle\Psi_{0}|\hat{O}|\Psi_{n}\rangle\langle\Psi_{n}|\partial_{\lambda}\hat{H}|\Psi_{0}\rangle}{(E_{0}-E_{n})^{2}},$

where the eigenstates and the eigenvalues are evaluated at $\lambda(t)$.

##  IV The Hamiltonian

In this work the above very general quantum-mechanical results will be adopted
to deal with an interacting $N$-electron system in the Born-Oppenheimer
approximation, where the nuclei are classical and the electronic time-
independent Schrödinger equation is

$\displaystyle\hat{H}|\Psi_{n}\rangle$ $\displaystyle=$ $\displaystyle
E_{n}|\Psi_{n}\rangle,\qquad\hat{H}=\frac{1}{2m}\sum_{i=1}^{N}\mbox{\boldmath$\pi$}_{i}^{2}+\hat{V}$
$\displaystyle\mbox{\boldmath$\pi$}_{i}$ $\displaystyle=$ $\displaystyle{\bf
p}_{i}+\frac{e}{c}{\bf A}({\bf r}_{i}),\qquad{\bf p}_{i}=-i\hbar\nabla_{{\bf
r}_{i}}$ (13)

where the potential $\hat{V}$ comprises one-body and two body terms, and is a
multiplicative operator in the Schrödinger representation. Both the potential
$\hat{V}$ and the vector potential ${\bf A}({\bf r})$ may depend
parametrically on $\lambda$.

I assume a singlet ground state and irrelevant spin variables are neglected.
Whenever Cartesian components need to be addressed, they are indicated with
Greek subscripts; sum over repeated Cartesian indices is implicitly
understood.

In the special case where $|\Psi_{0}\rangle$ is expressed as the Slater
determinant of doubly occupied single-particle orbitals (either Hartree-Fock
or Kohn-Sham), $\Omega(\kappa,\lambda)$ becomes a sum of single-particle
curvatures: the expression is presented in Appendix B.

##  V Microscopic current density and continuity equation

The most fundamental adiabatic response is the microscopic electrical current
density, which explicitly displays how the electronic charge is “dragged” by a
slow variation of the Hamiltonian. It is by definition the expectation value
of the operator

$\hat{\bf j}({\bf r})=-\frac{e}{2m}\sum_{i=1}^{N}\\{\delta({\bf
r}-r_{i})\mbox{\boldmath$\pi$}_{i}+\mbox{\boldmath$\pi$}_{i}\delta({\bf
r}-r_{i})\\};$ (14)

the quantum-chemical literature equivalently addresses the electronic flux,
whose operator differs from Eq. (14) by the $-e$ factor.

The lowest-order adiabatic current density is, from Eq. (12):

$\displaystyle{\bf j}(r,t)$ $\displaystyle=$
$\displaystyle\langle\Psi_{0}|\,\hat{{\bf j}}({\bf r})\,|\Psi_{0}\rangle$ (15)
$\displaystyle+$ $\displaystyle 2\hbar\dot{\lambda}(t)\,\mbox{Im}\sum_{n\neq
0}\frac{\langle\Psi_{0}|\hat{{\bf j}}({\bf
r})|\Psi_{n}\rangle\langle\Psi_{n}|\partial_{\lambda}\hat{H}|\Psi_{0}\rangle}{(E_{0}-E_{n})^{2}}.$

Expressions for the adiabatic current density—equivalent to this one—first
appeared in molecular physics in the 1980s, quite independently from the Niu-
Thouless result.Nafie83

Next I am going to show that Eq. (15) conserves continuity to leading order in
the adiabaticity parameter; if the Hamiltonian varies harmonically at
frequency $\omega$, continuity is conserved to first order in $\omega$.

Given that the current in a stationary state is divergenceless, Eq. (15)
yields

$\nabla\cdot{\bf j}(r,t)=2\hbar\dot{\lambda}(t)\,\mbox{Im}\sum_{n\neq
0}\frac{\langle\Psi_{0}|\nabla\cdot\hat{{\bf j}}({\bf
r})|\Psi_{n}\rangle\langle\Psi_{n}|\partial_{\lambda}\hat{H}|\Psi_{0}\rangle}{(E_{0}-E_{n})^{2}}.$
(16)

The expression for the operator $\nabla\cdot\hat{{\bf j}}({\bf r})$ is derived
in Appendix A, Eq. (63):

$\nabla\cdot\hat{{\bf j}}({\bf r})=\frac{i}{\hbar}[\,\hat{\rho}({\bf
r}),\hat{H}\,],$ (17)

where the time-dependent Hamiltonian $\hat{H}_{t}$ is replaced with Eq. (13),
evaluated at $\lambda(t)$. Therein, $\hat{\rho}({\bf r})$ is the charge-
density operator

$\hat{\rho}({\bf r})=-e\sum_{i=1}^{N}\delta({\bf r}-{\bf r}_{i}).$ (18)

By inserting Eq. (17) in Eq. (16) one gets

$\displaystyle\nabla\cdot{\bf j}(r,t)$ $\displaystyle=$
$\displaystyle-2\dot{\lambda}(t)\,\mbox{Re}\sum_{n\neq
0}\frac{\langle\Psi_{0}|\hat{\rho}({\bf
r})|\Psi_{n}\rangle\langle\Psi_{n}|\partial_{\lambda}\hat{H}|\Psi_{0}\rangle}{E_{0}-E_{n}}$
(19) $\displaystyle=$ $\displaystyle-2\dot{\lambda}(t)\,\mbox{Re
}\langle\Psi_{0}|\,\hat{\rho}({\bf r})\,|\partial_{\lambda}\Psi_{0}\rangle,$

where

$|\partial_{\lambda}\Psi_{0}\rangle=\sum_{n\neq
0}|\Psi_{n}\rangle\frac{\langle\Psi_{n}|\partial_{\lambda}\hat{H}|\Psi_{0}\rangle}{E_{0}-E_{n}}$
(20)

has been exploited.

The adiabatic time-derivative of the charge density obtains from the first
term in Eq. (12), given that the second term’s contribution is proportional to
$\ddot{\lambda}(t)$; equivalently, it is of order $\omega^{2}$ if the
Hamiltonian varies harmonically in time. One thus gets

$\frac{\partial\rho({\bf r},t)}{\partial t}=2\dot{\lambda}(t)\,\mbox{Re
}\langle\Psi_{0}|\,\hat{\rho}({\bf r})\,|\partial_{\lambda}\Psi_{0}\rangle,$
(21)

which in fact coincides with minus Eq. (19).

