PHYSICAL REVIEW B 77, 054438 共2008兲
Dichroic f-sum rule and the orbital magnetization of crystals
Ivo Souza1 and David Vanderbilt2
1Department
of Physics, University of California, Berkeley, California 94720, USA
of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854-8019, USA
共Received 14 September 2007; revised manuscript received 2 January 2008; published 28 February 2008兲
2Department
We consider the magnetic circular dichroism spectrum of a crystal with broken time-reversal symmetry in
the electric-dipole approximation. Using the Kubo-Greenwood formula for the absorptive part of the antisymmetric optical conductivity, its frequency integral is recast as a ground-state property. We show that in insulators this quantity is proportional to the circulation of the occupied Wannier orbitals around their centers 共more
precisely, to the gauge-invariant part thereof兲. This differs from the net circulation, or ground-state orbital
magnetization, which has two additional contributions: 共i兲 the remaining Wannier self-rotation, and 共ii兲 the
“itinerant” circulation arising from the center-of-mass motion of the Wannier orbitals, both on the surface and
in the interior of the sample. Contributions 共i兲 and 共ii兲 are not separately meaningful, since their individual
values depend on the particular choice of Wannier functions. Their sum is, however, gauge invariant, and can
be inferred from a combination of two experiments: a measurement of the magneto-optical spectrum over a
sufficiently wide range to evaluate the sum rule, and a gyromagnetic determination of the total orbital
magnetization.
DOI: 10.1103/PhysRevB.77.054438
PACS number共s兲: 75.10.Lp, 78.20.Ls, 73.43.⫺f
I. INTRODUCTION
Optical sum rules provide a link between excitation spectra and ground-state properties. The best-known example is
the f-sum rule of atomic physics.1 It relates the frequencyintegrated absorption of linearly polarized light to the number of valence electrons. In this work we consider the analogous result for circularly polarized light. For nonmagnetic
systems the circular f-sum rule is simply the average of the
f-sum rules for the two linearly polarized components of the
beam, again yielding the total number of electrons. If, however, the system is magnetized, either spontaneously or by an
applied field, this is no longer the case; there is a small
correction that flips sign when either the magnetization of the
sample or the helicity of the incident light is reversed. We are
interested in what information this correction to the circular
f-sum rule provides about the magnetization.
The differential absorption of right and left circularly polarized light by magnetic materials is known as magnetic
circular dichroism 共MCD兲. The object of interest in this work
can thus be viewed as a “dichroic” f-sum rule for the integrated MCD spectrum. Such a sum rule was first derived by
Hasegawa and Howard for the special case of a hydrogen
atom in a magnetic field.2 They showed that it is proportional
to the quantum-mechanical expectation value of the orbital
angular momentum operator, i.e., to the orbital moment. It
has been assumed that this conclusion generalizes trivially to
many-electron systems such as solids.3,4 This is not so,5 as
shown by Oppeneer, who obtained the correct sum rule for
that case.6 He observed that it yields a quantity that is subtly
different from the orbital magnetization Morb, and should
instead be viewed as one of two terms adding up to Morb.
In a separate development, a rigorous theory of orbital
magnetization in crystals was recently formulated.7–10 Interestingly, it also identifies two separate contributions to Morb.
One key result of the present work is to recast the dichroic
f-sum rule in the language of this modern theory, elucidating
1098-0121/2008/77共5兲/054438共13兲
its physical content. Conversely, the sum rule solves an open
problem in the theory of Refs. 7–10 as raised explicitly in
Ref. 9: whether the two gauge-invariant contributions to
Morb identified therein are separately measurable in principle. The present work answers this question in the affirmative.
Although we will mostly focus on crystalline solids, we
find it useful to start in Sec. II by discussing the sum rule in
the more general context of bounded samples under open
boundary conditions. The detailed treatment of periodic crystals is deferred until Sec. III. In both cases, special emphasis
will be placed on insulating systems, for which an intuitive
real-space picture in terms of occupied Wannier orbitals can
be given. We conclude in Sec. IV with a summary and outlook. In Appendices A–C we derive and elaborate on some
results quoted in the main text. In particular, Appendix A
discusses the relation between the dichroic f-sum rule and
three other known sum rules.
II. BOUNDED SAMPLES
A. Preliminaries
In this work we are interested in systems displaying broken time-reversal symmetry in the spatial wave functions. A
typical example would be a ferromagnet such as iron in
which the exchange interaction breaks time-reversal symmetry in the spin channel and this symmetry breaking is then
transmitted to the orbital degrees of freedom by the spinorbit interaction. Other examples include systems in applied
magnetic fields, and also certain spinless model Hamiltonians such as the Haldane model.11
We work in the independent-particle approximation. The
interaction with light will be treated in the electric-dipole
approximation, valid at not-too-high frequencies. This should
be adequate provided that the sum rule saturates before
higher-order contributions, such as electric quadrupole, and
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©2008 The American Physical Society
PHYSICAL REVIEW B 77, 054438 共2008兲
IVO SOUZA AND DAVID VANDERBILT
magnetic dipole terms become significant. The oscillator
strength for the transition between one-electron states n and
m is
共⑀ˆ 兲
f n→m
=
2me
円具m兩⑀ˆ · v̂兩n典円2 .
ប␻mn
共1兲
This expression, valid for a general polarization ⑀ˆ of light,
can be derived in the same way1 as the familiar oscillator
strength formula for linear polarization ⑀ˆ = x̂. For light propagating along ẑ with circular polarization
⑀ˆ ⫾ =
x̂ ⫾ iŷ
冑2
共⫾兲
⬘ + ␴⬘S,yy兴 ⫿ ␴A,xy
⬙ .
␴abs
= 21 关␴S,xx
Thus the difference in absorption between light with negative
and positive helicities is given by twice the imaginary part of
the antisymmetric optical conductivity,12
共−兲
共+兲
⬙ = 21 关␴abs
␴A,xy
− ␴abs
兴.
共⫾兲
f n→m
=
⬘
共2兲
B. Dichroic f-sum rule
With the notation
⬙ ,
+ f ⬘nm,yy兴 ⫿ f nm,xy
共3兲
where we have introduced the matrix
f nm,␣␤ =
2me
具n兩v̂␣兩m典具m兩v̂␤兩n典.
ប␻mn
共4兲
Here ␣ and ␤ label Cartesian directions, ប␻mn = Em − En, and
me is the electron mass. Note that the matrix f = f ⬘ + if ⬙ is
Hermitian in the Cartesian indices. Thus its real and imaginary parts are symmetric and antisymmetric, respectively.
According to Eq. 共3兲, the sum of the oscillator strengths
for the two circular polarizations ⑀ˆ + and ⑀ˆ − equals the sum of
the oscillator strengths for the two linear polarizations x̂ and
ŷ, and is related to f ⬘. The circular dichroism, i.e., the difference between the two circular oscillator strengths, is given
by f ⬙ as follows:
共−兲
共+兲
⬙ = 21 关f n→m
− f n→m
兴.
f nm,xy
具f典 ⬅
␴H = ␴S⬘ + i␴A⬙
共6兲
␴AH = ␴A⬘ + i␴S⬙ ,
共7兲
and
where the Cartesian indices and the frequency have been
omitted. The properties of ␴ can be summarized by noting
that the Hermitian part is dissipative while the antiHermitian part is reactive, and the symmetric part is “ordinary” while the antisymmetric part is “dichroic.” At T = 0 the
dissipative 共or absorptive兲 part is
兺m
f nm␦共␻ − ␻mn兲,
共8兲
where −e is the electron charge. The analog of Eq. 共3兲 in
terms of conductivities is
⬁
f共␻兲d␻ ,
共11兲
the dichroic f-sum rule relates the integrated MCD spectrum
具␴A,
⬙ ␣␤典 to a certain ground-state property of the system. To
see how, we begin by expressing ␴A,
⬙ ␣␤共␻兲 as the imaginary
part of the Kubo-Greenwood formula 共8兲. Combining with
Eq. 共4兲 and taking the integral,
occ empty
␲e2
⬙ ␣␤典 =
具␴A,
兺
បV n
兺m
冉
Im
冊
具n兩v̂␣兩m典具m兩v̂␤兩n典
.
␻mn
共12兲
Using the identity
具n兩v̂␣兩m典
= − i具n兩r̂␣兩m典
␻mn
共13兲
and defining the projector onto the empty states Q̂
empty
兩m典具m兩,
= 兺m
occ
␲e2
⬙ ␣␤典 = −
具␴A,
兺 具n兩r̂␣Q̂v̂␤兩n典 − 共␣ ↔ ␤兲.
2បV n
共14兲
Introducing the pseudovector ␴A,
⬙ ␥ = 共1 / 2兲⑀␣␤␥␴A,
⬙ ␣␤ and P̂
兩n典具n兩,
this
can
be
written
more
concisely
as
= 兺occ
n
⬙典 = −
具␴A
␲e2
Tr关P̂r̂ ⫻ Q̂v̂兴.
