Transport measurement of fractional charges in topological models

Cheng, Shu-guang; Wu, Yijia; Jiang, Hua; Sun, Qing-Feng; **e, X. C.

doi:10.1038/s41535-023-00558-7

Transport measurement of fractional charges in topological models

Article
Open access
Published: 01 June 2023

Volume 8, article number 30, (2023)
Cite this article

Download PDF

You have full access to this open access article

npj Quantum Materials

Transport measurement of fractional charges in topological models

Download PDF

Shu-guang Cheng¹,
Yijia Wu ORCID: orcid.org/0000-0003-4379-9863^2,3,
Hua Jiang ORCID: orcid.org/0000-0002-8626-735X^3,4,
Qing-Feng Sun^2,5 &
…
X. C. **e^2,3,6

1237 Accesses
3 Citations
Explore all metrics

Abstract

The static topological fractional charge (TFC) in condensed matter systems is related to the band topology and thus has potential applications in topological quantum computation. However, the experimental measurement of these TFCs in electronic systems is quite challenging. We propose an electronic transport measurement scheme in which both the charge amount and the spatial distribution of the TFC can be extracted from the differential conductance through a quantum dot coupled to the topological system being measured. For one-dimensional Su–Schrieffer–Heeger (SSH) model, both the e/2 charge of the TFC and its distribution can be verified. As for the disorder effect, it is shown that the Anderson disorder, which breaks certain symmetry related to the TFC, is significant in higher-dimensional systems while having little effect on the one-dimensional SSH chain. Nonetheless, our measurement scheme can still work well for specific higher-order topological insulator materials, for instance, the 2e/3 TFC in the breathing kagome model could be confirmed even in the presence of disorder effect. These conclusions about spatial dimension and disorder effect are quite universal, which also applies to other topological systems such as topological classic wave system.

Fractionalized topological insulators

Article 30 April 2015

Spin effects in edge transport in two-dimensional topological insulators

Article 01 December 2016

Non-local order parameters as a probe for phase transitions in the extended Fermi-Hubbard model

Article 26 July 2017

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 physics^{1,2,3,4,5,6,7,8,9}. 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) states^1,2,3,4. And the later one, can be found as topological edge states in many topological materials, such as one-dimensional systems with the Jackiw-Rebbi mechanism^5,6,7,8,9, topological crystalline insulators with disclination¹⁰, and higher-order topological insulators^{11,12,13,14,15,16,17}. Remarkably, both the moving^18,19,20,21 and the static fractional charges^{22,23,24,25,26} 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 noise^27,28. 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 oscillation^22,30. 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 systems^29,32,33,34, 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 nanoribbons^35,36,37,38. 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 interaction^39,40,41,42. In this way, the charge possessed by the target atom can be extracted from the shift of the differential conductance resonance peak^43,44,45. Then the amount and spatial distribution of the TFC possessed by the topological edge states can be extracted by measuring all the atoms involved.

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 ε_i is the i-th energy level for the QD and t_k 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 t_1/2 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 t_c 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, b_s}). 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 function^46,47

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

(2)

Here ${{{\Gamma }}}_{L/R}^{r}$ is the symmetric line-width function of the terminals and G^r 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 〈n_s〉 the total electron number below the Fermi level. G^r is numerically obtained from the Dyson equation for g^r 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 ε_i to ε_i + U〈n_s〉 due to the Coulomb blockade^{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 Q_n is the TFC at site n. Therefore, Q_n 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 Q_n is independent of U. Thus, the obtained Q_n is still correct even if there is a screening effect which only renormalizes the value of U.}

Figure 2c displays the spatial distribution of Q_n and its summation ∑_nQ_n. Such a summation is truncated when Δd_n decays into a sufficiently small value. For odd sites, Q_n decreases as the target site moves from the end to the bulk of the SSH chain and for even sites, Q_n 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 Q_n = 0 for the even site^50,51. For the present SSH chain model with parameters adopted from experiment³⁷, 10 sites are sufficient for the truncation in summation. In Fig. 2b ∑_nQ_n 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 results^50,51. 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 insulator^11,12, 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 w_i is uniformly distributed as w_i ∈ [ −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.**

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 Q_n and its summation ∑_nQ_n 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(Q_n) 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 insulator^11,12 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 Q_n 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 states^32,33,34. 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 lattice^{14,15,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 generation^56,57,58. 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 years^59,60. 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 charge^36,37. 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 G^r(E_F) 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 Σ^r the retarded self-energy and the matrix G^r(E) is in the basis {a₁, a₂, b_s}. The particle numbers n_1,2,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 HΨ_i = E_iΨ_i, 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, Q_n = 0 for all sites due to the charge neutral condition.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

