# Geometrical observables in condensed matter

** 1 The context **

All of the present review is in the framework of band-structure theory, which is –since the 1930s– the main theoretical tool for addressing both ground-state and excitation properties of a large class of materials: crystalline solids whose electrons are well described at a mean-field level. Band-structure theory amounts to considering noninteracting electrons in a lattice-periodical one-body potential $V (\mathbf{r})$. As for the choice of the appropriate $V (\mathbf{r})$, it has been mostly semiempirical until the late 1970s, and mostly first-principle (either Hartree-Fock or Kohn-Sham) afterwards.

The electronic ground state is uniquely defined in terms of the lowest one-electron orbitals,
occupied according to Pauli’s principle up to the Fermi level; several ground-state observables
have a simple expression in terms of these orbitals. There exist, however, ground-state
observables which for many years eluded solid-state theory: most notably macroscopic electrical
polarization and orbital magnetization. Even the very *definition* of what these observables
are is flawed, not implementable in practical
calculations.

A new paradigm appeared in the early 1990s, when it became clear what polarization really is, and how it can be computed. The modern theory of polarization is based on a geometric phase (Berry phase); the geometrical nature of other ground-state observables has been elucidated over the years. The adjective “geometrical” does not refer to the geometry of the ordinary (coordinate) space; it refers instead to the quantum geometry in the Hilbert space of the state vectors, as sketched in Box 1. In the cases discussed here, the relevant state vectors are the occupied one-body orbitals.

** 2 Geometry in Bloch space **

Owing to the lattice periodicity of the one-body Hamiltonian $\mathcal{H}$, the orbitals assume the Bloch form: $|\psi_{j\mathbf{k}}\rangle = e^{i\mathbf{k}\cdot \mathbf{r}} |u_{j\mathbf{k}} \rangle$, where $\mathbf{k}$ is the Bloch vector in the first Brillouin zone (BZ). The $|u_{j\mathbf{k}} \rangle$ orbitals are eigenstates of $\mathcal{H}_{\mathbf{k}} = e^{- i\mathbf{k}\cdot \mathbf{r}} \mathcal{H} e^{i\mathbf{k}\cdot \mathbf{r}}$ with eigenvalues $\epsilon_{j\mathbf{k}}$. The $|\psi_{j\mathbf{k}}\rangle$ at different $\mathbf{k}$ values are orthogonal, hence their quantum distance is infinite. We will consider instead the geometry of the $|u_{j\mathbf{k}} \rangle$, in the Hilbert space of the lattice-periodical functions. Both the distance and the connection are thus expressed (see Box 1) in terms of the scalar products $\langle u_{j \mathbf{k}}|u_{j' \mathbf{k'}} \rangle$; the corresponding differential forms are

(1) $ D^2_{\mathbf{k},\mathbf{k}+\mathrm{d}\mathbf{k}} = g_{\alpha\beta}(\mathbf{k}) \mathrm{d}k_\alpha \mathrm{d}k_\beta, \qquad \varphi_{\mathbf{k},\mathbf{k}+\mathrm{d}\mathbf{k}} = {\cal A}_{\alpha}(\mathbf{k}) \mathrm{d}k_{\alpha} $

where $g_{\alpha\beta}(\mathbf{k})$ is the quantum metric tensor and ${\cal A}_{\alpha}(\mathbf{k})$ is known as the Berry connection; summation over Cartesian indices is implicit (here and throughout).