2បV
共15兲
Equation 共15兲 is the dichroic f-sum rule, also obtained in
Ref. 6. Using the closure relation Q̂ = 1̂ − P̂, it becomes apparent that the right-hand side depends exclusively on the
occupied states, and is closely related to the total 共macroscopic兲 ground-state orbital magnetization Morb = ␥ Tr关P̂r̂
⫻ v̂兴, where ␥ = −共e / 2cV兲 in esu. Writing
共I兲
Morb = MSR
+ ⌬M,
共16兲
共I兲
= ␥ Tr关P̂r̂ ⫻ Q̂v̂兴
MSR
共17兲
⌬M = ␥ Tr关P̂r̂ ⫻ P̂v̂兴
共18兲
with
occ empty
␲e2
␴ H共 ␻ 兲 =
兺
2meV n
冕
0
共5兲
Consider now a macroscopic system 共e.g., a sample of
volume V cut from a bulk crystal兲 and decompose its optical
conductivity ␴␣␤共␻兲 in three different ways: 共i兲 real and
imaginary parts, ␴⬘ and ␴⬙; 共ii兲 symmetric and antisymmetric
parts, ␴S and ␴A; 共iii兲 Hermitian and anti-Hermitian parts, ␴H
and ␴AH. Then
共10兲
Like other magneto-optical effects, MCD vanishes for timereversal-invariant systems. This can be seen from the Onsager relation ␴␣␤共H , M兲 = ␴␤␣共−H , −M兲, which implies
␴A,␣␤ 共H = M = 0兲 = 0.
共“⫹” corresponds to positive helicity, or left-circular polarization兲,
1
2 关f nm,xx
共9兲
and
共the notation will be explained shortly兲, Eq. 共15兲 becomes
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DICHROIC f-SUM RULE AND THE ORBITAL…
⬙典 =
具␴A
␲ec 共I兲
MSR .
ប
共19兲
共I兲
with units
Hence the sum rule yields an orbital quantity MSR
of magnetization, but differing from the actual orbital magnetization by the remainder ⌬M.
Two of the three quantities in Eq. 共16兲 are independently
measurable. The left-hand side can be determined from gy共I兲
romagnetic experiments,13–15 while MSR
on the right-hand
side is obtainable from magneto-optical experiments via the
sum rule. Thus, their difference ⌬M can also be determined
共I兲
in principle. However, measuring MSR
and ⌬M independently will be of only limited interest unless some physical
meaning can be attached to each of them separately. With
this goal in mind we shall now make contact with the recent
theory of macroscopic orbital magnetization.
that minimizes the quadratic spread of the WFs, producing
so-called maximally localized WFs.16 Naturally, any physical
observable 共e.g., Morb兲 is necessarily invariant under a
change of gauge. This is unfortunately not the case for the
individual terms MSR and MIC in Eqs. 共22兲 and 共23兲, which
turn out to be gauge dependent. This is to be expected since
these quantities do not take the form of traces, unlike those
in the decomposition introduced earlier via Eqs. 共16兲–共18兲.
The two decompositions 共16兲–共18兲 and 共21兲–共23兲 are not
unrelated, however. To see this, we insert the identity 1̂ = Q̂
+ P̂ at the location of the cross product in Eq. 共22兲 to obtain
共I兲
共II兲
+ MSR
,
MSR = MSR
共I兲
is the quantity defined in Eq. 共17兲 共since 具wi兩Q̂
where MSR
= 0兲, and
冉
The results obtained so far are fairly general. To proceed
further we specialize to insulating samples. For the present
purposes “insulating” means that the ground-state wave
function can be written as a Slater determinant of welllocalized orthonormal molecular orbitals 兩wi典, which we will
generically refer to as Wannier functions 共WFs兲 even when
the sample does not have a crystalline interior.16 This definition encompasses a broad range of systems, both macroscopic and microscopic, but it excludes metals and Chern
insulators,17 which are not Wannier representable in the
above sense.
By invariance of the trace, the orbital magnetization can
be expressed in the Wannier representation as
i
occ
i,j⫽i
共25兲
In this way we have segregated the gauge dependence of
共II兲
共I兲
, isolating a gauge-invariant part MSR
MSR to the term MSR
which turns out to be precisely the quantity defined in Eq.
共17兲 and appearing in the sum rule 共19兲. When the gauge共II兲
dependent self-rotation MSR
is combined with the gaugedependent itinerant circulation MIC, it forms the gaugeinvariant quantity ⌬M of Eq. 共18兲. The relation between the
decompositions 共16兲–共18兲 and 共21兲–共23兲 can be summarized
by writing
I
II
Morb = MSR
+ MSR
+ MIC .
M
occ
共26兲
共20兲
There is a remarkable parallelism between the decomposition 共24兲 of the Wannier self-rotation 共22兲 and the
decomposition16
共21兲
⍀ = ⍀I + ⍀̃
i
In Ref. 7 this was decomposed as18
Morb = MSR + MIC ,
冊
occ
共II兲
= ␥ Tr关P̂r̂ ⫻ P̂v̂兴 − 兺 r̄i ⫻ v̄i = ␥ 兺 rij ⫻ v ji .
MSR
C. Relation to the orbital magnetization
Morb = ␥ 兺 具wi兩r̂ ⫻ v̂兩wi典.
共24兲
共27兲
of the Wannier spread
where
occ
occ
MSR = ␥ 兺 具wi兩共r̂ − r̄i兲 ⫻ v̂兩wi典
⍀ = 兺 具wi兩共r̂ − r̄i兲2兩wi典
共22兲
arises from the circulation of the occupied WFs around their
centers r̄i = 具wi兩r̂兩wi典 = rii 共“self-rotation”兲, while
into a gauge-invariant part
⍀I = 兺 Tr关P̂r̂␣Q̂r̂␣兴
occ
MIC = 兺 r̄i ⫻ 具wi兩v̂兩wi典
共28兲
i
i
共23兲
i
is the circulation arising from the motion of the centers of
mass of the WFs.
It is well known that the WFs of a given system are not
uniquely defined; unitary mixing among the WFs is allowed,
giving rise to a “gauge freedom” 共not to be confused with the
freedom to choose the electromagnetic gauge兲. In practice
one deals with this issue by choosing, among the infinitely
many possible gauges, a particular one that has certain desirable properties. A common strategy is to work in the gauge
␣
共29兲
and a gauge-dependent part
occ
⍀̃ =
兺 兩rij兩2 .
共30兲
i,j⫽i
The similarities between Eqs. 共17兲 and 共29兲, and between
Eqs. 共25兲 and 共30兲 are striking. Interestingly, the gaugeinvariant spread ⍀I is related to the ordinary absorption spectrum by a second sum rule, as discussed in Ref. 19 and
Appendix A. In addition, the interpretation of ⍀I as a measure of the quadratic quantum fluctuations, or “quantum
054438-3
PHYSICAL REVIEW B 77, 054438 共2008兲
IVO SOUZA AND DAVID VANDERBILT
spread,” of the many-electron center of mass19 is mirrored by
共I兲
MSR
having the meaning of a center-of-mass circulation, as
discussed in Appendix B.
First-principles calculations show that for maximally localized WFs, ⍀̃ is typically much smaller than ⍀I.16 Indeed,
the minimization of the spread acts precisely to reduce ⍀̃ as
much as possible. In general ⍀̃ cannot be made to vanish
exactly in two or higher dimensions, since the noncommutativity of P̂x̂P̂, P̂ŷP̂, and P̂ẑP̂ implies that the off-diagonal rij
cannot all be zero. In practice, however, they can become
共II兲
quite small. According to Eq. 共25兲, MSR
would also vanish if
all off-diagonal rij were precisely zero. Hence we expect the
self-rotation of maximally localized WFs to be dominated by
the gauge-invariant part as well.20
The fact that ⌬M is composed of self-rotation and
itinerant-circulation parts which are not separately gauge invariant means that angular momentum can be converted back
and forth between MSR and MIC via gauge transformations.
This will be discussed in more detail in Sec. III D; here we
simply note that the two parts are similar in that both originate from the spatial overlap between neighboring WFs. This
共II兲
is evident from the definition of MSR
, and for MIC it follows
from writing v̄i in terms of the “current donated from one
Wannier orbital to its neighbors” as in Ref. 7. ⌬M can therefore be interpreted as an interorbital contribution to Morb,
共I兲
even though it includes part of the self-rotation, while MSR
is
the purely intraorbital portion. 共Similarly, ⍀I and ⍀̃ are the
intraorbital and interorbital parts of the Wannier spread, respectively.兲
tallites兲 in Sec. III B. The thermodynamic limit of each term
is then taken, producing the reciprocal-space expressions of
Eqs. 共46兲 and 共47兲 共the details of the derivation can be found
in Appendix C兲. Interestingly, we find that our gauge共I兲
invariant terms MSR
and ⌬M differ from—but are simply
related to—those of the gauge-invariant decomposition of
Ref. 9. In the particular case of an insulator with a single
valence band, on the other hand, they reduce exactly to the
terms identified in Ref. 8, as will be discussed in Sec. III C.
Because the work of Ref. 8 is based on a semiclassical picture of wave packet dynamics, however, it is not easily generalized to a multiband gauge-invariant framework as is done
here.
