# Faraday law, oxidation numbers, and ionic conductivity:
The role of topology

Raffaele Resta<EMAIL_ADDRESS>Istituto Officina dei Materiali IOM-CNR,
Strada Costiera 11, 34151 Trieste, Italy http://www-dft.ts.infn.it/~resta/.
Donostia International Physics Center, 20018 San Sebastián, Spain

###### Abstract

Faraday’s experiment measures—within a modern view—the charge adiabatically
transported over a macroscopic distance by a given nuclear species in
insulating liquids: the reason why it is integer is deeply rooted in topology.
Whole numbers enter chemistry in a different form: atomic oxidation states.
They are not directly measurable, and are determined instead from an agreed
set of rules. Insulating liquids are a remarkable exception: Faraday’s
experiment indeed measures the oxidation numbers of each dissociated component
in the liquid phase, whose topological values are unambiguous. Ionic
conductivity in insulating liquids is expressed in terms of the
autocorrelation function of the fluctuating charge current at a given
temperature in zero electric field; topology plays a major role in this
important observable as well. The existing literature deals with the above
issues by adopting the independent-electron framework; here I provide the
many-body generalization of all the above findings, which furthermore allows
for compact and very transparent notations and formulas.

††preprint: AIP/123-QED

##  I Introduction

Faraday’s first law of electrolysis addresses insulating liquids in
electrolytic cells, and can be recast in modern terms as follows. When a
macroscopic number $N$ of nuclei of a given chemical species passes from one
electrode to the other, the transported electrical charge is an integer
multiple of $N$ times $e$. The law is additive: it concerns each different
chemical species in the cell independently. It addresses ionic conduction in
electrolytes, including molten salts; it does not apply to metallic
conduction, which is unrelated to the motion of nuclei.

Faraday’s discovery dates from the 1830s, decades before the existence of
atoms became accepted. Remarkably, he wrote:Faraday  “Although we know nothing
of what an atom is, yet we cannot resist forming some idea of a small
particle…”. What he actually measured were charge-to-mass ratios; when scaled
to the actual atomic masses, Faraday’s “equivalent numbers” can be regarded as
the archetypical definition of oxidation numbers (also known as oxidation
states) in insulating liquids. I also stress that such definition is not
static: Faraday’s numbers are—within a modern quantum-mechanical
view—dynamical properties in the adiabatic limit.

In modern chemistry, the oxidation number of an atom in a molecule or in a
solid is an integer determined by an agreed set of rules,Walsh18  and is often
related in a postdictiverap142 —not predictive—way to some measurable
properties, thus facilitating the interpretation of several experimental
observations.Walsh18 ; rap142 ; rap_a29 ; Sit11 ; Jiang12  Nowadays we know
“what an atom is”, but we also know that solids and liquids are not assemblies
if ions; they are assemblies of atoms, having ionic character only because
neighboring atoms have different electronegativity.Appelbaum76  Literally
dozens of electronegativity scales and/or definitions of atomic charges have
been proposed in the literature,Meister94  none of them yielding integer
values.

The reason why integer oxidation numbers in electrolytes are measurable
quantities is deeply rooted in topology and in 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
emphasized 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 oxidation
numbers in solids and liquids.rap_a29 ; rap142 ; Jiang12 ; Grasselli19  The
main message from topology is that the ionic charges behave as integer only
when adiabatically transported in a macroscopic system; no trace of
quantization may appear in a “snapshot” of either a solid or a liquid at a
given time. The quantized charge transport may occur either in a real
experiment (as in Faraday’s) or in a “gedanken” experiment, nowadays called a
“Thouless pump”.

An outstanding 2019 paper by Grasselli and BaroniGrasselli19  has extended the
scope of topology in the context of charge transport in insulating liquids.
Besides the amount of transported charge, another basic observable is the
value of ionic conductivity at a given temperature, which is provided within
classical statistical mechanics by the fluctuation-dissipation theorem, as
expressed in the Green-Kubo formula, based on the autocorrelation function of
the fluctuating charge current.Hansen ; Frenkel  Owing to the adiabatic
approximation, the nuclear motion is classical even within ab-initio molecular
dynamics, where the quantum nature of the electrons is fully accounted for.
The contribution to the current from the motion of each nucleus at a given
time is quite different from the one of a scalar integer charge—the
topological oxidation number times $e$–moving on the same trajectory.
According to Ref. Grasselli19, , topology warrants that, when the real current
is replaced with the fictitious current carried by the said integer charges
(along the ab-initio trajectories), the Green-Kubo formula yields the correct
ionic conductivity.

Charge transport in condensed matter has a close relationship to the theory of
polarization (not “modern” anymore), developed along the 1990s.rap73 ; King93
; Vanderbilt93 ; Ortiz94 ; rap100  In fact the current density (due to
electrons and nuclei) which flows in a macroscopic insulator while the nuclei
are adiabatically displaced equals by definition the time-derivative of
macroscopic polarization. The formal definition of oxidation numbers proposed
in Ref. Grasselli19,  for insulating liquids is indeed rooted in polarization
theory, in its independent-electron (i.e. density-functional)
version.Vanderbilt  At variance with the previous literature, here I
thoroughly address charge transport in insulating liquids in a many-body
framework, thus generalizing the main results of Refs. Pendry84,  and
Grasselli19,  beyond the independent-electron level. The present many-body
approach naturally yields a compact and very perspicuous formulation.

In Sec. II I introduce the closed system—made of classical nuclei and quantum
electrons—adopted throughout this work in order to represent an electrolytic
cell and charge transport therein. As an introduction to the following
formalism, in Sec. III I address classical transport, in order to show how its
essential features are conveniently recast in terms of a phase angle. The
quantum electronic transport is dealt with in Sec. IV; the theory is also
based on a phase angle,rap100 ; rap162  now a property of the many-body
wavefunction, which sometimes is dubbed “single-point Berry phase”. In Sec. V
the main findings of Refs. Pendry84, ; Grasselli19,  are retrieved in the
present compact many-body framework in order to arrive at an unambiguous
definition of oxidation numbers in the insulating liquid. The special case of
noninteracting electrons is dealt with in Sec. VI, which relies on the
Wannier-function formulation of polarization theory.Vanderbilt  The recent
Grasselli-Baroni major advance in conductivity theory is generalized in Sec.
VII. In Sec. VIII I discuss the topological oxidation numbers, as emerging
from the present analysis of Faraday’s experiment, in their relationship to
their more conventional definition in the chemistry literature; a remarkable
pitfall is discussed in the following Sec. IX. In Sec. X I present some
concluding remarks. In Appendix A I outline the main points of the theory of
the insulating state as adopted here;rap107 ; rap_a25 ; Stella11 ; rap_a33  in
Appendix B I present a different physical property whose theory partly
overlaps with the present theory: infrared absorption in insulating liquids. A
few somewhat advanced mathematical considerations have been confined to
Appendix C.

##  II Electrolytic cell as a Thouless pump

A liquid is an assembly of nuclei and electrons; when a nucleus takes off from
one electrode and lands on the other in the electrolytic cell it pumps some
electronic charge across the cell. The electronic current, driven by the
electromotive force, flows through the closed circuit which includes the
battery; owing to continuity, we may address transport by monitoring the
passage of nuclei and electrons across an ideal section of the cell.

We wish to describe the electrons in the liquid by means of a finite (and
closed) quantum system: in order to account for dc steady-state currents it is
then mandatory to adopt periodic boundary conditions (PBCs). We therefore
assume that the liquid is contained in a macroscopic cylindrical vessel of
section $A$ and length $L$ (along $x$). By adopting PBCs with period $L$, the
$x$ coordinate becomes equivalent to the angle $\varphi=2\pi x/L$. The system
comprises therefore a central cell and its replicas in infinite number, thus
acquiring the topology of a torus.

One cycle of the pump consists in adiabatic transporting one nucleus from a
given position in the central cell to its replica position in the nearby cell,
thus making one loop around the torus; the transported charge $Q$ is the bare
nuclear charge plus the time-integrated electronic current across the vessel’s
section. It will be shown next that, because of topological reasons, an
integer electronic charge is pumped: as said above, the process goes nowadays
under the name of Thouless pump.

The Thouless pump was first formulated in 1983 at the independent-electron
level.Thouless83  The present work is deeply rooted in its many-body
generalization, presented by Niu and ThoulessNiu84  in 1984. A comprehensive
review of related topics can be found in Ref. Xiao10, .

##  III Classical transport as a winding number

We start considering the simplest classical model of a binary molten salt:
point-like ions of charges $\pm qe$, interacting e.g. via two-body forces
(Coulomb and short-range).Hansen ; Hansen75  Clearly Faraday law requires
integer values of $q$, in which case it is trivially fulfilled by
construction. If ${\bf R}_{\ell}^{(+)}(t)$ and ${\bf R}_{\ell}^{(-)}(t)$ are
the instantaneous positions of the $N$ positive and $N$ negative ions, one
could write the macroscopic electrical current density at time $t$, for each
chemical species, as

${\bf j}^{(+)}(t)=\frac{qe}{LA}\sum_{\ell=1}^{N}\dot{{\bf
R}}_{\ell}^{(+)}(t),\quad{\bf
j}^{(-)}(t)=-\frac{qe}{LA}\sum_{\ell=1}^{N}\dot{{\bf R}}_{\ell}^{(-)}(t).$ (1)

In order to deal with the quantum-mechanical case addressed in the following,
it proves convenient instead to express the adiabatic electrical currents
density as the time derivative of macroscopic polarization ($x$-component
thereof): $j_{x}^{(\pm)}(t)=dP_{x}^{(\pm)}(t)/dt$.

