Introduction

The fractional charge, widely existing in a variety of topological systems, directly relates to the non-trivial topology of electronic states and is of great significance in the field of condensed matter physics1,2,3,4,5,6,7,8,9. There are mainly two categories of fractional charge: the moving type and the static type. The former kind acts as charge carrier in the fractional quantum Hall (FQH) states1,2,3,4. And the later one, can be found as topological edge states in many topological materials, such as one-dimensional systems with the Jackiw-Rebbi mechanism5,6,7,8,9, topological crystalline insulators with disclination10, and higher-order topological insulators11,12,13,14,15,16,17. Remarkably, both the moving18,19,20,21 and the static fractional charges22,23,24,25,26 follow the quantum statistics beyond the Boson/Fermion statistics and can be employed for topological quantum computation.

Back in 1997, the moving fractional charge in the FQH state was experimentally verified via the transport measurement of shot noise27,28. The detection of the static topological fractional charge (TFC), in contrast, is quite challenging in condensed matter systems. The direct measurement of TFC via scanning tunneling spectroscopy is hindered by the experimental resolution29. In addition to that, earlier theories have proposed an approach to detect the Jackiw-Rebbi zero mode through its π-period Aharanov–Bohm oscillation22,30. Although the presence of the zero mode could be certificated, the charge amount of the static TFC cannot be determined in such a proposal. In another earlier theoretical work31, a proposal is raised based on an effective model where the TFC trapped by the magnetic domains can be detected via the Coulomb blockade. Such a simplified scheme, in which the entire electronic structure as well as the disorder effect are excluded, has not been confirmed in a more realistic lattice model. Another drawback of this scheme is that the spatial distribution of the TFC cannot be obtained. It is worth noting that although the experimental measurement of TFC has recently been reported in classical wave systems29,32,33,34, the elusive disorder effect remains to be further investigated. Significantly, the TFC state here is occupied only when an input with certain frequency is provided. The absence of Fermi surface hinders the verification of the quantum statistics of the TFC in classical wave systems.

In recent years, remarkable progresses for the experimental realization of Su–Schrieffer–Heeger (SSH) model in condensed matter systems have been made by engineering graphene nanoribbons35,36,37,38. By precisely decorating the graphene nanoribbon edge profiles, both the topologically trivial and non-trivial states are manifested through scanning tunneling spectroscopy. Owing to these progress, the enthusiasm for the discrimination and measurement of TFC in topological materials has been highly raised.

In this theoretic work, the static TFC in topological systems is obtained through the electronic transport of a quantum dot (QD) coupled to the topological system. For the SSH model, both the e/2 TFC and its spatial distribution is obtained from transport results. The disorder effect, which is widely presented in topological systems supporting TFC but has not been thoroughly studied yet, is also intensively investigated in this work. The Anderson disorder breaking certain symmetry related to the TFC generally has significant effect on the TFC. However, such effect is greatly suppressed in one-dimensional system that for disordered SSH chain, the TFC amount measured is still in good agreement with the clean result. For certain higher-order topological materials, e.g. some armchair-edged breathing kagome material, the well-localized TFC possessing 2e/3 charge could also be confirmed by our transport scheme even under disorder.

Results

Model

Our transport system measuring TFC is shown in Fig. 1a that a QD (the red ball) with two energy levels is connected to two separated terminals (the grey ones). The QD can be moved to the vicinity of the target atom of the topological system (e.g. a SSH chain shown by blue balls) and weakly bonds to it, leading to a Coulomb interaction39,40,41,42. In this way, the charge possessed by the target atom can be extracted from the shift of the differential conductance resonance peak43,44,45. Then the amount and spatial distribution of the TFC possessed by the topological edge states can be extracted by measuring all the atoms involved.

Fig. 1: The schematic setup and related measurement features of TFC in SSH model.
figure 1

a Schematic plot of the transport device measuring the TFC in the SSH chain. b, c Band structure for non-trivial/trivial SSH model and the QD levels. The number of electron of the zero mode nt in b is the same as the increased portion of the bulk states' number of electron in c, as denoted by the orange areas. d–i Fermi level dependence of the conductance curves for d trivial end of the SSH chain with t1 = 0.6t, and t2 = 0.37t; and e–i non-trivial end of the same SSH chain with t1 = 0.37t, and t2 = 0.6t. The parameters are drawn from a practical experiment where the SSH chain is built from the graphene naoribbon37 and t = 1eV is used as the energy unit. Other parameters are U = 0.05t, tk = 0.1t, tc = 0.01t.

