Quantum weight

Yugo Onishi1 and Liang Fu1

1
Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
(Dated: January 26, 2024)
We introduce the concept of quantum weight as a fundamental property of insulating states of
matter that is encoded in the ground-state static structure and measures quantum fluctuation in
electrons’ center of mass. We find a sum rule that directly relates quantum weight—a ground
state property—with the negative-first moment of the optical conductivity above the gap frequency.
Building on this connection to optical absorption, we derive both an upper bound and a lower bound
on quantum weight in terms of electron density, dielectric constant, and energy gap. Therefore,
quantum weight constitutes a key material parameter that can be experimentally determined from
arXiv:2401.13847v1 [cond-mat.str-el] 24 Jan 2024

X-ray scattering.

It is well known that quantum states of matter with Using fluctuation-dissipation theorem [5], we show that
an energy gap have vanishing dc longitudinal conductiv- as a ground state property, the quantum weight defined
ity at zero temperature, while the optical conductivity here is directly related to the negative-first moment of
is generally nonzero at frequencies above the gap. Inter- optical conductivity through the Planck constant. This
estingly, the ground state property of an insulating state establishes the equivalence between the old and new defi-
still has important bearings on its optical conductivity. nition of quantum weight in terms of optical conductivity
Consider the real part of longitudinal optical conductiv- and static structure factor respectively.
ity Re σxx (ω), which determines the amount of optical Next, we derive lower and upper bounds on the quan-
absorption in the medium. It is well known that its ze- tum weight, in terms of the electron density, the energy
roth moment known as the optical spectral weight is re- gap, and the dielectric constant. These bounds apply to
lated to the electron density in the system through the any insulating system, and represent a universal relation
f -sum rule [1]. Higher order moments of optical conduc- between ground state property (quantum weight), op-
tivity are much less studied. The negative-second mo- tical response, thermodynamic response (dielectric con-
ment is directly related to electric susceptibility through stant), and the energy gap of the system. Remarkably,
the Kramers-Kronig relation [2]. Recently, we employed our bounds can provide a good estimate of the quantum
the negative-first moment of both longitudinal and Hall weight of real materials.
conductivities to derive a universal upper bound on the We start by considering generalized optical weights W i
energy gap of (integer or fractional) Chern insulators [3]. for insulating states [3]. W i is defined as the negative i-th
While the Hall conductivity only exists in the presence moment of the absorptive part of the optical conductivity
of time reversal symmetry breaking, the longitudinal con- σ(ω),
ductivity is present in all systems. In a pioneering early Z ∞ abs
i
σαβ (ω)
work [4], Souza, Willkins and Martin (SWM) showed that Wαβ ≡ dω , (1)
ω i
the negative-first moment of σxx (ω) is related to a ba- 0
sic property of quantum insulators that was expressed where σαβabs
≡ (σαβ + σβα ∗
)/2 is composed of the real
in terms of the quantum metric of many-body ground part of longitudinal optical conductivity and the imagi-
states over twisted boundary condition. This quantity nary part of optical Hall conductivity. For isotropic two-
was recently termed “quantum weight” because it con- abs
dimensional systems, σαβ = Re(σxx )δαβ + i Im(σxy )ϵαβ .
nects quantum metric and optical weight [3]. Despite its Throughout this work, we consider gapped systems so
ubiquitous presence in solids, the quantum weight has that the absorptive part of optical conductivity vanishes
not received adequate attention, and to our knowledge, at low frequency:
its value has not been experimentally determined for any
material. σ abs (ω) = 0 for ℏω ≤ Eg . (2)
In this work, we provide a new and general definition This condition defines the optical gap Eg which must be
of quantum weight for all insulators as the quadratic co- equal to or greater than the spectral gap ∆. We will
efficient of the ground-state static structure factor Sq at consider three optical weights: W 0 , W 1 and W 2 , and fo-
small q. For periodic systems, Sq at any q other than cus on their real part, which is associated with optical
reciprocal-lattice points vanishes in the classical limit longitudinal conductivity and is present in any solids.
(ℏ = 0), and is nonzero only because of quantum fluc- Re W 0 and Re W 2 are related to the charge density
tuation in electron position. Therefore, the new defini- and the electric susceptibility, respectively:
tion of quantum weight manifests its purely quantum-
mechanical origin, and allows it to be experimentally Re W 0 = πne2 /(2m), (3)
measured by X-ray scattering. Re W 2 = (π/2)χ. (4)
 2