When written in terms of the angles $2\pi x^{(+)}_{\ell}(t)/L$, the $x$
component of the cationic current is

$\displaystyle j_{x}^{(+)}(t)$ $\displaystyle=$
$\displaystyle\frac{dP^{(+)}(t)}{dt}=\frac{e}{2\pi A}\frac{d}{dt}\mbox{Im ln
}{\mathfrak{z}}^{(+)}(t)$ $\displaystyle{\mathfrak{z}}^{(+)}(t)$
$\displaystyle=$ $\displaystyle{\rm e}^{i\frac{2\pi
q}{L}\sum_{\ell=1}^{N}X_{\ell}^{(+)}(t)},$ (2)

where $X_{\ell}^{(+)}\equiv R_{\ell,x}^{(+)}$, and analogously for the anionic
current $j_{x}^{(-)}(t)$; Eq. (2) is clearly additive, given that

$\mbox{ln }{\rm e}^{i\frac{2\pi
q}{L}\sum_{\ell=1}^{N}X_{\ell}^{(+)}(t)}=\sum_{\ell=1}^{N}\mbox{ln }{\rm
e}^{i\frac{2\pi q}{L}X_{\ell}^{(+)}(t)}.$ (3)

We notice that according to Eq. (2) macroscopic polarization is a multivalued
quantity, as indeed mandated by the theory of polarization.Vanderbilt93 ;
Vanderbilt

The two complex numbers ${\mathfrak{z}}^{(\pm)}(t)$ trace a path on the unit
circle in the complex plane; the branch choice in “Im ln” is made in such way
that each current is a continuous function of $t$. In terms of the phase
angles $\gamma^{(\pm)}(t)=\mbox{Im ln }{\mathfrak{z}}^{(\pm)}(t)$, the charge
transported across a section of the toroidal vessel in time $T$ by the nuclei
of each chemical species is proportional to the accumulated angle:

$Q^{(\pm)}(T)=A\int_{0}^{T}dt\;j_{x}^{(\pm)}(t)=\frac{e}{2\pi}[\;\gamma^{(\pm)}(T)-\gamma^{(\pm)}(0)\;].$
(4)

In the special case where the trajectories ${\bf R}^{(\pm)}_{\ell}(t)$ are
periodic in time with period $T$, the net charges $Q^{(\pm)}(T)$ transported
across the section by the ions of each species coincides with the winding
numbers of ${\mathfrak{z}}^{(\pm)}(t)$ in the complex plane times $e$.

Finally, the total classical current which flows through the vessel can be
expressed in terms of the time-derivative of a phase angle:

$\displaystyle I_{x}(t)$ $\displaystyle=$ $\displaystyle
Aj_{x}(t)=\frac{e}{2\pi}\dot{\gamma}(t),$ (5) $\displaystyle\gamma(t)$
$\displaystyle=$ $\displaystyle\mbox{Im ln }{\rm e}^{i\frac{2\pi
q}{L}\sum_{\ell=1}^{N}[\,X_{\ell}^{(+)}(t)-X_{\ell}^{(-)}(t)\,]}.$ (6)

##  IV Quantum transport

The $\ell$-th nucleus, when infinitesimally moved, displaces an effective
charge due to the nucleus itself and to the polarization of the electronic
distribution. The concept is exactly cast into the (instantaneous) Born
effective charge tensorBaroni01 ; Vanderbilt  $eZ^{*}_{\ell,\alpha\beta}(t)$,
where Greek indices indicate Cartesian coordinates. The Born charges go also
under the name of infrared charges and (in quantum chemistry) of atomic polar
tensors; the $\ell$-th tensor depends on the environment at time $t$, it is
anisotropic, and is not even symmetric in general. Within the classical ionic
model discussed above $Z^{*}_{\ell,\alpha\beta}(t)$ must be identified with
$\pm q\delta_{\alpha\beta}$ at all times. If there are $N$ nuclei in the
vessel, the macroscopic electrical current which flows through the torus is

$I_{x}(t)=A\,j_{x}(t)=\frac{e}{L}\sum_{\ell=1}^{N}\sum_{\beta}Z^{*}_{\ell,x\beta}(t)\dot{R}_{\ell,\beta}(t);$
(7)

the same adiabatic current flows in average though a given section of the
vessel. The exact expression of Eq. (7) is additive, but clearly the charge
carried by each nucleus is neither scalar nor integer. Faraday’s law will
manifest itself only when a nucleus is carried over a macroscopic distance.

We start addressing the electronic term only. Within the adiabatic
approximation, the many-electron wavefunction obeys a time-dependent
Schrödinger equation, whose one-body potential is determined by the nuclear
trajectories. Let $|\Psi(t)\rangle$ be the instantaneous adiabatic ground
eigenstate at time $t$ of a system of $N^{(\rm el)}$ electrons, within the
same periodic boundary conditions as above: $|\Psi(t)\rangle$ is periodic with
period $L$ in the $x$ coordinate of each electron, independently. The
adiabatic electronic current density at time $t$ is:rap100 ; rap_a20

$\displaystyle j_{x}^{(\rm el)}(t)$ $\displaystyle=$
$\displaystyle\frac{e}{2\pi A}\frac{d}{dt}\lim_{L\rightarrow\infty}\mbox{Im ln
}{\mathfrak{z}}^{(\rm el)}(t)$ $\displaystyle{\mathfrak{z}}^{(\rm el)}(t)$
$\displaystyle=$ $\displaystyle\langle\Psi(t)|\,{\rm
e}^{-i\frac{2\pi}{L}\sum_{j=1}^{N^{(\rm el)}}x_{j}}\,|\Psi(t)\rangle;$ (8)

the matrix element is the expectation value of a unitary operator, ergo
${\mathfrak{z}}^{(\rm el)}(t)$ is a complex number whose modulus is no larger
than one. The limit will be omitted in the following formulas, given that both
$L$ and $A$ are macroscopic.

At this point we may add the contribution to the electrical current from the
$N$ classical point-like nuclei: it is enough to adopt Eq. (2), where the bare
nuclear charges $eZ_{\ell}$ replace the ionic charges $\pm qe$:

${\mathfrak{z}}(t)=\langle\Psi(t)|\,{\rm
e}^{i\frac{2\pi}{L}(\sum_{\ell=1}^{N}Z_{\ell}X_{\ell}-\sum_{j=1}^{N^{(\rm
el)}}x_{j})}\,|\Psi(t)\rangle,$ (9)

where, owing to charge neutrality, $N^{(\rm el)}=\sum_{\ell=1}^{N}Z_{\ell}$.
Analogously to the classical case of Eqs. (5) and (6), the total current is
expressed by means of the time-derivative of a phase angle:

$\displaystyle I_{x}(t)$ $\displaystyle=$
$\displaystyle\frac{e}{2\pi}\dot{\gamma}(t)=\frac{e}{2\pi}\frac{d}{dt}\mbox{Im
ln }{\mathfrak{z}}(t)$ (10) $\displaystyle\gamma(t)$ $\displaystyle=$
$\displaystyle\mbox{Im ln }\langle\Psi(t)|\,{\rm
e}^{i\frac{2\pi}{L}(\sum_{\ell=1}^{N}Z_{\ell}X_{\ell}-\sum_{j=1}^{N^{(\rm
el)}}x_{j})}\,|\Psi(t)\rangle.$ (11)

The phase $\gamma(t)$ (electronic term thereof) often goes in the literature
under the somewhat oxymoronic name of “single-point Berry phase”.rap_a20

We stress that Eqs. (10) and (11) are equivalent to Eq. (7), although in the
$L\rightarrow\infty$ limit only; the equivalence proves that Eq. (10), despite
not being additive in-form, is additive indeed. While $\gamma(t)$ itself is
not additive, its time-derivative is such, as a consequence of linear
response: the total current is the sum of the currents induced by
infinitesimally displacing each nucleus one at a time. This “dynamical”
additivity can be exploited by defining

$\displaystyle\dot{\gamma}_{\ell}(t)$ $\displaystyle=$
$\displaystyle\frac{2\pi}{L}\sum_{\beta}Z^{*}_{\ell,x\beta}(t)\dot{R}_{\ell,\beta}(t)$
$\displaystyle\gamma_{\ell}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}dt^{\prime}\;\dot{\gamma}_{\ell}(t^{\prime}),$ (12)

such that

$\gamma(T)-\gamma(0)=\sum_{\ell=1}^{N}\gamma_{\ell}(T).$ (13)

The total accumulated angle is therefore unambiguously decomposed into the sum
of the angles accumulated by each nucleus; each $\gamma_{\ell}(t)$,
nonetheless, depends on the history of the system via the trajectories ${\bf
R}_{\ell}(t)$.

The total electrical charge which crosses a vessel’s section in time $T$ is
thus in average

$Q(T)=\frac{e}{2\pi}[\gamma(T)-\gamma(0)];$ (14)

as in the classical case, if the nuclear coordinates ${\bf R}_{\ell}(t)$ are
periodic in time, the transported charge is integer, and is measured by the
winding number of ${\mathfrak{z}}(t)$ in the complex plane times $e$. The
$3N$-dimensional space of the ${\bf R}_{\ell}(t)$ coordinates is a torus (in
the $x$-coordinate), and the above condition means that all nuclear
trajectories are closed.