##  VI Molecules vs. condensed matter

The results presented so far apply to either an isolated molecule or to
condensed matter. Once the Hamiltonian is given, the eigenvalue problem is
uniquely determined when the adopted boundary conditions are specified, which
amounts to a choice of the Hilbert space where the state vectors are defined.

In the molecular case one addresses the bounded ground state of an
$N$-electron system. One thus requires that the wavefunction vanishes far away
from the sample: such boundary conditions are dubbed “open” in condensed-
matter theory jargon. The role of quantum geometry in some adiabatic response
functions of an isolated molecule has been recently elucidated in Refs.
rap166, ; rap168,  by adopting open boundary conditions and notations similar
to the present ones; here instead I am going to address condensed-matter
properties only.

One deals with solids and liquids by addressing an unbounded many-electron
system within Born-von-Kàrmàn periodic boundary conditions (PBCs): the many-
body wavefunctions are periodic with period $L$ over each electron Cartesian
coordinate $r_{i\alpha}$ independently. As said above, this determines the
Hilbert space where the solutions of Eq. (13) are defined.

In the condensed-matter case one thus considers a system of $N$ interacting
$d$-dimensional electrons in a cubic box—often called “supercell”—of volume
$L^{d}$, together with its periodic replicas. Both the vector potential ${\bf
A}({\bf r})$ and the many-body potential $\hat{V}$ obey PBCs; the latter
includes the classical nuclear-nuclear repulsion; the supercell is charge-
neutral. Therefore both electronic and nuclear coordinates are equivalent to
angles, i.e. they are defined on a torus.

The PBCs eigenstates are normalized to one over the supercell; the limit
$N\rightarrow\infty$, $L\rightarrow\infty$, $N/L^{d}=n$ constant is
understood. It is not required that the system is crystalline, only that it is
macroscopically homogeneous.

##  VII Macroscopic current density

In order to address the macroscopic current It is expedient to rewrite the
Hamiltonian in Eq. (13) as

$\hat{H}_{{\mbox{\boldmath$\kappa$}}}=\frac{1}{2m}\sum_{i=1}^{N}\left[{\bf
p}_{i}+\frac{e}{c}{\bf A}({\bf
r}_{i})+\hbar{\mbox{\boldmath$\kappa$}}\right]^{2}+\hat{V},$ (22)

where the vector potential ${\bf A}({\bf r})$ summarizes all intrinsic
T-breaking terms, as e.g. those due to a coupling to a background of local
moments. The “flux” $\kappa$—cast into inverse-length dimensions for
convenience—corresponds to perturbing the Hamiltonian with a vector potential
$\delta{\bf A}=\hbar c{\mbox{\boldmath$\kappa$}}/e$, constant in space. If
furthermore $\kappa$ is set to be constant even in time, then its occurrence
is a pure gauge transformation. As first realized by Kohn in 1964, PBCs
violate gauge-invarianceKohn64  and the energy may depend on $\kappa$;
actually, the energy is $\kappa$-independent in insulators and
$\kappa$-dependent in metals.

The macroscopic current density ${\bf j}(t)$ is by definition the supercell
average of the microscopic one, Eq. (15). The virtue of Eq. (22) is that it
allows writing the macroscopic current operator as the $\kappa$-derivative of
the Hamiltonian:

$\hat{\bf j}=-\frac{e}{mL^{d}}\sum_{i=1}^{N}\left[{\bf p}_{i}+\frac{e}{c}{\bf
A}({\bf r}_{i})+\hbar{\mbox{\boldmath$\kappa$}}\right]=-\frac{e}{\hbar
L^{d}}\partial_{{\mbox{\boldmath$\kappa$}}}\hat{H}_{{\mbox{\boldmath$\kappa$}}}.$
(23)

This expression requires the potential $\hat{V}$ to be a multiplicative
operator.

If one identifies the generic parameter $\kappa$ in Eq. (7) with a Cartesian
component of the flux $\kappa_{\alpha}$ in Eq. (23), then Eq. (10) immediately
yields

$j_{\alpha}(t)=-\frac{e}{\hbar
L^{d}}\partial_{\kappa_{\alpha}}E_{{\mbox{\boldmath$\kappa$}}}+\frac{e}{L^{d}}\Omega(\kappa_{\alpha},\lambda)\dot{\lambda}(t),$
(24)

where $E_{{\mbox{\boldmath$\kappa$}}}$ is the ground-state eigenvalue of
$\hat{H}_{{\mbox{\boldmath$\kappa$}}}$; both the energy and the curvature
depend parametrically on time.

The focus of this work is on physical properties which are well defined by
means of the adiabatic limit. While in insulators adiabaticity is obvious (the
spectrum is gapped), the metallic case—also dealt with in the
following—requires a crucial observation. Following Kohn, $\kappa$-derivatives
will be evaluated first, and the $L\rightarrow\infty$ limit taken afterwards.
At any finite size the spectrum is gapped even in metals; the procedure
warrants that the response of a metallic system to an infinitely slow
perturbation is indeed adiabatic.

Equivalently, one may consider time scales. The inverse of the finite-$L$ gap
sets the internal time scale of the many-electron system, while the
Hamiltonian evolves in time $t$. The adiabatic limit is the
$t\rightarrow\infty$ limit, and such limit is here performed before the
$L\rightarrow\infty$ one.

##  VIII Born effective charges in insulators

The Born (or dynamical) effective charge tensors $Z^{*}_{s,\alpha\beta}$ are a
staple in the theory of harmonic lattice dynamics of crystalline insulators
and in the theory of charge transport in insulating liquids.Grasselli19 ;
rap164  Focussing here on the former case, let us suppose that ${\bf R}_{s}$
are the equilibrium nuclear positions; the Born tensor then measures the
$\alpha$-component of macroscopic polarization ${\bf P}$ linearly induced by a
unit displacement of the $s$-th nucleus in direction $\beta$, under the
boundary condition that the macroscopic electric field is kept vanishing.
Notice that such condition is equivalent to imposing PBCs on the potential
$\hat{V}$ of the perturbed solid.

Here it is expedient to adopt an equivalent definition of
$Z^{*}_{s,\alpha\beta}$. Since the macroscopic current is the time derivative
of ${\bf P}$, the Born tensor also measures the $\alpha$-component of
macroscopic current linearly induced by displacing the $s$-th nucleus with
unit velocity in direction $\beta$, i.e. (in dimensionless form)

$Z^{*}_{s,\alpha\beta}=\frac{L^{d}}{e}\frac{\partial j^{(\rm
total)}_{\alpha}}{\partial v_{s,\beta}},\qquad{\bf v_{s}}=\dot{\bf R}_{s},$
(25)

where both electrons and nuclei contribute to ${\bf j}^{(\rm total)}$.