Full size image

There are two basic rules for this TFC measurement scheme: (i) Any topological edge state carries an integer number of electrons. (ii) A trivial insulator is charge neutral when the Fermi energy lies inside the gap. The first rule enables us to determine the Coulomb interaction strength, while the second rule enables us to count the charge by comparing the charge difference between a trivial insulator and its non-trivial counterpart. Remarkably, in the presence of disorder, whether the second rule is still valid depends on the symmetry of the disorder term and the dimension of the topological material.

We first apply our method to the SSH chain composed of an odd number of sites, in which the TFC is presented only in one end of the chain (non-trivial end), while another end of this chain (trivial end) behaves like a trivial insulator. In this way, the measurement and the charge difference comparison can be conducted in a single device. The total Hamiltonian of the system is in the form of:

$$H={H}_{{{{\rm{QD}}}}}+{H}_{{{{\rm{SSH}}}}}+{H}_{{{{\rm{c}}}}},$$
(1)

where the two-level QD (with level index i) is connected to the source and drain (with index α = L, R) as \({H}_{{{{\rm{QD}}}}}={\sum }_{k\alpha }{\varepsilon }_{k\alpha }{C}_{k\alpha }^{{\dagger} }{C}_{k\alpha }+{\sum }_{i}{\varepsilon }_{i}{a}_{i}^{{\dagger} }{a}_{i}+{\sum }_{k\alpha ,i}{t}_{k}({C}_{k\alpha }^{{\dagger} }{a}_{i}+h.c.)\) with \({C}_{k\alpha }^{{\dagger} }\) and \({a}_{i}^{{\dagger} }\) the creation operators in terminals and QD, respectively. Here εi is the i-th energy level for the QD and tk is the coupling between terminals and the QD. \({H}_{{{{\rm{SSH}}}}}={\sum }_{n}({t}_{1}{b}_{2n-1}^{{\dagger} }{b}_{2n}+{t}_{2}{b}_{2n+1}^{{\dagger} }{b}_{2n}+h.c.)\) describes the SSH chain and t1/2 is the alternative coupling between nearest sites. The coupling term reads \({H}_{{{{\rm{c}}}}}={\sum }_{i}[U{a}_{i}^{{\dagger} }{a}_{i}{b}_{s}^{{\dagger} }{b}_{s}+{t}_{c}({a}_{i}^{{\dagger} }{b}_{s}+{b}_{s}^{{\dagger} }{a}_{i})]\) with U the Coulomb interaction strength, and tc the direct tunneling between the QD and the target site (denoted by index s) of the SSH chain.

In the calculation, the on-site energy of the QD and its Coulomb interaction to the target site constitute the unperturbed Hamiltonian H0 (3 × 3 matrix in the basis of {a1, a2, bs}). The rest part of H [see Eq. (1)] is regarded as perturbations (see Methods section). In this way, the differential conductance of the system is obtained from the Green’s function46,47

$${{{\mathcal{G}}}}({E}_{F})=\frac{{e}^{2}}{h}{{{\rm{Tr}}}}[{{{\Gamma }}}_{L}^{r}{G}^{r}{{{\Gamma }}}_{R}^{r}{({G}^{r})}^{{\dagger} }].$$
(2)

Here \({{{\Gamma }}}_{L/R}^{r}\) is the symmetric line-width function of the terminals and Gr is the retarded Green’s function. The unperturbed retarded Green’s function for the n-th level in QDs from H0 reads: \({g}_{n}^{r}(E)=1/(E-{\varepsilon }_{n}-U\langle {n}_{s}\rangle +i\eta )\) with η a positive infinitesimal and 〈ns〉 the total electron number below the Fermi level. Gr is numerically obtained from the Dyson equation for gr and they share similar features. When the QD is weakly coupled to the topological system, the resonance peaks of the \({{{\mathcal{G}}}}({E}_{F})\) shift from εi to εi + U〈ns〉 due to the Coulomb blockade48,49. The weak coupling assumption is quite reasonable since the direct tunneling tc exponentially decays with the distance while the Coulomb interaction U is inversely proportional to the distance. In practice, U is a priorly unknown parameter which should also be extracted from the measurement.

