Why are insulators insulating and metals conducting?
insulating and metals conducting?
0 Raffaele Resta, University ofTrieste , Italy
A n insulator is distinguished from a metal by its vanishing dc .1'1. conductivity at low temperature. In contrast to what happens in metals, the electronic charge in insulators (and quite generally nonmetals) cannot flow freely under an applied dc field, but instead undergoes static polarization. Within classical physics, this qualitative difference is attributed to the nature of the electronic charge, as sketched in Fig. 1: either "bound" (Lorentz model for insulators) or"free" (Drude model for metals). In other words, electrons are localized in insulators and delocalized in metals. Switching to quantum physics, this dearcut distinction is apparently lost. In most textbooks [1], the insulating/metallic behaviour is explained by means of band structure theory, focussing on the position of the Fermi level of the given material: either in a band gap (insulators), or across a band (metals), as in Fig. 2. . Why do we need a theory of the insulating state differentand formally more complexthan the familiar one sketched in Fig. 2? The point is that such a picture applies only to a crystalline material, within the independentelectron approximation [1]: a very limited dass of insulators indeed. In some materials the insulating behaviour is dominated by disorder (Anderson insulators), in some it is dominated by electron correlation (Mott insulators): therefore, for a large number of insulators, the band picture is grossly inadequate. The present theory of the insulating state [2,3, 4] deals with all kinds of insulators on the same basis: either crystalline or disordered, either independentelectron or correlated.

The insulating/metallic state of matter is characterized by the
excitation spectrum, but the qualitative difference in dc conduc
tivity must also reflect a qualitative difference in the organization
of the electrons in their ground state: a concept first emphasized
byW Kohn in a milestone 1964 paper [
5
]. Its outstanding message
is that even within quantum mechanics the cause for insulating
behaviour is electron localization. Such localization, however,
manifests itself in a very subtle way: in fact the electrons in a con
densed system appear, from several viewpoints, about equally
delocalized in nonmetals and metals. For instance, the Bloch
orbitals in eit.\cr crystalline silicon or crystalline alu.mnum are
similarly delocalized, and do not reveal any sharp difference. The
challenge is to show how electron localization can be detected and
measured in the ground wavefunction of a condensedmanyelec
tron system. The difference between localized and delocalized
must be, in the thermodynamic limit, a sharp one. A solution to
this problem was provided by Kohn in his original 1964 paper. In
1999 the problem was reconsidered and a solution different from
Kohn'sand in many respects simplerwas found [
3
].
There is an outstanding phenomenologicallink between
macroscopic polarization and the insulating state of matter. Sup
pose we expose a finite macroscopic sample to an electric field, say
.. Fig. 1: Schematic view of
insulators and metals in
classical physics. Top sketch:
Lorentz model for insulators,
where each electron is tied
(by an harmonic force) to a
particular center. Bottom
sketch: Drude model for
metals. where electrons roam
freely over macroscopic
distances, hindered only by
atomic scattering potentia Is.
I
i
i
I
i
I
~~ ....._J
Insulator
Metal
I
I
i
i
1
••.•  ••••
Ji. Fig. 2: Traditional textbook view of the qualitative difference
between insulators and metals. The plots show the energy band
structures of crystalline (Le. ordered) materials, chosen one
dimensional for the sake of simplicity. The insulating/metallic
behaviour depends on the position of the Fermi level, which in
turn is determined by the number of electrons per cell. Afilled
band results in insulating behaviour, whi:le an halffilled one
results in metallic behaviour. For many insulating materials (e.g.
disordered and/or correlated) such a band picture is
inappropriate. The present theory ofthe insulating state is based
on quite different concepts, and applies on the same grounds to
any insulator.
inserting it in a capacitor and applying a voltage. Then the
induced macroscopic polarization is qualitatively different in
metals and insulators. In the former materials polarization is triv
ial: universal, materialindependent, due to surface phenomena
only (screening by free carriers). Therefore polarization in met
als is not a bulk phenomenon. The opposite is true for insulators:
macroscopic polarization is a nontrivial, materialdependent,
bulk phenomenon.