For an insulating solid with *n* occupied bands the metric and the connection are a geometrical
property of the occupied Bloch manifold; their explicit expressions are

(2) $ g_{\alpha\beta}(\mathbf{k}) = \mathrm{Re} \sum_{j=1}^n \langle \partial_{k_{\alpha}} u_{j\mathbf{k}} | \partial_{k_{\beta}} u_{j\mathbf{k}}\rangle - \sum_{jj'=1}^n \langle\partial_{k_{\alpha}} u_{j\mathbf{k}} | u_{j'\mathbf{k}}\rangle \langle u_{j'\mathbf{k}} | \partial_{k_{\beta}} u_{j\mathbf{k}} \rangle $

(3) $ \mathcal{A}_\alpha (\mathbf{k}) = i \sum_{j=1}^n \langle u_{j\mathbf{k}} | \partial_{k_{\alpha}} u_{j\mathbf{k}}\rangle $

A third important quantity in differential geometry is the curl of $ \mathcal{A} ( \mathbf{k} ) $, known as the Berry curvature:

(4) $ \Omega_{\alpha\beta} ( \mathbf{k} ) = \partial_{k_{\alpha}} \mathcal{A}_{\beta} ( \mathbf{k} ) - \partial_{k_{\beta}} \mathcal{A}_{\alpha} ( \mathbf{k} ) $

Since we will need the curvature even in metals, it is expedient to write its most general form as

(5) $ \Omega_{\alpha\beta} ( \mathbf{k} ) = i \sum_{\epsilon_{j\mathbf{k}} \leq \mu} ( \langle\partial_{k_{\alpha}} u_{j\mathbf{k}} | \partial_{k_{\beta}} u_{j\mathbf{k}} \rangle - \langle \partial_{k_{\beta}} u_{j\mathbf{k}} | \partial_{k_{\alpha}} u_{j{\bf k}} \rangle ) $

where $\mu$ is the Fermi level. It is worth remarking that all of the above forms are real; the connection is a 1-form and is gauge-dependent, while both the metric and the curvature are 2-forms and are gauge-invariant.

The quantum distance and metric are at the root of the modern theory of the insulating state, which addresses all insulators –particularly those where band-structure theory is inadequate– and is therefore beyond the scope of the present work. In the following we will instead address the connection, the curvature, and other related geometrical forms in band insulators and band metals.

** 3 Modern theory of polarization **

Macroscopic electrical polarization only makes sense for insulators which are charge-neutral in average, and is comprised of an electronic (quantum) term and a nuclear (classical) term. Each of the terms separately depends on the choice of the coordinate origin, while their sum is translationally invariant. The basic tenet of the modern theory is that the electronic term is a Berry phase, expressed as a BZ integral of the Berry connection. Such integral is implemented –in its discretized version– in most electronic structure codes, and is routinely used to compute the polarization of a large class of solids.

For the sake of simplicity, we outline the theory over the simple case of a quasi-one-dimensional
system (a stereoregular polymer), where the polarization *P* is the dipole per unit length and has
the dimensions of a pure charge. The Bloch vector *k* is one-dimensional, and the theory yields the
electronic term in *P* as

(6) $ P_{\rm el} = -e \frac{\gamma_{\rm el}}{2\pi}, \qquad \gamma_{\rm el} = 2 \int_{\rm BZ} \mathrm{d}k \mathcal{A}(k) $

where *e* is the unit charge and the factor of two accounts for double orbital occupancy. We have
already observed that $\mathcal{A}(k)$ is gauge-dependent; its BZ integral, instead, is gauge-invariant but is
multivalued: it is only defined modulo $2\pi$.

Even the nuclear term can be converted to a phase form; if the nuclei of charge $eZ_\ell$ in the unit cell sit at the positions $X_\ell$ along the polymer axis, it is expedient to define

(7) $ \gamma_{\rm nucl} = \mathrm{Im} \ln e^{-i\frac{2\pi}{a}\sum_{\ell} Z_{\ell} X_\ell} $

where *a* is the lattice constant. The total polarization is thus

(8) $ P = -e \frac{\gamma}{2\pi}, \qquad \gamma = \gamma_{\rm el} + \gamma_{\rm nucl} $

In the presence of inversion symmetry $P = –P$, hence $\gamma$ is either zero or $\pi$ (mod $2\pi$): this has clearly a one-to-one mapping to $\mathbb{Z}_2$, the additive group of the integers modulo two. The polarization of a centrosymmetric polymer is in fact topological; arguably, it is the simplest occurrence of a $\mathbb{Z}_2$ topological invariant in condensed-matter physics. Similar arguments lead to the quantization of the soliton charge in polyacetylene, whose topological nature was discovered by Su, Schrieffer, and Heeger back in 1979.

