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In this work, we provide a new and general definition of quantum weight for all insulators as the quadratic coefficient of the ground-state static structure factor S𝒒subscript𝑆𝒒S_{{\bf\it q}}italic_S start_POSTSUBSCRIPT bold_italic_q end_POSTSUBSCRIPT at small 𝒒𝒒{\bf\it q}bold_italic_q. For periodic systems, S𝒒subscript𝑆𝒒S_{{\bf\it q}}italic_S start_POSTSUBSCRIPT bold_italic_q end_POSTSUBSCRIPT at any 𝒒𝒒{\bf\it q}bold_italic_q other than reciprocal-lattice points vanishes in the classical limit (ℏ=0Planck-constant-over-2-pi0\hbar=0roman_ℏ = 0), and is nonzero only because of quantum fluctuation in electron position. Therefore, the new definition of quantum weight manifests its purely quantum-mechanical origin, and allows it to be experimentally measured by X-ray scattering.

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Due to the continuity equation ∂ρ∕∂t+∇⋅𝒋=0partial-derivative𝑡𝜌⋅∇𝒋0\partialderivative*{\rho}{t}+\nabla\cdot{\bf\it j}=0∕ start_ARG ∂ start_ARG italic_ρ end_ARG end_ARG start_ARG ∂ start_ARG italic_t end_ARG end_ARG + ∇ ⋅ bold_italic_j = 0, ΠΠ\Piroman_Π and σ𝜎\sigmaitalic_σ satisfies the following relation:

Our focus in this work is on the real part of the negative-first moment, ReW1superscript𝑊1\real W^{1}start_OPERATOR roman_Re end_OPERATOR italic_W start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT. As we will show, the optical weight ReW1superscript𝑊1\real W^{1}start_OPERATOR roman_Re end_OPERATOR italic_W start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT is directly related to the quantum fluctuation in electron position, even though this quantity itself can be defined even for classical systems. To motivate the discussion, let us first consider a system of highly localized electrons arising from strong potential and interaction effects. In the classical limit (ℏ=0Planck-constant-over-2-pi0\hbar=0roman_ℏ = 0), we can treat electrons as point charges and the ground state is obtained by minimizing the sum of the potential energy and the electron-electron interaction energy (which only depend on electron position):

Here, β=(kB⁢T)−1𝛽superscriptsubscript𝑘B𝑇1\beta=(k_{\mathrm{B}}T)^{-1}italic_β = ( italic_k start_POSTSUBSCRIPT roman_B end_POSTSUBSCRIPT italic_T ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT is the inverse temperature, Z𝑍Zitalic_Z is the partition function, |n⟩ket𝑛\ket{n}| start_ARG italic_n end_ARG ⟩ is the n𝑛nitalic_n-th energy eigenstate with energy εnsubscript𝜀𝑛\varepsilon_{n}italic_ε start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT. In particular, Eq. (31) implies S⁢(𝒒,ω)=0𝑆𝒒𝜔0S({\bf\it q},\omega)=0italic_S ( bold_italic_q , italic_ω ) = 0 for ω<0𝜔0\omega<0italic_ω < 0 at zero temperature.

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where we have used S⁢(𝒒,ω)=0𝑆𝒒𝜔0S({\bf\it q},\omega)=0italic_S ( bold_italic_q , italic_ω ) = 0 for ω<0𝜔0\omega<0italic_ω < 0 at zero temperature (for more details see Supplemental Materials). Taking 𝒒→0→𝒒0{\bf\it q}\to 0bold_italic_q → 0 limit, we obtain a relation between optical weight ReW1superscript𝑊1\real W^{1}start_OPERATOR roman_Re end_OPERATOR italic_W start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT at 𝒒=0𝒒0{\bf\it q}=0bold_italic_q = 0 and the quantum weight defined above as the quadratic coefficient of S𝒒subscript𝑆𝒒S_{{\bf\it q}}italic_S start_POSTSUBSCRIPT bold_italic_q end_POSTSUBSCRIPT:

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One can readily verify that ∑αZα2/L=nsubscript𝛼superscriptsubscript𝑍𝛼2𝐿𝑛\sum_{\alpha}Z_{\alpha}^{2}/L=n∑ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_Z start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / italic_L = italic_n with the electron density n𝑛nitalic_n, as expected from the f𝑓fitalic_f-sum rule.

with P⁢(𝒌)=|Ψ⁢(𝒌)⟩⟨Ψ⁢(𝒌)|𝑃𝒌Ψ𝒌Ψ𝒌P({\bf\it k})=\outerproduct{\Psi({\bf\it k})}{\Psi({\bf\it k})}italic_P ( bold_italic_k ) = | start_ARG roman_Ψ ( bold_italic_k ) end_ARG ⟩ ⟨ start_ARG roman_Ψ ( bold_italic_k ) end_ARG | the projection operator onto the Slater determinant of occupied states at wavevector 𝒌𝒌{\bf\it k}bold_italic_k: |Ψ⁢(𝒌)⟩=|u1⁢(𝒌)⁢…⁢us⁢(𝒌)|ketΨ𝒌subscript𝑢1𝒌…subscript𝑢𝑠𝒌\ket{\Psi({\bf\it k})}=|u_{1}({\bf\it k})\dots u_{s}({\bf\it k})|| start_ARG roman_Ψ ( bold_italic_k ) end_ARG ⟩ = | italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( bold_italic_k ) … italic_u start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( bold_italic_k ) |. Equivalently, g⁢(𝒌)𝑔𝒌g({\bf\it k})italic_g ( bold_italic_k ) is equal to the trace of non-Abelian quantum metric of occupied bands [10]. This quantity appears as the gauge-invariant term in the localization functional of the Wannier functions [11]. Thus, K𝐾Kitalic_K is related to the degree of localization of occupied electron states, consistent with our result on the quantum weight for strongly localized electron systems (ℏ→0→Planck-constant-over-2-pi0\hbar\rightarrow 0roman_ℏ → 0) given above. We also note a relation between electron localization length and energy gap for noninteracting disordered systems in one dimension [12], which is a special case of our general relation between the quantum weight and the energy gap for all insulators.