Measurement of e/2 charge in SSH models

For a QD consisting of two energy levels ε1, ε2, when both these two levels are below the fractionally-charged subgap zero mode of the SSH chain as ε1 < ε2 < 0 [Fig. 1b], the corresponding conductance peaks are separated by d1 = ε2 − ε1. Alternatively, by tuning the gate voltage Vg, these two levels can be elevated to \({\varepsilon }_{1}^{{\prime} }\) and \({\varepsilon }_{2}^{{\prime} }\) that \({\varepsilon }_{1}^{{\prime} } \,<\, 0\) and \({\varepsilon }_{2}^{{\prime} } \,>\, 0\) [Fig. 1b]. Since \({\varepsilon }_{1}^{{\prime} }={\varepsilon }_{1}+e{V}_{g}\) and \({\varepsilon }_{2}^{{\prime} }={\varepsilon }_{2}+e{V}_{g}+U{n}_{t}\), where nt denotes the amount of charge of the zero mode at the site being measured, now the conductance peaks are separated as \({d}_{2}={\varepsilon }_{2}^{{\prime} }-{\varepsilon }_{1}^{{\prime} }={d}_{1}+U{n}_{t}\). Thus, at this specific site, d2 − d1 = Unt, which measures the product of U and nt. Owing to the integer charge rule mentioned earlier (∑nt = 1), when all the sites that the TFC resides in are considered, we have ∑(d2 − d1) = U ∑ nt = U and consequently, the value of U is measured.

Figure 1e–i display the \({{{\mathcal{G}}}}-{E}_{F}\) relations for all the odd sites at the non-trivial end of the SSH chain for both (i) ε1, ε2 < 0; and (ii) \({\varepsilon }_{1}^{{\prime} } \,<\, 0\), \({\varepsilon }_{2}^{{\prime} } \,>\, 0\). The conductance peak shifts for even sites are trivial as d1 = d2 (see Supplementary Fig. 3). Figure 2a summarizes the site n dependence of Δdn = d2 − d1 extracted from Fig. 1e–i. For odd sites, Δdn decreases as the target site moves from the end to the bulk, and for all the even sites, Δdn = 0. Summing them up gives U0 = ∑Δdn = 0.052t, which is quite close to the input value of U = 0.05t and demonstrates that the priorly unknown parameter U0 can be obtained from the transport measurement. The small difference between U0 and U may come from three issues. (i) Small but non-zero direct tunnelings (tc = 0.01t, compared with tk = 0.1t) shifting the \({{{\mathcal{G}}}}\) peak position; (ii) Peak’s position of \({{{\mathcal{G}}}}\) is obtained via numerical treatment other than exact analytical derivation; and iii) \({{{\mathcal{G}}}}\) is obtained by iteration with a finite but accepatable accuracy.

Fig. 2: The TFC and its distribution in the SSH chain.
figure 2

a, c The shift of the conductance peaks' separation Δdn = d2 − d1 [drawn from Fig. 1e–i] and its summation ∑nΔdn. b, d The TFC distribution Qn and its summation ∑nQn.

Full size image

To figure out the amount and the distribution of the TFC, we need to measure the trivial end of the same SSH chain while keeping Vg unchanged. Compared with the non-trivial end, the bulk states’ charge amount in the trivial end is increased by nt [see Fig. 1c]. The conductance peaks now locate at \({\varepsilon }_{1}^{t}\) and \({\varepsilon }_{2}^{t}\), where \({\varepsilon }_{1}^{t}={\varepsilon }_{1}^{{\prime} }-{U}_{0}{Q}_{n}/e\) and Qn is the TFC at site n. Therefore, Qn can be extracted from the conductance peak shift as \({Q}_{n}=({\varepsilon }_{1}^{{\prime} }-{\varepsilon }_{1}^{t})e/{U}_{0}\) since U0 has been obtained previously. Note that Qn is independent of U. Thus, the obtained Qn is still correct even if there is a screening effect which only renormalizes the value of U.