References

Tsui, D. C., Stormer, H. L. & Gossard, A. C. Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett. 48, 1559 (1982).
Article ADS Google Scholar
Laughlin, R. B. Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett. 50, 1395 (1983).
Article ADS Google Scholar
Laughlin, R. B. Nobel lecture: fractional quantization. Rev. Mod. Phys. 71, 863 (1998).
Article ADS MathSciNet MATH Google Scholar
Arovas, D., Schrieffer, J. R. & Wilczek, F. Fractional statistics and the quantum Hall effect. Phys. Rev. Lett. 53, 722 (1984).
Article ADS Google Scholar
Jackie, R. & Rebbi, C. Solitons with fermion number 1/2. Phys. Rev. B 13, 3398 (1976).
ADS MathSciNet Google Scholar
Su, W. P., Schrieffer, J. R. & Heeger, A. J. Soliton excitations in polyacetylene. Phys. Rev. B 22, 2099 (1980).
Article ADS Google Scholar
Su, W. P., Schrieffer, J. R. & Heeger, A. J. Solitons in polyacetylene. Phys. Rev. Lett. 42, 1698 (1979).
Article ADS Google Scholar
Goldstone, J. & Wilczek, F. Fractional quantum numbers on solitons. Phys. Rev. Lett. 47, 986 (1981).
Article ADS MathSciNet Google Scholar
Kivelson, S. & Schrieffer, J. R. Fractional charge, a sharp quantum observable. Phys. Rev. B 25, 6447 (1982).
Article ADS Google Scholar
Li, T. H., Zhu, P. H., Benalcazar, W. A. & Hughes, T. L. Fractional disclination charge in two-dimensional Cn-symmetric topological crystalline insulators. Phys. Rev. B 101, 115115 (2020).
Article ADS Google Scholar
Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61 (2017).
Article ADS MathSciNet MATH Google Scholar
Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Electric multipole moments, topological multipole moment pum**, and chiral hinge states in crystalline insulators. Phys. Rev. B 96, 245115 (2017).
Article ADS Google Scholar
Lemut, G., Mierzejewski, M. & Bonča, J. Complete Many-Body Localization in the t-j model caused by a random magnetic field. Phys. Rev. Lett. 119, 246601 (2017).
Article ADS Google Scholar
Ezawa, M. Higher-order topological insulators and semimetals on the breathing kagome and pyrochlore lattices. Phys. Rev. Lett. 120, 026801 (2018).
Article ADS Google Scholar
van Miert, G. & Ortix, C. On the topological immunity of corner states in two-dimensional crystalline insulators. npj Quantum Mater. 5, 63 (2020).
Article ADS Google Scholar
Zeng, J., Liu, H. W., Jiang, H., Sun, Q. F. & **e, X. C. Multiorbital model reveals a second-order topological insulator in 1H transition metal dichalcogenides. Phys. Rev. B 104, L161108 (2021).
Article ADS Google Scholar
Qian, S. F., Liu, G. B., Liu, C. C. & Yao, Y. G. Cn-symmetric higher-order topological crystalline insulators in atomically thin transition metal dichalcogenides. Phys. Rev. B 105, 045417 (2022).
Article ADS Google Scholar
Moore, G. & Read, N. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Nucl. Phys. B 360, 362 (1991).
Article ADS Google Scholar
Ivanov, D. A. Non-Abelian statistics of half-quantum vortices in p-wave superconductors. Phys. Rev. Lett. 86, 268 (2001).
Article ADS Google Scholar
Wilczek, F. Magnetic flux, angular momentum, and statistics. Phys. Rev. Lett. 48, 1144 (1982).
Article ADS Google Scholar
Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083 (2008).
Article ADS MathSciNet MATH Google Scholar
Wu, Y. J., Liu, H. W., Liu, J., Jiang, H. & **e, X. C. Double-frequency Aharonov-Bohm effect and non-Abelian braiding properties of Jackiw-Rebbi zero-mode. Natl Sci. Rev. 7, 572 (2020).
Article Google Scholar
Wu, Y. J., Jiang, H., Liu, J., Liu, H. W. & **e, X. C. Non-Abelian braiding of Dirac Fermionic modes using topological corner states in higher-order topological insulator. Phys. Rev. Lett. 125, 036801 (2020).
Article ADS MathSciNet Google Scholar
Tan, Y., Huang, Z. H. & Liu, X.-J. Two-particle Berry phase mechanism for Dirac and Majorana Kramers pairs of corner modes. Phys. Rev. B 105, L041105 (2022).
Article ADS Google Scholar
Boross, P. K., Asbóth, J., Széchenyi, G., Oroszlány, L. & Pályi, A. Poor man’s topological quantum gate based on the Su-Schrieffer-Heeger model. Phys. Rev. B 100, 045414 (2019).
Article ADS Google Scholar
Fu, B., Zou, J. Y., Hu, Z., Wang, H. W. & Shen, S. Q. Quantum anomalous semimetals. npj Quantum Mater. 7, 94 (2022).
Article ADS Google Scholar
de-Picciotto, R. et al. Direct observation of a fractional charge. Nature 389, 162 (1997).
Article ADS Google Scholar
Saminadayar, L., Glattli, D. C., **, Y. & Etienne, B. Observation of the e/3 fractionally charged Laughlin quasiparticle. Phys. Rev. Lett. 79, 2526 (1997).
Article ADS Google Scholar
Ortix, C. Electrons broken into pieces at crystal defects. Nature 589, 356 (2021).
Article ADS Google Scholar
Rainis, D., Saha, A., Klinovaja, J., Trifunovic, L. & Loss, D. Transport signatures of fractional Fermions in Rashba nanowires. Phys. Rev. Lett. 112, 196803 (2014).
Article ADS Google Scholar