Here, α𝛼\alphaitalic_α is the principal axis of the material. The standard f𝑓fitalic_f-sum rule, W0=π⁢n⁢e2/(2⁢m)superscript𝑊0𝜋𝑛superscript𝑒22𝑚W^{0}=\pi ne^{2}/(2m)italic_W start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT = italic_π italic_n italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( 2 italic_m ) further leads to a universal bound on the quantum weight as

where we have used S⁢(𝒒,−ω)=0𝑆𝒒𝜔0S({\bf\it q},-\omega)=0italic_S ( bold_italic_q , - italic_ω ) = 0. Integrating over frequencies from 00 to ∞\infty∞, we will get

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ReW0superscript𝑊0\real W^{0}start_OPERATOR roman_Re end_OPERATOR italic_W start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT and ReW2superscript𝑊2\real W^{2}start_OPERATOR roman_Re end_OPERATOR italic_W start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT are related to the charge density and the electric susceptibility, respectively:

Assuming that the optical conductivity in the thermodynamic limit is insensitive to the choice of boundary condition θ𝜃\thetaitalic_θ or the particular ground state, one can show [3] that the optical weight ReW1superscript𝑊1\real W^{1}start_OPERATOR roman_Re end_OPERATOR italic_W start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT and hence the quantum weight K𝐾Kitalic_K are related to G𝐺Gitalic_G as

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To make the connection between polarization fluctuation and quantum weight, note that polarization is related to charge density through ∇⋅P=−ρ⋅∇𝑃𝜌\nabla\cdot P=-\rho∇ ⋅ italic_P = - italic_ρ. Therefore, one may identify −i⁢q⁢δ⁢P=ρq𝑖𝑞𝛿𝑃subscript𝜌𝑞-iq\delta P=\rho_{q}- italic_i italic_q italic_δ italic_P = italic_ρ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT and then quantum weight defined by Eq. (11) represents the polarization (or center-of-mass position) fluctuation, in the ground state: K=⟨(δ⁢P)2⟩/(2⁢π⁢e2⁢V)𝐾expectation-valuesuperscript𝛿𝑃22𝜋superscript𝑒2𝑉K=\expectationvalue{(\delta P)^{2}}/(2\pi e^{2}V)italic_K = ⟨ start_ARG ( italic_δ italic_P ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ⟩ / ( 2 italic_π italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_V ). In contrast to our treatment based on static structure factor, previous works treated polarization fluctuation in terms of the second cumulant moment of the position operator for systems with open or twisted boundary conditions [4, 7].

For noninteracting band insulators, the ground state is always unique and takes the form of a Slater determinant of occupied states over all wavevectors 𝒌𝒌{\bf\it k}bold_italic_k in the Brillouin zone. Then, the many-body quantum metric G𝐺Gitalic_G reduce to an integral in k𝑘kitalic_k-space:

Motivated by the case of strongly localized electron systems discussed above, we now establish a general relation between the optical weight ReW1superscript𝑊1\real W^{1}start_OPERATOR roman_Re end_OPERATOR italic_W start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT and the quantum weight K𝐾Kitalic_K encoded in ground-state static structure factor S𝒒subscript𝑆𝒒S_{{\bf\it q}}italic_S start_POSTSUBSCRIPT bold_italic_q end_POSTSUBSCRIPT, which captures quantum fluctuation in electrons’ center of mass of all insulators. Let us consider the response of an insulator to a time-dependent periodic potential Vextsubscript𝑉extV_{\rm ext}italic_V start_POSTSUBSCRIPT roman_ext end_POSTSUBSCRIPT with wavevector 𝒒𝒒{\bf\it q}bold_italic_q and frequency ω𝜔\omegaitalic_ω. The induced change in the density and the current response are characterized by the density-density response function Π⁢(𝒒,ω)Π𝒒𝜔\Pi({\bf\it q},\omega)roman_Π ( bold_italic_q , italic_ω ) and the conductivity tensor σ⁢(𝒒,ω)𝜎𝒒𝜔\sigma({\bf\it q},\omega)italic_σ ( bold_italic_q , italic_ω ) respectively:

We expand Hcsubscript𝐻𝑐H_{c}italic_H start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT up to the second order in the displacement of the electrons from the ground state position, which yields a spring constant matrix k𝑘kitalic_k: Hc=Ec+∑i,jki⁢j⁢δ⁢xi⁢δ⁢xj/2subscript𝐻𝑐subscript𝐸𝑐subscript𝑖𝑗subscript𝑘𝑖𝑗𝛿subscript𝑥𝑖𝛿subscript𝑥𝑗2H_{c}=E_{c}+\sum_{i,j}k_{ij}\delta x_{i}\delta x_{j}/2italic_H start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = italic_E start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_δ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_δ italic_x start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT / 2. Diagonalizing k𝑘kitalic_k, we obtain the normal modes xα′=∑icα⁢i⁢δ⁢xisubscriptsuperscript𝑥′𝛼subscript𝑖subscript𝑐𝛼𝑖𝛿subscript𝑥𝑖x^{\prime}_{\alpha}=\sum_{i}c_{\alpha i}\delta x_{i}italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_α italic_i end_POSTSUBSCRIPT italic_δ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT (with ∑icα⁢i⁢cβ⁢i=δα⁢βsubscript𝑖subscript𝑐𝛼𝑖subscript𝑐𝛽𝑖subscript𝛿𝛼𝛽\sum_{i}c_{\alpha i}c_{\beta i}=\delta_{\alpha\beta}∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_α italic_i end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_β italic_i end_POSTSUBSCRIPT = italic_δ start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT) and the spring constant kαsubscript𝑘𝛼k_{\alpha}italic_k start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT: Hc=∑αkα⁢xα′2/2subscript𝐻𝑐subscript𝛼subscript𝑘𝛼superscriptsubscriptsuperscript𝑥′𝛼22H_{c}=\sum_{\alpha}k_{\alpha}{x^{\prime}_{\alpha}}^{2}/2italic_H start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2. Correspondingly, we can rewrite the total kinetic energy HKsubscript𝐻𝐾H_{K}italic_H start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT with the momentum conjugate to the normal modes, pα′=∑icα⁢i⁢pisubscriptsuperscript𝑝′𝛼subscript𝑖subscript𝑐𝛼𝑖subscript𝑝𝑖p^{\prime}_{\alpha}=\sum_{i}c_{\alpha i}p_{i}italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_α italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, as ∑αpα′2/(2⁢m)subscript𝛼superscriptsubscriptsuperscript𝑝′𝛼22𝑚\sum_{\alpha}{p^{\prime}_{\alpha}}^{2}/(2m)∑ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / ( 2 italic_m ). Then we obtain a collection of independent harmonic oscillators, one for each normal mode,

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 K𝐾Kitalic_K is given by the ratio of the intra-atomic position fluctuation to the area of the unit cell A0subscript𝐴0A_{0}italic_A start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT:

While the Hall conductivity only exists in the presence of time reversal symmetry breaking, the longitudinal conductivity is present in all systems. In a pioneering early work [4], Souza, Willkins and Martin (SWM) showed that the negative-first moment of σx⁢x⁢(ω)subscript𝜎𝑥𝑥𝜔\sigma_{xx}(\omega)italic_σ start_POSTSUBSCRIPT italic_x italic_x end_POSTSUBSCRIPT ( italic_ω ) is related to a basic property of quantum insulators that was expressed in terms of the quantum metric of many-body ground states over twisted boundary condition. This quantity was recently termed “quantum weight” because it connects quantum metric and optical weight [3]. Despite its ubiquitous presence in solids, the quantum weight has not received adequate attention, and to our knowledge, its value has not been experimentally determined for any material.

This condition defines the optical gap Egsubscript𝐸𝑔E_{g}italic_E start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT which must be equal to or greater than the spectral gap ΔΔ\Deltaroman_Δ. We will consider three optical weights: W0,W1superscript𝑊0superscript𝑊1W^{0},W^{1}italic_W start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT , italic_W start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT and W2superscript𝑊2W^{2}italic_W start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, and focus on their real part, which is associated with optical longitudinal conductivity and is present in any solids.

leads to quantum fluctuation in electron position, which we treat below by quantizing electron’s motion around the ground state configuration.

Eq.(3) is the well-known f𝑓fitalic_f sum rule [6] that relates the full optical spectral weight to the electron density n𝑛nitalic_n and mass m𝑚mitalic_m. Eq.(4) was recently derived from the relation between conductivity and polarizability using the Kramers-Kronig relation [2]. Here, χ𝜒\chiitalic_χ describes the polarization induced by a static external electric field, and is directly related to the dielectric constant χ=ϵ0⁢(ϵ−1)𝜒subscriptitalic-ϵ0italic-ϵ1\chi=\epsilon_{0}(\epsilon-1)italic_χ = italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_ϵ - 1 ) with ϵ0subscriptitalic-ϵ0\epsilon_{0}italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT the vacuum permittivity.

This inequality should hold for any system regardless of the dimensionality of the system. Both the lower and upper bounds are saturated when optical conductivity is nonzero only at single frequency ω=±Eg/ℏ𝜔plus-or-minussubscript𝐸𝑔Planck-constant-over-2-pi\omega=\pm E_{g}/\hbaritalic_ω = ± italic_E start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT / roman_ℏ, which we shall call single frequency absorption. We also note that the inequality (21) remains valid even when the optical gap Egsubscript𝐸𝑔E_{g}italic_E start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT is replaced with the spectral gap, although the bounds would generally become less tight.

Interestingly, ReW1superscript𝑊1\real W^{1}start_OPERATOR roman_Re end_OPERATOR italic_W start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT is related through the Planck constant to the quantum fluctuation in the polarization of our system, which arises from electron’s zero-point motion and is obtained by quantization: pα′→−i⁢ℏ⁢∂α′→subscriptsuperscript𝑝′𝛼𝑖Planck-constant-over-2-pisubscriptsuperscript′𝛼p^{\prime}_{\alpha}\rightarrow-i\hbar\partial^{\prime}_{\alpha}italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT → - italic_i roman_ℏ ∂ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT. Noting that the total polarization is given by δ⁢P=e⁢∑iδ⁢xi=e⁢∑αZα⁢xα′𝛿𝑃𝑒subscript𝑖𝛿subscript𝑥𝑖𝑒subscript𝛼subscript𝑍𝛼superscriptsubscript𝑥𝛼′\delta P=e\sum_{i}\delta x_{i}=e\sum_{\alpha}Z_{\alpha}x_{\alpha}^{\prime}italic_δ italic_P = italic_e ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_δ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_e ∑ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_Z start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and ⟨xα′⁢xβ′⟩=δα⁢β⁢ℏ/(2⁢m⁢ωα)expectation-valuesuperscriptsubscript𝑥𝛼′superscriptsubscript𝑥𝛽′subscript𝛿𝛼𝛽Planck-constant-over-2-pi2𝑚subscript𝜔𝛼\expectationvalue{x_{\alpha}^{\prime}x_{\beta}^{\prime}}=\delta_{\alpha\beta}% \hbar/(2m\omega_{\alpha})⟨ start_ARG italic_x start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG ⟩ = italic_δ start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT roman_ℏ / ( 2 italic_m italic_ω start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ), we can rewrite ReW1superscript𝑊1\real W^{1}start_OPERATOR roman_Re end_OPERATOR italic_W start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT as

The frequency for each mode is ωα=kα/msubscript𝜔𝛼subscript𝑘𝛼𝑚\omega_{\alpha}=\sqrt{k_{\alpha}/m}italic_ω start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT = square-root start_ARG italic_k start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT / italic_m end_ARG, leading to the energy gap of ℏ⁢ωαPlanck-constant-over-2-pisubscript𝜔𝛼\hbar\omega_{\alpha}roman_ℏ italic_ω start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT after quantization.