When all the nuclei are translated with the same velocity ${\bf v}$, Eq. (25)
yields to linear order

$j^{(\rm
total)}_{\alpha}=\frac{e}{L^{d}}\left(\sum_{s}Z^{*}_{s,\alpha\beta}\right)v_{\beta}.$
(26)

In the insulating case no current may flow for a rigid translation of the
whole solid, whence the basic relationship $\sum_{s}Z^{*}_{s,\alpha\beta}=0$,
called the acoustic sum rule.PCM  In the metallic case some current instead
flows: this is discussed in Sec. XI.

The geometrical expression for $Z^{*}_{s,\alpha\beta}$ obtains from the
generic expression of Eq. (24) by identifying $\lambda$ with a nuclear
coordinate $R_{s,\beta}$. The first term therein vanishes (no current flows in
a static lattice); by adding the nuclear current one gets

$Z^{*}_{s,\alpha\beta}=Z_{s}\delta_{\alpha\beta}+\Omega(\kappa_{\alpha},R_{s,\beta}),$
(27)

where $Z_{s}$ is the atomic number of nucleus $s$. The curvature is evaluated
at ${\mbox{\boldmath$\kappa$}}=0$ and at the equilibrium atomic positions.

The present many-body formulation yields a very compact geometrical
expression, Eq. (27). Nonetheless it is worth reminding that the Berry-
curvature formulation for a generic polarization derivative—albeit at the
independent-electron level—first appeared in the milestone 1993 paper by King-
Smith and Vanderbilt.King93

##  IX Faraday transport in electrolytes and topological oxidation numbers

I consider here liquid electrolytes, whose simplest example is a molten salt.
The crucial requirement is that the liquid is insulating: if the nuclei are
ideally kept clamped, no electronic current flows. All of charge transport
occurs then by means of an associated mass transport: indeed the “equivalent
numbers” originally tabulated by FaradayFaraday  are charge-to-mass ratios
(decades before the atoms’ existence became universally accepted).

In modern terms Faraday’s first law of electrolysis can be recast as follows:
when a macroscopic number of nuclei of a given chemical species passes from
one electrode to the other, the transported electrical charge is an integer
multiple of such number times $e$. The law is additive: it concerns each
different chemical species in the cell independently.

Therefore Faraday’s experiment yields the oxidation number of the given
chemical species: this looks quite intuitive, but intuition here is a gross
oversimplification. In the general case oxidation numbers are determined by an
agreed set of rules.Walsh18  They are often related to some measurable
quantities in a postdictive—not predictive—way,rap142 ; Walsh18 ; rap142 ;
rap_a29 ; Sit11  but cannot be direcly measured; Faraday’s experiment is a
special case, whose theory is deeply rooted—as shown below—in quantum
topology.

Suppose we ideally take a “snapshot” of a liquid (or even of a solid) at a
given time: partitioning the total electronic charge into subsets “belonging”
to the different nuclei is clearly a very ambiguous task. And in fact
literally dozens of definitions for the atomic static charges have been
proposed in the literature,Meister94  none of them yielding integer values.
Faraday’s experiment is not a static one, it is a dynamical one in the
adiabatic limit: like the other cases dealt with in the present work the
transported charge (electronic term thereof) is a geometrical property of the
many-electron wavefunction. Furthermore the property from geometrical becomes
topological owing to the fact that a macroscopic number of nuclei is
transported over a macroscopic distance.

At the root of topological quantization of charge transport is a milestone
1983 paper by D. Thouless;Thouless83  while this paper only addresses solid-
state issues, its relevance for understanding quantized charge transport in
ionic liquids was pointed out shortly after in a very little quoted—and
presumably also little known—paper by Pendry and Hodges.Pendry84  Since then,
a few authors have built upon their result and endorsed the topological nature
of Faraday’s law.rap_a29 ; rap142 ; Grasselli19 ; rap164

Suppose the parameter $\lambda(t)$ entering the potential $\hat{V}$ is
identified with the set of the nuclear coordinates $\\{{\bf R}_{s}(t)\\}$; we
start by investigating the amount of charge transported in time $T$ in a given
direction across the electrolytic cell, whose circuit is closed through the
battery. Here the electrons in the liquid are described as a closed quantum
system (no electrodes, no battery); in order to allow for dc steady-state
currents it is then mandatory to adopt PBCs. Both nuclear and electronic
coordinates are therefore equivalently defined on a torus. Given the
Hamiltonian of Eq. (22) and the periodic supercell of volume $L^{3}$, we
address the total amount of charge transported across the supercell, say in
the $x$-direction, in time $T$. The transported nuclear charge $Q^{(\rm
nucl)}$is obviously integer; we focus on the electronic one for the time
being; from Eq. (24) one gets

$\displaystyle Q^{(\rm el)}$ $\displaystyle=$ $\displaystyle
L^{2}\int_{0}^{T}dt\;j^{(+)}_{x}(t)=\frac{e}{L}\int_{0}^{T}dt\;\Omega(\kappa_{x},\lambda)\dot{\lambda}(t)$
(28) $\displaystyle=$
$\displaystyle\frac{e}{L}\int_{\lambda_{1}(0)}^{\lambda_{2}(T)}d\lambda\;\Omega(\kappa_{x},\lambda),$

Therein the curvature is evaluated at ${\mbox{\boldmath$\kappa$}}=0$, while
the large-$L$ limit is implicit: one may therefore replace

$\left.\Omega(\kappa_{x},\lambda)\right|_{{\mbox{\boldmath$\kappa$}}=0}\simeq\frac{L}{2\pi}\int_{0}^{\frac{2\pi}{L}}d\kappa_{x}\;\Omega(\kappa_{x},\lambda)$
(29) $Q^{(\rm
el)}=\frac{e}{2\pi}\int_{0}^{\frac{2\pi}{L}}d\kappa_{x}\int_{\lambda_{1}(0)}^{\lambda_{2}(T)}d\lambda\;\Omega(\kappa_{x},\lambda).$
(30)

Given the adopted PBCs, in general some of the nuclei have likely crossed the
$x=0$ and $x=L$ boundaries of the home supercell and have moved to some of its
replicas, thus dragging some electronic charge with them; the averaged current
is then $I=Q/T=(Q^{(\rm el)}+Q^{(\rm nucl)})/T$.

