Abstract
Correlation and frustration play essential roles in physics, giving rise to novel quantum phases1,2,3,4,5,6. A typical frustrated system is correlated bosons on moat bands, which could host topological orders with long-range quantum entanglement4. However, the realization of moat-band physics is still challenging. Here, we explore moat-band phenomena in shallowly inverted InAs/GaSb quantum wells, where we observe an unconventional time-reversal-symmetry breaking excitonic ground state under imbalanced electron and hole densities. We find that a large bulk gap exists, encompassing a broad range of density imbalances at zero magnetic field (B), accompanied by edge channels that resemble helical transport. Under an increasing perpendicular B, the bulk gap persists, and an anomalous plateau of Hall signals appears, which demonstrates an evolution from helical-like to chiral-like edge transport with a Hall conductance approximately equal to e2/h at 35 tesla, where e is the elementary charge and h is Planck’s constant. Theoretically, we show that strong frustration from density imbalance leads to a moat band for excitons, resulting in a time-reversal-symmetry breaking excitonic topological order, which explains all our experimental observations. Our work opens up a new direction for research on topological and correlated bosonic systems in solid states beyond the framework of symmetry-protected topological phases, including but not limited to the bosonic fractional quantum Hall effect.
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Data availability
All data needed to evaluate the conclusions in the paper are included in this paper. Additional data that support the plots and other analyses in this work are available from the corresponding authors upon request.
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Acknowledgements
This work at Nanjing University was supported by the National Key R&D Program of China (grant no. 2022YFA1403601) and the National Natural Science Foundation of China (grant nos. 12274206, 12074177 and 12034014), Program for Innovative Talents and Entrepreneur in Jiangsu and the Xiaomi Foundation. The work at Peking University was supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (grant no. XDB28000000), National Key R&D Program of China (grant no. 2019YFA0308400) and National Natural Science Foundation of China (grant no. 11921005). The InAs/GaSb quantum well structures were prepared by molecular beam epitaxy by G. Sullivan. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by the National Science Foundation Cooperative Agreement no. DMR-1157490 and the State of Florida.
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R.W., T.A.S. and B.W. conceived the theoretical project. R.W. developed the theoretical model and performed the calculations supervised by B.W., R.-R.D. and L.-J.D. conceived the experimental project. L.-J.D. fabricated devices and performed transport experiments. L.-J.D. and R.-R.D. analysed the data. R.W., L.-J.D. and R.-R.D. co-wrote the manuscript with input from other authors. All authors discussed the results. R.-R.D. provided overall coordination of the whole project.
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Extended data figures and tables
Extended Data Fig. 1 Extraction of energy gap Δ under 0 T.
(a). Bulk conductance of the Corbino device σc as a function of Vf under 0 T at different temperatures. (b) and (c). Arrhenius plot of the conductance σc at Vf from −1.8 V to −2.4 V, and from −2.5 V to −3.2 V, respectively. The data can be fit by σc ∝ exp (−Δ/2kbT) to obtain Δ. Dashed lines are guides to the eye. For Vf≥−2.1 V or Vf≤−2.9 V, the σc exhibits non-activated temperature dependence, which is also plotted in panel (b) for comparison. Note that according to the ETO theory, as the e-h density imbalance increases, the gap that opens around the Fermi level would decrease towards zero, which could describe the data in Vf≥−2.1 V or Vf≤−2.9 V.
Extended Data Fig. 2 Estimation of carrier densities.
(a). B⊥/eRxy versus Vf in a 50 μm × 50 μm Hall-bar at 300 mK under B⊥= 1 T. Inset shows a schematic of the asymmetric Hall-bar, and the dashed box region is covered by the front gate. The yellow dashed line is obtained from the integration of measured capacitance over Vf, representing the net-carrier density |ne − np| (see ref. 12). In the linear regime of B⊥/eRxy, np is negligible with |ne − np| ≈ ne, and the yellow line represents ne (blue arrow). B⊥/eRxy deviates from the yellow line due to the increased np in the two-carrier transport. In this regime, Vf would tune both ne and np. Since the geometry capacitance remains constant and dominates the measured capacitance, |ne − np| would still change linearly with Vf and follow the yellow line. The green dashed line represents the Vf dependence of ne (blue arrow) in the two-carrier transport regime12. The difference between the green and yellow lines corresponds to np (red arrow). At Vf = −2 V, B⊥/eRxy drops because the topological edge occurs and contributes a large Rxx component on the Rxy due to the asymmetrical Hall-bar geometry12. The edge state persists in a voltage range from −2 V to −3 V (topological EI regime). At Vf = −2 V, ne ≈1.1 × 1011 cm−2 and np ≈4.5 × 1010 cm−2. From −2 V to −2.5 V, the e-h density imbalance regime with electrons as the majority carrier overlaps the topological EI regime. Symmetrically, a similar case occurs from −2.5 V to −3 V but with holes as the major carrier. (b). Calculated ne and np using a two-carrier model (Methods). The calculated results agree well with those obtained in (a).
Extended Data Fig. 3 Summary of edge resistance against edge length measured in the topological EI from shallowly inverted InAs/GaSb QWs with different device geometries.