Figure 2c displays the spatial distribution of Qn and its summation ∑nQn. Such a summation is truncated when Δdn decays into a sufficiently small value. For odd sites, Qn decreases as the target site moves from the end to the bulk of the SSH chain and for even sites, Qn is nearly zero. These are in good agreement with the analytical result: \({Q}_{n}=(e/2){({t}_{1}/{t}_{2})}^{n-1}[1-{({t}_{1}/{t}_{2})}^{2}]\) for the odd site, and Qn = 0 for the even site50,51. For the present SSH chain model with parameters adopted from experiment37, 10 sites are sufficient for the truncation in summation. In Fig. 2b ∑nQn approaches e/2, confirming the 1/2 charge quantization of the TFC in the clean SSH model. Figure 2c, d displays the TFC obtained from the above scheme for another set of parameters t1 = 0.2t and t2 = 0.6t (see Supplementary Fig. 3). In such a case, U0 = 0.052t [Fig. 2c] can be drawn from the transport data. The measurement scheme also shows that the TFC now becomes more localized at the end of the SSH chain, while the TFC amount is still close to e/2 [Fig. 2d]. All these are in perfect match with the analytical results50,51. Such transport measurement scheme is also adopted for obtaining the spatial distribution and verifying the e/2 amount of the TFC carried by the topological corner state of the quadrupole insulator11,12, where the latter is regarded as the two-dimensional analogy of the SSH chain (see Supplementary Note 6).

Though the measurement scheme above is based on a comparison between trivial and non-trivial SSH chain, it also works for the SSH chain with only non-trivial end states. This is because the electron density distribution in the sites far away from the end of the SSH chain is the same for both non-trivial and trivial SSH chain. Hence the sites far away from the bulk of the non-trivial SSH chain can be treated as a trivial SSH chain end because of the translational invariance. So far, there being just only a single value of U is required in our proposal. For another SSH-type model in which each site supports two orbitals so that there are two different U’s52, the current version of our measurement scheme will not work well. Nevertheless, when the separation between the QD and the target atom is large compared to the lattice constant of the SSH chain, our treatment is a good approximation and the amount of TFC can still be obtained (see Supplementary Note 5).

The influence of disorder effects

Disorder effect including bond disorder and Anderson disorder is widely presented in practical experiments, which may induce charge fluctuation and thus hinder the identification of the genuine fractional charge. The bond disorder, for instance, in the form of \({\sum }_{i}{w}_{i}({b}_{i}^{{\dagger} }{b}_{i+1}+h.c.)\) in the SSH model where wi is uniformly distributed as wi ∈ [ −W/2, W/2] and W is the disorder strength53, preserves the chiral symmetry obeyed by the clean SSH model. Hence the charge neutral rule for the trivial insulator remains valid, and the fluctuation only comes from the redistribution of the TFC in the non-trivial state. As shown in Fig. 3b, under bond disorder, although the distribution of the TFC deviates from the clean SSH model, the TFC amount still approaches e/2.

Fig. 3: Features of TFC under disorders.
figure 3

a–d The TFC distribution in the disordered SSH model as a, b bond disorder with W = 0.2t; c, d Anderson disorder with W = 0.2t. All other parameters are the same as those in Fig. 1. e, f Standard deviation of the TFC σn at site n, and standard deviation of the total TFC inside the area concerned \({\sigma }_{{{\Sigma }}}^{n}\) under Anderson disorder for e SSH chain, and f quadrupole insulator. These two models are sketched in g, where the red rectangles indicate the area being concerned. Both these two models have the same band gap (~0.4t), sample length and disorder strength W = 0.5t. In e, f, the standard deviations are drawn from 104 disordered configurations.

Full size image

In contrast, the Anderson disorder \({\sum }_{i}{w}_{i}{b}_{i}^{{\dagger} }{b}_{i}\)54 in the SSH model breaks the chiral symmetry so that the charge neutral rule is only satisfied in an average manner. The TFC fluctuation is now ascribed to the redistribution of the electron density of both the bulk states (in the trivial and non-trivial SSH chain) and the topological edge state (in the non-trivial SSH chain). The transport data confirms that the position of the conductance peaks in each site of the trivial SSH chain also fluctuates under Anderson disorder (see Supplementary Fig. 3), which is in stark contrast to the bond disorder condition. As a result, though the amount of the TFC is approximately e/2, both the TFC distribution Qn and its summation ∑nQn shows significant fluctuation [Fig. 3d] even for the sites far away from the end of the SSH chain.