We are now ready to perform a gedanken experiment, first devised—at the
independent-electron level—by Pendry and Hodges.Pendry84  We pick only one of
the nuclei, say the $s$-th, and we drive it to the next replica cell;
equivalently we drive it once round the torus. We assume that the other nuclei
move out of the way in order to minimize the instantaneous energy, and that
the other nuclei do not cross the supercell boundaries (except for two-way
crossings). For insulators $\kappa_{x}$ and $\kappa_{x}+2\pi/L$ are equivalent
points;Xiao10 ; Kohn64  if furthermore the final configuration
$\lambda_{2}(T)$ coincides with the initial one $\lambda_{1}(0)$, then the
integral in Eq. (30) is over a closed surface (a torus) and

$\frac{1}{2\pi}\int_{0}^{\frac{2\pi}{L}}d\kappa_{x}\int_{\lambda_{1}(0)}^{\lambda_{2}(T)}d\lambda\;\Omega(\kappa_{x},\lambda)=C_{1},$
(31)

where $C_{1}\in{\mathbb{Z}}$ is a Chern number; the Chern theorem requires the
system to be insulating in the integration domain. We thus arrive at the
outstanding result

$Q_{s}^{(\rm el)}=e\,C_{1}.$ (32)

The Pendry-Hodges gedanken experiment is a realization of what nowadays is
generally called a “Thouless pump”, whose general theory owes (in the many-
electron case) to Niu and Thouless.Niu84 ; Xiao10

Some further logical steps, first spelled out—at the independent-electron
level—in Ref. Grasselli19, , are needed in order to identify
$Q_{s}/e=C_{1}+Z_{s}$ with the topological oxidation number of species $s$.

First we notice that two nuclei having the same atomic number accumulate the
same Chern number in the period $T_{\rm c}$ of one pump cycle. In fact Chern
theorem only requires the configuration $\\{{\bf R}_{s}(T_{c})\\}$ to be equal
to $\\{{\bf R}_{s}(0)\\}$; it does not require the individual coordinates
${\bf R}_{s}(T_{c})$ to return to themselves; exchanges of identical nuclei
are allowed.

A liquid configuration returning to itself is clearly a rare event; but for a
macroscopic time $T$ one needs to consider a large number $n_{\rm c}$ of pump
cycles, such that the averaged current is

$I=\frac{en_{\rm c}(C_{1}+Z_{s})}{T}.$ (33)

Consider next the general case where the nuclear trajectory $\\{{\bf
R}_{s}(t)\\}$ ends in an arbitrary configuration with $\\{{\bf R}_{s}(T)\\}$
different from $\\{{\bf R}_{s}(0)\\}$. We can always imagine a further
evolution such that $\\{{\bf R}_{s}(T+\Delta T)\\}=\\{{\bf R}_{s}(0)\\}$,
where $T$ is macroscopic and $\Delta T$ is microscopic; the transported charge
$\Delta Q$ is microscopic as well, and the averaged current is

$I=\frac{en_{\rm c}(C_{1}+Z_{s})+\Delta Q}{T+\Delta T}.,$ (34)

which displays Faraday’s quantization in the large-$T$ limit.

##  X Longitudinal dc conductivity

The (linear) frequency-dependent conductivity tensor is by definition

$\sigma_{\alpha\beta}(\omega)=\sigma^{(+)}_{\alpha\beta}(\omega)+\sigma^{(-)}_{\alpha\beta}(\omega)=\frac{\partial
j_{\alpha}(\omega)}{\partial{\cal E}_{\beta}(\omega)},$ (35)

where ${\mathbfcal E}(\omega)$ is the macroscopic electric field; it is here
partitioned into its symmetric (longitudinal) and antisymmetric (transverse)
components, $\sigma^{(+)}$ and $\sigma^{(-)}$, respectively. Each of them
comprises a real and an imaginary part, related to each other by the Kramers-
Kronig relationship.

The present work only addresses dc conductivity, in which case the adiabatic
response theory is enough, as it will be shown in the following. It is also
worth reminding that—for the reason given above (end of Sec. VII)—the
adiabatic limit is legitimate despite the fact that the spectrum of a metal is
gapless in the $L\rightarrow\infty$ limit.

Next the $\lambda$ parameter in Eq. (24) is identified with $\kappa_{\beta}$.
The second term therein is proportional to $\dot{\mbox{\boldmath$\kappa$}}$,
i.e. to the electric field

${\mathbfcal E}=-\frac{1}{c}\frac{d}{dt}\delta{\bf
A}=-\frac{\hbar}{e}\dot{\mbox{\boldmath$\kappa$}}$ (36)

by means of an antisymmetric coefficient: it therefore accounts for transverse
conductivity, discussed in Sec. XII. For the time being I address the first
term in Eq. (24), indicated as $j^{(+)}_{\alpha}(t)$ in the following. Taking
its time derivative at ${\mbox{\boldmath$\kappa$}}=0$ one gets

$\partial_{t}j^{(+)}_{\alpha}(t)=-\frac{e}{\hbar
L^{d}}\frac{\partial^{2}E_{0}}{\partial
t\partial\kappa_{\alpha}}=\frac{e^{2}}{\hbar^{2}L^{d}}\frac{\partial^{2}E_{0}}{\partial\kappa_{\alpha}\partial\kappa_{\beta}}{\cal
E}_{\beta}:$ (37)

this clearly shows that the many-electron system in a constant field undergoes
free acceleration (in absence of extrinsic dissipation mechanisms).

In order to make contact with the existing literature Eq. (37) is rewritten as

$\partial_{t}j^{(+)}_{\alpha}(t)=\frac{D_{\alpha\beta}}{\pi}{\cal
E}_{\beta},\qquad D_{\alpha\beta}=\frac{\pi
e^{2}}{\hbar^{2}L^{d}}\frac{\partial^{2}E_{0}}{\partial\kappa_{\alpha}\partial\kappa_{\beta}},$
(38)

where $D_{\alpha\beta}$ is the Drude weight, alternatively also called
adiabatic charge stiffness; clearly $D_{\alpha\beta}$ measures the inverse
inertia of the many-electron system. In the special case of an electron gas in
a flat background potential (either interacting or non interacting) the Drude
weight is

$D_{\alpha\beta}=D_{\rm free}\delta_{\alpha\beta},\qquad D_{\rm
free}=\frac{\pi e^{2}n}{m},$ (39)

where $n=N/L^{d}$ is the density. In the general case one may recast
$D_{\alpha\beta}$ as

$D_{\alpha\beta}=\frac{\pi e^{2}n^{*}_{\alpha\beta}}{m},$ (40)

where $n^{*}$ acquires the meaning of the effective electron density
contributing to the adiabatic current.Scalapino93 ; rap157

Suppose instead that ${\mathbfcal E}=0$ and all the nuclei are rigidly
translated with constant velocity ${\bf v}=\dot{\bf u}$. If $\lambda$ in Eq.
(24) is identified with $u_{\beta}$ the total current carried by nuclei and
electrons is

$j^{(\rm
total)}_{\alpha}=\frac{e}{L^{d}}\left[N\delta_{\alpha\beta}+\Omega(\kappa_{\alpha},u_{\beta})\right]v_{\beta},$
(41)

since by charge neutrality the sum of the atomic numbers $Z_{s}$ equals the
total number of electrons $N=nL^{d}$.