Eq.(3) is the well-known f sum rule [6] that relates the cillator:
full optical spectral weight to the electron density n and 1 X πZα2 e2
mass m. Eq.(4) was recently derived from the relation be- Re σ(ω) = δ(ω − ωα ), (8)
L α 2m
tween conductivity and polarizability using the Kramers-
Kronig relation [2]. Here, χ describes the polarization with Zα e =
P
i cαi e the effective charge of the nor-
induced by a static external electric field, and is directly mal mode α. Here, for simplicity we consider a one-
related to the dielectric constant χ = ϵ0 (ϵ − 1) with ϵ0 dimensional system with length L to illustrate the essen-
the vacuum permittivity. tial physics. It then follows
Our focus in this work is on the real part of the
negative-first moment, Re W 1 . As we will show, the op- 1 X πZα2 e2
Re W 1 = . (9)
tical weight Re W 1 is directly related to the quantum L α 2mωα
fluctuation in electron position, even though this quan-
One can readily verify that α Zα2 /L = n with the elec-
P
tity itself can be defined even for classical systems. To
motivate the discussion, let us first consider a system of tron density n, as expected from the f -sum rule.
highly localized electrons arising from strong potential Interestingly, Re W 1 is related through the Planck con-
and interaction effects. In the classical limit (ℏ = 0), we stant to the quantum fluctuation in the polarization of
can treat electrons as point charges and the ground state our system, which arises from electron’s zero-point mo-
is obtained by minimizing the sum of the potential en- tion and is obtained by quantization: p′α → −iℏ∂α′ .
ergy and the electron-electron interaction energy (which Noting that the total polarization
D E is given by δP =
′ ′ ′
P P
only depend on electron position): e i δxi = e α Zα xα and xα xβ = δαβ ℏ/(2mωα ), we
X X can rewrite Re W 1 as
Hc = V (xi ) + U (xi − xj ), (5) ℏ 1
i i,j Re W 1 = (δP )2 , (10)
π L
with V the potential and U the two-body interaction. On where ⟨. . .⟩ is the expectation value in the ground state.
the other hand, the kinetic energy This relation Eq. (10) is noteworthy for two reasons.
The optical weight Re W 1 in the left-hand side is finite
X p2 even for classical systems, while the polarization fluctua-
i
H0 = , (6)
i
2m tion in the right-hand side is purely quantum-mechanical
in nature and vanishes in ℏ → 0 limit. The left hand side
leads to quantum fluctuation in electron position, which involves optical conductivity at all frequencies, while the
we treat below by quantizing electron’s motion around right hand side is a ground state property.
the ground state configuration. In order to properly define polarization fluctuation for
We expand Hc up to the second order in the dis- insulators in general, let us consider the static struc-
placement of the electrons from the ground state posi- ture factor that measures equal-time density-density cor-
which yields a spring constant matrix k: Hc =
tion, P relation in the ground state: Sq ≡ (1/V )(⟨ρ̂q ρ̂−q ⟩ −
Ec + i,j kij δxi δxj /2. k, we obtain the ⟨ρ̂q ⟩ ⟨ρ̂−q ⟩). In particular, we focus on the quadratic co-
P Diagonalizing P
normal modes x′α = c αi δxi (with c c = δαβ ) efficient of Sq at small nonzero q:
i P i αi2 βi
and the spring constant kα : Hc = α kα x′α /2. Corre- e2
spondingly, we can rewrite the total kinetic energy HK Sq = Kαβ qα qβ + . . . , (11)
2π
with the
P momentum P conjugate to the normal modes,
with ρ̂q = dr e−iq·r ρ̂(r) the charge density operator
2
R
p′α = i cαi pi , as α p′α /(2m). Then we obtain a col-
lection of independent harmonic oscillators, one for each with wavevector q and V the volume of the system. We
normal mode, call the quadratic coefficient K defined by Eq. (11) quan-
tum weight. Importantly, K can be nonzero only because
H = H0 + Hc of quantum fluctuation. In the classical limit (ℏ = 0), the
X p′ 2 ground state of a periodic system is a periodic array of
1 2
≈ α
+ kα x′α + . . . . (7) electron point charges, and correspondingly, the static
α
2m 2 structure factor is composed of δ functions centered at
p reciprocal lattice vectors and vanishes everywhere else,
The frequency for each mode is ωα = kα /m, leading to leading to K = 0. Hence quantum weight measures the
the energy gap of ℏωα after quantization. degree of “quantumness” in the insulating ground state.
For this system of strongly localized electrons, a uni- To make the connection between polarization fluctu-
form electric field
P couples to the center of mass displace- ation and quantum weight, note that polarization is re-
ment i δxi = α Zα x′α , and the real part of the optical
P
lated to charge density through ∇ · P = −ρ. There-
conductivity is well known from that of the harmonic os- fore, one may identify −iqδP = ρq and then quantum
 3