** 4 Multivalued nature of polarization **

Macroscopic polarization $\mathbf{P}$ is phenomenologically defined by addressing bounded samples of an insulating homogenous material. It is the electrical dipole divided by the sample volume, in the large-sample limit:

(9) $ \mathbf{P} = \frac{\mathbf{d}}{V} = \frac{1}{V} \int \mathrm{d}\mathbf{r} \mathbf{r} \rho (\mathbf{r} ) $

The integral in eq. (9) is clearly dominated by boundary contributions, and does not make any sense for an unbounded sample, where $\rho ( \mathbf{r} ) $ is lattice periodical and extends over all space: this is the reason why the polarization problem remained unsolved until the 1990s, and wrong statements continue to appear even afterwards. According to the modern theory, the bulk value of $\mathbf{P}$ is a geometric phase of the crystalline orbitals (plus the nuclear contribution). The density $\rho ( \mathbf{r} ) $ obtains instead from the square modulus of the crystalline orbitals: therein, any phase information is obliterated.

In the simple case of a quasi-one-dimensional system the bulk polarization, eq. (8), has a modulo
*e* ambiguity. This may appear a disturbing mathematical artefact; it is instead a key feature of the
real world. Band-structure theory addresses unbounded samples, and the modulo ambiguity is
fixed only after the sample termination is specified. We are going to show this in detail on the
paradigmatic example of polyacetylene, where the modern theory yields $P = 0$ mod *e*: it is a
topological $\mathbb{Z}_2$-even case.

We consider two differently terminated samples of trans-polyacetylene, as shown in fig. 1: notice
that in both cases the molecule as a whole is not centrosymmetric, although the bulk is. The dipoles
of such molecules have been computed for several lengths from the Hartree-Fock ground state, as
provided by a standard quantum-chemistry code. The dipoles per monomer are plotted in fig. 2:
for small lengths both dipoles are nonzero, as expected, while in the large-chain limit they clearly
converge to a quantized value. Since the lattice constant is $a = 4.67$ a.u., the phenomenological
definition of eq. (9) yields $P = 0$ and $P = e$ for the two cases; we have obviously replaced the
volume *V* with the length, *i.e.* the number of monomers times $a$.

The results in fig. 2 are in perspicuous agreement with the modern theory: in the two bounded
realizations of the same quasi-one-dimensional periodic system the dipole per unit length
assumes –in the large-system limit– two of the values provided by the theory. Insofar as the
system is unbounded the modulo *e* ambiguity in the $P$ value cannot be removed.

** 5 Anomalous Hall conductivity **

Edwin Hall discovered the eponymous effect in 1879; two years later he discovered the anomalous Hall effect in ferromagnetic metals. The latter is, by definition, the Hall effect in the absence of a macroscopic $\mathbf{B}$ field. Nonvanishing transverse conductivity requires breaking of time-reversal symmetry: in the normal Hall effect the symmetry is broken by the applied $\mathbf{B}$ field; in the anomalous one it is spontaneously broken, for instance by the development of ferromagnetic order. The theory of anomalous Hall conductivity in metals has been controversial for many years; since the early 2000s it became clear that, besides extrinsic effects, there is also an intrinsic contribution, which can be expressed as a geometrical property of the occupied Bloch manifold in the pristine crystal.

We indicate with $\sigma^{( - )}_{\alpha \beta} ( 0 ) $ the intrinsic contribution only to dc transverse conductivity: its
expression is proportional to the Fermi-volume integral of the Berry curvature, *i.e.*

(10) $ \sigma^{(-)}_{\alpha\beta}(0) = - \frac{e^2}{\hbar} \int_{\rm BZ} \frac{\mathrm{d}\mathbf{k}}{(2\pi)^d} \Omega_{\alpha\beta}(\mathbf{k}) $

where $\Omega_{\alpha\beta}(\mathbf{k}) $ is given in eq. (5), and *d* is the dimension ($d = 2$ or $3$). This formula, as well as all of
the following ones, is given for single-orbital occupancy.