In the reference frame of the nuclei the current is carried by the electrons
only, all moving with velocity $-{\bf v}$, and whose effective density is
$n^{*}_{\alpha\beta}$; therefore from Eq. (40)

$j^{(\rm
total)}_{\alpha}=e\,n^{*}_{\alpha\beta}v_{\beta}=\frac{mD_{\alpha\beta}}{\pi
e}v_{\beta}.$ (42)

We thus arrive at the equivalent Berry-curvature form of the Drude weight:

$D_{\alpha\beta}=D_{\rm free}\delta_{\alpha\beta}.+\frac{\pi
e^{2}}{mL^{d}}\Omega(\kappa_{\alpha},u_{\beta}),$ (43)

where the curvature is evaluated at ${\mbox{\boldmath$\kappa$}}=0$ and ${\bf
u}=0$. A geometrical form for the Drude weight (somewhat different from the
present one) was first suggested in 2013 by Hetènyi.Hetenyi13  The equivalence
of Eq. (43) with Eq. (38) is double-checked below, from Eq. (51) onwards; it
requires T-invariance at ${\mbox{\boldmath$\kappa$}}=0$, i.e. a vanishing
${\bf A}({\bf r})$ in Eq. (22).

In order to convert the above results into the $\omega$ domain, one writes the
longitudinal term in Eq. (35) as

$\sigma^{(+)}_{\alpha\beta}(\omega)=\frac{\partial
j^{(+)}_{\alpha}(\omega)}{\partial{\cal E}_{\beta}(\omega)}=\frac{\partial
j^{(+)}_{\alpha}(\omega)}{\partial\kappa_{\beta}(\omega)}\frac{\partial\kappa_{\beta}(\omega)}{\partial{\cal
E}_{\beta}(\omega)};$ (44)

the rightmost factor is addressed first.

From $\dot{\mbox{\boldmath$\kappa$}}(t)=-e{\mathbfcal E}/\hbar$ one gets
${\mbox{\boldmath$\kappa$}}(t)=-e{\mathbfcal E}t/\hbar+\mbox{const}$;
switching to the Fourier transforms ${\mathbfcal
E}(\omega)=i\omega\hbar{\mbox{\boldmath$\kappa$}}(\omega)/e$, whose inversion
is

${\mbox{\boldmath$\kappa$}}(\omega)=\frac{e}{\hbar}\left[-\frac{i}{\omega}+\mbox{const}\times\delta(\omega)\right]{\mathbfcal
E}(\omega).$ (45)

The integration constant, as usual, is determined by imposing causality:

$\displaystyle\frac{\partial\kappa_{\alpha}(\omega)}{\partial{\cal
E}_{\beta}(\omega)}$ $\displaystyle=$
$\displaystyle-\frac{e}{\hbar}\left(\frac{i}{\omega+i\eta}\right)\delta_{\alpha\beta}$
(46) $\displaystyle=$
$\displaystyle-\frac{e}{\hbar}\left[\frac{i}{\omega}+\pi\delta(\omega)\right]\delta_{\alpha\beta},$

where $\eta\rightarrow 0^{+}$ is understood.

Replacing in Eq. (44) one gets

$\sigma^{(+)}_{\alpha\beta}(\omega)=-\frac{e}{\hbar}\frac{\partial
j^{(+)}_{\alpha}(\omega)}{\partial\kappa_{\beta}(\omega)}\frac{i}{\omega+i\eta}.$
(47)

At finite frequency, $\partial
j_{\alpha}(\omega)/\partial\kappa_{\beta}(\omega)$ obtains from time-dependent
linear response theory (i.e. Kubo formulas). Here instead we limit ourselves
to an adiabatic perturbation, hence the $\kappa$-derivative is taken with
respect to a static $\kappa$. We thus get from the first term in Eq. (24) the
dc contribution to longitudinal conductivity as:

$\frac{\partial j^{(+)}_{\alpha}}{\partial\kappa_{\beta}}=-\frac{e}{\hbar
L^{d}}\frac{\partial^{2}E_{0}}{\partial\kappa_{\alpha}\partial\kappa_{\beta}},$
(48) $\sigma^{(\rm
Drude)}_{\alpha\beta}(\omega)=D_{\alpha\beta}\left[\delta(\omega)+\frac{i}{\pi\omega}\right].$
(49)

This expression was first obtained in 1964 by Kohn by manipulating the Kubo
formula;Kohn64  the present adiabatic derivation owes to Ref. rap165, . The
two terms in square parenthesis obey the Kramers-Kronig relationship; the
response is causal but nondissipative.

##  XI Born effective charges in metals

As said above, the “primary” definition of Born charges is by means of
polarization derivatives, which does not make any sense in metals. The
alternative definition, Eqs. (25) and (27), can instead be adopted as it
stands even in the metallic case

The key difference from the insulating case is that—when all the nuclei are
translated with the same velocity—the electrons are left behind and Eq. (26)
does not vanish. By summing Eq. (27) over all the nuclei one gets

$\displaystyle\sum_{s}Z^{*}_{s,\alpha\beta}$ $\displaystyle=$ $\displaystyle
N\delta_{\alpha\beta}+\Omega(\kappa_{\alpha},u_{\beta})$ (50) $\displaystyle=$
$\displaystyle N\delta_{\alpha\beta}-2\,\mbox{Im
}\langle\partial_{\kappa_{\alpha}}\Psi_{0}|\partial_{u_{\beta}}\Psi_{0}\rangle,$

consistently with Eq. (41). From Eq. (42) one immediately gets the outstanding
sum rule

$\sum_{s}Z^{*}_{s,\alpha\beta}=\frac{mL^{d}}{\pi
e^{2}}D_{\alpha\beta}=\frac{m}{\hbar^{2}}\frac{\partial^{2}E_{0}}{\partial\kappa_{\alpha}\partial\kappa_{\beta}},$
(51)

recently found by Dreyer, Coh, and Stengel Dreyer22  for the special case of a
crystalline system of noninteracting electrons. It is also worth noticing that
the Born charges in metals are dubbed “nonadiabatic” in Ref. Dreyer22,  and in
some related papers (very much in contrast with the viewpoint adopted here).