The topological nature of the transported charge is spelled out as follows.
Suppose that we continuosly deformate these closed trajectories: the
corresponding ${\mathfrak{z}}(t)$ curve in the complex plane is accordingly
deformated, but its winding number cannot vary insofar as
${\mathfrak{z}}(t)\neq 0$. The topological invariant is “protected” by the
condition

$|{\mathfrak{z}}(t)|=|\langle\Psi(t)|\,{\rm
e}^{-i\frac{2\pi}{L}\sum_{j=1}^{N^{(\rm el)}}x_{j}}\,|\Psi(t)\rangle|\neq 0$
(15)

at all times and in the $L\rightarrow\infty$ limit. Not surprisingly, this is
precisely the most general condition which characterizes the insulating state
of matter;rap107 ; rap_a25 ; Stella11 ; rap_a33  furthermore
$|{\mathfrak{z}}|=0$ even at finite $L$ for independent-electron (i.e. Kohn-
Sham) metals. See Appendix A for more details. The above findings are nothing
else than a reformulation of Thouless theoremThouless83  in a many-body
framework; a somewhat different proof of the same result was provided by Niu
and Thouless long ago.Niu84 ; Xiao10

The condition that ${\bf R}_{\ell}(T)={\bf R}_{\ell}(0)$ for all $\ell$ is a
kind of “rare event”, which would require a very long simulation time.
Nonetheless the above topological argument holds for a much larger class of
trajectories. It is enough to require that the path of ${\mathfrak{z}}(t)$ in
the complex plane remains a closed curve, i.e.
${\mathfrak{z}}(T)={\mathfrak{z}}(0)$, an event that definitely may happen in
a much shorter simulation time; furthermore, since both $L$ and $T$ are
macroscopic, we may replace at any time the instantaneous current $I_{x}(t)$
with its average $Q/T$. An additional argument—originally due to Grasselli and
BaroniGrasselli19 —further validates the quantization of charge transport in
the present gedanken experiment.

The argument, exploited in Sec. VII below, is as follows. Suppose we start
from a given nuclear configuration $\\{{\bf R}_{\ell}(0)\\}$, and we then
follow the adiabatic time-evolution of the system along a path ending in a
final configuration $\\{{\bf R}_{\ell}(T)\\}$ different from the initial one.
For a macroscopic system, after a macroscopic time $T$, the accumulated angle
will be

$\gamma(T)-\gamma(0)\gg 2\pi.$ (16)

We can always ideally concatenate to the previous path another (short) path
which brings back the actual configuration $\\{{\bf R}_{\ell}(T)\\}$ into the
initial one $\\{{\bf R}_{\ell}(0)\\}$, by displacing all nuclei inside a
single cell, and whose accumulated angle $\Delta\gamma$ is therefore smaller
than $2\pi$. From the above arguments the total accumulated phase is

$\gamma(T)-\gamma(0)+\Delta\gamma=2\pi W,$ (17)

with integer $W$ (winding number). The average current is

$I_{x}=\frac{Q(T)}{T}=\frac{e}{2\pi
T}[\gamma(T)-\gamma(0)]=\frac{e}{T}\left(W-\frac{\Delta\gamma}{2\pi}\right),$
(18)

the $\Delta\gamma$ contribution vanishes il the large-$T$ (and large-$W$)
limit: $Q(T)$ becomes integer.

##  V Topological oxidation numbers

Next we perform the following gedanken experiment. We pick only one of the
nuclei, say the $\ell$-th, and we drive it round the torus (actually, to the
nearby replica cell), thus crossing our ideal section once; the other $N-1$
nuclei move out of the way, in order to minimize the energy at any time. We
assume that the system returns to the same identical configuration at the end
of the cycle, and that all the other $N-1$ nuclei have not crossed the ideal
section (except for two-way crossings). As shown above, if the electronic
wavefunction stays insulating at all times—as from Eq. (15)—then the
electrical charge $Q_{\ell}$ which has flown through the section in the
process equals the winding number of ${\mathfrak{z}}(t)$ in the complex plane
(times $e$).

While there is no integer scalar charge to speak of at any given time $t$, the
transport of a single nucleus along a closed path on the macroscopic torus is
equivalent to transporting a time-independent scalar integer charge
$Q_{\ell}$. This gedanken experiment has been proposed much earlier at the
independent-electron level by Pendry and Hodges.Pendry84  As said above,
nowadays a cyclic phenomenon like the one discussed here is called a Thouless
pump, whose many-body generalization owes to Niu and Thouless.Niu84 ; Xiao10
Based on it, we have generalized the Pendry-Hodges argument to the case of
interacting electrons.

The transported charge is path-independent. Consider in fact a couple of
closed paths having the same initial and end points: they can be continuously
deformed and made coincident without varying their winding numbers if no
metallic state is crossed in the process (see also Sec. IX). The two winding
numbers must therefore be equal.

In order to identify $Q_{\ell}/e$ with the oxidation number $q_{\ell}$ of the
chemical species of the $\ell$-th nucleus, one has to additionally show that
two nuclei having the same atomic number $Z$ transport the same charge
$Q_{\ell}$ in a single cycle of the Thouless pump. To this aim, I generalize
once more an argument due to Grasselli and Baroni,Grasselli19  based on
additivity. If the adiabatic time $T$ is the period of the Thouless pump, then
Eq. (13) yields $\gamma_{\ell}(T)=2\pi q_{\ell}$, with integer $q_{\ell}$
(winding number), while instead $\gamma_{\ell^{\prime}}(T)=0$ for all
$\ell^{\prime}\neq\ell$. Suppose next that two nuclei, labeled with $\ell=1$
and $\ell=2$, having the same atomic number $Z$, are driven along two paths:
nucleus 1 from ${\bf R}_{1}(0)$ to ${\bf R}_{1}(T)$ with winding number
$q_{1}$ in the complex ${\mathfrak{z}}$ plane. and nucleus 2 from ${\bf
R}_{2}(0)$ to ${\bf R}_{2}(T)$ with winding number $q_{2}$. If the initial
positions ${\bf R}_{1}(0)$ and ${\bf R}_{2}(0)$ can be exchanged without
crossing a metallic state, then necessarily $q_{1}=q_{2}$.

It is worth stressing that all topological definitions based on the previous
Refs. Pendry84, ; rap_a29, ; Jiang12, ; Grasselli19, , as well as the one
adopted here, are not static: topological oxidation numbers are dynamical
quantities, in the adiabatic limit.

##  VI Independent electrons and Wannier functions

The case of independent electrons (in the mean-field sense) was dealt with
before;Pendry84 ; Grasselli19  here we retrieve it as a special case of the
general many-body formulation; some more details are given in Appendix C. We
consider our system and its periodic replicas as a quasi-one-dimensional
“crystal” of macroscopic lattice constant $L$, with $N$ nuclei and
$N^{\rm(el)}$ electrons per supercell. In the noninteracting (e.g. Kohn-Sham)
case the insulating singlet ground state $|\Psi(t)\rangle$ can be written as a
Slater determinant of $N^{(\rm el)}/2$ doubly occupied Bloch orbitals, or
equivalently of $N^{(\rm el)}/2$ doubly occupied Wannier functions (WFs). In
the latter case, the electronic density is partitioned into localized
contributions: it is then tempting to regard the electrolyte as an assembly of
closed shell entities, each of them constituted by the nucleus and by the WFs
closest to it. This viewpoint would yield a definition of the oxidation
numbers; nonetheless, while this picture may reasonably hold in a strongly
ionic case (e.g. molten alkali halides), partitioning the electronic charge on
the basis of WF proximity leads in general to difficulties and ambiguities.
The issue is discussed in Ref. Jiang12, , which provides a solid-state
definition of oxidation numbers, based on topology (not on WF proximity), and
similar in spirit to the approach of Ref. Grasselli19, , generalized here.

We observe that, while the total electronic density is gauge-invariant, its
partitioning into WF contributions is gauge-dependent, hence strongly
nonunique. Nonetheless the contribution to macroscopic polarization from the
WF centers in the central cell—as displayed next—is gauge-invariant and
unique.Vanderbilt  Let us indicate as $X_{j}^{(\rm W)}$ the $x$-coordinates of
the $N^{(\rm el)}/2$ centers. As outlined in Appendix C the single-point Berry
phase of Eq. (11) transforms (in the large-$L$ limit) into the Berry phase of
the Bloch orbitals; the latter in turn can be equivalently recast in terms of
WFs.Vanderbilt  The polarization assumes then a somewhat transparent
expression in terms of the WF centers; when the nuclear contribution is
included, the result isVanderbilt

$\displaystyle j_{x}(t)$ $\displaystyle=$
$\displaystyle\dot{P}_{x}(t)=\frac{e}{2\pi A}\dot{\gamma}(t)$ (19)
$\displaystyle=$
$\displaystyle\frac{e}{LA}\left[\sum_{\ell=1}^{N}Z_{\ell}\dot{X}_{\ell}(t)-2\sum_{j=1}^{N^{\rm(el)}/2}\dot{X}_{j}^{(\rm
W)}(t)\right].$

This expression is similar in form to the classical case of Eqs. (1) and (6),
and therefore $\gamma(t)$ be recast in a similar way:

$\gamma(t)=\mbox{Im ln }{\rm
e}^{i\frac{2\pi}{L}[\,\sum_{\ell=1}^{N}Z_{\ell}X_{\ell}(t)-2\,\sum_{j=1}^{N^{\rm(el)}/2}X_{j}^{(\rm
W)}(t)\,]}.$ (20)

Notice that the correspondence between nuclear coordinates and WF centers is
ambiguous, and therefore in general $\gamma(t)$ is not additive (only its
nuclear term is such); nonetheless, as observed above, its time-derivative is
additive.

The same gedanken experiment described above—when specialized to the
independent-electron case and formulated in terms of WFs—amounts to
transporting one nucleus which “drags” some of the Kohn-Sham WFs, thus
transporting an even number of electrons: the oxidation number of a given
chemical species obtains then from WF counting. The same occurs in the solid-
state topological definition of oxidation numbers; only double-occupancy has
been considered here, while in some circumstances spin-polarized ground states
must be considered as well.Jiang12

##  VII Ionic conductivity

The Green-Kubo formula for the static ionic conductivity is:Hansen

$\sigma=\frac{1}{LA\,k_{\rm B}T}\int_{0}^{\infty}dt\;\langle
I_{x}(t)\,I_{x}(0)\rangle,$ (21)

where the bracketed quantity is the autocorrelation function of the
fluctuating charge current in absence of the electric field; the fluctuations
are averaged over an equilibrium ensemble and the thermodynamic limit is
understood.