weight defined by Eq. (11) represents the polarization (or where we have used S(q, ω) = 0 for ω < 0 at zero tem-
center-of-mass position) fluctuation, in the ground state: perature (for more details see Supplemental Materials).
K = (δP )2 /(2πe2 V ). In contrast to our treatment Taking q → 0 limit, we obtain a relation between optical
based on static structure factor, previous works treated weight Re W 1 at q = 0 and the quantum weight defined
polarization fluctuation in terms of the second cumulant above as the quadratic coefficient of Sq :
moment of the position operator for systems with open
∞
Re σ abs (ω) e2
Z
or twisted boundary conditions [4, 7].
Re W 1 = dω = K. (18)
Motivated by the case of strongly localized electron 0 ω 2ℏ
systems discussed above, we now establish a general re-
lation between the optical weight Re W 1 and the quan- Eq. (17), together with its q → 0 limit Eq. (18), is a key
tum weight K encoded in ground-state static structure result of our work. It constitutes a new optical sum rule
factor Sq , which captures quantum fluctuation in elec- relating the negative first moment of longitudinal optical
trons’ center of mass of all insulators. Let us consider conductivity to ground-state structure factor through the
the response of an insulator to a time-dependent peri- Planck constant, which is a generalization of the f sum
odic potential Vext with wavevector q and frequency ω. rule relating the optical spectral weight to the electron
The induced change in the density and the current re- density.
sponse are characterized by the density-density response The static structure factor Sq can be experimentally
function Π(q, ω) and the conductivity tensor σ(q, ω) re- obtained from X-ray scattering experiments, while the
spectively: optical weight Re W 1 can be determined from the ex-
perimentally measured optical conductivity. The ratio
ρ(q, ω) = −Π(q, ω)Vext (q, ω), (12) between Sq /q 2 and Re W 1 yields a fundamental physical
j(q, ω) = σ(q, ω)E(q, ω), (13) constant, the Planck constant. This provides a way of
determining the Planck constant by optical spectroscopy
where E(q, ω) = −iqVext (q, ω) is the external electric measurements of basic material properties.
field. Due to the continuity equation ∂ρ/∂t + ∇ · j = 0, We have shown that quantum weight is both a funda-
σ and Π are directly related: mental ground state property of insulators and an impor-
tant material parameter related to optical properties. We
qα qβ σαβ (q, ω) now further derive lower and upper bounds on quantum
Π(q, ω) = i . (14)
ω weight in terms of common material parameters: elec-
It is important to note that there is no singularity at tron density, energy gap and dielectric constant. First,
small ω and at small q in Eq. (14) for insulators [8]. noting that the real part of the longitudinal optical con-
Correspondingly, the negative-first moment of the real ductivity is always non-negative and can be finite only
part of optical conductivity at q = 0 is related to the for |ω| ≥ Eg /ℏ, the following inequality among optical
imaginary part of density response Π(q, ω) at small q weights always holds:
integrated over the frequency: i+1 i
Z ∞ Re Wαα ≤ ℏ Re Wαα /Eg . (19)
1
dω Im Π(q, ω) = qα qβ Re(Wαβ ) + .... (15) Combining Eq. (19) with i = 0, 1 and the expressions for
0
optical weights W 0 , W 1 , W 2 shown in Eq. (3), (4) and
By the fluctuation-dissipation theorem [5], Im Π(q, ω) is (18), we obtain the bound on the quantum weight:
directly related to the dynamical structure factor:
π 2ℏ2 Wαα
0
Im Π(q, ω) = S(q, ω)/(2ℏ), (16) Eg χ αα ≤ K αα ≤ . (20)
e2 e2 Eg
where the dynamical
R∞ structural factor is defined as Here, α is the principal axis of the material. The stan-
S(q, ω) = V1 −∞ dt eiωt ⟨ρ̂q (t)ρ̂−q (0)⟩. Here, ρ̂q (t) = dard f -sum rule, W 0 = πne2 /(2m) further leads to a
dr e−iq·r ρ̂(r, t) is the density operator with wavevec-
R
universal bound on the quantum weight as
tor q in the Heisenberg picture and V is the volume of
the system. π πnℏ2
Eg χ αα ≤ K αα ≤ . (21)
Combining Eq. (14), (15) and (16), we obtain a general e2 mEg
relation between optical conductivity and ground state
static structural factor: This inequality should hold for any system regardless of
Z ∞ abs Z ∞ the dimensionality of the system. Both the lower and
Re σαβ (q, ω) 1 upper bounds are saturated when optical conductivity is
qα qβ dω = dω S(q, ω)
0 ω 2ℏ −∞ nonzero only at single frequency ω = ±Eg /ℏ, which we
π shall call single frequency absorption. We also note that
= Sq (17)
ℏ the inequality (21) remains valid even when the optical
 4