On the theoretical side, the concept itself of macroscopic polar
ization in quantum physics has long evaded even a correct
definition: most textbooks contain incorrect statements [I]. The
modern theory of polarization [
2,6
], based on a Berry phase [7],
was developed a decade ago: it revolutionizes both the very defi
nition of the relevant bulk observable, and the ways to compute it
in real solids. This theory has been the subject of a previous arti
cle in this journal [
8
], and starts making its way in elementary
textbooks [
9
]. The recent advances about the insulating state of
matter [
3,4
] are deeply rooted in the modern theory of polariza
tion: in fact polarization and localization can be regarded as two
aspects of the same phenomenon, and stem from the same for
malism.
In order to provide an oversimplified treatment, here I only
deal with a system of N onedimensional electrons, chosen spin
less (or parallelspin) for the sake of simplicity. The manybody
ground wavefunction is then q'(Xl,x2,... Xj, ... XN), and all the elec
trons are confined to a segment of length 1. Eventually, we will be
interested in the thermodynamic limit, defined as the limit N ., co
and L ., co, while the density N/L is kept constant. For practical
purposes, this limit is well approximated when L is much larger
than a typical atomic dimension. A crucial role in our treatment is
played by the boundary conditions chosen for the wavefunction:
we adopt hereas almost mandatory in condensed matter
physics [I]Bornvon Karman periodic boundary conditions,
which amount to imposing that the wavefunction q' is periodic,
with period L, over each electronic variable Xj separately. Equiva
lently, one can imagine the electrons to be confined in a circular
rail of length L: the coordinates Xj are then proportional to the
angles 2 'TtXj IL, defined modulo 2 1t.
Following Refs. [
2,3
], the key quantity needed to deal with both
polarization and localization is the groundstate expectation
value
ZN = (wlUlw) = f dx\ ... foL dXNlw(x\, ... XNWU(Xt> ... XN),
where the unitary operator U, called the "manybody phase
operator" or "twist operator", is defined as
and clearly obeys periodic boundary conditions. The expecta
tion value ZN is a dimensionless complex number, whose modulus
is no larger than one.
The electronic contribution to the macroscopic polarization
of the system can be expressed in the very compact form [
2,4
]:
P,I = 2e7r }~.~ IIII log z""
where e is the electron charge. Notice that, for a onedimensional
system, the polarization has the dimensions of a charge (dipole
per unit length). The essential ingredient in Eq. (3) is Im 10gzN, i.e.
the phase of the complex number ZN. This phase, which is a rather
peculiar kind of Berry phase [
7
], is ill defined whenever ZN van
ishes. And here comes the key message [
3, 4
]: what differentiates
very sharply metals from insulators is the behaviour of the mod
ulus of ZN in the thermodynamic limit: in the former materials it
goes to zero, while in the latter it goes to one. We find therefore, in
agreement with the above phenomenological considerations,
that macroscopic polarization is well defined in insulators and ill
defined in metals.
The modulus of ZN can be used to measure the localization of
the manybody wavefunction, thus providing a quantitative
Insulator Metal
o
o
o
o
o
o 0
o
o
0
0
0
o
o
o 0
o
o
(1)
(2)
(3)
Ji. Fig. 3: Same energy band structure as in fig. 2, for a finite one
dimensional system with periodic boundary conditions. In
drawing the figure, the period Lhas been taken as 14 times the
lattice constant. In the msulating case the band is filled, cmd tile
ground wavefunction '¥ is the antisymmetrized product (Slater
determinant) of 14 Bloch orbitals, whose kvectors and
corresponding energies are indicated by dots. Notice that only
one of the states at the BriIJouinzone boundaries must be
occupied to avoid double counting. In the metallic case the band
is halffilled, and the ground wavefunction is the antisymmetrized
product of the 7 Bloch orbitals whose energy is below the Fermi
level, indicated by dots.
assessment of Kohn's [5) main idea. To this aim, we have intro
duced [3) the intensive quantity
(x2)e =  Nh.tm001V1" (L21)r2 log IZN!2 ,
(4)
having the dimensions of a squared length. It can be proved [4)
that, in insulators, the modulus of ZN differs from one by the
order of 1/N, hence (x?)c is finite, while in metals it is divergent:
this finding clearly vindicates the classical viewpoint of Fig. 1.