共II兲
In Eq. 共26兲 of Sec. II the decomposition ⌬M = MSR
+ MIC for insulating systems was obtained by working in the
Wannier representation. For insulating crystallites MIC can
共surf兲
be divided further into a “surface” part MIC
and an “inte共int兲
rior” part MIC . The interplay between the resulting three
contributions to ⌬M will be the focus of the final two subsections. Single-band insulators are discussed in Sec. III C.
The general case of multiband insulators is considered in
Sec. III D, where the gauge-transformation properties of
those terms are analyzed.
A. Dichroic f-sum rule
The first step is to rewrite the Kubo-Greenwood formula
共8兲 in a form appropriate for periodic crystals, where dipole
transitions connect valence and conduction Bloch states with
the same crystal momentum k. Equation 共4兲 becomes, dropping the index k for conciseness,
f nm,␣␤ = − 共2me␻mn/ប兲具un兩⳵␣um典具um兩⳵␤un典,
III. CRYSTALLINE SOLIDS
In this section we apply the general formalism of Sec. II
to crystalline solids, recasting the relevant quantities in the
form of Brillouin zone integrals. We start in Sec. III A by
rederiving the dichroic f-sum rule for Bloch electrons. In the
remaining subsections we explore the connections between
this bulk reformulation and the theory of orbital magnetization in crystals.7–10
A somewhat unsatisfying aspect of that theory as developed in Ref. 9 is the lack of consistency in the way the
orbital magnetization was decomposed, in the following
sense. One partition 共Morb = MLC + MIC in their notation18兲
was made for bounded samples, after which the thermodynamic limit was taken for each term separately. The resulting
k-space expressions were then combined to form the total
Morb. Finally, working in k space, a different partition
共Morb = M̃LC + M̃IC兲 was identified whose individual terms
were gauge invariant, unlike those of the original decomposition. In the process, however, the intuitive real-space interpretation of the original decomposition was lost, and the
separate meanings of the two terms in the gauge-invariant
decomposition were left unclear.
Here, instead, we shall work from the very beginning with
共I兲
and ⌬M, which afford a
the two gauge-invariant terms MSR
simple real-space interpretation in terms of WFs. They are
first identified for fragments with a crystalline interior 共crys-
共31兲
where ⳵␣ ⬅ ⳵ / ⳵k␣ and we have used the relation21 vnm,␣
= ␻mn具un 兩 ⳵␣um典 for m ⫽ n, with 兩un典 a cell-periodic Bloch
state. Equation 共8兲 now reads
␲e2
␴ H共 ␻ 兲 =
2me
冕
occ empty
dk
兺
共2␲兲3 n
兺m
f nm␦共␻ − ␻mn兲.
共32兲
Consider the frequency integral of ␴H共␻兲,
␲e2
具 ␴ H典 =
2me
冕
occ empty
dk
兺
共2␲兲3 n
兺m
f nm .
共33兲
The dichroic f-sum rule will be obtained from the imaginary
part of this complex quantity, while the real part yields the
ordinary f-sum rule 共see Appendix A兲.
Using Eq. 共31兲 to expand the summation,
occ empty
兺n 兺m
occ empty
f nm,␣␤ = −
2me
兺
ប2 n
=−
2me
兺
ប2 n
兺m
具un兩⳵␣um典共Em − En兲具um兩⳵␤un典
occ empty
=
兺m
具⳵␣un兩um典共En − Em兲具um兩⳵␤un典
2me
共gk,␣␤ − hk,␣␤兲,
ប2
共34兲
where we have introduced a set of notations as follows:
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occ
bk,␣␤ = 兺 具˜⳵␣un兩˜⳵␤un典,
共35兲
n
occ
gk,␣␤ = 兺 具˜⳵␣un兩Ĥ兩˜⳵␤un典,
共36兲
n
and
occ
hk,␣␤ = 兺 En具˜⳵␣un兩˜⳵␤un典.
共37兲
Chern insulators is subtle, but the understanding emerging
from Refs. 8–10 is that the appropriate generalization is obtained by making the replacements Ĥ → Ĥ − ␮ and En → En
− ␮ in Eqs. 共36兲 and 共37兲, where ␮ is the electron chemical
potential. Clearly gk − hk, and with it the sum rule 共39兲, are
insensitive to these substitutions.23
Comparing Eqs. 共40兲 and 共43兲 for extended crystals with
Eqs. 共16兲 and 共19兲 for bounded samples, it appears plausible
that the two partitions 共16兲 and 共40兲 of Morb ought to be
related by
n
The symbol ˜⳵ denotes the covariant derivative,9,22 defined as
empty
兩umk典具umk兩. The imagi兩˜⳵␣unk典 = Q̂k兩⳵␣unk典, where Q̂k = 兺m
nary part of bk,␣␤ is essentially the Berry curvature while its
real part is related to the quantum metric 共Appendix C of
Ref. 16; we discuss the physical content of bk,␣␤ in Appendix
A兲. Quantities gk,␣␤ and hk,␣␤ are similar to bk,␣␤ except that
they carry an extra factor of Hamiltonian or energy. Note that
bk,␣␤ corresponds to the quantity f k,␣␤ in Ref. 9, while gk,␣␤
and hk,␣␤ are the same as in that work.
With these definitions Eq. 共33兲 becomes
具␴H,␣␤典 =
冕
␲e2
ប2
dk
共gk,␣␤ − hk,␣␤兲.
共2␲兲3
␲e2
ប2
冕
dk
共g⬙ − hk⬙ 兲.
共2␲兲3 k
where
M̃LC =
e
បc
M̃IC =
e
បc
冕
冕
or explicitly,
共I兲
=
MSR
e
បc
⌬M =
冕
dk
共g⬙ − hk⬙ 兲,
共2␲兲3 k
共46兲
2e
បc
冕
共47兲
dk
h⬙ .
共2␲兲3 k
B. Magnetization of an insulating crystallite
共40兲
dk
g⬙ ,
共2␲兲3 k
共41兲
dk
h⬙ .
共2␲兲3 k
共42兲
␲ec
共M̃LC − M̃IC兲
ប
共45兲
共39兲
We thus arrive at our main result
⬙典 =
具␴A
⌬M = 2M̃IC ,
The correctness of these identities is demonstrated in Appendix C by taking the thermodynamic limit of results derived in
the next subsection.
This is the dichroic f-sum rule in the Bloch representation.
We can now compare this result with the decomposition
obtained in Ref. 9, where the ground-state orbital magnetization was partitioned into two gauge-invariant terms as
Morb = M̃LC + M̃IC ,
共44兲
共38兲
The imaginary part reads, in vector form,
⬙典 =
具␴A
共I兲
= M̃LC − M̃IC ,
MSR
共43兲
relating the integrated MCD spectrum to the components of
the orbital magnetization. Note that the sum rule is proportional to the difference between the gauge-invariant contributions of Ref. 9. By independently measuring the sum of M̃LC
and M̃IC via gyromagnetic experiments13,14 and the difference via the magneto-optical sum rule, the value of each
individual term can indeed be measured, in principle, resolving an open problem posed in Ref. 9.
Strictly speaking, Eqs. 共40兲–共42兲 as written are valid for
conventional insulators only. The extension to metals and
To gain a better understanding of the bulk expressions
derived in the previous section, we now specialize the results
obtained for bounded samples in Sec. II C to the case that the
sample has a crystalline interior. Working in the Wannier
representation, we are then able to establish connections between the k-space and Wannier viewpoints and associate a
local physical picture with the various terms appearing in the
bulk orbital magnetization.
Following Refs. 7 and 9, we divide our crystallite into
surface and interior regions. This division is largely arbitrary,
and it only needs to satisfy two requirements: 共i兲 the border
between the two regions should be placed sufficiently deep
inside the sample where the local environment is already
crystalline, and 共ii兲 the surface region should occupy a nonextensive volume in the thermodynamic limit. The Wannier
orbitals spanning the ground state are assigned to each region. Those in the interior converge exponentially to the bulk
WFs 兩Rn典 共R is a lattice vector兲, and those on the surface
will be denoted by 兩ws典.
We first divide the orbital magnetization into self-rotation
共SR兲 and itinerant-circulation 共IC兲 contributions according to
Eqs. 共21兲–共23兲. In the thermodynamic limit the SR part,
which only involves the relative coordinate r̂ − r̄i, is dominated by the interior region. Invoking translational invariance,
MSR = ␥c 兺 关具0n兩r̂ ⫻ v̂兩0n典 − r̄n ⫻ v̄n兴,
共48兲
n
where ␥c = −e / 共2cVc兲, with Vc the cell volume, r̄n
= 具0n兩r̂兩0n典, and v̄n = 具0n 兩 v̂ 兩 0n典. Henceforth summations over
054438-5
PHYSICAL REVIEW B 77, 054438 共2008兲
IVO SOUZA AND DAVID VANDERBILT
bandlike indices span the valence-band states.
Next we break down the self-rotation as in Eq. 共24兲, setting P̂ = 兺R兺n兩Rn典具Rn兩 as follows:
共I兲
= ␥c Re trc关P̂r̂ ⫻ Q̂v̂兴,
MSR
冉
共II兲
= ␥c Re trc关P̂r̂ ⫻ P̂v̂兴 − 兺 r̄n ⫻ v̄n
MSR
= ␥c 兺
n
n
共49兲
冊
兺 Re兵具0n兩r̂兩Rm典 ⫻ 具Rm兩v̂兩0n典其.