symmetry breaking or topological order):

r
1X
Gαβ (θ) ≡ Re ⟨∂α Ψiθ |(1 − Pθ )|∂β Ψiθ ⟩ . (22)
r i=1

Here, |Ψiθ ⟩ is the i-th ground state under the twisted

boundary condition specified by θ: Ψiκ (r1 , . . . , rn +
Lµ , . . . , rN ) = eiθµ Ψiκ (r1 , . . . , rn , . . . , rN ) where Lµ =
(0, . . . , Lµ , . . . , 0) with Lµ the system size in µ-direction,
and Pθ = |Ψθ ⟩⟨Ψθ | is the projector operator associated
with the ground state.
Assuming that the optical conductivity in the thermo-
dynamic limit is insensitive to the choice of boundary
condition θ or the particular ground state, one can show
FIG. 1. The bound on the quantum weight K for real materi- [3] that the optical weight Re W 1 and hence the quantum
als. As the energy gap, we used the gap obtained from optical weight K are related to G as
measurements. The details of the used parameters are given
dd θ
Z
in Ref. [2].
K = 2π G(θ). (23)
(2π)d

gap Eg is replaced with the spectral gap, although the For noninteracting band insulators, the ground state is
bounds would generally become less tight. always unique and takes the form of a Slater determinant
of occupied states over all wavevectors k in the Brillouin
We calculated the bound on the quantum weight for
zone. Then, the many-body quantum metric G reduce
real materials, and the results are shown in Fig. 1. Our
to an integral in k-space:
bound (21) gives a fairly good estimate of the quantum
weight only from the electric susceptibility (or equiva- dd k
Z
lently the dielectric constant), the energy gap, and the Kαβ = 2π gαβ (k). (24)
(2π)d
electron density. The most remarkable case is cubic
boron nitride (c-BN). c-BN is an indirect gap insulator where g(k) is a quantum metric in k-space defined as:
with direct gap 14.5 eV and the dielectric constant ϵ =
4.46 [9]. From the dielectric constant, the electric suscep- gαβ (k) = Re[ ⟨∂α Ψ(k)|(1 − P (k))|∂β Ψ(k)⟩], (25)
tibility χ = ϵ0 (ϵ − 1) is given by χ = 3.1 × 10−11 F m−1 . with P (k) = |Ψ(k)⟩⟨Ψ(k)| the projection operator onto
The cubic unit cell of c-BN has a lattice constant 3.62 Å the Slater determinant of occupied states at wavevector
and contains 4 boron and nitrogen atoms, each having k: |Ψ(k)⟩ = |u1 (k) . . . us (k)|. Equivalently, g(k) is equal
5 and 7 electrons, hence the total electron density is to the trace of non-Abelian quantum metric of occupied
n = 1.02 × 1024 cm−3 . Then the full spectral weight bands [10]. This quantity appears as the gauge-invariant
is given by Waa 0
= 4.5 × 1022 m−1 s−1 Ω−1 . Since K term in the localization functional of the Wannier func-
and χ are isotropic in c-BN, we find that the quantum tions [11]. Thus, K is related to the degree of localization
weight Kaa is bounded to a remarkably narrow interval: of occupied electron states, consistent with our result
−1
0.9 Å ≤ Kaa ≤ 1.7 Å. It should be emphasized that on the quantum weight for strongly localized electron
our analysis applies to any electronic systems, includ- systems (ℏ → 0) given above. We also note a relation
ing disordered and/or interacting systems. This result between electron localization length and energy gap for
demonstrates that our analysis is powerful in understand- noninteracting disordered systems in one dimension [12],
ing quantum materials. which is a special case of our general relation between the
Finally, for the sake of completeness, we discuss the quantum weight and the energy gap for all insulators.
relation of the quantum weight K defined by static struc- To conclude, we have established quantum weight, de-