In *2d* the integral is dimensionless, and in the insulating case the Fermi volume coincides with
the whole BZ: a compact orientable manifold in mathematical speak. The celebrated Gauss-Bonnet-Chern theorem (1945) guarantees that in this case $ \sigma^{(-)}_{xy}( 0 )$ is proportional to a topological
integer, called the first Chern number $C_1 \in \mathbb{Z}$. In detail:

(11) $ C_1 = \frac{1}{2\pi} \int_{\mathrm{BZ}} \mathrm{d}\mathbf{k} \Omega_{xy} ( \mathbf{k} ) , \qquad \sigma^{ ( - ) }_{xy} ( 0 ) = - \frac{e^2}{h} C_1 $

Notice that $h/e^{2}$ is the natural resistance unit: one klitzing, equal to about $2.6 \cdot 10^{4}\Omega$.

The possible existence of *2d* insulators with nonzero (and quantized) transverse conductivity
was pointed out by Haldane in 1988 (Nobel prize 2016); actual synthesis of materials which
realize the QAHE (quantum anomalous Hall effect) was first achieved in 2013.

Finally, we stress the crucially different role of the impurities in metals and in *2d* insulators: in
the former case –given that longitudinal conductivity is nondivergent– there must necessarily
be extrinsic effects, while in the latter case extrinsic effects are ruled out. In insulators the dc
longitudinal conductivity is zero, and, as a basic tenet of topology, impurities and disorder have
no effect on transverse conductivity insofar as the system remains insulating.

** 6 Orbital magnetization **

In analogy to anomalous Hall conductivity, even spontaneous macroscopic magnetization requires breaking of time-reversal symmetry, for instance by the development of ferromagnetic order. Magnetization is comprised of two terms: a spin and an orbital one, which can be experimentally resolved. As for the former term, in the present context it is trivial, simply proportional to the cell-averaged spin density. The latter term accounts (in common materials) for 5-10% of the total, and is a nontrivial geometrical ground-state property. We address here the orbital term only; the moment per unit volume is (in Gaussian units)

(12) $ \mathbf{M} = \frac{\mathbf{m}}{V} = \frac{1}{2cV} \int \mathrm{d}\mathbf{r} \mathbf{r} \times {\bf j} ( \mathbf{r} ) $

in the large-system limit, where $\mathbf{j} ( \mathbf{r} ) $ is the microscopic orbital current. This expression has the same drawbacks as the analogous eq. (9): it does not make any sense in the framework of band-structure theory. We also observe that, while $\mathbf{P}$ is defined for insulators only, $\mathbf{M}$ is a property of both insulators and metals.

Within the modern theory of orbital magnetization, completed in 2006, M is the Fermi volume integral of a geometrical integrand, closely related to the Berry curvature:

(13) $ M_\gamma = - \frac{ie}{2 \hbar c} \varepsilon_{\gamma\alpha\beta} \sum_{\epsilon_{j\mathbf{k}}\leq \mu} \int_{\rm BZ} \frac{\mathrm{d} \mathbf{k}}{(2\pi)^d} \langle\partial_{k_{\alpha}} u_{j\mathbf{k}}| ( {\cal H}_{\mathbf{k}} + \epsilon_{j\mathbf{k}} - 2 \mu )| \partial_{k_{\beta}} u_{j\mathbf{k}} \rangle $

where $\epsilon_{\gamma \alpha \beta}$ is the antisymmetric tensor. The similarity of the two expressions for $\mathbf{M}$ and for the anomalous Hall conductivity, eqs. (5) and (10), is evident. In the latter case the integrand is the Berry curvature, whose only entries are the $\mathbf{k}$-derivatives of the orbitals; in the former case, the Hamiltonian $\mathcal{H}_\mathbf{k} = e^{-i \mathbf{k} \cdot \mathbf{r}} \mathcal{H} e^{i \mathbf{k} \cdot \mathbf{r}}$ and its eigenvalues also enter the formula. Both integrands are gauge-invariant 2-forms.