Rm⫽0n
surf
I
II
int
Morb = MSR
+ MSR
+ MIC
+ MIC
,
共50兲
The symbol trc denotes the trace per unit cell. Note that we
have taken the real part of the traces explicitly; this was not
needed in Eqs. 共17兲 and 共25兲 for bounded samples, where the
traces were automatically real.
Now we turn to the IC term 共23兲 in Eq. 共21兲. Unlike MSR,
in the thermodynamic limit it generally has contributions
from both interior and surface regions:7,9
共int兲
共surf兲
MIC = MIC
+ MIC
.
共51兲
The interior part becomes
共int兲
MIC
= ␥c 兺 r̄n ⫻ v̄n ,
different choice it is possible to convert between “intracell”
共int兲
共surf兲
MIC
and “intercell” MIC
. For this and other reasons to be
detailed in Sec. III D, the interior and surface parts of Morb
are in general not physically well defined, even in crystalline
insulators. Collecting terms, the full orbital magnetization
reads
共52兲
M
which is similar to Eq. 共26兲 except that the IC term has been
separated into interior and surface parts.
This Wannier-based decomposition of the magnetization
of a crystallite follows closely that of Ref. 9. Two differences
are worth noting. First, we have emphasized the distinction
between Wannier self-rotation and itinerant circulation. In
Ref. 9 the emphasis was more on the separation between the
共surf兲
共denoted by MIC in that work兲
surface contribution MIC
and the interior contribution MLC = ␥c trc关P̂r̂ ⫻ v̂兴 containing
the net magnetic dipole density of the WFs in a crystalline
cell. This “local circulation” includes all of the self-rotation
as well as the intracell part of the itinerant circulation. In the
present notation the decomposition of Ref. 9 reads
I
II
int
surf
Morb = MSR
+ MSR
+ MIC
+ MIC
.
n
where it was necessary to use
兺 v̄n = 0
共56兲
共57兲
MLC
共53兲
n
when exploiting the translational invariance. Equation 共53兲
expresses the fact that no macroscopic current, or dynamic
polarization,22 flows through the bulk in a stationary state.
Because of this constraint, the quantity 共52兲 necessarily vanishes for insulators with a single valence band. In multiband
insulators it takes the form of an intracell itinerant circulation: the WF centers in each cell can have a net circulation
while their collective center of mass remains at rest.
Finally, the surface contribution is
共int兲
MIC
= 0,
Note that for one-band insulators
in which case the
interior contribution coincides with the self-rotation, and the
surface part with the itinerant circulation.7 Second, by identifying a gauge-invariant part of the self-rotation, we have
been able to organize the four resulting terms into the two
gauge-invariant groups indicated in Eq. 共56兲.
The present viewpoint appears to be more useful for arriving at a simple physical picture for the sum rule. It has the
additional advantage of being applicable to disordered and
microscopic systems, for which the distinction between interior and surface contributions loses meaning.
Ns
共surf兲
= ␥ 兺 r̄s ⫻ v̄s .
MIC
共54兲
C. One-band insulators
s=1
It was shown in Refs. 7 and 9 that in the thermodynamic
limit this can be recast as
共surf兲
= − ¯␥c Im
MIC
兺 R ⫻ 具m0兩r̂兩nR典具nR兩Ĥ兩m0典,
共55兲
mnR
where ¯␥c = ␥c / ប. This result is remarkable in that it expresses
a circulation in the surface region solely in terms of matrix
elements between the interior WFs, in a way that does not
depend on the precise location of the boundary between the
two regions 共provided that the boundary satisfies the two
criteria mentioned earlier兲. We emphasize that it holds for
crystalline insulators only.
共int兲
is an intracell-like term, in the bulk form
Whereas MIC
共surf兲
共55兲 MIC is seen to have an intercell character, vanishing in
the “Clausius-Mossotti” limit of zero overlap between WFs
belonging to different cells. The assignment of the bulk WFs
to specific cells is, however, not unique, and by making a
We begin our discussion of Morb in insulators with a
single valence band by considering the remainder ⌬M. We
saw in Sec. III B that, of the three terms into which it is
naturally decomposed in the Wannier representation, one of
them vanishes if there is only one WF per cell,
共int兲
= 0.
MIC
共58兲
Remarkably, the two surviving terms become identical,
共II兲
共surf兲
= MIC
=
MSR
⌬M
,
2
共59兲
and thus individually gauge invariant. This follows from Eqs.
共44兲 and 共45兲 in the one-band limit. Indeed, the quantities
M̃LC and M̃IC therein were defined in Ref. 9 in such a way
that for one-band insulators they reduce to the quantities
共surf兲
MLC = MSR and MIC
in Eq. 共57兲. It is then seen that Eqs.
共44兲 and 共45兲 correspond to the first and second equalities in
054438-6
PHYSICAL REVIEW B 77, 054438 共2008兲
DICHROIC f-SUM RULE AND THE ORBITAL…
Eq. 共59兲, respectively. We emphasize that Eqs. 共58兲 and 共59兲
only hold for crystalline WFs which respect the full translational symmetry of the crystal. If, for instance, a larger unit
cell is used 共effectively folding the Brillouin zone and turning the system into a multiband insulator兲, the additional
gauge freedom can be used to construct WFs for which Eqs.
共58兲 and 共59兲 no longer hold.
Consider now the full orbital magnetization. For one-band
insulators the reciprocal-space expressions 共46兲 and 共47兲 reduce to
共I兲
=
MSR
e
Im
2បc
冕
(a)
e
បc
冕
2
4
3
dk
具⳵kuk兩 ⫻ 共Ĥk − Ek兲兩⳵kuk典 共60兲
共2␲兲3
R2
and
⌬M =
1
(b)
dk
Ek Im具⳵kuk兩 ⫻ 兩⳵kuk典.
共2␲兲3
4
Their sum Morb is given by the right-hand side of Eq. 共60兲
with −Ek replaced by +Ek, which is the expression originally
obtained in Refs. 7 and 8. Moreover, the individual contribu共I兲
tions MSR
and ⌬M coincide with those identified in Ref. 8.
Instead, the derivation of Ref. 7 leads to the alternative—but,
for one-band insulators, also gauge invariant—partition into
共I兲
the interior and surface parts MLC = MSR = MSR
+ ⌬M / 2 and
共surf兲
MIC = ⌬M / 2.
共I兲
While the individual terms MSR
and ⌬M agree, for
single-band insulators, with those of Ref. 8, we interpret
them somewhat differently here. In Ref. 8, Eq. 共60兲 had the
meaning of an intrinsic magnetic moment associated with the
self-rotation of the carrier wave packets. According to the
present derivation, that term is only part of the Wannier selfrotation. As for Eq. 共61兲, in the derivation of Ref. 8 it was
seen to arise from a Berry-phase correction to the electronic
density of states, and was subsequently claimed to be associated with a boundary current circulation.24 Instead, according to the present viewpoint only half of it originates in the
共surf兲
itinerant circulation MIC
of the surface WFs, while the
共II兲
other half is ascribed to the remaining self-rotation MSR
of
the WFs in the bulk.
D. Gauge transformations for multiband insulators
共II兲
共int兲
In multiband insulators all three terms MSR
, MIC
, and
can be nonzero. However, their individual values are
not physically meaningful, since a gauge transformation can
redistribute the total ⌬M among them. In particular, it is
interesting to consider gauge transformations that shift the
location of a WF by a lattice vector.
A general gauge transformation takes the form16
共surf兲
MIC
兩unk典 → 兺 兩umk典Umnk ,
1
2
共61兲
共62兲
m
where Uk is an Nb ⫻ Nb unitary matrix in the band indices.
We assume that a transformation of this kind has already
been applied to transform from the Hamiltonian eigenstates
at each k to a set of states that are smooth in k from which
the WFs are to be constructed. We can then interpose an
additional diagonal gauge transformation
3
FIG. 1. Schematic model of a molecular crystal with one molecule per cell and four WFs per molecule. The range of the orbitals
is indicated by the overlapping circles, and their center-of-mass
velocities v̄n are denoted by arrows. The two panels show in bold
two possible choices of “Wannier basis.”
兩unk典 → e−ik·Rn兩unk典,
共63兲
where Rn is a real-space lattice vector; this has the effect of
shifting the location of WF n by Rn. For a one-band insulator, or if Rn is the same for all bands, this amounts to shifting
the choice of the “home” unit cell. However, in the multiband case different WFs can be shifted differently, corresponding to the freedom in choosing which WFs “belong” to
the home unit cell.
For example, Fig. 1 shows four cells of a model twodimensional crystal consisting of “molecular magnets” disposed on a square lattice with lattice constant a. Before the
transformation 共63兲, the home unit cell contains the four WFs
shown in bold in panel 共a兲. Applying the transformation with
R2 = ax̂ and Rm = 0 for all other WFs changes the selection of
the “basis” of WFs belonging to the home cell to be that
shown in panel 共b兲.