Qi, X.-L., Hughes, T. L. & Zhang, S.-C. Fractional charge and quantized current in the quantum spin Hall state. Nat. Phys. 4, 273 (2008).
Article Google Scholar
Peterson, C. W. et al. A fractional corner anomaly reveals higher-order topology. Science 368, 1114 (2020).
Article ADS Google Scholar
Peterson, C. W. et al. Trapped fractional charges at bulk defects in topological insulators. Nature 589, 376 (2021).
Article ADS Google Scholar
Liu, Y. et al. Bulk-disclination correspondence in topological crystalline insulators. Nature 589, 381 (2021).
Article ADS Google Scholar
Franke, K. J. & Von Oppen, F. Designer topology in graphene nanoribbons. Nature 560, 175 (2018).
Article ADS Google Scholar
Rizzo, D. J. et al. Topological band engineering of graphene nanoribbons. Nature 560, 204 (2018).
Article ADS Google Scholar
Gröning, O. et al. Engineering of robust topological quantum phases in graphene nanoribbons. Nature 560, 209 (2018).
Article ADS Google Scholar
Zhu, Z. et al. A tunable and unidirectional one-dimensional electronic system Nb_2n+1Si_nTe_4n+2. npj Quantum Mater. 5, 35 (2020).
Article ADS Google Scholar
Johnson, A. C., Marcus, C. M., Hanson, M. P. & Gossard, A. C. Coulomb-modified Fano resonance in a one-Lead quantum dot. Phys. Rev. Lett. 93, 106803 (2004).
Article ADS Google Scholar
Beenakker, C. W. J. Theory of Coulomb-blockade oscillations in the conductance of a quantum dot. Phys. Rev. B 44, 1646 (1991).
Article ADS Google Scholar
Groshev, A. Coulomb blockade of resonant tunneling. Phys. Rev. B 42, 5895 (1990).
Article ADS Google Scholar
Baines, D. Y. et al. Transport through side-coupled double quantum dots: From weak to strong interdot coupling. Phys. Rev. B 85, 195117 (2012).
Article ADS Google Scholar
Berkovits, R., von Oppen, F. & Gefen, Y. Theory of charge sensing in quantum-dot structures. Phys. Rev. Lett. 94, 076802 (2005).
Article ADS Google Scholar
Kano, S., Tada, T. & Majima, Y. Nanoparticle characterization based on STM and STS. Chem. Soc. Rev. 43, 3957 (2014).
Google Scholar
Chiang, C. L., Xu, C., Han, Z. & Ho, W. Real-space imaging of molecular structure and chemical bonding by single-molecule inelastic tunneling probe. Science 344, 885 (2014).
Article ADS Google Scholar
Güclü, A. D., Sun, Q. F. & Guo, H. Kondo resonance in a quantum dot molecule. Phys. Rev. B 68, 245323 (2003).
Article ADS Google Scholar
Sun, Q. F., **ng, Y. X. & Shen, S. Q. Double quantum dot as detector of spin bias. Phys. Rev. B 77, 195313 (2008).
Article ADS Google Scholar
Liu, J., Sun, Q. F. & **e, X. C. Enhancement of the thermoelectric figure of merit in a quantum dot due to the Coulomb blockade effect. Phys. Rev. B 81, 245323 (2010).
Article ADS Google Scholar
Haug, H. & Jauho, A. P., Quantum kinetics in transport and optics of semiconductors (Springer-Verlag, Berlin, 1998).
Peng, Y., Bao, Y. M. & von Oppen, F. Boundary Green functions of topological insulators and superconductors. Phys. Rev. B 95, 235143 (2017).
Article ADS Google Scholar
Meier, E. J., Alex, An,F. & Gadway, B. Observation of the topological soliton state in the Su-Schrieffer-Heeger model. Nat. Commun. 7, 13986 (2016).
Article ADS Google Scholar
Nelson, A., Neupert, T., Bzdušek, T. & Alexandradinata, A. Multicellularity of delicate topological insulators. Phys. Rev. Lett. 126, 216404 (2021).
Article ADS Google Scholar
Song, J. T. et al. Dependence of topological Anderson insulator on the type of disorder. Phys. Rev. B 85, 195125 (2012).
Article ADS Google Scholar
Qiao, Z. H., Jung, J., Niu, Q. & MacDonald, A. H. Electronic highways in bilayer graphene. Nano Lett. 11, 3453 (2011).
Article ADS Google Scholar
Zhang, Z. Q., Wu, B. L., Chen, C. Z. & Jiang, H. Global phase diagram of disordered higher-order Weyl semimetals. Phys. Rev. B 104, 014203 (2021).
Article ADS Google Scholar
Vasyukov, D. et al. A scanning superconducting quantum interference device with single electron spin sensitivity. Nat. Nanotechnol. 8, 639 (2013).
Article ADS Google Scholar
Marguerite, A. et al. Imaging work and dissipation in the quantum Hall state in graphene. Nature 575, 628 (2019).
Article ADS Google Scholar
Aharon-Steinberg, A. et al. Long-range nontopological edge currents in charge-neutral graphene nature. Nature 593, 528 (2021).
Article ADS Google Scholar
Cheng, Z. et al. Direct imaging of molecular orbitals of metal phthalocyanines on metal surfaces with an O₂-functionalized tip of a scanning tunneling microscope. Nano Res. 4, 523 (2011).
Article Google Scholar
Bian, K. et al. Scanning probe microscopy. Nat. Rev. Methods Prim. 1, 36 (2021).
Article Google Scholar
Volk, C. et al. Loading a quantum-dot based “Qubyte" register. npj Quantum Inf. 5, 29 (2019).
Article ADS Google Scholar
Sun, Q. F. & Guo, H. Kondo resonance in a multiprobe quantum dot. Phys. Rev. B 64, 153306 (2001).
Article ADS Google Scholar