Our previous results lead to recasting $\sigma$ as

$\sigma=\frac{L}{A\,k_{\rm
B}T}\frac{e^{2}}{4\pi^{2}}\int_{0}^{\infty}dt\;\langle\dot{\gamma}(t)\,\dot{\gamma}(0)\rangle,$
(22)

where $\gamma(t)$ is given in Eq. (6) for a classical molten salt, in Eq. (11)
for a system of interacting electrons and classical nuclei, and in Eq. (20) in
the mean-field (e.g. Kohn-Sham) approximation. In all three cases $\sigma$ is
given by the autocorrelation function of the derivative of $\gamma(t)$; about
the autocorrelation function of $\gamma(t)$ itself see Appendix B.

It has been stressed above that the ab-initio current can be written in two
formally equivalent ways (in the large-system limit); for the sake of clarity
we rewrite the two expressions here

$I_{x}(t)=\frac{e}{2\pi}\dot{\gamma}(t)=\frac{e}{L}\sum_{\ell=1}^{N}\sum_{\beta}Z^{*}_{\ell,x\beta}(t)\dot{R}_{\ell,\beta}(t).$
(23)

Actual simulations have implemented the Berry phase expression,Cavazzoni99
the equivalent Wannier-function formulation,Rozsa18  and the alternative
expression in terms of the Born tensor, where the latter have been computed on
the fly from density-functional perturbation theory.Baroni01 ; Giannozzi09
The single-point Berry phase $\gamma(t)$ is implemented in a different (though
similar) context: see Appendix B.

We pause to observe that actual simulations implement the fluctuation formulas
in a different geometry from the one adopted here for presentation purposes.
In fact they are routinely performed in a cubic supercell of volume $L^{3}$
and Eq. (21) is equivalently cast as

$\sigma=\frac{L^{3}}{3k_{\rm B}T}\int_{0}^{\infty}dt\;\langle{\bf
j}(t)\cdot{\bf j}(0)\rangle,$ (24)

where PBCs with period $L$ are adopted on all Cartesian coordinates; the
result is converged whenever $L$ is larger than both correlation lengths and
diffusion lengths. Eq. (24) is implemented in either classical molecular
dynamics simulations (where the particles in the cell are ions)Frenkel ;
Hansen ; Hansen75  or in ab-initio molecular dynamics (where the particles in
the cell are electrons and nuclei, and the adiabatic approximation is
adopted).French11 ; Grasselli19

A surprising discovery was heuristically made in 2011 by French, Hamel, and
Redmer,French11  who addressed the conductivity of partially dissociated
water: upon replacing the Born tensors $Z^{*}_{\ell,\alpha\beta}$ with the
corresponding oxidation numbers (i.e. $-2$ for oxygen and $+1$ for hydrogen)
along the ab-initio trajectories, the ab-initio $\sigma$ value was reproduced
to high accuracy. Remarkably, the average of $Z^{*}_{\ell,\alpha\beta}$ along
the trajectories was found to be a scalar differing by about 30% from the
above integer values. The reason for such counterintuitive result was found
much later by Grasselli and Baroni,Grasselli19  and is rooted once more in
topology.

Here I am going to adopt the compact many-body formalism of the present work
in order to present the major result of Ref. Grasselli19, , which is based on
the Einstein-Helfand relationship.Helfand60  Within the geometry adopted here
the Green-Kubo formula, Eqs. (21) and (22), is equivalent to the Einstein-
Helfand formula

$\sigma=\frac{L}{Ak_{\rm
B}T}\frac{e^{2}}{4\pi^{2}}\,\lim_{t\rightarrow\infty}\frac{\langle\gamma^{2}(t)\rangle}{2t}.$
(25)

From the dynamical partitioning of Eq. (13) we get

$\gamma(t)=\gamma(0)+\sum_{\ell=1}^{N}\gamma_{\ell}(t),$ (26)

where each $\gamma_{\ell}(t)$ is a function of the $\ell$-th trajectory. We
also define the auxiliary phase angles $\tilde{\gamma}_{\ell}(t)$, where the
Born effective charge tensors in Eq. (12) are replaced by the corresponding
topological oxidation numbers, over the same trajectories:

$\tilde{\gamma}_{\ell}(t)=\frac{2\pi}{L}q_{\ell}X_{\ell}(t).$ (27)

The same reasoning leading to Eqs. (16) and (18), concerning the total phase
$\gamma(t)$, can be adopted here for the $\ell$-th phase: in the large-$t$
limit both $\gamma_{\ell}(t)$ and $\tilde{\gamma}_{\ell}(t)$ are unbounded,
and they differ by a bounded term. Their sums differ by a bounded term as
well; therefore if in Eq. (26) we replace $\gamma_{\ell}(t)$ with
$\tilde{\gamma}_{\ell}(t)$ for all $\ell$ the limit in Eq. (25) is unaffected.

I have provided therefore the many-body generalization of the Grasselli-Baroni
theorem,Grasselli19  who also validated it by means of ab-initio molecular
dynamics simulations on molten KCl; as said above, the theorem is also
validated by the previous serendipitous discovery of Ref. French11, .

I take the present occasion for stressing an outstanding implication of the
Grasselli-Baroni theorem. Let us consider for instance liquid (undissociated)
water under normal conditions: it has a finite static dielectric constant
($\epsilon_{0}\simeq 80$). Hence the Faraday experiment would give a negative
result, in agreement with the fact that the oxidation number of a neutral
molecule as a whole is zero: it is then kind of obvious that ionic
conductivity must vanish. This fact is apparently less obvious when analyzed
from the fluctuation-dissipation-theorem viewpoint. Water is strongly
infrared-active,Silvestrelli97  the Born charge tensor of the molecule as a
whole is quite sizable,rap116  and the fluctuating equilibrium current is
therefore conspicuously nonzero. Yet, given that the topological charge of a
neutral undissociated molecule is zero, topology also guarantees that the
time-integrated equilibrium autocorrelation function of the current vanishes,
as indeed needed on physics considerations.

##  VIII More about oxidation numbers

In inorganic chemistry the oxidation numbers (also called oxidation states)
are determined via a postulatory method based on an agreed set of rules. In
simple cases—when no $d$ shells are involved—the oxidation numbers are simply
determined by the octet rule: atoms are assigned an octet in order of
decreasing electronegativity until all valence electrons are
distributed,Walsh18  except for atoms which are obviously assigned a pair (H,
He, and Li). The rule is unambiguous, given that the ordering of the (simple)
elements as provided by different ionicity scales is the same. Oxidation
numbers are formal quantitites not related to the wavefunction or to any
quantum-mechanical operator; in molecules and in crystalline solids they are
not measurable, yet they correlate to several measurable properties in a
postdictive way.Walsh18 ; rap142 ; rap_a29 ; Sit11 ; Jiang12

Given the above, it is quite remarkable that oxidation numbers in insulating
liquids are instead directly measurable quantities thanks to Faraday’s
experiment. The fundamental difference which makes the oxidation numbers
nonmeasurable in crystalline insulators and measurable in liquid insulators is
that in the latter case the nuclei travel over macroscopic distances: as
emphasized by Pendry and HodgesPendry84  back in 1984, integer charges
manifest themselves only when transported, owing to Thouless
theorem;Thouless83 ; Niu84 ; Xiao10  static charges do not have a first-
principle definition in molecules,Meister94  in solids, and in liquids.

So far, we have tacitly addressed ionic liquids where the oxidation
numbers—upon chemical intuition—are quite robust in the given compound and do
not depend on the fluctuating environment: this happens, for instance, when
the octet rule is enough to determine the conventional oxidation numbers.
There are challenging cases, though, well known in the solid state, like e.g.
charge-ordering phenomena in transition metal oxides. In such cases a given
element may assume different oxidation numbers depending on the ligands and on
coordination. As an example, Mn can be labeled with up to seven oxidation
numbers, from Mn(I) to Mn(VII), distinguished from their distinct
spectroscopic and magnetic signatures.Walsh18  It is easy to guess that when
the (conventionally defined) oxidation numbers do depend on the environment
the liquid is not insulating and topology cannot be invoked. A challenging
case of different kind is dealt with in the next Section.

The “insulating” qualification deserves some comments. The main
phenomenological quantity which discriminates insulators from metals is
$\varepsilon_{\infty}$, called “high frequency” dielectric constant:Ashcroft
it is finite in insulators and formally infinite in metals. Under the action
of a macroscopic electric field, and when all nuclei are ideally “clamped”,
the electrons in a metal undergo free acceleration, while in an insulator they
display a finite polarization. Equivalently, the inverse inertia of the many-
electron system is finite in metals and vanishing in insulators (see Appendix
A).

In a liquid all intensive properties—including $\varepsilon_{\infty}$—are
ensemble averages; the fact that $\varepsilon_{\infty}$ is finite does not
always guarantee that all configurations visited by the system are insulating,
as required by topology. Indeed, a dielectric medium with metallic inclusions
(nanodomains or mesodomains) behaves macroscopically as an insulator. One
could imagine that in some circumstances metallic nanodomains may be present
even in the fluctuating configurations of the insulating liquid, in which case
ionic transport would violate Faraday law.

Faraday’s measurement of oxidation numbers in insulating liquids is possible
owing to mass transport over macroscopic distances; the same may occur in
solid-state ionic conductors. In a common crystalline solid, instead, no mass
transport may occur and oxidation numbers are not measurable in a real
experiment.

Notwithstanding, it is possible to devise a gedanken experiment sharing the
same topological features as Faraday’s experiment. In fact, back in 2012 Jiang
et al.Jiang12  have proposed a rigorous quantum-mechanical definition of
oxidation numbers in crystalline insulators based on the seminal work of Ref.
Pendry84,  and on polarization theory: their topological approach is
conceptually related to the present one. In a crystal, when a sublattice is
displaced by a lattice constant along an insulating path, the trasported
charge is an integer multiple of $e$, thus defining the oxidation number of
the given chemical species in the given solid. The approach is demonstrated on
a few test cases by means of density-functional calculations, where—as
observed above—evaluating the oxidation numbers amounts to counting the number
of transported Kohn-Sham WFs. One test case is very simple: rocksalt LiH. Not
surprisingly the topological approach yields for LiH the same result as the
octet rule (oxidation numbers $-1$ for H and $+1$ for Li). The other two test
cases concern a given transition metal atom which assumes different oxidation
numbers depending on the crystalline environment: Ba either $+3$ or $+5$, Fe
either $+2$ or $+3$. In all cases explicitly dealt with, the topological
theory of Ref. Jiang12,  yields oxidation numbers in agreement with
conventional wisdom and with previous work.