I stress that (x?)c is an intensive property characterizing the
localization of the manybody wavefunction qt as a whole, which
applies on the same grounds to ordered/disordered and correlat
ed/uncorrelated manyelectron systems. It is now expedient to
focus on a special case: ordered and uncorrelated, i.e. a crys
talline system of independent electrons, as in the bandstructure
picture of Fig. 2, where a single band is considered. In this case the
ground manybodywavefunction qt(Xl,x2,•.• XN) is a Slater deter
minant (i.e. antisymmetrized product) of N singleparticle
orbitals, which are usually chosen in the Bloch form both for met
als and insulators. But there is an outstanding difference,
illustrated in Fig. 3: only one half of the band is used to build the
Slater determinant for the metal, while the whole band is used for
the insulator. These two Slater determinants are therefore quali
tatively very different, despite the fact that both are built of
(delocalized) Bloch orbitals. Their difference is probed very
sharply by the manybody phase operator U, Eq. (2): in fact it can
be proved [3,4) that in the metallic case the expectation value ZN,
Eq. (1), vanishes, thus leading to a formally infinite value of (x?)c,
while in the insulating case (x2)c assumes a finite value. Actual
values of (x?)c in nonmetallic materials, as computed e.g. for the
IIIV semiconductors, are of the order of a few bohr2• I stress
that, in the present analysis, no use is made of the spectrum of the
system: the metallic/insulating behaviours reflect a different orga
nization of the electrons in the ground state.
~ Fig. 4: The ground wavefunction qt of an insulator can be
equivalently written as an antisymmetrized product of either Bloch
orbitals (Fig. 3, left panel) or Wannier orbitals. Both Bloch and
Wannier orbitals obey periodic boundary co'nditions overthe period
L, which we have taken as 14 times the lattice constant (same as in
Fig. 3). For the sake of clarity, we have plotted (red and green) only 2
of the 14 Wannier orbitals which are needed to build 'f'.ln the
thermodynamic (L 4 00) limit the Wannier orbitals are exponemially
localized and the second cumulant moment oftheir distribution
(x~)  (x» is finite: in fact it is equal to (x~)" Eq. (4).The situation is
completely different in the metallic case (Fig. 3, right panel): one
cannot write 'P as an antisymmetrized product of orbitals whose
second moment is finite in the thermodynamic limit. Notice that
such a sharp qualitative difference reflects a different organization of
the electrons in the ground state, and makes no reference to either
the excitation spectrum or conductivity properties.
The intensive quantity (x?)" measuring electron localization in
the manybodywavefunction, has the meaning of a second cumu
lant moment. Once more, it is expedient to illustrate this for the
special case of a crystalline system of independent electrons, as
in Figs. 2 and 3. Since a determinant is invariant under unitary
transformations, we can perform any unitary transformation on
the N singleparticle occupied orbitals without affecting the
ground Nparticle wavefunction '1', and therefore leaving ZN and
(x2)c invariant. Starting from orbitals of the Bloch form, hence
delocalized throughout the crystal, we may look for a unitary
transformation leading to orbitals which are localized around
some crystalline sites. One such transformation, namely, the Wan
nier transformation, is well known in solidstate physics: this is
illustrated in Fig. 4. According to our theory, (x2)c is the mini
mum possible value for the averaged second cumulant moment
(x?)  (x?) of the electron distribution of the localized orbitals, in
the N 4 00 limit. One outstanding implication is that, for insula
tors, the manybodywavefunction can be written as a Slater
determinant oflocalized singleparticle orbitals, whose distribu~
tions have finite second moments. More precisely, the averaged
second moment of the singleparticle orbitals can be made as
small as (x?)c with a suitable choice of the unitary transforma
tion. Suppose, instead, that we attempt a localizing transformation
on the occupied Bloch orbitals of a metal. Then, since (x2)c
diverges, it is impossible that all of the transformed orbitals have
a finite second moment in the thermodynamic limit.
In conclusion, the present theory of the insulating state sharply
discriminates between an insulator and a metal without actually
looking either at the excitation spectrum or at conductivity
properties. Instead, it is enough to probewith an elegant tool, the
manybody phase operator ofEq. (2)the organization of the elec
trons in the ground state. Once our simple definition of
. localization is adopted, electrons are localized in any insulator and
delocalized in any metal. Localization in the ground electronic
wavefullction is the key reason v.hy insulators sustain bulk dielec
tric polarization. In the present treatment, localization and
polarization appear as two aspects of the same phenomenon, and
are naturally described by the same formalism.
Raffaele Resta is professor of Struttura della Materia, University of
Trieste, and a coordinator in the INFM DEMOCRITOS National
Simulation Centre in Trieste. Previously, he was a professor at the
University of Pisa and at SISSA, Trieste. He has also worked for
many years at EPFL (Swiss Institute of Technology, Lausanne).
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