How does this affect the individual terms composing
⌬M? Clearly the self-rotation 共48兲 is not affected. According
共int兲
to Eq. 共52兲, MIC
changes by ␥cR2 ⫻ v̄2. To preserve the
共surf兲
overall invariance of ⌬M the remaining term MIC
must
change by an equal and opposite amount. Let us see in more
detail how this comes about.
We begin with a formal derivation. The k-space expres共surf兲
sion for MIC
is given by9
054438-7
PHYSICAL REVIEW B 77, 054438 共2008兲
IVO SOUZA AND DAVID VANDERBILT
FIG. 2. A finite sample cut from the bulk crystal of Fig. 1. With
the choice of Wannier basis of Fig. 1共a兲, interior and surface WFs
lie inside and outside the dashed line, respectively; with that of Fig.
1共b兲, they are denoted by solid and open circles, respectively. Right
panel: open and solid arrows show the extra “itinerant currents” 关
共surf兲
共int兲
MIC
and MIC
, respectively兴 associated with the latter choice.
共surf兲
=
MIC
e
Im 兺
2បc
mn
冕
dk
Emnk具⳵kunk兩 ⫻ 兩⳵kumk典.
共2␲兲3
共64兲
A few steps of algebra show that under the transformation
共surf兲
changes by
共63兲 MIC
e
Rn ⫻ 兺
បc
m
冕
dk
Re兵具un兩⳵kum典具um兩Ĥk兩un典其.
共2␲兲3
共65兲
Replacing 具un 兩 ⳵kum典 by −具⳵kun 兩 um典 allows us to identify a
term P̂kĤk = Ĥk in the above expression, which becomes
−
e
Rn ⫻
បc
冕
dk
Re兵具un兩Ĥk兩⳵kun典其.
共2␲兲3
共66兲
Comparing with the Wannier velocity22
v̄n = −
2Vc
ប
冕
dk
Re兵具un兩Ĥk兩⳵kun典其
共2␲兲3
共67兲
and setting n = 2 then produce the desired result −␥cR2 ⫻ v̄2
共surf兲
for the change in MIC
.
Coming back to the example in Fig. 1, the intramolecular
orbital overlap gives rise to the nonzero velocities v̄n indicated by the arrows. With the choice of Wannier basis of
panel 共a兲, the collective circulation of the Wannier centers in
共int兲
共surf兲
each cell results in a finite MIC
, while from Eq. 共55兲 MIC
vanishes, since there is negligible intercell overlap. When the
共surf兲
configuration of panel 共b兲 is chosen, MIC
becomes −␥cR2
⫻ v̄2. From the present viewpoint this nonzero value is made
possible by the intramolecular 共but now intercell兲 overlap
between the second WF of each cell with WFs one, three,
and four from the cell shifted by R2.
共surf兲
To view MIC
as a surface contribution rather than a bulk
intercell term, we consider now a finite sample of the same
crystal 共Fig. 2兲, which has been divided into surface and
interior regions. In deciding which WFs are “interiorlike”
and which are “surfacelike” we shall require that all WFs
assigned to the same cell must belong to the same region. If
the Wannier basis of Fig. 1共a兲 is used, the surface region can
be chosen to comprise the outermost layer of molecules, so
that the border between the two regions is given by the
dashed line. The four WFs on each molecule form a unit with
some internal IC but zero center-of-mass velocity. The total
sample magnetization is the sum of all such internal circulations, which in the large-sample limit is interior dominated,
共surf兲
so that MIC
→ 0.
If the Wannier basis of Fig. 1共b兲 is chosen instead, the
upper and lower surface regions are still composed of the
outermost layer of molecules. However, the left surface now
contains, in addition, one WF from each molecule in the
second layer. Those lone surface WFs carry a downward particle “IC current” which extends along the left surface and is
indicated by an open arrow on the right panel. A corresponding IC current appears on the right surface, and together they
共surf兲
= −共e / 2ca兲兩v̄2兩ẑ, which agrees with the result
yield MIC
共surf兲
MIC = −␥cR2 ⫻ v̄2 found earlier using a purely bulk argument 关in this example ␥c = −e / 共2ca2兲兴. A change of gauge
should not change any physical quantity, such as the actual
current flowing on the left surface. Since it appears to change
by adding the open arrow, there must be another equal and
opposite contribution 共the adjacent solid arrow兲. This contribution is the interior IC current carried by the remaining
three WFs 共filled circles兲 on the molecules of the second
layer.
The situation just described is reminiscent of the “quantum of polarization” in the theory of dielectric polarization,25
where a change of Wannier basis like that leading from Fig.
1共a兲 to Fig. 1共b兲 shifts the polarization by a quantum and also
changes the surface charge by one electron per surface cell
area. This might suggest that the full gauge invariance of the
interior and surface parts of Morb discussed in Sec. III C for
single-band insulators becomes, in multiband insulators, a
gauge-invariance modulo ␥cRn ⫻ v̄n. While true for this particular example, this is generally not so.26 Even for this
model it will cease to be true as soon as the molecules start
overlapping significantly. When this happens, the value of
共surf兲
MIC
can be tuned continuously using other types of gauge
transformations, e.g., the continuous diagonal transformation
兩unk典 → ei␪nk兩unk典
共68兲
共surf兲
given by
with ␪n,k+G = ␪nk. This produces a change in MIC
Eq. 共66兲 with Rn therein replaced by a factor of −⳵k␪nk in the
integrand. Since both r̄n and v̄n remain invariant 关the former
was shown in Ref. 21 and the latter follows from Eq. 共53兲
together with the fact that all other v̄m are unaffected兴, so
共int兲
共surf兲
does MIC
. The change in MIC
must therefore be absorbed
共II兲
by MSR .
To summarize, the transformation 共63兲 transfers discrete
amounts of itinerant circulation between the interior and surface regions, while the transformation 共68兲 converts continuously between interior self-rotation and surface itinerant circulation. Finally, under the most general transformation 共62兲
all three gauge-dependent terms in Eq. 共56兲 can be affected
simultaneously, so that only their sum ⌬M is unique and
physically meaningful.
IV. SUMMARY AND OUTLOOK
We have presented an exact sum rule for the MCD spectrum, elucidated its physical interpretation, and related it to
054438-8
PHYSICAL REVIEW B 77, 054438 共2008兲
DICHROIC f-SUM RULE AND THE ORBITAL…
the recent rigorous formulation of orbital magnetization in
crystals. In insulating systems the sum rule probes the gauge共I兲
invariant part MSR
of the self-rotation of the occupied Wannier orbitals. The total orbital magnetization has a second,
less obvious contribution ⌬M, arising from the overlap between neighboring WFs. It comprises both self-rotation 共SR兲
and itinerant-circulation 共IC兲 parts in proportions which depend on the precise choice of WFs, while ⌬M itself has a
unique value. Although the intuitive interpretation in terms
of the occupied WFs is restricted to Wannier-representable
共I兲
and
systems such as conventional insulators, the terms MSR
⌬M are, in fact, well defined for all electron systems, including metals and Chern insulators.
The practical importance of the sum rule is that it allows
us to break down Morb into physically meaningful parts, using a combination of gyromagnetic and magneto-optical
measurements. This should provide valuable information on
the intraorbital 共or localized兲 versus interorbital 共or itinerant兲
character of orbital magnetism. For example, it has been suggested 共Ref. 27, Appendix B兲 that the anomalously large
g-factors of Bi might be caused by itinerant circulations very
much like the ones discussed here. On the basis of the
present work it should now be possible to test this conjecture.
In the past decade and a half a sum rule for the x-ray
MCD 共XMCD兲 spectrum4 has been used extensively to obtain site-specific information about orbital magnetism in solids. The resulting orbital moments have been compared with
gyromagnetic measurements of Morb.28 If a significant itinerant contribution ⌬M is present, one may expect a discrepancy between the XMCD orbital moments and the total Morb
inferred from gyromagnetics. It would therefore be of great
interest to find such systems defying the conventional wisdom about the connection between the MCD spectrum and
orbital magnetization.
The ideas discussed in this work should be most relevant
for materials displaying appreciable orbital magnetism and,
in particular, appreciable interorbital effects which might en共I兲
兩. These criteria do not favor band
hance the ratio 兩⌬M / MSR
ferromagnets. First, their orbital magnetization tends to be
relatively small. In the transition metal ferromagnets Fe, Co,
and Ni, for example, it accounts for less than 10% of the
spontaneous magnetization.13,14 共For comparison, the fieldinduced orbital magnetization of the d paramagnetic metals
can be as large as the induced spin magnetization. This has
been established both from gyromagnetic experiments15 and
from first-principles calculations.29兲 Secondly, ferromagnetism is favored by narrow bands and localized orbitals, for
which interorbital effects are expected to be relatively minor.
Finally, the spin-orbit-induced Morb of ferromagnets is believed to be an essentially atomic phenomenom largely confined to a small core region close to the nucleus,30,31 and one
might therefore expect ⌬M to be small. Paramagnets and
diamagnets, with relatively wide bands 共e.g., the s-p metals
and semiconductors兲 and additional contributions to Morb unrelated to spin-orbit, therefore appear to be more promising
candidates. Among ferromagnets, the “zero magnetization
ferromagnets,”32 whose orbital magnetization is so large as
to cancel the spin magnetization, might be particularly interesting.