ture factor to quantum geometry. As first shown by fined through static structure factor, as a key material
SWM, the first-negative moment of optical conductiv- parameter that is connected to a variety of physical ob-
ity Re W 1 is related to the many-body quantum metric servables. The quantum weight represents the quantum
defined over the twisted boundary condition [4]. Then, fluctuation in electrons’ center of mass. We derived its
based on the relation Eq. (18), the quantum weight K general relation to optical conductivity, dielectric con-
is also related to the many-body quantum metric. Here, stant, quantum geometry, and energy gap. Our results
we generalize the work of SWM which considers systems apply to all insulators, including strongly correlated sys-
with a unique ground state and introduce the many-body tems. Experimental determination by X-ray scattering
quantum metric G for the general case of r-fold degener- as well as first-principles calculation of quantum weight
ated ground states (related to each other by spontaneous for real materials [13] are called for.
 5

This work was supported by National Science Founda- Mechanical Theory of Irreversible Processes. II. Response
tion (NSF) Convergence Accelerator Award No. 2235945. to Thermal Disturbance, Journal of the Physical Society
YO is grateful for the support provided by the Funai of Japan 12, 1203 (1957).
Overseas Scholarship. LF was partly supported by the [7] R. Resta, Polarization Fluctuations in Insulators and
Metals: New and Old Theories Merge, Physical Review
David and Lucile Packard Foundation. Letters 96, 137601 (2006), publisher: American Physical
Society.
[8] We note that the static structure factor for metallic
systems at small q is dominated by |q|-linear term, in
contrast to insulators where the leading order term is
[1] R. Kubo, Statistical-Mechanical Theory of Irreversible quadratic.
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Physical Society of Japan 12, 570 (1957). [10] S. Peotta and P. Törmä, Superfluidity in topologically
[2] Y. Onishi and L. Fu, Universal relation between en- nontrivial flat bands, Nature Communications 6, 8944
ergy gap and dielectric constant (2024), arXiv:2401.04180 (2015), number: 1 Publisher: Nature Publishing Group.
[cond-mat]. [11] N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and
[3] Y. Onishi and L. Fu, Fundamental bound on topological D. Vanderbilt, Maximally localized Wannier functions:
gap (2023), arXiv:2306.00078 [cond-mat]. Theory and applications, Reviews of Modern Physics 84,
[4] I. Souza, T. Wilkens, and R. M. Martin, Polarization and 1419 (2012), publisher: American Physical Society.
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Physical Society. solitons, Physical Review B 26, 4269 (1982), publisher:
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eralized Noise, Physical Review 83, 34 (1951), publisher: [13] B. Ghosh, Y. Onishi, S.-Y. Xu, H. Lin, L. Fu, and
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Supplemental Material

Quantum weight in atomic insulators

To gain insight into the quantum weight, let us consider a solid made of an array of atoms that are far away
from each other, so that the hopping between the atoms is negligible leading to trivial flat bands. In such atomic
insulators, optical absorption comes from electric dipole transitions between occupied and unoccupied energy levels
within individual atoms. It is straightforward to show that the quantum weight K is given by the ratio of the
intra-atomic position fluctuation to the area of the unit cell A0 :