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where Π⁢(𝒒,ω)Π𝒒𝜔\Pi({\bf\it q},\omega)roman_Π ( bold_italic_q , italic_ω ) describes the density response to the external potential Vextsubscript𝑉extV_{\rm ext}italic_V start_POSTSUBSCRIPT roman_ext end_POSTSUBSCRIPT, as defined in the main text. We will also use the following relation between S⁢(𝒒,ω)𝑆𝒒𝜔S({\bf\it q},\omega)italic_S ( bold_italic_q , italic_ω ) and S⁢(−𝒒,−ω)𝑆𝒒𝜔S(-{\bf\it q},-\omega)italic_S ( - bold_italic_q , - italic_ω ):

Combining Eq. (14), (15) and (16), we obtain a general relation between optical conductivity and ground state static structural factor:

Finally, for the sake of completeness, we discuss the relation of the quantum weight K𝐾Kitalic_K defined by static structure factor to quantum geometry. As first shown by SWM, the first-negative moment of optical conductivity ReW1superscript𝑊1\real W^{1}start_OPERATOR roman_Re end_OPERATOR italic_W start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT is related to the many-body quantum metric defined over the twisted boundary condition [4]. Then, based on the relation Eq. (18), the quantum weight K𝐾Kitalic_K is also related to the many-body quantum metric. Here, we generalize the work of SWM which considers systems with a unique ground state and introduce the many-body quantum metric G𝐺Gitalic_G for the general case of r𝑟ritalic_r-fold degenerated ground states (related to each other by spontaneous symmetry breaking or topological order):

Here, |Ψi⁢θ⟩ketsubscriptΨ𝑖𝜃\ket{\Psi_{i\theta}}| start_ARG roman_Ψ start_POSTSUBSCRIPT italic_i italic_θ end_POSTSUBSCRIPT end_ARG ⟩ is the i𝑖iitalic_i-th ground state under the twisted boundary condition specified by 𝜽𝜽{\bf\it\theta}bold_italic_θ: Ψi⁢𝜿⁢(𝒓1,…,𝒓n+𝑳μ,…,𝒓N)=ei⁢θμ⁢Ψi⁢𝜿⁢(𝒓1,…,𝒓n,…,𝒓N)subscriptΨ𝑖𝜿subscript𝒓1…subscript𝒓𝑛subscript𝑳𝜇…subscript𝒓𝑁superscript𝑒𝑖subscript𝜃𝜇subscriptΨ𝑖𝜿subscript𝒓1…subscript𝒓𝑛…subscript𝒓𝑁\Psi_{i{\bf\it\kappa}}({\bf\it r}_{1},\dots,{\bf\it r}_{n}+{\bf\it L}_{\mu},% \dots,{\bf\it r}_{N})=e^{i\theta_{\mu}}\Psi_{i{\bf\it\kappa}}({\bf\it r}_{1},% \dots,{\bf\it r}_{n},\dots,{\bf\it r}_{N})roman_Ψ start_POSTSUBSCRIPT italic_i bold_italic_κ end_POSTSUBSCRIPT ( bold_italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , bold_italic_r start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT + bold_italic_L start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT , … , bold_italic_r start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ) = italic_e start_POSTSUPERSCRIPT italic_i italic_θ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_Ψ start_POSTSUBSCRIPT italic_i bold_italic_κ end_POSTSUBSCRIPT ( bold_italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , bold_italic_r start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , … , bold_italic_r start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ) where 𝑳μ=(0,…,Lμ,…,0)subscript𝑳𝜇0…subscript𝐿𝜇…0{\bf\it L}_{\mu}=(0,\dots,L_{\mu},\dots,0)bold_italic_L start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT = ( 0 , … , italic_L start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT , … , 0 ) with Lμsubscript𝐿𝜇L_{\mu}italic_L start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT the system size in μ𝜇\muitalic_μ-direction, and P𝜽=|Ψ𝜽⟩⟨Ψ𝜽|subscript𝑃𝜽subscriptΨ𝜽subscriptΨ𝜽P_{{\bf\it\theta}}=\outerproduct{\Psi_{{\bf\it\theta}}}{\Psi_{{\bf\it\theta}}}italic_P start_POSTSUBSCRIPT bold_italic_θ end_POSTSUBSCRIPT = | start_ARG roman_Ψ start_POSTSUBSCRIPT bold_italic_θ end_POSTSUBSCRIPT end_ARG ⟩ ⟨ start_ARG roman_Ψ start_POSTSUBSCRIPT bold_italic_θ end_POSTSUBSCRIPT end_ARG | is the projector operator associated with the ground state.