Under the T-invariance hypothesis one has

$|\partial_{\bf u}\Psi_{0}\rangle=-\sum_{i=1}^{N}|\partial_{{\bf
r}_{i}}\Psi_{0}\rangle=-\frac{i}{\hbar}\sum_{i=1}^{N}{\bf
p}_{i}|\Psi_{0}\rangle=-\frac{im}{\hbar^{2}}(\partial_{{\mbox{\boldmath$\kappa$}}}\hat{H}_{{\mbox{\boldmath$\kappa$}}})|\Psi_{0}\rangle,$

and Eq. (50) is recast as

$\sum_{s}Z^{*}_{s,\alpha\beta}=N\delta_{\alpha\beta}+\frac{2m}{\hbar^{2}}\mbox{Re
}\langle\partial_{\kappa_{\alpha}}\Psi_{0}|\,\partial_{\kappa_{\beta}}\hat{H}_{{\mbox{\boldmath$\kappa$}}}\,|\Psi_{0}\rangle.$
(52)

In order to compare this with the rightmost expression in Eq. (51) I start
from the Hellmann-Feynman theorem at an arbitrary $\kappa$:

$\partial_{\kappa_{\beta}}E_{{\mbox{\boldmath$\kappa$}}}=\langle\Psi_{0}|\,{\partial_{\kappa_{\beta}}\hat{H}}_{{\mbox{\boldmath$\kappa$}}}\,|\Psi_{0}\rangle.$
(53)

Taking another derivative at ${\mbox{\boldmath$\kappa$}}=0$ one gets:

$\frac{m}{\hbar^{2}}\frac{\partial^{2}E_{0}}{\partial\kappa_{\alpha}\partial\kappa_{\beta}}=N\delta_{\alpha\beta}+\frac{2m}{\hbar^{2}}\mbox{Re
}\langle\partial_{\kappa_{\alpha}}\Psi_{0}|\,{\partial_{\kappa_{\beta}}\hat{H}}_{{\mbox{\boldmath$\kappa$}}}\,|\Psi_{0}\rangle;$
(54)

comparison to Eq. (52) completes the proof of Eq. (51).

In insulators the Born effective charges manifest themselves in the phonon
spectra: they generate a depolarization field ${\mathbfcal E}$ which
contributes to the restoring force at the zone center.Huang50 ; Cochran62  No
such contribution may exist in metals in the adiabatic limit, since
macroscopic fields are screened therein (Faraday-cage effect). A few
calculations of Born charges in metals are available;Dreyer22  their possible
experimental manifestations—e.g. in the case of a binary metal—are presently
under discussion.Hickox23

##  XII Anomalous Hall conductivity

As anticipated above, the second term in the r.h.s. of Eq. (24) yields the
transverse (or Hall) conductivity at $\omega=0$, in both insulators and
metals. In the latter case extrinsic effects are also relevant;Nagaosa10  they
are not addressed here (analogously they were not addressed in the
longitudinal case). Both intrinsic and extrinsic Hall conductivity can be
nonzero only if the system breaks T-invariance in absence of an applied
magnetic field (hence the adjective “anomalous”).

One identifies once more $\lambda$ in Eq. (24) with $\kappa_{\beta}$. Since
for a static field $\dot{\mbox{\boldmath$\kappa$}}=-e{\mathbfcal E}/\hbar$,
the second term in Eq. (24) immediately yields

$\sigma^{(-)}_{\alpha\beta}(0)=-\frac{e^{2}}{\hbar
L^{d}}\Omega(\kappa_{\alpha},\kappa_{\beta}),$ (55)

where the curvature is evaluated at ${\mbox{\boldmath$\kappa$}}=0$. Eq. (55)
holds for $d=2,3$, and for both insulators and metals; it can be alternatively
derived from Kubo formulas.rap165

In the special case of an insulator and $d=2$ the Hall conductivity is
topologically quantized. This famous result was first established at the
independent-electron level,Thouless82  and later extended to an interacting
many-electron system.Niu85  The original work did not address the anomalous
case, whose theory is indeed simpler (as e.g. in Ref. Xiao10, ). The form of
Eq. (55) is not quantized at finite $L$ (see Fig. 2 in Ref. rap135, ); the
quantization occurs in the large-$L$ limit only. More explicitly, Eq. (55)
yields:

$\sigma^{(-)}_{xy}(0)=-\frac{2\pi
e^{2}}{h}\lim_{L\rightarrow\infty}\frac{1}{L^{2}}\left.\Omega(\kappa_{x},\kappa_{y})\right|_{{\mbox{\boldmath$\kappa$}}=0}=-\frac{e^{2}}{h}C_{1},$
(56)

where $C_{1}\in{\mathbb{Z}}$ is a Chern number.

In the standard formulation of Ref. Xiao10,  (Sec. IIIC) the many-body Chern
number is written as

$C_{1}=\frac{1}{2\pi}\int_{0}^{\frac{2\pi}{L}}d\kappa_{x}\int_{0}^{\frac{2\pi}{L}}d\kappa_{y}\;\Omega(\kappa_{x},\kappa_{y}),$
(57)

which holds at any $L$; notice that $\kappa_{\alpha}$ and
$\kappa_{\alpha}+2\pi/L$ are equivalent points only if the system is
insulating.Kohn64  Then the mean-value theorem yields

$\displaystyle\lim_{L\rightarrow\infty}$
$\displaystyle\\!\\!\\!\frac{1}{L^{2}}\left(\frac{L}{2\pi}\right)^{2}\int_{0}^{\frac{2\pi}{L}}d\kappa_{x}\int_{0}^{\frac{2\pi}{L}}d\kappa_{y}\;\Omega(\kappa_{x},\kappa_{y})$
(58) $\displaystyle=$
$\displaystyle\lim_{L\rightarrow\infty}\frac{1}{L^{2}}\left.\Omega(\kappa_{x},\kappa_{y})\right|_{{\mbox{\boldmath$\kappa$}}=0},$

which proves Eq. (56).