In Sec. VI we have assumed a singlet ground state with doubly occupied Kohn-
Sham WFs; it is then worth noticing that—in order to account for elements
which may display even and odd oxidation numbers—one needs a spin-polarized
ground state where some of the WFs are singly occupied, as assumed in Ref.
Jiang12, . Common quantum-chemistry wisdomSzaboOstlund  holds that spin-
polarized Slater determinants may display the qualitatively correct behavior
(at the price of breaking spin symmetry) in cases where double-occupancy
determinants would be inadequate.

##  IX A pitfall

We have emphasized above the fact that charge transport in a metal occurs
independently from mass transport; one may wonder whether even in some
insulating liquids charge transport (over a macrocopic distance) may occur
without any associated mass transport. A case study in this class has been
investigated long ago: a dilute metal-alkali halide solution, i.e. a
nonstoichiometric molten salt. The simulations, of an hybrid quantum-classical
type, have been performed for a system of 32 K atoms and 30 Cl atoms.Fois88
The octet ruleWalsh18  would assign the oxidation number $-1$ to all Cl atoms,
while 30 of the K atoms would get $+1$, and two of them zero. But such
conventional partitioning is grossly inadequate: snapshots from the simulation
show that a singlet lone pair (dubbed bipolaron) typically sits at a Cl
vacancy; the solvated electron pair moves over macroscopic distances without
any associated mass transport, driven by the wandering vacancy.

A recent paper by Pegolo et al.Pegolo20  deals—among other things—with the
same case study as above, by means of ab-initio molecular dynamics
simulations: the aim is to assess which of the conditions warranting
topological transport—and ultimately Faraday law—are actually violated. After
checking that the liquid is indeed insulating, the authors perform a revealing
computer experiment based on a Thouless pump. A K nucleus is adiabatically
driven along two different paths beginning and ending with the same nuclear
configuration; only one nucleus is driven along the loop (i.e. to the nearest
cell), while the other nuclei remain in the central cell. The measured
topological charge is not the same: $Q=+e$ on one path, and $Q=-e$ on the
other. This means that the two paths cannot be continuously deformed and made
coincident without crossing a metallic region, thus violating one of the
conditions detailed above in Sec. V. It is worth further specifying that—at
the independent-electron level—a configuration is by definition metallic
whenever the spectral gap closes. In the most general case we have seen from
Eq. (15) above that a configuration is metallic whenever $|{\mathfrak{z}}|$
vanishes in the large-$L$ limit (see also Appendix A).

Remarkably, the existence of metallic regions in the configuration space
manifests itself even if the actual trajectories do not visit such regions.
The situation is quite similar to the well known case of conical intersections
in molecules, which affect the nuclear behavior even when the nuclear
trajectories do not actually visit the degeneracy points (or regions): the
effect goes under the name of "molecular Aharonov–Bohm effect".Mead80 ; Mead92

##  X Conclusions

Faraday clearly endorsed the concept of an atom as a “small particle” much in
advance with the common wisdom of his time.Faraday  Nowadays the atomistic
nature of Faraday’s law is obvious; nonetheless it is too simplistic to assume
that an electrolyte is a liquid where some ions are endowed with an integer
charge: a condensed matter system—either solid or liquid—is an assembly of
nuclei and electrons: there is no way of detecting integer-charged ions in a
“snapshot” of the system at a given time, because the electronic density in
the liquid is a continuous function. Only isolated ions in vacuo do have a
static integer charge.

There is another occurrence where integer numbers appear in modern chemistry
as a label for atoms: these are the oxidation numbers (also known as oxidation
states). They are nonmeasurable quantities, determined according to an agreed
set of rules.Walsh18  It is therefore worth stressing that Faraday’s
experiment is the only known instance where oxidation numbers are directly
measured. This may happen because of two outstanding reasons: (i) the charges
measured by Faraday’s experiment are not static, they are dynamical charges in
the adiabatic regime; (ii) charges are transported over macroscopic distances,
a class of phenomena where topology plays a major role.Thouless83 ; Niu84 ;
Xiao10

Over the years, only a very small number of papers was devoted to the
topological interpretation of Faraday’s experiment,Pendry84 ; rap_a29 ;
Grasselli19  as well as to the consequent relationship between Faraday’s
“equivalent numbers” and oxidation numbers as defined in modern chemistry:
this work is generalized here, by adopting a many-body formulation which
additionally prompts for very compact notations.

Besides the time-integrated transported charge, another key observable in
insulating liquids is the value of ionic conductivity at a given temperature,
which obtains from the fluctuation-dissipation theorem.Hansen ; Frenkel  Here
I have adopted the same logic as in Grasselli and BaroniGrasselli19  in a
compact many-body framework: it is shown that when the real current is
replaced with the fictitious current carried by the topological oxidation
numbers (times $e$) along the real nuclear trajectories, topology warrants
that the conductivity value is correctly reproduced. This interesting result
was serendipitously obtained, with no explanation, in a previous ab-initio
simulation on dissociated water at extreme conditions.French11

An outstanding corollary of the Grasselli-Baroni theorem (and of its present
generalization) addresses molecular undissociated liquids. Given that the
topological oxidation number of a molecule as a whole is zero, the
fluctuation-dissipation theoremHansen ; Frenkel  yields a vanishing
conductivity, even in presence of quite sizable fluctuating currents.

## Acknowledgments

I have discussed thoroughly the present findings with F. Grasselli, S. Baroni,
and A. Marrazzo; their invaluable contribution is gratefully acknowledged.
Work supported by the Office of Naval Research (USA) Grant No.
N00014-20-1-2847.

## Data availability

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

##  Appendix A The insulating state of matter

The electronic state vector $|\Psi(t)\rangle$ addressed throughout in the
present work is the instantaneous ground eigenstate of the many-body
Hamiltonian

$\hat{H}(t)=\frac{1}{2m}\sum_{i=1}^{N^{(\rm el)}}p_{i}^{2}+\hat{V}(t),$ (28)

where $\hat{V}(t)$ comprises the nuclear potential and the electron-electron
interaction. In our geometry the many-body wavefunction is bounded in the
$y_{i},z_{i}$ coordinates, and obeys PBCs with period $L$ in the $x_{i}$
coordinates; we neglect the irrelevant $t$-dependence in the following. It is
expedient, following a famous paper of Kohn’s,Kohn64  to generalize Eq. (28)
by including a constant vector potential (i.e. a pure gauge) along $x$:

$\hat{H}_{\kappa}=\frac{1}{2m}\sum_{i=1}^{N^{(\rm el)}}(\,{\bf
p}_{i}+\hbar\kappa{\bf e}_{1}\,)^{2}+\hat{V},$ (29)

where ${\bf e}_{1}$ is the unit vector in the $x$ direction, and $\kappa$ is a
scalar with the dimensions of an inverse length. Let $|\Psi_{\kappa}\rangle$
be the ground PBC eigenstate of $\hat{H_{\kappa}}$ with energy $E_{\kappa}$.
If we define $\hat{x}=\sum_{i}x_{i}$, the quantity ${\rm
e}^{-i\kappa\hat{x}}|\Psi_{0}\rangle$ is a solution of Schrödinger equation
with energy $E_{0}$, but it does not obeys PBCs (for a generic $\kappa$):
hence it is not an eigenstate. The genuine PBCs ground eigenstate
$|\Psi_{\kappa}\rangle$ has a nontrivial $\kappa$ dependence.

The kinetic-energy term in Eq. (29) yields the many-body velocity operator
(along $x$):

$\hat{v}_{\kappa}=\frac{1}{m}\sum_{i=1}^{N^{(\rm
el)}}({p}_{ix}+\hbar\kappa)=\frac{1}{\hbar}\frac{d}{d\kappa}\hat{H}_{\kappa},$
(30)

and the stationary electronic current which crosses a vessel’s section is

$I_{\kappa}^{(\rm el)}=-\frac{e}{\hbar
L}\langle\Psi_{\kappa}|\,\frac{d}{d\kappa}\hat{H}_{\kappa}\,|\Psi_{\kappa}\rangle=-\frac{e}{\hbar
L}\frac{dE_{\kappa}}{d\kappa}.$ (31)

Given that no stationary current may flow in an insulator when the nuclei are
at rest, the ground state energy is $\kappa$-independent. Matters are
different in metals: it has been shown by KohnKohn64  that
$d^{2}E_{\kappa}/d\kappa^{2}$ is proportional to the Drude weight. We remind
that the Drude weight measures the inverse inertia of the many-electron system
in response to a constant electric field; in absence of dissipation, the
electrons in a metal undergo free acceleration.rap157

Whenever $\kappa$ is an integer multiple of $2\pi/L$ the state vector ${\rm
e}^{-i\kappa\hat{x}}|\Psi_{0}\rangle$ obeys PBCs, besides being a solution of
Schrödinger equation with energy $E_{0}$: it is therefore the ground
eigenstate of $\hat{H}_{\kappa}$. We pick one of these values:
$\kappa_{1}=2\pi/L$. According to Kohn’s prescription, the state vector
$|\Psi_{\kappa_{1}}\rangle$ obtains by adiabatically following
$|\Psi_{0}\rangle$ when $\kappa$ is adiabatically turned on. Because of what
has been said above, $|\Psi_{\kappa_{1}}\rangle$ is orthogonal to ${\rm
e}^{-i\kappa_{1}\hat{x}}|\Psi_{0}\rangle$ in metals, while it coincides with
it in insulators (apart for an arbitrary phase factor), ergo:

$\displaystyle|\langle\Psi_{\kappa_{1}}|\,{\rm
e}^{-i\frac{2\pi}{L}\sum_{j=1}^{N^{(\rm el)}}x_{j}}\,|\Psi_{0}\rangle|$
$\displaystyle=$ $\displaystyle 0\qquad\mbox{in metals};$ (32)
$\displaystyle|\langle\Psi_{\kappa_{1}}|\,{\rm
e}^{-i\frac{2\pi}{L}\sum_{j=1}^{N^{(\rm el)}}x_{j}}\,|\Psi_{0}\rangle|$
$\displaystyle=$ $\displaystyle 1\qquad\mbox{in insulators}.$ (33)

By approximating $|\Psi_{\kappa_{1}}\rangle$ with $|\Psi_{0}\rangle$ one gets
$|{\mathfrak{z}}|$ as defined in Eq. (15):

$|\langle\Psi_{\kappa_{1}}|\,{\rm e}^{-i\frac{2\pi}{L}\sum_{j=1}^{N^{(\rm
el)}}x_{j}}\,|\Psi_{0}\rangle|\simeq|\langle\Psi_{0}|\,{\rm
e}^{-i\frac{2\pi}{L}\sum_{j=1}^{N^{(\rm
el)}}x_{j}}\,|\Psi_{0}\rangle|=|{\mathfrak{z}}|.$ (34)

The approximation is to leading order in $1/N^{(\rm el)}$, i.e. to leading
order in $1/L$; in the large system limit $|{\mathfrak{z}}|$ tends to one from
below in all insulators, and to zero in all metals, as stated in the main
text. It is worth noticing that in the special case of noninteracting
electrons (either Hartree-Fock or Kohn-Sham) a selection rule guarantees that
for metals ${\mathfrak{z}}=0$ even at finite size;rap107 ; rap_a33  the large-
size limit is instead needed for interacting electrons.Stella11

##  Appendix B Infrared spectra

This paper is devoted to two static properties (namely, charge transport and
dc ionic conductivity), emphasizing their topological nature. For the sake of
completeness, I discuss here an important $\omega$-dependent property which is
closely related to them, and is instead merely geometrical.

Infrared spectra are expressed in terms of the imaginary part of the
dielectric function $\varepsilon(\omega)$. In order to show the close
relationship to the previous results, I adopt the same cylindrical geometry
and the same notations as above, in which case the known expressionGuillot91
can be recast in the form

$\displaystyle\mbox{Im }\varepsilon(\omega)$ $\displaystyle=$
$\displaystyle\frac{2\pi LA\omega}{k_{\rm B}T}\int_{-\infty}^{\infty}dt\;{\rm
e}^{i\omega t}\langle P_{x}(t)\,P_{x}(0)\rangle$ (35) $\displaystyle=$
$\displaystyle\frac{e^{2}L\omega}{2\pi Ak_{\rm
B}T}\int_{-\infty}^{\infty}dt\;{\rm e}^{i\omega
t}\langle\gamma(t)\gamma(0)\rangle;$

as above, the linear dimensions of the system must be larger than both
correlation lengths and diffusion lengths. The main entry in Eq. (35) is the
power spectrum of the autocorrelation function of $\gamma(t)$ itself, while we
remind that instead the main entry in the expression of ionic conductivity is
the autocorrelation function of the time derivative of $\gamma(t)$, as in Eq.
(22).

The single-point Berry phase $\gamma(t)$, as defined in Eq. (11), is actually
implemented since long time in ab-initio simulations in order to compute the
infrared spectra of liquids and amorphous solids from Eq. (35).Silvestrelli97
; Debernardi97  Equivalently the WF expression, Eq. (20), has been adopted in
the literature in the paradigmatic case of liquid water,rap129 ; rap140  where
hydrogen bonding is at the root of unusual dielectric properties.rap134

##  Appendix C Winding number as a Chern number

As said in Sec. VI, in the independent-electron case (Hartree-Fock or Kohn-
Sham) our macroscopic vessel is regarded as a quasi-one-dimensional “crystal”
of macroscopic constant $L$, with $N$ nuclei and $N^{\rm(el)}$ electrons per
supercell. The insulating ground state can be written as a Slater determinant
of doubly occupied Bloch orbitals $|\psi_{jk}\rangle$, with a gapped spectrum
and $N^{(\rm el)}/2$ filled bands; the quasi-one-dimensional orbitals

$|u_{jk}\rangle={\rm e}^{-ikx}|\psi_{jk}\rangle,\quad k\in[0,2\pi/L),\quad
j=1,\dots N^{(\rm el)}$ (36)

are periodic in $x$ and bounded in $(y,z)$.

It is shown elsewhererap100 ; rap_a20 ; rap162  that the single-point Berry-
phase

$\gamma^{(\rm el)}(t)=\mbox{Im ln }\langle\Psi(t)|\,{\rm
e}^{-i\frac{2\pi}{L}\sum_{j=1}^{N^{(\rm el)}}x_{j}}\,|\Psi(t)\rangle$ (37)

yields—in the crystalline case—the standard Berry phase of polarization
theory:Vanderbilt

$\gamma^{(\rm el)}(t)=2\;i\int_{0}^{2\pi/L}dk\sum_{j=1}^{N^{(\rm
el)}/2}\langle u_{jk}(t)|\frac{\partial}{\partial k}u_{jk}(t)\rangle,$ (38)

where $|\psi_{jk}\rangle$ is chosen within the periodic gauge, ergo

$|u_{j,k+G}\rangle={\rm e}^{-iGx}|u_{jk}\rangle,\qquad G=2\pi/L.$ (39)

It is expedient to define the (dimensionless) differential form

${\cal A}=i\\!\\!\sum_{j=1}^{N^{(\rm el)}/2}\\!\\![\langle
u_{jk}(t)|\frac{\partial}{\partial k}u_{jk}(t)\rangle dk+\langle
u_{jk}(t)|\frac{\partial}{\partial t}u_{jk}(t)\rangle dt],$ (40)

known in mathematical speak as a Chern-Simons 1-form, and its exterior
derivative $\Omega=d\wedge{\cal A}$ (first Chern form)

$\displaystyle\Omega$ $\displaystyle=$ $\displaystyle\Omega_{k}(t)\;dk\,dt,$
$\displaystyle\Omega_{k}(t)$ $\displaystyle=$
$\displaystyle-2\;\mbox{Im}\sum_{j=1}^{N^{(\rm
el)}/2}\langle\frac{\partial}{\partial k}u_{jk}(t)|\frac{\partial}{\partial
t}u_{jk}(t)\rangle,$ (41)

where $\Omega_{k}(t)$ is a Berry curvature in the $(k,t)$ variables.

As shown in the main text, whenever the Hamiltonian is periodic in $t$ (with
period $T$), then

$W^{(\rm el)}=\frac{1}{2\pi}[\gamma^{(\rm el)}(T)-\gamma^{(\rm el)}(0)]$ (42)

is integer (many-electron winding number). With reference to Fig. 1, it is
clear that $2\pi W^{(\rm el)}$ is equal to twice the integral of ${\cal A}$,
Eq. (40), on the top horizontal segment, minus its analogue over the bottom
segment. Furthermore, the two integrals of ${\cal A}$ over the two vertical
segments cancel owing to Eq. (39), and we thus get

$\frac{1}{2}W^{(\rm el)}=-\frac{1}{2\pi}\oint{\cal A},$ (43)

where the (counterclockwise) loop is the contour of the rectangle in the
$(k,t)$ plane. Owing to Stokes theorem then

$\frac{1}{2}W^{(\rm
el)}=-\frac{1}{2\pi}\int_{0}^{2\pi/L}dk\int_{0}^{T}dt\;\Omega_{k}(t)=C_{1},$
(44)

where $C_{1}\in{\mathbb{Z}}$ is the first Chern number; in mathematical speak
the integration domain is a compact orientable manifold (a torus, owing the
the periodicity in both $k$ and $t$).

Figure 1: The many-band Chern number $C_{1}$ is the integral of Eq. (41) over
the rectangular domain in the $(k,t)$ plane, divided by $2\pi$. Its value is
$L$-independent, and in the $L\rightarrow\infty$ limit it coincides with the
winding number $W$ (electronic term thereof) thoroughly addressed in this
work.

The second equality in Eq. (44) is a statement of the celebrated Thouless
theorem;Thouless83 ; Vanderbilt  here $C_{1}$ is the sum of the individual
Chern numbers of the $N^{(\rm el)}/2$ occupied bands. Since $N^{(\rm el)}$ is
proportional to $L$, the $C_{1}$ value is $L$-independent, i.e. it is
independent of the area of the integration domain in Fig. 1. Notice also that
in the above formulas $k$ is a continuous variable, ergo the large-system
limit is implicit even at finite L; instead the single-point Berry phase
acquires its exact physical meaning only in the $L\rightarrow\infty$ limit.
One might therefore say that the electronic term $W^{(\rm el)}$ in the winding
number is nothing else than (twice) the many-band Thouless’ Chern number
$C_{1}$ in the limit where the two vertical segments in Fig. 1 contract to a
single segment.

The present results are worth a further interpretation. We notice that the
large-$L$ limit of Eq. (44) yields

$W^{(\rm el)}\longrightarrow-\frac{2}{L}\int_{0}^{T}dt\;\Omega_{0}(t);$ (45)

comparing to Eq. (42), the (extensive) Berry curvature $\Omega_{0}(t)$ is
related to the single-point Berry phase as

$2\,\Omega_{0}(t)=\frac{L}{2\pi}\dot{\gamma}^{(\rm el)}(t).$ (46)

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The proof of Theorem 25 relies on a new lemma concerning tree-partitions. We
say that a rooted $T$-partition $(B_{\\_}x\colon x\in V(T))$ of a graph $G$ is
_detached_ if for every non-root node $y\in V(T)$ with parent $x\in V(T)$,
each vertex in $B_{\\_}y$ is adjacent to at most one component of
$G[B_{\\_}x]$.