A question then arises that whether the fluctuation of the total TFC becomes more significant if more sites are taken into consideration. This issue is essential because in higher-dimensional systems, the site number involved for the topological edge state is proportional to the power of the localization length of the edge state. For the one-dimensional SSH chain, the standard deviation of the TFC at each site σn ≡ σ(Qn) and the standard deviation of the total TFC inside the area concerned \({\sigma }_{{{\Sigma }}}^{n}\equiv \sigma ({\sum }_{i\in \square }{Q}_{i})\) is investigated ["□” indicates the area concerned, shown by the red rectangles in Fig. 3g]. Significantly, both σn and \({\sigma }_{{{\Sigma }}}^{n}\) are in the same order for the one-dimensional SSH chain [Fig. 3e]. As a comparison, as shown in Fig. 3f, g, for an Anderson-disordered two-dimensional quadrupole insulator11,12 whose band gap and disorder strength are both the same as the one-dimensional SSH chain, although σn is nearly independent of n [n refers the index of the grey dots in Fig. 3g], \({\sigma }_{{{\Sigma }}}^{n}\) quickly increases in the fashion proportional to \(\sqrt{n}\). Such fluctuation behavior is certainly detrimental to the TFC measurement.

In case of Anderson disorder, the fluctuations of Qn in adjacent sites are correlated so that the fluctuation of the total TFC \({\sigma }_{{{\Sigma }}}^{n}\) inside the area concerned is only determined by the charge fluctuation at the boundaries. For one-dimensional model like SSH chain, such a boundary is a single site, while for higher-dimensional systems like quadrupole insulator, the number of sites at the boundary increases with the area being concerned. Consequently, \({\sigma }_{{{\Sigma }}}^{n}\) does not increase with n for one-dimensional topological system, while it quickly increases with n for higher-dimensional system. In other words, for higher-dimensional topological materials supporting TFC, the disorder-induced fluctuation of the TFC is reduced for a better-localized topological edge state. The results of dimension and disorder effects on TFC measurement are universal. They hold true not only for topological electronic systems, but also for topological classic wave systems. For classic wave systems, the TFC is measured via integrating the local density of states32,33,34. Though the experiments have shown specific spatial crystal symmetries related to the TFC are broken by the inevitable disorder effect, the TFC could still be observed since the topological corner states here are well localized.

Measurement of 2e/3 charge in breathing kagome lattice

Recently, a TFC of 2e/3 is reported in breathing kagome lattice14,15,16,17, though such material is two-dimensional, the topological corner states therein can be well localized (e.g. monolayer MoS216,17), so that our transport measurement scheme is still applicable. It is worth noting that in addition to the topological corner state, the zigzag-edged breathing kagome lattice also possesses a metallic one-dimensional edge state, hence the TFC here can not be detected by our scheme. Therefore, we first turn to investigate the armchair-edged breathing kagome lattice whose one-dimensional edge state is insulating.

Figure 4a shows a corner of a triangular armchair-edged breathing kagome lattice that the three supercells near the corner being mainly concerned are highlighted by colored disks. Figure 4b exhibits the conductance curves obtained from our measurement scheme for the representative 15 of the 27 sites inside these three supercells. The conductance peak shifts Δdn are listed in Fig. 4c, and the TFC Qn as well as its summation are shown in Fig. 4d. The Qn obtained is in good agreement with the numerical result \({Q}_{n}^{num}\) from the diagonalization treatment55. It is shown that the TFC is mainly distributed in the corner sites [see inset of Fig. 4c]. At the edge, for example, sites 1, 2, and 8 in the pale red supercell contributes nearly zero net charge after summation. Finally, a quantized 2e/3 TFC is obtained as expected.

Fig. 4: TFC in breathing kagome models.
figure 4

a Schematic plot of the armchair-edged breathing kagome lattice that the red (blue) bond indicates hopping amplitude t1 (t2). The sites are numbered in each individual supercell as highlighted by colored disks. b The conductance curves in each site for both the trivial (t1 = t, t2 = 0.3t) and the non-trivial (t1 = 0.3t, t2 = t) cases. c The conductance peak shift Δdn drawn from b. Inset: the TFC is mainly distributed at the two sites marked by the red balls. d The TFC distribution Qn and ∑nQn drawn from c, where the colored numbers in the horizontal axis refer to the site indices in a. e The same TFC distribution for zigzag-edged breathing kagome model. Other parameters are U = 0.1t, tk = 0.1t, and tc = 0.01t.