An important direction for future work is to carry out
共I兲
first-principles calculations of MSR
and ⌬M for real materials. Such calculations would test the validity of the assumption that orbital magnetism in solids is atomiclike in
nature.30,31 While plausible, that assumption was made in the
past partly out of practical necessity, since a rigorous bulk
definition of Morb in terms of the extended Bloch states was
not available. Confronting experiment with a full calculation
of Morb within spin-density functional theory 共SDFT兲, including the itinerant terms, would clarify whether SDFT can
adequately describe orbital magnetism in solids, or whether
an extended framework 共e.g., LSD+ U including “orbital polarization” terms30,31 or current- and spin-density functional
theory10兲 is needed.
In Appendix A we place the dichroic f-sum rule in the
broader context of other known sum rules. We note, in particular, that by taking different inverse-frequency moments,
the interband MCD spectrum can be related to two other
phenomena resulting from broken time-reversal symmetry,
namely, orbital magnetism and intrinsic anomalous Hall effect. These are generally expected to coexist, and this is indeed the case for ferromagnets, where all three occur spontaneously. In the case of Pauli paramagnets, however, the
intrinsic Hall mechanism of Karplus and Luttinger has received little if any attention. On the other hand, it is known
that Pauli paramagnets can display a field-induced MCD
spectrum.33,34 This raises the question as to what role the
Berry curvature may play in their “ordinary” 共field-induced兲
Hall effect. Such a “dissipationless” contribution is undoubtedly present in principle by virtue of the sum rule 共A7兲.
First-principles calculations of this effect will be presented in
a future Communication.35
To conclude, we have described the orbital magnetization
共I兲
of crystals in terms of localized 共MSR
兲 and itinerant 共⌬M兲
parts, and shown how to relate them to magneto-optical and
gyromagnetic observables. This should allow one to probe
more deeply into the nature of magnetism in condensedmatter systems than previously possible.
ACKNOWLEDGMENT
This work was supported by NSF Grant No. DMR0549198.
APPENDIX A: OTHER SUM RULES
In this appendix we derive three additional sum rules for
Bloch electrons and discuss their relation to the dichroic
f-sum rule. All four involve inverse-frequency moments
具␻−p␴H典 关in the notation of Eq. 共11兲兴 of the absorption spectrum 共6兲. They are given by p = 0 and p = 1, and in each case
two sum rules are obtained by taking the real and imaginary
parts: one ordinary and the other dichroic, respectively.
We first consider p = 0. From the imaginary part of Eq.
共38兲 we obtained the dichroic f-sum rule 共43兲. To discuss the
real part we revert from 共38兲 to the form 共33兲,
⬘ ␣␤典 =
具␴S,
␲e2
2me
Since f nm,␣␤ = −关f mn,␣␤兴*,
054438-9
冕
occ empty
dk
兺
共2␲兲3 n
兺m
⬘ ␣␤ .
f nm,
共A1兲
PHYSICAL REVIEW B 77, 054438 共2008兲
IVO SOUZA AND DAVID VANDERBILT
occ empty
兺n 兺m
occ
⬘ ␣␤ = 兺
f nm,
n
occ
⬘ ␣␤ = 兺
兺 f nm,
m⫽n
n
冋 冉 冊 册
␦␣␤ −
me
me*
,
n,␣␤
共A2兲
where the second equality is the effective-mass theorem.
Hence we find
⬘ ␣␤典 =
具␴S,
␲e2
2me
冕
occ
冋 冉 冊 册
me
dk
3 兺 ␦␣␤ −
共2␲兲 n
me*
n,␣␤
,
共A3兲
the modified f-sum rule36 for the ordinary spectrum.
To obtain the two sum rules for p = 1 we again start from
Eq. 共32兲, but now replace Eq. 共34兲 by
occ empty
兺n 兺m
f nm,␣␤ 2me
=
bk,␣␤ ,
␻mn
ប
共A4兲
where bk,␣␤ was defined in Eq. 共35兲. Thus
具␻−1␴H典 =
冕
␲e2
ប
dk
bk .
共2␲兲3
共A5兲
For the dichroic part, noting that ⍀k = −2bk⬙ is the Berry
curvature summed over the occupied states at k, and comparing with the “intrinsic” Karplus-Luttinger Hall
conductivity37
e2
ប
␴A⬘ 共␻ = 0兲 = −
冕
dk
⍀k ,
共2␲兲3
共A6兲
one finds the Hall sum rule,
⬙典 =
具␻−1␴A
␲
␴⬘ 共␻ = 0兲.
2 A
V
N
冕
dk
b⬘ ,
共2␲兲3 k,␣␤
共A8兲
where N / V is the electron density. Hence we recover the
electron localization sum rule19
⬘ ␣␤典 =
具␻−1␴S,
␲ e 2N
⌳␣␤ .
បV
⬙ ␣␤典 =
具␻−1␴A,
共A7兲
This is the ␻ → 0 limit of the Kramers-Kronig relation for the
antisymmetric conductivity.12 Since only the interband part
of the optical conductivity was included on the left-hand
side, the intrinsic dc Hall conductivity was obtained on the
right-hand side. Extrinsic contributions to the latter 共e.g.,
skew scattering兲 presumably arise from intraband terms in
the former.
Finally, consider the ordinary 共real兲 part of Eq. 共A5兲. The
quantity bk,
⬘ ␣␤ is the quantum metric.16 It is related to the
localization tensor ⌳␣␤ of insulators by38
⌳␣␤ =
Besides emerging from a unified formalism, the four sum
rules display certain similarities. For instance, it will be
shown in AppendixB that the dichroic f-sum rule yields the
expectation value of the many-electron center-of-mass circulation operator, while the trace of the localization tensor
yields the spread of the center-of-mass quantum distribution.
Moreover, in a one-particle picture each quantity can be
viewed as the gauge-invariant part of the corresponding
property 共self-rotation or spread兲 of the Wannier orbitals, as
discussed in Sec. II C for bounded systems. There is, however, one important difference between the behavior of the
two quantities in the thermodynamic limit. While the centerof-mass circulation remains well defined for metals, the trace
of the localization tensor is only meaningful for insulators,
diverging in metals.19,38 Interestingly, the delocalization of
electrons in metals is also responsible for a correction to the
f-sum rule. Contrary to the canonical f-sum rule for atoms,1
the modified f-sum rule 共A3兲 does not yield the number density of valence electrons in a metal, due to the presence of
the last term on the right-hand side. This term appears because the Bloch states are extended and do not vanish at
infinity.36 The fact that the correction term nevertheless vanishes for insulators is a consequence of the localized nature
of insulating many-body wave functions in configuration
space.39
We conclude by noting that Eq. 共A7兲 provides an extreme
example of how sum rules from atomic physics can change
qualitatively when applied to extended systems. Indeed, the
corresponding sum rule for bound systems produces a vanishing result,40
共A9兲
In summary, we have in Eqs. 共38兲 and 共A5兲 two general
sum rules for the zeroth and first inverse frequency moments
of the optical absorption, respectively. Taking imaginary and
real parts of Eq. 共38兲 gives the dichroic f-sum rule 共43兲 and
the modified ordinary f-sum rule 共A3兲, while taking imaginary and real parts of Eq. 共A5兲 gives the Hall sum rule 共A7兲
and the electron localization sum rule 共A9兲.
␲e2
Im Tr关P̂r̂␣Q̂r̂␤兴 = 0.
បV
共A10兲
In contrast, the bulk formula 共A7兲 produces for Chern insulators a quantized Hall conductivity, and it also describes the
intrinsic anomalous Hall conductivity of ferromagnetic
metals.37 This apparent contradiction highlights the subtleties
associated with the process of taking the thermodynamic
limit and switching from open to periodic boundary conditions for non-Wannier-representable systems. Such issues are
still not fully resolved in the theory of orbital magnetization.
While a general derivation of the bulk formula for Morb has
been given working from the outset with a periodic crystal,10
derivations which start from finite crystallites and take them
to the thermodynamic limit 共Refs. 7 and 9 and Appendix C兲
are presently restricted to conventional insulators.
APPENDIX B: DICHROIC f-SUM RULE AND THE MANYBODY WAVE FUNCTION
In the main text we interpreted the dichroic f-sum rule,
and the associated decomposition 共16兲 of Morb, in an
independent-particle picture based on WFs. It is also possible
to relate these quantities directly to properties of the manyelectron wave function, without invoking any particular
single-particle representation. In preparation for that, let us
first discuss a one-electron system 共e.g., a hydrogen atom in
a magnetic field兲. Its absorption spectrum is composed of
sharp lines, and is more conveniently described in terms of
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PHYSICAL REVIEW B 77, 054438 共2008兲
DICHROIC f-SUM RULE AND THE ORBITAL…
共I兲
MSR
= ␥ 兺 rij ⫻ v ji共具⌿0兩c jc†j 兲共c†i ci兩⌿0典兲
an oscillator strength rather than an optical conductivity. Taking the imaginary part of Eq. 共4兲 and using the relation 共13兲
to replace one of the velocity matrix elements,
m
⬙ ␣␤ = − e 关具n兩r̂␣兩m典具m兩v̂␤兩n典 − 共␣ ↔ ␤兲兴.
f nm,
ប
me
具n兩r̂ ⫻ v̂兩n典.