######  Lemma 26.

Every graph $G$ has a detached $T$-partition of width at most
$90(\operatorname{tw}(G)+1)\Delta(G)$, for some tree $T$ with
$\Delta(T)\leqslant 15\Delta(G)$

The proof of Lemma 26 builds on a clever argument due to a referee of a paper
by Ding and Oporowski [7] showing that graphs with bounded treewidth and
bounded maximum degree have tree-partitions of bounded width (see also [40,
9]); see Appendix A for the details.

###### Proof of Theorem 25.

By Lemma 26, $G$ has a detached $T$-partition $(B_{\\_}x:x\in V(T))$ with
width at most $90(\operatorname{tw}(G)+1)\Delta(G)$ for some tree $T$ with
$\Delta(T)\leqslant 15\Delta(G)$ and root $z\in V(G)$. Let
$(V_{\\_}0,V_{\\_}1,\dots)$ be a bfs-layering of $G$ where $V_{\\_}0=\\{z\\}$.
We say a part $B_{\\_}x$ is in level $i$ if $x\in V_{\\_}i$. Colour the edges
of $G$ as follows: each edge with two ends in one part $B_{\\_}x$ is coloured
red, and each edge with one end in a part $B_{\\_}x$ at level $i$ and one end
in a part $B_{\\_}y$ at level $i+1$ is coloured red if $i$ is odd and blue if
$i$ is even. Let $\mathcal{R}$ be the connected partition of $G$ where each
part is the vertex-set of a component of the spanning subgraph of $G$
consisting of the red edges. Observe that the vertex-sets of the components of
$G[B_{\\_}z]$ are in $\mathcal{R}$. Moreover, for every other part
$X\in\mathcal{R}$, there is a node $x\in V(T)$ with children
$y_{\\_}1,\dots,y_{\\_}{\deg_{\\_}T(x)-1}\in V(T)$ such that $X\subseteq
B_{\\_}x\cup B_{\\_}{y_{\\_}1}\cup\dots\cup
B_{\\_}{y_{\\_}{\deg_{\\_}T(x)-1}}$. Since every node in
$V(T)\setminus\\{z\\}$ has at most $15\Delta(G)-1$ children, it follows that
each part in $\mathcal{R}$ has at most
$(15\Delta(G))\cdot(90(\operatorname{tw}(G)+1)\Delta(G))\leqslant
1350(\operatorname{tw}(G)+1)(\Delta(G))^{2}$ vertices.

For the sake of contradiction, suppose $G$ contains an $\mathcal{R}$-clean
path $P$ of length at least $3$. Since $P$ is $\mathcal{R}$-clean, its edges
are blue and so all edges of $P$ are between levels $i$ and $i+1$ for some
even $i$, and all the vertices of $P$ at level $i$ belong to one part
$B_{\\_}x$. Since each edge of $P$ is blue, the vertices of $P$ alternate
between vertices in $B_{\\_}x$ and vertices that belong to parts that are
indexed by the children of $x$. Since $P$ has length at least $3$, $P$ has an
internal vertex $w$ that belongs to $B_{\\_}y$ for some child $y$ of $z$.
Since $P$ is $\mathcal{R}$-clean, $w$ is adjacent to at least two components
of $G[B_{\\_}x]$, contradicting $(B_{\\_}x:x\in V(T))$ being a detached tree-
partition. So every $\mathcal{R}$-clean path in $G$ has length at most $2$, as
required. ∎

##  9 Open Problems

We conclude with some open problems.

######  Open Problem 27.

What is the minimum integer $\ell$ for which there exists a function $f$ such
that every planar graph $G$ has an $\ell$-blocking partition with width at
most $f(\Delta(G))$? We have proved that $\ell\leqslant 222$, although we have
chosen to simplify our proof rather than optimise the constant.

######  Open Problem 28.

Can Theorem 5 be proved with $f$ bounded by a polynomial function of $d,r,s$?
Our proof gives $f(d,r,s)\leqslant(sd)^{O(r!)}$.

Consider the following open problems for $k$-planar graphs.

######  Open Problem 29.

What is the minimum integer $c$ such that there is a function $f$ for which
every $k$-planar graph $G$ is contained in $H\boxtimes P\boxtimes
K_{\\_}{f(k)}$where $\operatorname{tw}(H)\leqslant c$? We know that
$3\leqslant c\leqslant 15\,288\,899$.

######  Open Problem 30.

Is there a constant $c$ and a polynomial function $f$ such that every
$k$-planar graph $G$ is contained in $H\boxtimes P\boxtimes K_{\\_}{f(k)}$
where $\operatorname{tw}(H)\leqslant c$? Our proof gives $f(k)\leqslant
2^{O(\lfloor k/2\rfloor!)}$.

Questions analogous to Open Problems 29 and 30 can be asked for other natural
classes.

Finally, consider what other graph classes have an $\ell$-blocking partitions?

######  Open Problem 31.

Does there exist integers $\ell,c\geqslant 1$ such that every graph with
maximum degree at most $3$ has an $\ell$-blocking partition of width at most
$c$?

######  Open Problem 32.

For every $t\in\mathbb{N}$, does there exist $k_{\\_}t\in\mathbb{N}$ and a
function $f_{\\_}t$ such that every $K_{\\_}t$-minor-free graph $G$ has a
$k_{\\_}t$-blocking partition with width at most $f_{\\_}t(\Delta(G))$?

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##  Appendix A Detached Tree-Partitions

This appendix is devoted to the proof of Lemma 26. Recall that a rooted tree-
partition $(B_{\\_}x\colon x\in V(T))$ of a graph $G$ is _detached_ if for
every non-root node $y\in V(T)$ with parent $x\in V(T)$, each vertex in
$B_{\\_}y$ is adjacent to at most one component of $G[B_{\\_}x]$.

######  Lemma 33.

For any graph $G$, for any non-empty set $S\subseteq V(G)$, there exists a set
$X$ such that:

  * •

$S\subseteq X\subseteq V(G)$;

  * •

$|X|\leqslant 2|S|-1$; and

  * •

each vertex in $G-X$ is adjacent to at most one component of $G[X]$.

###### Proof.

Consider the following algorithm: Initialise $i:=0$ and $S_{\\_}0:=S$. While
there is a vertex $v$ in $G-S_{\\_}i$ adjacent to at least two components of
$G[S_{\\_}i]$, let $S_{\\_}{i+1}:=S_{\\_}i\cup\\{v\\}$ and $i:=i+1$.

Say this algorithm stops at $i=m$. Let $X:=S_{\\_}m$. Then each vertex in
$G-X$ is adjacent to at most one component of $G[X]$. Let $c_{\\_}j$ be the
number of components of $G[S_{\\_}j]$. By construction, $|S_{\\_}j|=|S|+j$ and
$c_{\\_}j\leqslant c_{\\_}0-j\leqslant|S|-j$ for each $j\in\\{0,\dots,m\\}$.
In particular, if $m\geqslant|S|-1$ then $c_{\\_}{|S|-1}=1$. Thus
$m\leqslant|S|-1$ and $|X|\leqslant|S|+m\leqslant 2|S|-1$. ∎

The following lemma is the core of the proof of Lemma 26.

######  Lemma 34.

For $k,d\in\mathbb{N}$, for any graph $G$ with $\operatorname{tw}(G)\leqslant
k-1$ and $\Delta(G)\leqslant d$, for any set $S\subseteq V(G)$ with
$5k\leqslant|S|\leqslant 30kd$, there exists a detached tree-partition
$(B_{\\_}x:x\in V(T))$ of $G$ with root $z\in V(T)$ such that:

  * •

$\Delta(T)\leqslant 15d$;

  * •

$|B_{\\_}x|\leqslant 90kd$ for each $x\in V(T)$;

  * •

$S\subseteq B_{\\_}z$;

  * •

$|B_{\\_}z|\leqslant 3|S|-5k$; and

  * •

$\deg_{\\_}T(z)\leqslant\frac{|S|}{2k}-1$.

###### Proof.

We proceed by induction on $|V(G)$|.

Case 1. $|V(G-S)|\leqslant 90kd$: Let $T$ be the tree with $V(T)=\\{y,z\\}$
and $E(T)=\\{yz\\}$. Note that $\Delta(T)=1\leqslant 15d$ and
$\deg_{\\_}T(z)=1\leqslant\frac{|S|}{2k}-1$. By Lemma 33, there exists a set
$B_{\\_}z\subseteq V(G)$ such that $S\subseteq B_{\\_}z$, $|B_{\\_}z|\leqslant
2|S|-1\leqslant 3|S|-5k\leqslant 90kd$ and every vertex in $V(G)-B_{\\_}z$ is
adjacent to at most two components of $G[B_{\\_}z]$. Set
$B_{\\_}y:=V(G)-B_{\\_}z$. Then $|B_{\\_}y|\leqslant|V(G)-S|\leqslant 90kd$
and every vertex in $B_{\\_}y$ is adjacent to at most one component of
$G[B_{\\_}z]$. Hence $(B_{\\_}x:x\in V(T))$ is the desired detached tree-
partition of $G$.

Now assume that $|V(G-S)|\geqslant 90kd$.