##  XIII Conclusions

The observables addressed in this work measure the adiabatic response of a
condensed-matter system to a slow variation of the Hamiltonian, driven by a
parameter $\lambda(t)$ entering it. The many-body setting chosen for this work
allows for compact and meaningful formulas, dealing with crystalline and
noncrystalline systems on the same ground, with either interacting or
noninteracting electrons.

All of the addressed observables are expressed by means of the appropriate
Berry curvature; they are exact for $\dot{\lambda}(t)\rightarrow 0$, i.e. for
an infinitely slow perturbation. Higher order terms, proportional to
$\ddot{\lambda}(t)$ and to higher derivatives are neglected within the Niu-
ThoulessNiu84 ; Xiao10  approach, adopted here for all the cases discussed.

As a first step I have addressed the microscopic charge and current densities
in order to fix the apparent paradox as outlined in the Introduction. If the
density $\rho({\bf r})$ is evaluated as the expectation value of the
corresponding operator over the instantaneous ground state, its time
derivative is proportional to $\dot{\lambda}(t)$. In order to warrant
continuity the instantaneous value of the current operator is clearly not
enough: one needs the current ${\bf j}({\bf r})$ (an its divergence) to the
same order $\dot{\lambda}(t)$, i.e. in the adiabatic limit.

The observables addressed in the rest of this work fall in two classes: Born
and Faraday charges, and dc linear conductivities (longitudinal and
transverse). Adiabatic response theory is adopted throughout for both
insulators and metals, despite the absence of a spectral gap (in the
thermodynamic limit) in the latter case. The reasons are given at the end of
Sec. VII.

In the case of Born and Faraday charges the parameter $\lambda(t)$ is
identified with a nuclear coordinate, whose motion is infinitely slow: if the
nuclear motion is harmonic at frequency $\omega$, the adiabatic results are
exact to order $\omega$, and do not include corrections of order $\omega^{2}$
and higher. One important point is worth stressing: in the present formulation
the Born charges have the same formal expression in insulators and in metals,
and are adiabatic on the same ground (contrary to some opposite claimsDreyer22
). Faraday’s charges can be regarded as Born charges averaged over macroscopic
trajectories in a liquid sample. Owing to this they are topologically
quantized: their electronic contribution is a Chern numberPendry84 ; rap_a29 ;
Grasselli19  (it was alternatively cast as a winding number in Ref. rap164, ).

In the case of dc conductivities—longitudinal and transverse—the $\lambda(t)$
parameter has a different meaning. The perturbation induced by a macroscopic
electric field enters the Hamiltonian via a vector potential, ${\bf
r}$-independent and linear in time; this is because the scalar-potential gauge
violates Born-von-Kàrmàn PBCs. Therefore $\lambda(t)$ is identified with a
Cartesian component of the vector potential (times a trivial constant); since
$\ddot{\lambda}(t)\equiv 0$ for a constant ${\mathbfcal E}$ field, adiabatic
response theory is exact to linear order in ${\mathbfcal E}$. Kubo formulas
are not required to express linear conductivities (although they eventually
lead to the same expressionsrap165 ).

One major message of the present work is that the longitudinal responses of a
metallic solid to either a macroscopic electric field or to a rigid lattice
translation measure the same observable—i.e. the inverse inertia of the many-
electron system—and are expressed by the same Berry curvature.

## Acknowledgments

I have thoroughly discussed some key points of the present paper with Stefano
Baroni; his contribution is gratefully acknowledged. Work supported by the
Office of Naval Research (USA) Grant No. N00014-20-1-2847.

##  Appendix A Continuity (exact formulation)

In this Appendix we consider a many-body Schrödinger equation and one of its
exact solutions:

$\hat{H}_{t}|\Psi\rangle=\left(\frac{1}{2m}\sum_{i=1}^{N}\mbox{\boldmath$\pi$}_{i}^{2}+\hat{V}\right)|\Psi\rangle=i\hbar|\dot{\Psi}\rangle,$
(59)

where both the potential $\hat{V}$ and the vector potential ${\bf A}({\bf r})$
may depend on time.

In order to address the divergence of the current operator, Eq. (14), one
needs the ${\bf r}$-gradient of $\delta({\bf r}-{\bf r}_{i})$. It is safer to
evaluate how this operates on any state vector $|\Psi\rangle$ at its right:

$\displaystyle\nabla_{\bf r}\delta({\bf r}-{\bf
r}_{i})|\Psi\rangle=-\\{\nabla_{{\bf r}_{i}}\delta({\bf r}-{\bf
r}_{i})\\}|\Psi\rangle=[\delta({\bf r}-{\bf r}_{i}),\nabla_{{\bf
r}_{i}}]|\Psi\rangle$ (60) $\displaystyle=$
$\displaystyle\frac{i}{\hbar}[\delta({\bf r}-{\bf r}_{i}),{\bf
p}_{i}]|\Psi\rangle=\frac{i}{\hbar}[\delta({\bf r}-{\bf
r}_{i}),\mbox{\boldmath$\pi$}_{i}]|\Psi\rangle.$

One therefore gets

$\displaystyle\nabla_{\bf r}\cdot\hat{\bf j}({\bf r})$ $\displaystyle=$
$\displaystyle-\frac{ie}{2\hbar m}\sum_{i=1}^{N}\\{[\,\delta({\bf r}-{\bf
r}_{i}),\mbox{\boldmath$\pi$}_{i}\,]\mbox{\boldmath$\pi$}_{i}+\mbox{\boldmath$\pi$}_{i}[\,\delta({\bf
r}-{\bf r}_{i}),\mbox{\boldmath$\pi$}_{i}\,]\\}$ (61) $\displaystyle=$
$\displaystyle-\frac{ie}{2\hbar m}\sum_{i=1}^{N}[\,\delta({\bf r}-{\bf
r}_{i}),\mbox{\boldmath$\pi$}_{i}^{2}\,],$

Next we notice that for $i\neq j$ the operators $\delta({\bf r}-{\bf r}_{i})$
and $\mbox{\boldmath$\pi$}_{j}$ commute; ergo the sum may be replaced with a
double sum:

$\nabla_{\bf r}\cdot\hat{\bf j}({\bf
r})=-\frac{ie}{\hbar}\sum_{i=1}^{N}\left[\,\delta({\bf r}-{\bf
r}_{i}),\frac{1}{2m}\sum_{j=1}^{N}\mbox{\boldmath$\pi$}_{j}^{2}\,\right].$
(62)

Since the potential $\hat{V}$ is multiplicative, one eventually arrives at

$\nabla_{\bf r}\cdot\hat{\bf j}({\bf
r})=-\frac{ie}{\hbar}\sum_{i=1}^{N}[\,\delta({\bf r}-{\bf
r}_{i}),\hat{H}_{t}\,]=\frac{i}{\hbar}[\,\hat{\rho}({\bf r}),\hat{H}_{t}\,],$
(63)

where $\hat{\rho}({\bf r})$ is the charge-density operator, Eq. (18).