ប
共B2兲
共B3兲
so that Eq. 共19兲 continues to hold. From this many-body
perspective the difference ⌬M with respect to the full Morb is
N
ˆ 共2兲 − ⌳
ˆ 共1兲.
r̂i ⫻ v̂ j in ⌳
seen to arise from the cross terms 兺i,j⫽i
To recover from Eq. 共B3兲 the independent-particle expression 共17兲 we specialize to the case where 兩⌿m典 is a single
Slater determinant. In second-quantized notation r̂
= 兺ijrijc†i c j, v̂ = 兺ijvijc†i c j, and 兩⌿0典 = c†1 . . . cN† 兩0典, where i and j
label orthogonal one-particle states. Then Eq. 共B3兲 becomes
ij
冊 冉兺
rijc†i c j ⫻
kl
兺j
rij ⫻ v ji ,
共B5兲
where ni is the state occupancy. Clearly the expression on the
right-hand side is equivalent to that in Eq. 共16兲.
A similar analysis can be made for the other sum rules
presented in Appendix A. For example, the counterpart of the
Hall sum rule for a bounded many-electron system reads
f⬙
兺 0m = −
m⫽0 ␻m0
共I兲
ˆ 共2兲兩⌿ 典
= ␥具⌿n兩⌳
MSR
m
冉兺
i
ij
共B1兲
This is the original dichroic f-sum rule of Hasegawa and
Howard,2 with the orbital angular momentum appearing on
the right-hand side; in the notation of Sec. II B it reads
具␴A
⬙ 典 = 共␲ec / ប兲Morb 共since here ⌬M = 0兲.
We now generalize the discussion to N-electron systems.
N
N
r̂i and v̂ = 兺i=1
v̂i, and it is crucial to
In this context r̂ = 兺i=1
ˆ 共1兲
make a distinction between the one-particle operator ⌳
N
共2兲
ˆ
= 兺i=1r̂i ⫻ v̂i and the two-particle operator ⌳ = r̂ ⫻ v̂ appearing in Eq. 共B2兲, as emphasized in Ref. 41. The former is
related to the electronic angular momentum and orbital magnetization, while the latter is related to a many-electron
center-of-mass circulation. 关In the classical context, for example, a pair of electrons orbiting 180° out of phase in the
same circular orbit would have ⌳共1兲 ⫽ 0 but ⌳共2兲 = 0.兴
The derivation of the dichroic sum rule for the N-electron
case proceeds as before, except that the velocity matrix elements in Eq. 共4兲 become vnm = 具⌿n兩v̂兩⌿m典, where 兩⌿m典 are
now many-body eigenstates. The result is still given by Eq.
共B2兲, with 兩n典 replaced by 兩⌿n典. Indeed, it is natural to define
the many-body generalization of Eq. 共17兲 as
共I兲
MSR
= ␥共具0兩cN . . . c1兲
occ empty
= ␥ 兺 rij ⫻ v ji共1 − n j兲ni = ␥ 兺
Summing over m ⫽ n and using the closure relation together
with 具n兩v̂兩n典 = 0 one finds, in vector notation,
⬙ =−
兺 fnm
m⫽n
ij
ime
具⌿0兩r̂ ⫻ r̂兩⌿0典 = 0,
ប
共B6兲
which was termed in Ref. 3 the Kuhn sum rule. The
independent-particle form 共A10兲 can be recovered from Eq.
共B6兲 along the lines of Eqs. 共B4兲 and 共B5兲. As for the electron localization sum rule, it yields the second cumulant moment of the quantum distribution of the many-electron center
of mass.19 In the independent-particle limit this reduces to
⌳␣␤ = 共1 / N兲Tr关P̂r̂␣Q̂r̂␤兴, whose trace yields the gaugeinvariant WF spread 共29兲. The bulk formula 共A8兲 for insulating crystals can be recovered in the thermodynamic limit
following the strategy described below for the orbital magnetization.
APPENDIX C: THERMODYNAMIC LIMIT
In this appendix we start from the expressions 共17兲 and
共I兲
共18兲 for MSR
and ⌬M of insulating crystallites and, by taking the thermodynamic limit in the Wannier representation,
turn them into the reciprocal-space expressions 共46兲 and
共47兲.
Before proceeding, recall that the quantities gk and hk
entering Eqs. 共46兲 and 共47兲 were defined in Eqs. 共36兲 and
共37兲 in the context of the “Hamiltonian gauge” in which n
labels a Bloch energy eigenstate. Here, we work with a generalized Wannier representation as in Sec. III B, where n
labels a Wannier function and 兩unk典 is the state of Bloch
symmetry 共generally not an energy eigenstate兲 constructed
from that Wannier function.16 The two representations are
related by a k-dependent unitary rotation as in Eq. 共62兲. Then
Eq. 共36兲 remains valid in the present context, since it already
takes the form of a trace, while Eq. 共37兲 is now replaced by
hk,␣␤ = 兺 Enmk具˜⳵␣umk兩˜⳵␤unk典,
共C1兲
nm
冊
vklc†k cl 共c†1 . . . cN† 兩0典兲.
共B4兲
Terms in which the indices do not pair can immediately be
eliminated. Furthermore, pairings of the form 共k = l, i = j兲 give
occ
no contribution, since this leads to 共兺occ
i rii兲 ⫻ 共兺k vkk兲 which
vanishes because 具⌿0兩v̂兩⌿0典 = 0. The only surviving terms are
those with 共j = k, i = l兲, yielding
where Enmk = 具unk兩Ĥk兩umk典. With gk and hk written as traces
in this way, it is evident that each is a gauge-invariant
quantity.9
„I…
1. Gauge-invariant self-rotation MSR
For insulating crystallites in the thermodynamic limit, Eq.
共17兲 can be replaced by Eq. 共49兲. Thus we need to establish
the equivalence between Eqs. 共49兲 and 共46兲. Using
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PHYSICAL REVIEW B 77, 054438 共2008兲
IVO SOUZA AND DAVID VANDERBILT
i
v̂ = − 关r̂,Ĥ兴
ប
共C2兲
and specializing to the z component of Eq. 共49兲,
共I兲
=
M SR,z
e
Im trc关P̂x̂Q̂ĤQ̂ŷ − P̂ĤP̂x̂Q̂ŷ兴.
បcVc
共C3兲
共C11兲 into a form where the unbounded operators x̂ and ŷ are
sandwiched between P̂ and Q̂, as in Eq. 共C3兲. That ensures
that ill-defined diagonal position matrix elements between
the extended Bloch states do not occur. We will make use of
the following rules for finite-dimensional Hermitian matrices
A, B, C, and D:
The second term above may be expanded as a trace in the
Wannier representation as
trc关P̂ĤP̂x̂Q̂ŷ兴 = 兺 兺 具0m兩Ĥ兩Rn典具Rn兩x̂Q̂ŷ兩0m典. 共C4兲
R mn
Then using the identities
具0m兩Ĥ兩Rn典 = Vc
具Rn兩x̂Q̂ŷ兩0m典 = Vc
冕
冕
Im Tr关ABCD兴 = Im Tr关DABC兴,
共C12兲
Im Tr关ABCD兴 = − Im Tr关DCBA兴,
共C13兲
Im Tr关AB兴 = 0,
共C14兲
and, if any two of the matrices A, B, and C commute,
dk −ik·R
Emnk ,
e
共2␲兲3
dk ik·R
e 具˜⳵xunk兩˜⳵yumk典,
共2␲兲3
we obtain
1
trc关P̂ĤP̂x̂Q̂ŷ兴 =
Vc
冕
dk
hk,␣␤ .
共2␲兲3
Im Tr关ABC兴 = 0.
共C6兲
Rules 共C12兲 and 共C13兲 result from elementary properties of
the trace. Rule 共C14兲 is a consequence of rules 共C12兲 and
共C13兲, and rule 共C15兲 follows from rule 共C14兲. Replacing the
first P̂ in Eq. 共C11兲 by 1̂ − Q̂ and applying rule 共C15兲 to the
term containing 1̂ 共关x̂ , ŷ兴 = 0 and P̂ĤP̂ is Hermitian兲, we obtain
共C7兲
Using a similar argument, it follows that
1
trc关P̂x̂Q̂ĤQ̂ŷ兴 =
Vc
冕
dk
gk,␣␤ .
共2␲兲3
共⌬M兲z = − 4¯␥ Im Tr关P̂ĤP̂x̂Q̂ŷ兴,
共C8兲
2. Gauge-invariant remainder ⌬M
To take the thermodynamic limit of ⌬M we start from Eq.
共18兲 and apply it to a large crystallite to arrive at Eq. 共47兲.