Case 2. $5k\leqslant|S|\leqslant 15k$: By Lemma 33, there exists a set
$B_{\\_}z\subseteq V(G)$ such that $S\subseteq B_{\\_}z$, $|B_{\\_}z|\leqslant
2|S|\leqslant\min\\{3|S|-5k,30k\\}$ and every vertex in $V(G)-B_{\\_}z$ is
adjacent to at most one component of $G[B_{\\_}z]$. Let
$S^{\prime}:=\bigcup\\{N_{\\_}G(v)\setminus B_{\\_}z:v\in B_{\\_}z\\}$. So
$|S^{\prime}|\leqslant d|B_{\\_}z|\leqslant 30kd$. If $|S^{\prime}|<5k$ then
add $5k-|S^{\prime}|$ vertices from $V(G-B_{\\_}z-S^{\prime})$ to
$S^{\prime}$, so that $|S^{\prime}|=5k$. This is well-defined since
$|V(G-B_{\\_}z)|\geqslant 90kd-30k\geqslant 5k$, implying
$|V(G-B_{\\_}z-S^{\prime})|\geqslant 5k-|S^{\prime}|$. By induction, there
exists a detached tree-partition $(B_{\\_}x:x\in V(T^{\prime}))$ of
$G-B_{\\_}z$ with root $z^{\prime}\in V(T^{\prime})$ such that:

  * •

$|B_{\\_}x|\leqslant 90kd$ for each $x\in V(T^{\prime})$;

  * •

$\Delta(T^{\prime})\leqslant 15d$;

  * •

$S^{\prime}\subseteq B_{\\_}{z^{\prime}}$;

  * •

$|B_{\\_}{z^{\prime}}|\leqslant 3|S^{\prime}|-5k\leqslant 90kd$; and

  * •

$\deg_{\\_}{T^{\prime}}(z^{\prime})\leqslant\frac{|S^{\prime}|}{2k}-1\leqslant
15d-1$.

Let $T$ be the rooted tree obtained from $T^{\prime}$ by adding a new root $z$
adjacent to $z^{\prime}$. So $(B_{\\_}x:x\in V(T))$ is a tree-partition of $G$
with width at most
$\max\\{90kd,|B_{\\_}z|\\}\leqslant\max\\{90kd,30k\\}=90kd$. By construction,
$\deg_{\\_}T(z)=1\leqslant\frac{|S|}{2k}-1$ and
$\deg_{\\_}{T}(z^{\prime})=\deg_{\\_}{T^{\prime}}(z^{\prime})+1\leqslant(15d-1)+1=15d$.
Every other vertex in $T$ has the same degree as in $T^{\prime}$. Hence
$\Delta(T)\leqslant 15d$, as desired. Finally, since $(B_{\\_}x:x\in
V(T^{\prime}))$ is detached and every vertex in $V(G)-B_{\\_}z$ is adjacent to
at most one component of $G[B_{\\_}z]$, it follows that $(B_{\\_}x:x\in V(T))$
is also detached.

Case 3. $15k\leqslant|S|\leqslant 30kd$: By the separator lemma of Robertson
and Seymour [37, (2.6)], there are induced subgraphs $G_{\\_}1$ and $G_{\\_}2$
of $G$ with $G_{\\_}1\cup G_{\\_}2=G$ and $|V(G_{\\_}1\cap G_{\\_}2)|\leqslant
k$, where $|S\cap V(G_{\\_}i)|\leqslant\frac{2}{3}|S|$ for each
$i\in\\{1,2\\}$. Let $S_{\\_}i:=(S\cap V(G_{\\_}i))\cup V(G_{\\_}1\cap
G_{\\_}2)$ for each $i\in\\{1,2\\}$.

We now bound $|S_{\\_}i|$. For a lower bound, since $|S\cap
V(G_{\\_}1)|\leqslant\frac{2}{3}|S|$, we have $|S_{\\_}2|\geqslant|S\setminus
V(G_{\\_}1)|\geqslant\frac{1}{3}|S|\geqslant 5k$. By symmetry,
$|S_{\\_}1|\geqslant 5k$. For an upper bound,
$|S_{\\_}i|\leqslant\frac{2}{3}|S|+k\leqslant 20kd+k\leqslant 30kd$. Also note
that $|S_{\\_}1|+|S_{\\_}2|\leqslant|S|+2k$.

We have shown that $5k\leqslant|S_{\\_}i|\leqslant 30kd$ for each
$i\in\\{1,2\\}$. Thus we may apply induction to $G_{\\_}i$ with $S_{\\_}i$ the
specified set. Hence there exists a detached tree-partition
$(B^{i}_{\\_}x:x\in V(T_{\\_}i))$ of $G_{\\_}i$ with root $z_{\\_}i\in
V(T_{\\_}i)$ such that:

  * •

$|B^{i}_{\\_}x|\leqslant 90kd$ for each $x\in V(T_{\\_}i))$;

  * •

$\Delta(T_{\\_}i)\leqslant 15d$;

  * •

$S_{\\_}i\subseteq B_{\\_}{z_{\\_}i}$;

  * •

$|B_{\\_}{z_{\\_}i}|\leqslant 3|S_{\\_}i|-5k$; and

  * •

$\deg_{\\_}{T_{\\_}i}(z_{\\_}i)\leqslant\frac{|S_{\\_}i|}{2k}-1$.

Let $T$ be the rooted tree obtained from the disjoint union of $T_{\\_}1$ and
$T_{\\_}2$ by identifying $z_{\\_}1$ and $z_{\\_}2$ into a new root vertex
$z$. Let $B_{\\_}z:=B^{1}_{\\_}{z_{\\_}1}\cup B^{2}_{\\_}{z_{\\_}2}$. Let
$B_{\\_}x:=B^{i}_{\\_}x$ for each $x\in V(T_{\\_}i)\setminus\\{z_{\\_}i\\}$.
Since $G=G_{\\_}1\cup G_{\\_}2$ and $V(G_{\\_}1\cap G_{\\_}2)\subseteq
B^{1}_{\\_}{z_{\\_}1}\cap B^{2}_{\\_}{z_{\\_}2}\subseteq B_{\\_}z$, we have
that $(B_{\\_}x:x\in V(T))$ is a tree-partition of $G$. By construction,
$S\subseteq B_{\\_}z$ and since $V(G_{\\_}1\cap G_{\\_}2)\subseteq
B^{i}_{\\_}{z_{\\_}i}$ for each $i$,

$\displaystyle|B_{\\_}z|$
$\displaystyle\leqslant|B^{1}_{\\_}{z_{\\_}1}|+|B^{2}_{\\_}{z_{\\_}2}|-|V(G_{\\_}1\cap
G_{\\_}2)|$
$\displaystyle\leqslant(3|S_{\\_}1|-5k)+(3|S_{\\_}2|-5k)-|V(G_{\\_}1\cap
G_{\\_}2)|$ $\displaystyle=3(|S_{\\_}1|+|S_{\\_}2|)-10k-|V(G_{\\_}1\cap
G_{\\_}2)|$ $\displaystyle\leqslant 3(|S|+2|V(G_{\\_}1\cap
G_{\\_}2)|)-10k-|V(G_{\\_}1\cap G_{\\_}2)|$ $\displaystyle\leqslant
3|S|+5|V(G_{\\_}1\cap G_{\\_}2)|-10k$ $\displaystyle\leqslant 3|S|-5k$
$\displaystyle<90kd.$

Every other part has the same size as in the tree-partition of $G_{\\_}1$ or
$G_{\\_}2$. So this tree-partition of $G$ has width at most $90kd$. Note that

$\displaystyle\deg_{\\_}T(z)=\deg_{\\_}{T_{\\_}1}(z_{\\_}1)+\deg_{\\_}{T_{\\_}2}(z_{\\_}2)$
$\displaystyle\leqslant(\tfrac{|S_{\\_}1|}{2k}-1)+(\tfrac{|S_{\\_}2|}{2k}-1)$
$\displaystyle=\tfrac{|S_{\\_}1|+|S_{\\_}2|}{2k}-2$
$\displaystyle\leqslant\tfrac{|S|+2k}{2k}-2$ $\displaystyle=\tfrac{|S|}{2k}-1$
$\displaystyle<15d.$

Every other node of $T$ has the same degree as in $T_{\\_}1$ or $T_{\\_}2$.
Thus $\Delta(T)\leqslant 15d$. So it remains to show that $(B_{\\_}x\colon
x\in V(T))$ is detached. By induction, it follows that for every node $x\in
V(T)\setminus\\{z\\}$ with child $y$, every vertex in $B_{\\_}y$ is adjacent
to at most one component of $G[B_{\\_}x]$. Now suppose that a vertex $v\in
V(G)-B_{\\_}z$ is adjacent to at least two components of $G[B_{\\_}z]$. Let
$u,w\in B_{\\_}z$ be neighbours of $v$ in $G$ that belong to distinct
components of $G[B_{\\_}z]$. Since $(B^{i}_{\\_}x\colon x\in V(T_{\\_}i))$ is
a detached tree-partition of $G_{\\_}i$, it follows that either $u\in
V(G_{\\_}1)\setminus V(G_{\\_}2)$ and $w\in V(G_{\\_}2)\setminus V(G_{\\_}1)$,
or $u\in V(G_{\\_}2)\setminus V(G_{\\_}1)$ and $w\in V(G_{\\_}1)\setminus
V(G_{\\_}2)$. As such, $v\in V(G_{\\_}1)\cap V(G_{\\_}2)$, but this is a
contradiction since $V(G_{\\_}1)\cap V(G_{\\_}2)\subseteq B_{\\_}z$. So
$(B_{\\_}x\colon x\in V(T))$ is detached, which completes the proof. ∎

###### Proof of Lemma 26.

First suppose that $|V(G)|<5(\operatorname{tw}(G)+1)$. Let $T$ be the 1-vertex
tree with $V(T)=\\{x\\}$, and let $B_{\\_}x:=V(G)$. Then $(B_{\\_}x:x\in
V(T))$ is the desired detached tree-partition, since
$|B_{\\_}x|=|V(G)|<5(\operatorname{tw}(G)+1)\leqslant
90(\operatorname{tw}(G)+1)\Delta(G)$ and $\Delta(T)=0\leqslant 15\Delta(G)$.
Now assume that $|V(G)|\geqslant 5(\operatorname{tw}(G)+1)$. The result
follows from Lemma 34, where $S$ is any set of $5(\operatorname{tw}(G)+1)$
vertices in $G$. ∎