Full size image

As a comparison [see Fig. 4e], we also show that our transport measurement scheme fails to determine the TFC in the zigzag-edged breathing kagome lattice14, because the “charge neutral” rule, one of the two basic rules of our measurement scheme, is broken by its metallic one-dimensional topological edge state17 (see Supplementary Fig. 7). Finally, it is worth noting that the spin degeneracy has been ignored from the beginning. When the spin doubling is also taken into consideration, for instance, for the quadrupole insulator, one may be confused by the “integer” TFC of 2(e/2)29. In contrast, the 2e/3 TFC is always fractional even after considering the spin doubling, which serves as an additional advantage for the breathing kagome materials.

Discussion

We have presented the measurement scheme as well as the suitable materials. Now we turn to discuss feasible platforms supporting the measurement circuit, as well as the possible materials supporting TFC. In some pioneering experimental works, a superconducting quantum interference device on a tip (SOT) is used to study the spatial distribution of topological states and heat generation56,57,58. The SOT, serving as an mobile nanocircuit, moves precisely in a controlled manner that the separation between the tip and the sample can keep constant. Modified tips with a single molecule or specific clusters in scanning tunnelling microscopy were achieved in past years59,60. By attaching the SOT with a QD (a molecule or a cluster), it forms a moveable QD and it does not need to work below the superconducting critical temperature. Such a proposed measurement apparatus can be used in a graphene-nanoribbon-based SSH model to study the fractional charge36,37. Alternatively, in experiment a series of QDs has been fabricated in two-dimensional electron gas61. It can also be used for building suitable samples supporting TFC by tuning the gate voltages properly (see Supplementary Note 8).

In summary, a transport measurement scheme is proposed to measure the amount and the spatial distribution of TFC in topological materials. Through such a scheme, the e/2 amount of the TFC in the SSH model as well as its spatial distribution has been verified. The bond disorder preserving chiral symmetry will only slightly modify the spatial profile of the TFC. It implies that seeking a material in which the symmetry related to the TFC is quite robust will facilitate the experimental identification of the quantized TFC. In the presence of Anderson disorder breaking chiral symmetry, the fluctuation of the TFC amount is largely suppressed in one-dimensional systems. Meanwhile, for Anderson-disordered higher-dimensional topological materials such as breathing kagome lattice, the amount and the distribution of the TFC can still be obtained for the well-localized topological corner states. It indicates that in specific condensed matter materials, the difficulty of experimentally distinguishing the genuine fractional charge and the disorder-induced fluctuation can be circumvented.

Methods

The solving of the retarded Green’s function

The retarded Green’s function Gr(EF) needs to be solved by iteration for non-zero U62

$$\begin{array}{r}{{{{\bf{G}}}}}^{{{{\bf{r}}}}}(E)={\left[E-\left(\begin{array}{ccc}{\varepsilon }_{1}+U{n}_{s}&0&0\\ 0&{\varepsilon }_{2}+U{n}_{s}&0\\ 0&0&U({n}_{1}+{n}_{2})\\ \end{array}\right)-{{{{\boldsymbol{\Sigma }}}}}^{r}\right]}^{-1}\end{array}$$
(3)

with Σr the retarded self-energy and the matrix Gr(E) is in the basis {a1, a2, bs}. The particle numbers n1,2,s in equation (3) refers to electron number in the QD level ε1, ε2 and in the target site s, respectively. They are obtained through solving the integral equation self-consistently by iteration, e.g.\({n}_{s}=-\frac{1}{\pi }\int\nolimits_{-\infty }^{{E}_{F}}{{{\rm{Im}}}}[{{{{\bf{G}}}}}_{(3,3)}^{{{{\bf{r}}}}}(E)]dE\).

Numerical solution for TFC

In Figs. 3e, f and 4c, TFC is obtained via the tight-binding models of finite-size samples. Solving the eigen-equations HΨi = EiΨi, the total charge \({{{{\mathcal{Q}}}}}_{n}=e{\sum }_{i\in occ}| {{{\Psi }}}_{i}(n){| }^{2}\) is obtained by summing up the states below the Fermi level of site n. The net charge is obtained by \({Q}_{n}={Q}_{0}-{{{{\mathcal{Q}}}}}_{n}\) with Q0 the charge of the nucleus. In a clean trivial insulating system, Qn = 0 for all sites due to the charge neutral condition.