Focusing on the z component,
Tr关P̂ĤP̂x̂Q̂ŷ兴 = 兺 具w j兩ĤP̂共x̂ − x̄ j兲Q̂共ŷ − ȳ j兲兩w j典
j
共C17兲
共note that P̂r̄Q̂ = 0兲. Since only the relative coordinate appears, the contribution from the surface orbitals is nonextensive, vanishing in the thermodynamic limit. We are then left
with a bulklike expression
Tr关P̂ĤP̂x̂Q̂ŷ兴 → 兺
共C9兲
共C18兲
共C10兲
where we defined ¯␥ = ␥ / ប and replaced P̂Ĥ by the more symmetrical form P̂ĤP̂. Using −i Tr关Ô − O†兴 = 2 Im Tr关Ô兴, this
becomes
共⌬M兲z = 4¯␥ Im Tr关x̂P̂ŷ共P̂ĤP̂兲兴.
共C11兲
At this point we are still considering a bounded sample.
To obtain a bulk expression we first need to manipulate Eq.
J. Sakurai, Modern Quantum Mechanics 共Addison-Wesley,
Reading, MA, 1994兲.
2 H. Hasegawa and R. E. Howard, J. Phys. Chem. Solids 21, 179
共1961兲.
3 D. Y. Smith, Phys. Rev. B 13, 5303 共1976兲.
1 J.
兺 具Rm兩Ĥ兩R⬘n典具R⬘n兩x̂Q̂ŷ兩Rm典.
Rm R⬘n
Now use Eq. 共C2兲 to obtain
共⌬M兲z = − 2i¯␥ Tr关x̂P̂ŷP̂ĤP̂ − ŷP̂x̂P̂ĤP̂兴,
共C16兲
which has the desired form.
Now we invoke Wannier representability to write
Combining the above two equations with Eq. 共C3兲 then
yields Eq. 共46兲.
共⌬M兲z = ␥ Tr关P̂x̂P̂v̂y兴 − ␥ Tr关P̂ŷP̂v̂x兴.
共C15兲
共C5兲
Both matrix elements on the right-hand side depend on R
and R⬘ only through R⬘ − R, and therefore, comparing with
Eq. 共C4兲,
1
Tr关P̂ĤP̂x̂Q̂ŷ兴 → trc关P̂ĤP̂x̂Q̂ŷ兴,
Nc
共C19兲
where Nc is the number of crystalline cells in the sample.
Combining Eqs. 共C7兲, 共C16兲, and 共C19兲 one obtains Eq. 共47兲,
which concludes the proof.
4 B.
T. Thole, P. Carra, F. Sette, and G. van der Laan, Phys. Rev.
Lett. 68, 1943 共1992兲.
5 The many-electron generalization given in Eq. 共70兲 of Ref. 3 关and
reproduced in Eq. 共1兲 of Ref. 4兴 of the Hasegawa-Howard sum
rule appears to be incorrect. The angular momentum operator on
054438-12
PHYSICAL REVIEW B 77, 054438 共2008兲
DICHROIC f-SUM RULE AND THE ORBITAL…
the right-hand side should be replaced by its many-body counterpart, as discussed in Appendix B. Moreover, even for the case
of a single electron in the presence of an external magnetic field
and/or spin-orbit interaction, one must distinguish between the
ˆ , which is gauge covariant with
kinetic angular momentum r̂ ⫻ ␲
respect to the choice of electromagnetic gauge, and the canoniˆ = −共ime / ប兲
cal angular momentum r̂ ⫻ p̂, which is not. Here ␲
⫻关r̂ , Ĥ兴 ⬅ mev̂ is the 共gauge-covariant兲 kinetic linear momentum, related to the canonical momentum p̂ = −iប⵱ 共generator of
ˆ = p̂ + 共e / c兲A共r兲 + 共ប / 4mec2兲␴ ⫻ ⵱U共r兲 关A共r兲
translations兲 by ␲
is the vector potential, ␴ is the vector of Pauli matrices, and
U共r兲 is the potential energy兴. If, as in the present work, the
gauge-covariant definition of angular momentum is adopted, the
second and third terms inside the brackets on the right-hand side
of Eq. 共70兲 of Ref. 3 are absorbed into the first term.
6 P. M. Oppeneer, J. Magn. Magn. Mater. 188, 275 共1998兲.
7 T. Thonhauser, D. Ceresoli, D. Vanderbilt, and R. Resta, Phys.
Rev. Lett. 95, 137205 共2005兲.
8
D. Xiao, J. Shi, and Q. Niu, Phys. Rev. Lett. 95, 137204 共2005兲.
9 D. Ceresoli, T. Thonhauser, D. Vanderbilt, and R. Resta, Phys.
Rev. B 74, 024408 共2006兲.
10 J. Shi, G. Vignale, D. Xiao, and Q. Niu, Phys. Rev. Lett. 99,
197202 共2007兲.
11 F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 共1988兲.
12 H. S. Bennett and E. A. Stern, Phys. Rev. 137, A448 共1965兲.
13
C. Kittel, Introduction to Solid State Physics, 1st ed 共Wiley, New
York, 1953兲.
14 G. G. Scott, Rev. Mod. Phys. 34, 102 共1962兲.
15 R. Huguenin, G. P. Pells, and D. N. Baldock, J. Phys. F: Met.
Phys. 1, 281 共1971兲.
16 N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847 共1997兲.
17
T. Thonhauser and D. Vanderbilt, Phys. Rev. B 74, 235111
共2006兲.
18
The quantity MSR was called MLC in Ref. 7. One reason for using
a different notation here is that in Ref. 9 MLC was defined 共for
crystals兲 in a different way, by replacing the Wannier center r̄i in
Eq. 共22兲 by the lattice vector R labeling the cell to which the WF
belongs. The two definitions turn out to be equivalent for singleband insulators 共the case of interest in Ref. 7兲. In Sec. III B we
discuss the precise relation between our MSR and the MLC of
Ref. 9 in the multiband case.
19 I. Souza, T. Wilkens, and R. M. Martin, Phys. Rev. B 62, 1666
共2000兲.
20
However, since the off-diagonal rij enter the self-rotation as a first
power and the spread as a second power, the noninvariant part
may play a larger role in the former quantity than in the latter.
R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651
共1993兲.
22 I. Souza, J. Íñiguez, and D. Vanderbilt, Phys. Rev. B 69, 085106
共2004兲.
23 For metals the low-frequency part of the absorption spectrum
contains intraband 共Drude兲 contributions in addition to the interband absorption. Only the latter was taken into account in the
derivation of the dichroic f-sum rule for crystals.
24 D. Xiao, Y. Yao, Z. Fang, and Q. Niu, Phys. Rev. Lett. 97,
026603 共2006兲.
25 D. Vanderbilt and R. D. King-Smith, Phys. Rev. B 48, 4442
共1993兲.
26
The quantum of polarization can ultimately be traced to the quantization of the charge in a Wannier function. On the contrary,
neither the self-rotation nor the center-of-mass velocity of a
Wannier function is quantized; these can change continuously
under a multiband gauge transformation.
27 Y. Yafet, Solid State Phys. 14, 1 共1963兲.
28 C. T. Chen, Y. U. Idzerda, H.-J. Lin, N. V. Smith, G. Meigs, E.
Chaban, G. H. Ho, E. Pellegrin, and F. Sette, Phys. Rev. Lett.
75, 152 共1995兲.
29 A. Hjelm, J. Trygg, O. Eriksson, B. Johansson, and J. M. Wills,
Int. J. Mod. Phys. B 9, 2735 共1995兲.
30 I. V. Solovyev, A. I. Liechtenstein, and K. Terakura, Phys. Rev.
Lett. 80, 5758 共1998兲.
31 I. V. Solovyev, Phys. Rev. Lett. 95, 267205 共2005兲.
32 H. Adachi and H. Ino, Nature 共London兲 401, 148 共1999兲.
33
A. N. Yaresko, L. Uba, S. Uba, A. Y. Perlov, R. Gontarz, and V.
N. Antonov, Phys. Rev. B 58, 7648 共1998兲.
34
H. Ebert and S. Man’kovsky, Phys. Rev. Lett. 90, 077404 共2003兲.
35 J. R. Yates, D. Vanderbilt, and I. Souza 共unpublished兲.
36 N. F. Mott and H. Jones, The Theory of the Properties of Metals
and Alloys 共Clarendon, Oxford, 1936兲.
37 Y. Yao, L. Kleinman, A. H. MacDonald, J. Sinova, T. Jungwirth,
D.-S. Wang, E. Wang, and Q. Niu, Phys. Rev. Lett. 92, 037204
共2004兲.
38
R. Resta, J. Phys.: Condens. Matter 14, R625 共2002兲.
39 W. Kohn, in Many-Body Physics, edited by C. DeWitt and R.
Balian 共Gordon and Breach, New York, 1968兲, p. 351.
40 To see that the middle expression in Eq. 共A10兲 vanishes, write
Im Tr关P̂r̂␣Q̂r̂␤兴 as Im Tr关共P̂兲共r̂␣r̂␤兲兴 − Im Tr关共P̂r̂␣ P̂兲共P̂r̂␤ P̂兲兴, note
that P̂, r̂␣r̂␤, P̂r̂␣ P̂, and P̂r̂␤ P̂ are all Hermitian, and use Eq.
共C14兲.
41 J. Kunes and P. M. Oppeneer, Phys. Rev. B 61, 15774 共2000兲.
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