Download references

Acknowledgements

We thank Z.J., L.H.W., and Z.Z.Q. for fruitful discussions. This work is financially supported by the National Basic Research Program of China (Grants Nos. 2019YFA0308403 and 2022YFA1403700), the National Natural Science Foundation of China (Grant Nos. 11874298 and 11822407), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB28000000), China Postdoctoral Science Foundation (Grant No. 2021M690233), and Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2019JM-187).

Author information

Authors and Affiliations

Department of Physics, Northwest University, **’an, China
Shu-guang Cheng
International Center for Quantum Materials, School of Physics, Peking University, Bei**g, China
Yijia Wu, Qing-Feng Sun & X. C. **e
Institute for Nanoelectronic Devices and Quantum Computing, Fudan University, Shanghai, China
Yijia Wu, Hua Jiang & X. C. **e
Institute for Advanced Study, Soochow University, Suzhou, China
Hua Jiang
CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Bei**g, China
Qing-Feng Sun
Hefei National Laboratory, Hefei, China
X. C. **e

Authors

Shu-guang Cheng
View author publications
You can also search for this author in PubMed Google Scholar
Yijia Wu
View author publications
You can also search for this author in PubMed Google Scholar
Hua Jiang
View author publications
You can also search for this author in PubMed Google Scholar
Qing-Feng Sun
View author publications
You can also search for this author in PubMed Google Scholar
X. C. **e
View author publications
You can also search for this author in PubMed Google Scholar

Contributions

H.J. initiated the idea from discussions with Y.J.W., and X.C.X. C.S.G. performed the calculation with the suggestions from Q.F.S. and Y.J.W., H.J. and X.C.X. supervised the project. All authors were involved in analyzing the data and writing the manuscript.

Corresponding author

Correspondence to Hua Jiang.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and permissions

About this article

Cite this article

Cheng, Sg., Wu, Y., Jiang, H. et al. Transport measurement of fractional charges in topological models. npj Quantum Mater. 8, 30 (2023). https://doi.org/10.1038/s41535-023-00558-7

Download citation

Received: 16 November 2022
Accepted: 09 May 2023
Published: 01 June 2023
DOI: https://doi.org/10.1038/s41535-023-00558-7
Springer Nature Limited

This article is cited by

Emerging topological bound states in Haldane model zigzag nanoribbons
- Simone Traverso
- Maura Sassetti
- Niccolò Traverso Ziani
npj Quantum Materials (2024)

Transport measurement of fractional charges in topological models

Abstract