Given the analogy between their phenomenological expressions, eqs. (9) and (12), one would expect $\mathbf{P}$ and $\mathbf{M}$ to be expressed by similar geometrical observables; surprisingly, this is not the case. $\mathbf{P}$ is a multivalued quantity, obtained as the BZ integral of a gauge-dependent integrand, while $\mathbf{M}$ is single-valued, and obtains as a Fermi volume (BZ in insulators) integral of a gauge-invariant integrand. The profound reasons for this fundamental difference are commented below.

** 7 Magneto-optical sum rule **

An incorrect belief, widespread in the synchrotron-physics community, holds that orbital
magnetization is experimentally accessible via measurements of magnetic circular dichroism.
The belief is based on a popular 1992 paper, which unfortunately is flawed. The
so-called dichroic *f*-sum rule for the integrated spectrum measures indeed a geometrical
ground-state property of the solid, but this property *does not coincide* with the orbital
magnetization $\mathbf{M}$.

The difference in absorption between light with negative and positive helicities is given by twice the imaginary part of the antisymmetric optical conductivity $\sigma_{\alpha\beta}^{ ( - )} ( \omega )$. The dichroic $f$-sum rule reads

(14) $ \mathrm{Im } \int_{0}^{\infty} \mathrm{d}\omega \sigma^{(-)}_{\alpha\beta}(\omega) = - \frac{i\pi e^2}{\hbar^2} \sum_{\epsilon_{j\mathbf{k}}\leq \mu} \int_{\rm BZ} \frac{\mathrm{d} \mathbf{k}}{(2\pi)^3} \langle\partial_{k_{\alpha}} u_{j\mathbf{k}}| ({\cal H}_{\mathbf{k}} - \epsilon_{j\mathbf{k}})| \partial_{k_{\beta}} u_{j\mathbf{k}} \rangle $

comparison to eq. (13) perspicuously shows the difference between the two geometrical ground-state observables.

** 8 Chern forms and Chern-Simons forms **

We have addressed here four geometrical observables of the electronic ground state; it is clear from the above that they belong to two very different classes. $\mathbf{P}$ makes sense in insulators only, is multivalued, and is expressed as the $\mathbf{k}$-space integral of a gauge-dependent integrand. The other three geometrical observables, instead, make sense in both insulators and metals, are single-valued, and are expressed as $\mathbf{k}$-space integrals of gauge-invariant integrands.

The alert reader may also have noticed that the main entry of $\mathbf{P}$ is a 1-form (the Berry
connection), while in all the other cases the main entries are 2-forms (the Berry curvature
and related quantities). It is also worth observing that, at the bare-bone level, polarization is
essentially a 1*d* phenomenon: it can be defined and understood even for 1*d* electrons. The other
observables, instead, require at least dimension-two: in the magnetization case, for instance, one
may address 2*d* electrons in the *xy* plane and $\mathbf{M}$ normal to it.

In order to clearly understand the profound difference between the two classes, in the
following we focus only on the electronic polarization of a *1d* system, eq. (6), and on the
anomalous Hall conductivity of a *2d* insulator, eq. (11). The former is the *1d* BZ integral of the
Berry connection, the latter is the *2d* BZ integral of the Berry curvature. Expressions like these
were well known to mathematicians well before becoming endowed with physical content.

The calculus of differential forms has a long history, related to the names of Poincaré, Cartan,
Pontryagin, Weyl, and culminated with the work of Chern (since the 1940s) and Simons (1970s).
The key tools are the generalized vector product (wedge product) and the generalized curl
(exterior derivative). A basic tenet of differential geometry is that spaces of even and odd
dimensions behave quite differently. In dimension 2*n* one may define the *n*-th Chern form,
whose integral over a compact orientable manifold is quantized: eq. (11) is just a manifestation
of the theorem for $n = 1$. Furthermore, the exterior derivative of a Chern form vanishes; by
Poincaré’s lemma, this guarantees that the Chern form is (locally) the exterior derivative of a
$2n – 1$ form, called a Chern-Simons form. Clearly, in our case the Chern-Simons 1-form is the
Berry connection.