AI 2π
Kαβ = ⟨Φ|(rα − ⟨rα ⟩)(rβ − ⟨rβ ⟩)|Φ⟩ (26)
A0
where |Φ⟩P denotes the ground state of an atom, ⟨rα ⟩ = ⟨Φ|rα |Φ⟩ is the expectation value of rα in the ground state,
and rα = i rα,i is the sum of the position of each electron. Notably, the right-hand side of Eq. (26) is precisely the
polarization (=center-of-mass) fluctuation in an array of atoms, as discussed in the main text.

Detailed derivation of the relation between the quantum weight and the optical weight

Here we derive the following relation between the quantum weight K and the negative-first moment of optical
conductivity W 1 for general insulators:
e2
Re W 1 = K, (27)
2ℏ
where W 1 is defined as
∞
σ abs (ω)
Z
1
W = dω (28)
0 ω
 6

with the absorptive part of the conductivity tensor σ abs (ω) = (σ(ω) + σ(ω)† )/2 and the quantum weight is defined as
the quadratic coefficient in q of the static structure factor Sq = (1/V ) ⟨ρq ρ−q ⟩ as

e2
Sq = Kαβ qα qβ + . . . . (29)
2π
To show Eq. (27), we use the dissipation fluctuation theorem:

Im Π(q, ω) = S(q, ω)/(2ℏ), (30)

where Π(q, ω) describes the density response to the external potential Vext , as defined in the main text. We will also
use the following relation between S(q, ω) and S(−q, −ω):

S(q, ω) = eβℏω S(−q, −ω). (31)

This relation can be shown as follows:

1 ∞
Z
S(q, ω) = dt eiωt ⟨ρ̂q (t)ρ̂−q (0)⟩ (32)
V −∞
1 X ∞ e−βεn
Z
= dt eiωt ⟨n|ρ̂q (t)|m⟩ ⟨m|ρ̂−q (0)|n⟩ (33)
V n,m −∞ Z
1 X ∞ e−βεn
Z
= dt ei(ω+(εn −εm )/ℏ)t ⟨n|ρ̂q |m⟩ ⟨m|ρ̂−q |n⟩ (34)
V n,m −∞ Z
1 X εn − εm e−βεn
= 2πδ(ω + ) ⟨n|ρ̂q |m⟩ ⟨m|ρ̂−q |n⟩ (35)
V n,m ℏ Z
1 X εn − εm e−βεm
= 2πδ(ω − ) ⟨m|ρ̂q |n⟩ ⟨n|ρ̂−q |m⟩ (36)
V n,m ℏ Z
1 X εn − εm e−βεn
= eβℏω 2πδ(−ω + ) ⟨n|ρ̂−q |m⟩ ⟨m|ρ̂q |n⟩ (37)
V n,m ℏ Z
= eβℏω S(−q, −ω). (38)

Here, β = (kB T )−1 is the inverse temperature, Z is the partition function, |n⟩ is the n-th energy eigenstate with
energy εn . In particular, Eq. (31) implies S(q, ω) = 0 for ω < 0 at zero temperature.
Due to the continuity equation ∂ρ/∂t + ∇ · j = 0, Π and σ satisfies the following relation:

qα qβ σαβ (q, ω)
Π(q, ω) = i . (39)
ω
Noting that only the symmetric part of σ contribute to Π, Eq. (30) and Eq. (39) yield
abs
qα qβ Re σαβ (q, ω) S(q, ω) + S(q, −ω)
= for ω > 0 (40)
ω 2ℏ
where we have used S(q, −ω) = 0. Integrating over frequencies from 0 to ∞, we will get
∞ abs ∞
qα qβ Re σαβ (q, ω)
Z Z
S(q, ω) π
dω = dω = Sq . (41)
0 ω −∞ 2ℏ ℏ

Expanding in q around q = 0, we obtain

1 π e2
qα qβ Re Wαβ = Kαβ qα qβ , (42)
ℏ 2π
which implies Eq. (27).