Rm is the measured resistance. Rq denotes quantized values calculated by the LB formula, that is, h/2e2 (Hall bar, circle)23, h/4e2 (H bar, up-triangle), h/2e2 (two terminal bar, down-triangle)23, and h/4e2 (π bar, square)23. The edge lengths are depicted between black arrows in the devices. White arrows show the direction of the current. The transport of the edge state in the mesoscopic devices with length <λ is quantized. In the macroscopic devices, the backscattering occurs between two channels through virtual contacts42 where channels equilibrate over λ; within λ, the helical-like edge transport is ballistic and gives the value of a quantum resistor h/2e2. The up-left inset shows that the quantized resistance in the π bar device and the extracted coherence length λ from the macroscopic Hall bars are independent of temperatures up to 4 K. The lower-left inset shows the schematic plot of counter-propagating edge channels with a bulk EI state and arrows on the perimeters indicate the helical-like feature of the edge channels.
Extended Data Fig. 4 Longitudinal (top panel) and Hall (bottom panel) conductivities under magnetic fields up to 18 T.
(a–g) Longitudinal (top panel) and Hall (bottom panel) conductivities, σxx and σxy, converted from Rxx, Rxy measured in a 50 μm × 50 μm Hall-bar device C as a function of Vf at 20 mK under B⊥ up to 18 T. Magneto-transport below 8 T repeat that in device A (Fig. 2). The purple marks the transition regime from the QH states to topological EI. At 4 T, a σxx peak starts to emerge in the transition regime, and σxy collapses. The σxx peaks become more visible for higher B⊥. Meanwhile, σxy in the topological EI regime increases because the edge channels start to separate under B⊥. At 18 T, σxy between the v = 1 QH state and topological EI is close and the σxy collapse is replaced by a bump due to two-carrier transport. The corresponding σxx peak is smeared due to the residual conductance in the topological EI regime. \({\sigma }_{{\rm{xx}}}^{{\rm{EI}}}\) (in the topological EI regime) starts to decrease at 18 T and approaches zero at 35 T (Fig. 3c). The shrinking of the inner loop has two results: the inner loop is separated from the outer loop (at low B⊥); the inner loop is separated from the contacts and the outer loop (at high B⊥). The two results have opposite effects on \({\sigma }_{{\rm{xx}}}^{{\rm{EI}}}\) in a macroscopic Hall-bar (\({\sigma }_{{\rm{xx}}}^{{\rm{EI}}}\) < e2/2h at 0 T): the former makes \({\sigma }_{{\rm{xx}}}^{{\rm{EI}}}\) increase towards e2/2h due to reduced backscattering between the edge channels; the latter makes \({\sigma }_{{\rm{xx}}}^{{\rm{EI}}}\) decrease to zero since backscattering between the inner loop and the contacts decreases, and the outer loop works like a single chiral edge.
Extended Data Fig. 5 Schematic plot illustrating the ETO edge state on one side of a Hall bar.
Left and Middle, B⊥ = 0: The bulk of the imbalanced e–h bilayer opens a many-body gap, with its edge state formed by a pair of the weakly-bound particle (−e) and hole (+e) channels, as demonstrated by the theory of ETO; the robustness of the edge states is protected by a BCS-like bulk gap. Considering the quasiparticle-quasihole symmetry near the Fermi level, the excitonic edge state is viewed as equivalent to a helical-like state containing two oppositely propagating (−e) channels (Sec. V of Supplementary Information). Right, when B⊥ is applied, the two channels are spatially separated due to the Lorentz force, forming inner and outer loops in a Hall bar. The dotted lines denote the scattering between the two channels, which decays with increasing B⊥ but always exists in realistic samples, explaining why the Hall resistance does not exactly quantize in chiral-like transport.
Extended Data Fig. 6 Calculation of the ETO edge transport under B⊥.
To calculate the edge resistance, we consider a six-terminal Hall bar with edge separation under B⊥ and virtual contacts (Methods). (a) and (b) respectively show the calculated Rxx and Rxy as functions of B⊥ compared with the experimental data. Data Rxx in (a) is measured from the Hall-bar device C (see Extended Data Fig. 4), and data in (b) is extracted from Rxy in Fig. 2b. The inset to (a) schematically demonstrates the effect of B⊥ on the two edge channels. The error bars in (b) come from the uncertainties in the experimental Rxy in the topological EI regime.
Extended Data Fig. 7 Summary of the ETO properties.
We make a conceptual analogy of ETO formed in zero magnetic field to the electronic 1/m Laughlin liquid that is formed in an external magnetic field.
Extended Data Fig. 8 The symmetric term \({{\boldsymbol{R}}}_{{\bf{xy}}}^{{\bf{s}}}\) from \({{\boldsymbol{R}}}_{{\bf{xy}}}^{{\bf{EI}}}\).
\({R}_{{\rm{xy}}}^{{\rm{s}}}\) is shown as a function of Vf at different B⊥ in device B.
Extended Data Fig. 9 Gap Energies under perpendicular magnetic fields B⊥.
(a) The Arrhenius plot of the Corbino conductance σc at the CNP with temperatures under fixed B⊥. The Corbino conductance is normalized to the values at 5 K. The EI gap energy under a magnetic field is deduced through σc ∝ exp (−Δ/2kbT), their values are displayed in (b). The error bars come from the uncertainty in the extraction of gap energy from the Arrhenius plot. (c) shows the voltage ranges where the state in the topological EI regime is insulating (σc < 0.2 μS) at B⊥. Obviously, as B⊥ increases, the EI gap becomes larger, but the voltage ranges of the insulating state in the topological EI regime remain nearly the same.
Supplementary information
Supplementary Information
Supplementary Sections I–VII, Figs. 1–14 and References.
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Wang, R., Sedrakyan, T.A., Wang, B. et al. Excitonic topological order in imbalanced electron–hole bilayers. Nature 619, 57–62 (2023). https://doi.org/10.1038/s41586-023-06065-w
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DOI: https://doi.org/10.1038/s41586-023-06065-w
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