In eqs. (3) and (4) we have defined the connection (Chern-Simons 1-form) and the curvature
(first Chern form) starting from the scalar products $\langle u_{j \mathbf{k}} | u_{j' \mathbf{k'}} \rangle$. Using the same ingredients, the
mathematical literature provides the explicit expression for the Chern-Simons 3-form: like its
1-form analogue, it is gauge-dependent but its BZ integral is gauge-invariant and multivalued.
Therefore in a 3*d* crystalline insulator the Chern-Simons 3-form defines a geometrical
ground-state property, which can be expressed as an angle $\theta_{CS}$: a kind of “higher order” Berry
phase. In presence of some symmetries, $\theta_{CS}$ is equal to either zero or $\pi$, mod $2\pi$, and becomes
therefore a topological $\mathbb{Z}_2$ index, in full analogy with the phase $\gamma$ in centrosymmetric 1d systems.

Last but not least: which physical observable is defined by $\theta_{CS}$? The answer was provided by Qi, Hughes, and Zhang in 2008: $\theta_{CS}$ yields one of the terms –called the “axion” term– in the magnetoelectric response of a crystalline insulator. The bulk magnetoelectric response is multivalued, and the ambiguity is fixed only after the sample termination is specified (in analogy, again, with polarization).

** 9 Local vs. nonlocal observables **

Several bulk observables in condensed matter admit
a local definition as an intensive property, which may
vary over macroscopic lengths: textbook examples are
temperature and pressure. In the quantum-mechanical
realm, macroscopic spin-magnetization density is a local
intensive property, which *e.g.* may assume different values in
different homogeneous regions of a heterostructure. We may
thus address the issue whether even the above geometrical
observables can be defined locally in $\mathbf{r}$ space and thus admit
a “density”.

The case of polarization $\mathbf{P}$ is obvious after the discussion
in sect. 4: “polarization density” is an ill-defined concept.
Considering the two cases in fig. 1, it is clear that by
accessing the central region only of the polymer the $\mathbf{P}$ value
remains undetermined: it crucially depends on the sample
termination. On this count, $\mathbf{M}$ and the other two observables
addressed above are profoundly different: in fact their bulk
value is singlevalued and *not* multivalued. We focus on $\mathbf{M}$
for the time being: when we consider a bounded sample,
tinkering with its boundaries cannot alter the $\mathbf{M}$ value: this
is in striking contrast with eq. (12), which would indicate a
dominant role of boundary currents.

Because of the above, the possibility of defining $\mathbf{M}$ locally, or equivalently of defining an “orbital magnetization density” in $\mathbf{r}$ space, is not ruled out. But all the geometrical observables, as presented so far, have been defined as reciprocal space integrals in the framework of band-structure theory, which by definition deals with macroscopically homogenous solids and lacks spatial resolution. In recent years it has been shown that the single-valued geometrical observables can be alternatively formulated locally in $\mathbf{r}$-space. For the sake of completeness, we quote here that even the quantum metric can be made local.

** 10 Orbital magnetization as a local property **

In order to illustrate the main point, we consider bounded
samples (flakes) of a computational 2*d* material based on
the famous Haldane’s model Hamiltonian: it can be
regarded as a model for hexagonal boron nitride –a.k.a.
gapped graphene– plus “some magnetism” (needed to break
time-reversal symmetry). The system is either insulating or
metallic according to where the Fermi level $\mu$ is set. A typical
“Haldanium” flake is shown in fig. 3; we have addressed flakes of increasing size (at constant
aspect ratio), and we have computed $\mathbf{M}$ via several different formulæ.