We are now equipped for addressing the time derivative of the charge density:

$\displaystyle\frac{\partial\rho(r)}{\partial t}$ $\displaystyle=$
$\displaystyle\langle\Psi|\,\hat{\rho}({\bf
r})\,|\dot{\Psi}\rangle+\langle\dot{\Psi}|\,\hat{\rho}({\bf r})\,|\Psi\rangle$
(64) $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\langle\Psi|\,[\,\hat{\rho}({\bf
r}),\hat{H}_{t}\,]\,|\Psi\rangle;$

as required by continuity, Eq. (64) coincides indeed with minus the
expectation value of Eq. (63). Finally I observe that the hypothesis that
$\hat{V}$ is a multiplicative operator is essential in order to warrant
continuity.

##  Appendix B Independent-electron Berry curvature

In the mean-field case (either Hartree-Fock or Kohn-Sham) the singlet ground
state is a Slater determinant of doubly occupied single-particle orbitals
$|\psi_{j}\rangle$ with eigenvalues $\epsilon_{j}$. Given that the two-body
terms in $\hat{V}$ are parameter-independent, the $\kappa$-derivative entering
Eq. (2) is the sum of one-body operators:

$\partial_{\kappa}\hat{H}=\sum_{i=1}^{N}O_{1}({\bf r}_{i}),$ (65)

and analogously for $\partial_{\lambda}\hat{H}$. By the Slater-Condon rules
the matrix elements in Eq. (2) are converted into one-body elements of $O_{1}$
and $O_{2}$ between occupied and unoccupied orbitals; the energy differences
are easily expressed as well, and Eq. (2) becomes

$\Omega(\kappa,\lambda)=2\,i\sum_{\stackrel{{\scriptstyle
j=\mbox{occupied}}}{{j^{\prime}=\mbox{unoccupied}}}}\frac{\langle\psi_{j}|O_{1}|\psi_{j^{\prime}}\rangle\langle\psi_{j^{\prime}}|O_{2}|\psi_{j}\rangle}{(\epsilon_{j}-\epsilon_{j^{\prime}})^{2}}+\mbox{cc},$
(66)

where the factor of two accounts for spin, and “cc’ stays for complex
conjugate. Straightforward manipulations yield the independent-electron
version of Eq. (1) as

$\Omega(\kappa,\lambda)=2\,\tilde{\Omega}(\kappa,\lambda),$ (67)
$\tilde{\Omega}(\kappa,\lambda)=-2\,\mbox{Im
}\sum_{j}\theta(\mu-\epsilon_{j})(\,\langle\partial_{\kappa}\psi_{j}|\partial_{\lambda}\psi_{j}\rangle\,),$
(68)

where $\mu$ is the Fermi level.

$\tilde{\Omega}$ can be expressed in terms of the one-body ground-state
projector

$P=\sum_{j}\theta(\mu-\epsilon_{j})|\psi_{j}\rangle\langle\psi_{j}|:$ (69)
$\tilde{\Omega}(\kappa,\lambda)=-i\,\mbox{Tr
}\\{P\,[\,\partial_{\kappa}P,\partial_{\lambda}P\,]\\}.$ (70)

This shows that $\tilde{\Omega}$ enjoys a generalized form of gauge
invariance: it is in fact invariant by any unitary mixing of the occupied
orbitals between themselves.

##  Appendix C Band-structure formulas

For a crystalline system of Kohn-Sham electrons the many-body wavefunction is
the Slater determinant of Bloch orbitals $|\psi_{j{\bf k}}\rangle={\rm
e}^{i{\bf k}\cdot{\bf r}}|u_{j{\bf k}}\rangle$ with band energies
$\epsilon_{j{\bf k}}$; the orbitals are normalized to one over the crystal
cell. The discrete ${\bf k}$ vectors become a continuous variable after the
$L\rightarrow\infty$ limit is taken; the observables are then expressed as
Brillouin-zone (BZ) integrals in insulators, and as Fermi-volume integrals in
metals.

It is easy to prove that

$\lim_{L\rightarrow\infty}\frac{E_{{\mbox{\boldmath$\kappa$}}}}{L^{d}}=\sum_{j}\int_{\rm
BZ}\frac{d{\bf k}}{(2\pi)^{d}}f_{j}({\bf k})\epsilon_{j,{\bf
k}\\!+\\!{\mbox{\boldmath$\kappa$}}},$ (71)
$\displaystyle\lim_{L\rightarrow\infty}$
$\displaystyle\left.\frac{1}{L^{d}}\Omega(\kappa_{\alpha},\lambda)\right|_{{\mbox{\boldmath$\kappa$}}=0}$
(72) $\displaystyle=$ $\displaystyle\sum_{j}\int_{\rm BZ}\frac{d{\bf
k}}{(2\pi)^{d}}f_{j}({\bf k})\,\tilde{\Omega}_{j,\alpha\lambda}({\bf k}),$

where $f_{j}({\bf k})$ is the Fermi factor at $T=0$; in Eq. (72)
$\tilde{\Omega}_{j,\alpha\lambda}({\bf k})$ is the Berry curvature of band
$j$, defined as:Vanderbilt

$\tilde{\Omega}_{j,\alpha\lambda}({\bf k})=-2\,\mbox{Im
}\langle\partial_{k_{\alpha}}u_{j{\bf k}}|\partial_{\lambda}u_{j{\bf
k}}\rangle.$ (73)

Eqs. (71) and (72) are given per spin channel (or for “spinless electrons”).

All of the formulas provided so far in a many-body setting apply as they stand
to the independent-electron case, by simply adopting expressions of Eqs. (71)
and (72). As an example, the Drude weight becomes, from Eqs. (38) and (71),

$D_{\alpha\beta}=\pi e^{2}\sum_{j}\int_{\rm BZ}\frac{d{\bf
k}}{(2\pi)^{d}}f_{j}({\bf k})\,m^{-1}_{j,\alpha\beta}({\bf k}),$ (74)

where

$m^{-1}_{j,\alpha\beta}({\bf
k})=\frac{1}{\hbar^{2}}\frac{\partial^{2}\epsilon_{j{\bf k}}}{\partial
k_{\alpha}\partial k_{\beta}}$ (75)

is the effective inverse mass tensor of the $j$-th band.rap157

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