We start by considering generalized optical weights Wisuperscript𝑊𝑖W^{i}italic_W start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT for insulating states [3]. Wisuperscript𝑊𝑖W^{i}italic_W start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT is defined as the negative i𝑖iitalic_i-th moment of the absorptive part of the optical conductivity σ⁢(ω)𝜎𝜔\sigma(\omega)italic_σ ( italic_ω ),

Using fluctuation-dissipation theorem [5], we show that as a ground state property, the quantum weight defined here is directly related to the negative-first moment of optical conductivity through the Planck constant. This establishes the equivalence between the old and new definition of quantum weight in terms of optical conductivity and static structure factor respectively.

We calculated the bound on the quantum weight for real materials, and the results are shown in Fig. 1. Our bound (21) gives a fairly good estimate of the quantum weight only from the electric susceptibility (or equivalently the dielectric constant), the energy gap, and the electron density. The most remarkable case is cubic boron nitride (c-BN). c-BN is an indirect gap insulator with direct gap 14.5 eVtimes14.5electronvolt14.5\text{\,}\mathrm{eV}start_ARG 14.5 end_ARG start_ARG times end_ARG start_ARG roman_eV end_ARG and the dielectric constant ϵ=4.46italic-ϵ4.46\epsilon=4.46italic_ϵ = 4.46 [9]. From the dielectric constant, the electric susceptibility χ=ϵ0⁢(ϵ−1)𝜒subscriptitalic-ϵ0italic-ϵ1\chi=\epsilon_{0}(\epsilon-1)italic_χ = italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_ϵ - 1 ) is given by χ=3.1×10−11 F m−1𝜒times3.1E-11timesfaradmeter1\chi=$3.1\text{\times}{10}^{-11}\text{\,}\mathrm{F}\text{\,}{\mathrm{m}}^{-1}$italic_χ = start_ARG start_ARG 3.1 end_ARG start_ARG times end_ARG start_ARG power start_ARG 10 end_ARG start_ARG - 11 end_ARG end_ARG end_ARG start_ARG times end_ARG start_ARG start_ARG roman_F end_ARG start_ARG times end_ARG start_ARG power start_ARG roman_m end_ARG start_ARG - 1 end_ARG end_ARG end_ARG. The cubic unit cell of c-BN has a lattice constant 3.62 Åtimes3.62angstrom3.62\text{\,}\mathrm{\SIUnitSymbolAngstrom}start_ARG 3.62 end_ARG start_ARG times end_ARG start_ARG roman_Å end_ARG and contains 4 boron and nitrogen atoms, each having 5 and 7 electrons, hence the total electron density is n=1.02×1024 cm−3𝑛times1.02E24centimeter3n=$1.02\text{\times}{10}^{24}\text{\,}{\mathrm{cm}}^{-3}$italic_n = start_ARG start_ARG 1.02 end_ARG start_ARG times end_ARG start_ARG power start_ARG 10 end_ARG start_ARG 24 end_ARG end_ARG end_ARG start_ARG times end_ARG start_ARG power start_ARG roman_cm end_ARG start_ARG - 3 end_ARG end_ARG. Then the full spectral weight is given by Wa⁢a0=4.5×1022 m−1 s−1 Ω−1subscriptsuperscript𝑊0𝑎𝑎times4.5E22timesmeter1second1ohm1W^{0}_{aa}=$4.5\text{\times}{10}^{22}\text{\,}{\mathrm{m}}^{-1}\text{\,}{% \mathrm{s}}^{-1}\text{\,}{\mathrm{\SIUnitSymbolOhm}}^{-1}$italic_W start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a end_POSTSUBSCRIPT = start_ARG start_ARG 4.5 end_ARG start_ARG times end_ARG start_ARG power start_ARG 10 end_ARG start_ARG 22 end_ARG end_ARG end_ARG start_ARG times end_ARG start_ARG start_ARG power start_ARG roman_m end_ARG start_ARG - 1 end_ARG end_ARG start_ARG times end_ARG start_ARG power start_ARG roman_s end_ARG start_ARG - 1 end_ARG end_ARG start_ARG times end_ARG start_ARG power start_ARG roman_Ω end_ARG start_ARG - 1 end_ARG end_ARG end_ARG. Since K𝐾Kitalic_K and χ𝜒\chiitalic_χ are isotropic in c-BN, we find that the quantum weight Ka⁢asubscript𝐾𝑎𝑎K_{aa}italic_K start_POSTSUBSCRIPT italic_a italic_a end_POSTSUBSCRIPT is bounded to a remarkably narrow interval: 0.9 Å−1≤Ka⁢a≤1.7 Åtimes0.9angstrom1subscript𝐾𝑎𝑎times1.7angstrom$0.9\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-1}$\leq K_{aa}\leq$1.7\text{\,% }\mathrm{\SIUnitSymbolAngstrom}$start_ARG 0.9 end_ARG start_ARG times end_ARG start_ARG power start_ARG roman_Å end_ARG start_ARG - 1 end_ARG end_ARG ≤ italic_K start_POSTSUBSCRIPT italic_a italic_a end_POSTSUBSCRIPT ≤ start_ARG 1.7 end_ARG start_ARG times end_ARG start_ARG roman_Å end_ARG. It should be emphasized that our analysis applies to any electronic systems, including disordered and/or interacting systems. This result demonstrates that our analysis is powerful in understanding quantum materials.

with Zα⁢e=∑icα⁢i⁢esubscript𝑍𝛼𝑒subscript𝑖subscript𝑐𝛼𝑖𝑒Z_{\alpha}e=\sum_{i}c_{\alpha i}eitalic_Z start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_e = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_α italic_i end_POSTSUBSCRIPT italic_e the effective charge of the normal mode α𝛼\alphaitalic_α. Here, for simplicity we consider a one-dimensional system with length L𝐿Litalic_L to illustrate the essential physics. It then follows

where |Φ⟩ketΦ\ket{\Phi}| start_ARG roman_Φ end_ARG ⟩ denotes the ground state of an atom, ⟨rα⟩=⟨Φ|rα|Φ⟩expectation-valuesubscript𝑟𝛼expectation-valuesubscript𝑟𝛼ΦΦ\expectationvalue{r_{\alpha}}=\matrixelement{\Phi}{r_{\alpha}}{\Phi}⟨ start_ARG italic_r start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_ARG ⟩ = ⟨ start_ARG roman_Φ end_ARG | start_ARG italic_r start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_ARG | start_ARG roman_Φ end_ARG ⟩ is the expectation value of rαsubscript𝑟𝛼r_{\alpha}italic_r start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT in the ground state, and rα=∑irα,isubscript𝑟𝛼subscript𝑖subscript𝑟𝛼𝑖r_{\alpha}=\sum_{i}r_{\alpha,i}italic_r start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT italic_α , italic_i end_POSTSUBSCRIPT 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.