The results are shown in fig. 4 for the insulating case. The circles are obtained by means of
eq. (12) (where the flake area *A* replaces *V*): $\mathbf{M}$ perspicuously converges to a finite asymptotic
value, which coincides with the bulk value, shown as the horizontal dashed line. The latter has
been computed for the unbounded sample by means of the (discretized) $\mathbf{k}$-space integral,
eq. (13). We have not proven the locality yet: equation (12) is an $\mathbf{r}$-space expression, but it is
strongly nonlocal and dominated by boundary contributions. We arrived at an
explicit expression for the orbital-magnetization density $\mathfrak{M} (\mathbf{r} )$ (see Box 2). The identity

(15) $ \frac{\mathbf{m}}{A} = \frac{1}{2cA} \int \mathrm{d}\mathbf{r} \mathbf{r} \times {\bf j} ( \mathbf{r} ) = \frac{1}{A} \int \mathrm{d}\mathbf{r} \mathfrak{M} ( \mathbf{r} ) $

holds at any flake size, but the *integrands* therein are quite different. This is similar in spirit to
what happens when integrating a function by parts, thus reshuffling the integrand in different
space regions.

The virtue of $ \mathfrak{M} ( \mathbf{r} ) $ is that it behaves like an honest density, *i.e.* it can be integrated over an
inner region of the sample (and divided by the area of that region). Figure 4 shows precisely
this: diamonds and squares were obtained from integrating $ \mathfrak{M} ( \mathbf{r} ) $ over the central cell (two sites)
and over the “bulk” region (1/4 of the sites), respectively, as displayed in fig. 3. The conventional
formula, eq. (12), converges like the inverse linear dimension of the flake (in both the insulating
and the metallic case) to the asymptotic value in the large-flake limit. Remarkably, our novel
approach converges much faster; in the insulating case shown here the convergence is
exponential.

** 11 Conclusions **

We have addressed some of the known geometrical bulk observables within band-structure theory. All of them are expressed as reciprocal-space integrals of a geometrical integrand, but they can be partitioned in two very different classes. The observables of class (i) only make sense for insulators, and are defined modulo $2\pi$ (in dimensionless units), while the observables of class (ii) are defined for both insulators and metals, and are single-valued.

As for class (i), only two observables are known: electrical
polarization (in dimension 1, 2, or 3), and the “axion” term in
magnetoelectric response (in dimension 3). Mathematics-wise, the former is rooted in a Chern-Simons 1-form, the latter
in the corresponding Chern-Simons 3-form. For both cases
the modulo ambiguity of the bulk observable is fixed only
after the termination of the insulating sample is specified.
Notice that not only the bulk, but also the boundary must be
insulating; the latter condition is always verified in a quasi-1*d* system (like the one discussed here), since its boundary
is zero-dimensional. In the presence of some protecting
symmetry, an observable of class (i) becomes a topological $\mathbb{Z}_2$
index: a $\mathbb{Z}_2$-even crystalline insulator cannot be “continuously
deformed” into a $\mathbb{Z}_2$-odd one without closing the gap.

Several observables in class (ii) are known. Here we have
addressed the time-reversal odd ones, *i.e.* those which are
nonvanishing only in the absence of time-reversal symmetry.

They are: intrinsic anomalous Hall conductivity; orbital magnetization; and a popular magneto-optical sum rule. We have shown that, contrary to a widespread belief, magneto-optical measurements cannot access orbital magnetization.

The geometrical observables of class (ii), when evaluated
for a bounded sample, do not depend on the sample
termination: this key feature owes to the fact that their
$\mathbf{k}$-space expression is single-valued. Therefore these
observables admit a dual representation in $\mathbf{r}$-space which is
local in nature, *i.e.* they admit a density; the case of orbital
magnetization has been discussed in some detail.

** Acknowledgments **

The author acknowledges very useful discussions with R. Bianco, A. Marrazzo, I. Souza, and D. Vanderbilt, over many years. Work supported by the Office of Naval Research (USA) under grant No. N00014-17-1-2803.