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where σα⁢βabs≡(σα⁢β+σβ⁢α*)/2subscriptsuperscript𝜎abs𝛼𝛽subscript𝜎𝛼𝛽subscriptsuperscript𝜎𝛽𝛼2\sigma^{\rm abs}_{\alpha\beta}\equiv(\sigma_{\alpha\beta}+\sigma^{*}_{\beta% \alpha})/2italic_σ start_POSTSUPERSCRIPT roman_abs end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT ≡ ( italic_σ start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT + italic_σ start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_β italic_α end_POSTSUBSCRIPT ) / 2 is composed of the real part of longitudinal optical conductivity and the imaginary part of optical Hall conductivity. For isotropic two-dimensional systems, σα⁢βabs=Re(σx⁢x)⁡δα⁢β+i⁢Im(σx⁢y)⁡ϵα⁢βsubscriptsuperscript𝜎abs𝛼𝛽subscript𝜎𝑥𝑥subscript𝛿𝛼𝛽𝑖subscript𝜎𝑥𝑦subscriptitalic-ϵ𝛼𝛽\sigma^{\rm abs}_{\alpha\beta}=\real(\sigma_{xx})\delta_{\alpha\beta}+i% \imaginary(\sigma_{xy})\epsilon_{\alpha\beta}italic_σ start_POSTSUPERSCRIPT roman_abs end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT = start_OPERATOR roman_Re end_OPERATOR ( italic_σ start_POSTSUBSCRIPT italic_x italic_x end_POSTSUBSCRIPT ) italic_δ start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT + italic_i start_OPERATOR roman_Im end_OPERATOR ( italic_σ start_POSTSUBSCRIPT italic_x italic_y end_POSTSUBSCRIPT ) italic_ϵ start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT. Throughout this work, we consider gapped systems so that the absorptive part of optical conductivity vanishes at low frequency:

Eq. (17), together with its 𝒒→0→𝒒0{\bf\it q}\rightarrow 0bold_italic_q → 0 limit Eq. (18), is a key result of our work. It constitutes a new optical sum rule relating the negative first moment of longitudinal optical conductivity to ground-state structure factor through the Planck constant, which is a generalization of the f𝑓fitalic_f sum rule relating the optical spectral weight to the electron density.

By the fluctuation-dissipation theorem [5], ImΠ⁢(𝒒,ω)Π𝒒𝜔\imaginary\Pi({\bf\it q},\omega)start_OPERATOR roman_Im end_OPERATOR roman_Π ( bold_italic_q , italic_ω ) is directly related to the dynamical structure factor:

This relation Eq. (10) is noteworthy for two reasons. The optical weight ReW1superscript𝑊1\real W^{1}start_OPERATOR roman_Re end_OPERATOR italic_W start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT in the left-hand side is finite even for classical systems, while the polarization fluctuation in the right-hand side is purely quantum-mechanical in nature and vanishes in ℏ→0→Planck-constant-over-2-pi0\hbar\to 0roman_ℏ → 0 limit. The left hand side involves optical conductivity at all frequencies, while the right hand side is a ground state property.

In order to properly define polarization fluctuation for insulators in general, let us consider the static structure factor that measures equal-time density-density correlation in the ground state: S𝒒≡(1/V)⁢(⟨ρ^𝒒⁢ρ^−𝒒⟩−⟨ρ^𝒒⟩⁢⟨ρ^−𝒒⟩)subscript𝑆𝒒1𝑉expectation-valuesubscript^𝜌𝒒subscript^𝜌𝒒expectation-valuesubscript^𝜌𝒒expectation-valuesubscript^𝜌𝒒S_{{\bf\it q}}\equiv(1/V)(\expectationvalue{\hat{\rho}_{{\bf\it q}}\hat{\rho}_% {-{\bf\it q}}}-\expectationvalue{\hat{\rho}_{{\bf\it q}}}\expectationvalue{% \hat{\rho}_{-{\bf\it q}}})italic_S start_POSTSUBSCRIPT bold_italic_q end_POSTSUBSCRIPT ≡ ( 1 / italic_V ) ( ⟨ start_ARG over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT bold_italic_q end_POSTSUBSCRIPT over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT - bold_italic_q end_POSTSUBSCRIPT end_ARG ⟩ - ⟨ start_ARG over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT bold_italic_q end_POSTSUBSCRIPT end_ARG ⟩ ⟨ start_ARG over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT - bold_italic_q end_POSTSUBSCRIPT end_ARG ⟩ ). In particular, we focus on the quadratic coefficient of S𝒒subscript𝑆𝒒S_{{\bf\it q}}italic_S start_POSTSUBSCRIPT bold_italic_q end_POSTSUBSCRIPT at small nonzero 𝒒𝒒{\bf\it q}bold_italic_q:

with the absorptive part of the conductivity tensor σabs⁢(ω)=(σ⁢(ω)+σ⁢(ω)†)/2superscript𝜎abs𝜔𝜎𝜔𝜎superscript𝜔†2\sigma^{\rm abs}(\omega)=(\sigma(\omega)+\sigma(\omega)^{\dagger})/2italic_σ start_POSTSUPERSCRIPT roman_abs end_POSTSUPERSCRIPT ( italic_ω ) = ( italic_σ ( italic_ω ) + italic_σ ( italic_ω ) start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ) / 2 and the quantum weight is defined as the quadratic coefficient in 𝒒𝒒{\bf\it q}bold_italic_q of the static structure factor S𝒒=(1/V)⁢⟨ρ𝒒⁢ρ−𝒒⟩subscript𝑆𝒒1𝑉expectation-valuesubscript𝜌𝒒subscript𝜌𝒒S_{{\bf\it q}}=(1/V)\expectationvalue{\rho_{{\bf\it q}}\rho_{-{\bf\it q}}}italic_S start_POSTSUBSCRIPT bold_italic_q end_POSTSUBSCRIPT = ( 1 / italic_V ) ⟨ start_ARG italic_ρ start_POSTSUBSCRIPT bold_italic_q end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT - bold_italic_q end_POSTSUBSCRIPT end_ARG ⟩ as

with ρ^𝒒=∫d𝒓⁢e−i⁢𝒒⋅𝒓⁢ρ^⁢(𝒓)subscript^𝜌𝒒𝒓superscript𝑒dot-product𝑖𝒒𝒓^𝜌𝒓\hat{\rho}_{{\bf\it q}}=\int\differential{{\bf\it r}}e^{-i{\bf\it q}% \dotproduct{\bf\it r}}\hat{\rho}({\bf\it r})over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT bold_italic_q end_POSTSUBSCRIPT = ∫ roman_d start_ARG bold_italic_r end_ARG italic_e start_POSTSUPERSCRIPT - italic_i bold_italic_q ⋅ bold_italic_r end_POSTSUPERSCRIPT over^ start_ARG italic_ρ end_ARG ( bold_italic_r ) the charge density operator with wavevector 𝒒𝒒{\bf\it q}bold_italic_q and V𝑉Vitalic_V the volume of the system. We call the quadratic coefficient K𝐾Kitalic_K defined by Eq. (11) quantum weight. Importantly, K𝐾Kitalic_K can be nonzero only because of quantum fluctuation. In the classical limit (ℏ=0Planck-constant-over-2-pi0\hbar=0roman_ℏ = 0), the ground state of a periodic system is a periodic array of electron point charges, and correspondingly, the static structure factor is composed of δ𝛿\deltaitalic_δ functions centered at reciprocal lattice vectors and vanishes everywhere else, leading to K=0𝐾0K=0italic_K = 0. Hence quantum weight measures the degree of “quantumness” in the insulating ground state.

We have shown that quantum weight is both a fundamental ground state property of insulators and an important material parameter related to optical properties. We now further derive lower and upper bounds on quantum weight in terms of common material parameters: electron density, energy gap and dielectric constant. First, noting that the real part of the longitudinal optical conductivity is always non-negative and can be finite only for |ω|≥Eg/ℏ𝜔subscript𝐸𝑔Planck-constant-over-2-pi\absolutevalue{\omega}\geq E_{g}/\hbar| start_ARG italic_ω end_ARG | ≥ italic_E start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT / roman_ℏ, the following inequality among optical weights always holds:

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where 𝑬⁢(𝒒,ω)=−i⁢𝒒⁢Vext⁢(𝒒,ω)𝑬𝒒𝜔𝑖𝒒subscript𝑉ext𝒒𝜔{\bf\it E}({\bf\it q},\omega)=-i{\bf\it q}V_{\rm ext}({\bf\it q},\omega)bold_italic_E ( bold_italic_q , italic_ω ) = - italic_i bold_italic_q italic_V start_POSTSUBSCRIPT roman_ext end_POSTSUBSCRIPT ( bold_italic_q , italic_ω ) is the external electric field. Due to the continuity equation ∂ρ∕∂t+∇⋅𝒋=0partial-derivative𝑡𝜌⋅∇𝒋0\partialderivative*{\rho}{t}+\nabla\cdot{\bf\it j}=0∕ start_ARG ∂ start_ARG italic_ρ end_ARG end_ARG start_ARG ∂ start_ARG italic_t end_ARG end_ARG + ∇ ⋅ bold_italic_j = 0, σ𝜎\sigmaitalic_σ and ΠΠ\Piroman_Π are directly related:

It is well known that quantum states of matter with an energy gap have vanishing dc longitudinal conductivity at zero temperature, while the optical conductivity is generally nonzero at frequencies above the gap. Interestingly, the ground state property of an insulating state still has important bearings on its optical conductivity. Consider the real part of longitudinal optical conductivity Reσx⁢x⁢(ω)subscript𝜎𝑥𝑥𝜔\real\sigma_{xx}(\omega)start_OPERATOR roman_Re end_OPERATOR italic_σ start_POSTSUBSCRIPT italic_x italic_x end_POSTSUBSCRIPT ( italic_ω ), which determines the amount of optical absorption in the medium. It is well known that its zeroth moment known as the optical spectral weight is related to the electron density in the system through the f𝑓fitalic_f-sum rule [1]. Higher order moments of optical conductivity are much less studied. The negative-second moment is directly related to electric susceptibility through the Kramers-Kronig relation [2]. Recently, we employed the negative-first moment of both longitudinal and Hall conductivities to derive a universal upper bound on the energy gap of (integer or fractional) Chern insulators [3].

The static structure factor S𝒒subscript𝑆𝒒S_{{\bf\it q}}italic_S start_POSTSUBSCRIPT bold_italic_q end_POSTSUBSCRIPT can be experimentally obtained from X-ray scattering experiments, while the optical weight ReW1superscript𝑊1\real W^{1}start_OPERATOR roman_Re end_OPERATOR italic_W start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT can be determined from the experimentally measured optical conductivity. The ratio between S𝒒/q2subscript𝑆𝒒superscript𝑞2S_{{\bf\it q}}/q^{2}italic_S start_POSTSUBSCRIPT bold_italic_q end_POSTSUBSCRIPT / italic_q start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT and ReW1superscript𝑊1\real W^{1}start_OPERATOR roman_Re end_OPERATOR italic_W start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT yields a fundamental physical constant, the Planck constant. This provides a way of determining the Planck constant by optical spectroscopy measurements of basic material properties.

Here we derive the following relation between the quantum weight K𝐾Kitalic_K and the negative-first moment of optical conductivity W1superscript𝑊1W^{1}italic_W start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT for general insulators:

For this system of strongly localized electrons, a uniform electric field couples to the center of mass displacement ∑iδ⁢xi=∑αZα⁢xα′subscript𝑖𝛿subscript𝑥𝑖subscript𝛼subscript𝑍𝛼subscriptsuperscript𝑥′𝛼\sum_{i}\delta x_{i}=\sum_{\alpha}Z_{\alpha}x^{\prime}_{\alpha}∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_δ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_Z start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT, and the real part of the optical conductivity is well known from that of the harmonic oscillator:

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To conclude, we have established quantum weight, defined through static structure factor, as a key material parameter that is connected to a variety of physical observables. The quantum weight represents the quantum fluctuation in electrons’ center of mass. We derived its general relation to optical conductivity, dielectric constant, quantum geometry, and energy gap. Our results apply to all insulators, including strongly correlated systems. Experimental determination by X-ray scattering as well as first-principles calculation of quantum weight for real materials [13] are called for.

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Combining Eq. (19) with i=0,1𝑖01i=0,1italic_i = 0 , 1 and the expressions for optical weights W0,W1,W2superscript𝑊0superscript𝑊1superscript𝑊2W^{0},W^{1},W^{2}italic_W start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT , italic_W start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , italic_W start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT shown in Eq. (3), (4) and (18), we obtain the bound on the quantum weight:

It is important to note that there is no singularity at small ω𝜔\omegaitalic_ω and at small 𝒒𝒒{\bf\it q}bold_italic_q in Eq. (14) for insulators 111We note that the static structure factor for metallic systems at small 𝒒𝒒{\bf\it q}bold_italic_q is dominated by |q|𝑞\absolutevalue{q}| start_ARG italic_q end_ARG |-linear term, in contrast to insulators where the leading order term is quadratic. . Correspondingly, the negative-first moment of the real part of optical conductivity at 𝒒=0𝒒0{{\bf\it q}}=0bold_italic_q = 0 is related to the imaginary part of density response Π⁢(𝒒,ω)Π𝒒𝜔\Pi({{\bf\it q}},\omega)roman_Π ( bold_italic_q , italic_ω ) at small 𝒒𝒒{{\bf\it q}}bold_italic_q integrated over the frequency:

Next, we derive lower and upper bounds on the quantum weight, in terms of the electron density, the energy gap, and the dielectric constant. These bounds apply to any insulating system, and represent a universal relation between ground state property (quantum weight), optical response, thermodynamic response (dielectric constant), and the energy gap of the system. Remarkably, our bounds can provide a good estimate of the quantum weight of real materials.

where the dynamical structural factor is defined as S⁢(𝒒,ω)=1V⁢∫−∞∞dt⁢ei⁢ω⁢t⁢⟨ρ^𝒒⁢(t)⁢ρ^−𝒒⁢(0)⟩𝑆𝒒𝜔1𝑉superscriptsubscript𝑡superscript𝑒𝑖𝜔𝑡expectation-valuesubscript^𝜌𝒒𝑡subscript^𝜌𝒒0S({\bf\it q},\omega)=\frac{1}{V}\int_{-\infty}^{\infty}\differential{t}e^{i% \omega t}\expectationvalue{\hat{\rho}_{{\bf\it q}}(t)\hat{\rho}_{-{\bf\it q}}(% 0)}italic_S ( bold_italic_q , italic_ω ) = divide start_ARG 1 end_ARG start_ARG italic_V end_ARG ∫ start_POSTSUBSCRIPT - ∞ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT roman_d start_ARG italic_t end_ARG italic_e start_POSTSUPERSCRIPT italic_i italic_ω italic_t end_POSTSUPERSCRIPT ⟨ start_ARG over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT bold_italic_q end_POSTSUBSCRIPT ( italic_t ) over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT - bold_italic_q end_POSTSUBSCRIPT ( 0 ) end_ARG ⟩. Here, ρ^𝒒⁢(t)=∫d𝒓⁢e−i⁢𝒒⋅𝒓⁢ρ^⁢(𝒓,t)subscript^𝜌𝒒𝑡𝒓superscript𝑒dot-product𝑖𝒒𝒓^𝜌𝒓𝑡\hat{\rho}_{{\bf\it q}}(t)=\int\differential{{\bf\it r}}e^{-i{\bf\it q}% \dotproduct{\bf\it r}}\hat{\rho}({\bf\it r},t)over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT bold_italic_q end_POSTSUBSCRIPT ( italic_t ) = ∫ roman_d start_ARG bold_italic_r end_ARG italic_e start_POSTSUPERSCRIPT - italic_i bold_italic_q ⋅ bold_italic_r end_POSTSUPERSCRIPT over^ start_ARG italic_ρ end_ARG ( bold_italic_r , italic_t ) is the density operator with wavevector 𝒒𝒒{\bf\it q}bold_italic_q in the Heisenberg picture and V𝑉Vitalic_V is the volume of the system.

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 X-ray scattering.