Abstract
Correlation and frustration play essential roles in physics, giving rise to novel quantum phases^{1,2,3,4,5,6}. A typical frustrated system is correlated bosons on moat bands, which could host topological orders with longrange quantum entanglement^{4}. However, the realization of moatband physics is still challenging. Here, we explore moatband phenomena in shallowly inverted InAs/GaSb quantum wells, where we observe an unconventional timereversalsymmetry 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 helicallike to chirallike edge transport with a Hall conductance approximately equal to e^{2}/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 timereversalsymmetry 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 symmetryprotected topological phases, including but not limited to the bosonic fractional quantum Hall effect.
This is a preview of subscription content, access via your institution
Access options
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our bestvalue onlineaccess subscription
$29.99 / 30 days
cancel any time
Subscribe to this journal
Receive 51 print issues and online access
$199.00 per year
only $3.90 per issue
Buy this article
 Purchase on SpringerLink
 Instant access to full article PDF
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
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.
References
Bardeen, J., Cooper, L. N. & Schrieffer, J. R. Theory of superconductivity. Phys. Rev. 108, 1175–1204 (1957).
Knox, R. S. Theory of excitons. Solid State Phys. Suppl. 5, 100 (1963).
Keldysh, L. V. & Kopaev, Y. V.Possible instability of the semimetallic state with respect to coulombic interaction. Fiz. Tverd. Tela 6, 2791–2798 (1964). [Sov. Phys. Solid State 6, 2219–2224 (1965)].
Wen, X.G. Choreographed entanglement dances: topological states of quantum matter. Science 363, eaal3099 (2019).
Tsui, D. C., Stormer, H. L. & Gossard, A. C. Twodimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett. 48, 1559–1562 (1982).
Balents, L. Spin liquids in frustrated magnets. Nature 464, 199–208 (2010).
Sedrakyan, T. A., Glazman, L. I. & Kamenev, A. Absence of Bose condensation on lattice with moat bands. Phys. Rev. B 89, 201112(R) (2014).
Sedrakyan, T. A., Kamenev, A. & Glazman, L. I. Composite fermion state of spinorbitcoupled bosons. Phys. Rev. A 86, 063639 (2012).
Sedrakyan, T. A., Glazman, L. I. & Kamenev, A. Spontaneous formation of a nonuniform chiral spin liquid in a moatband lattice. Phys. Rev. Lett. 114, 037203 (2015).
Sedrakyan, T. A., Galitski, V. M. & Kamenev, A. Statistical transmutation in Floquet driven optical lattices. Phys. Rev. Lett. 115, 195301 (2015).
Kalmeyer, V. & Laughlin, R. B. Equivalence of the resonatingvalencebond and fractional quantum Hall states. Phys. Rev. Lett. 59, 2095–2098 (1987).
Du, L. et al. Evidence for a topological excitonic insulator in InAs/GaSb bilayers. Nat. Commun. 8, 1971 (2017).
Jérome, D., Rice, T. M. & Kohn, W. Exciton insulator. Phys. Rev. 158, 462–475 (1967).
Fulde, P. & Ferrell, R. A. Superconductivity in a strong spinexchange field. Phys. Rev. 135, A550–A563 (1964).
Larkin, A. I. & Ovchinnikov, Y. N. Inhomogeneous state of superconductors. Sov. Phys. JETP 20, 762–769 (1965).
Casalbuoni, R. & Nardulli, G. Inhomogeneous superconductivity in condensed matter and QCD. Rev. Mod. Phys. 76, 263–320 (2004).
Buzdin, A. I. & Kachkachi, H. Generalized Ginzburg–Landau theory for nonuniform FFLO superconductors. Phys. Lett. A 255, 341–348 (1997).
Varley, J. R. & Lee, D. K. K. Structure of exciton condensates in imbalanced electronhole bilayers. Phys. Rev. B 94, 174519 (2016).
Efimkin, D. K. & Lozovik, Yu. E. Electronhole pairing with nonzero momentum in a graphene bilayer. JETP 113, 880–886 (2011).
Parish, M. M., Marchetti, F. M. & Littlewood, P. B. Supersolidity in electronhole bilayers with a large density imbalance. EPL 95, 27007 (2011).
Seradjeh, B. Topological exciton condensation of imbalanced electrons and holes. Phys. Rev. B 85, 235146 (2012).
Brazovskii, S. A. Phase transition of an isotropic system to a nonuniform state. Zh. Eksp. Teor. Fiz. 68, 175–185 (1975).
Du, L., Knez, I., Sullivan, G. & Du, R.R. Robust helical edge transport in gated InAs/GaSb bilayers. Phys. Rev. Lett. 114, 096802 (2015).
Büttiker, M. Absence of backscattering in the quantum Hall effect in multiprobe conductors. Phys. Rev. B 38, 9375–9389 (1988).
Li, T., Wang, P., Sullivan, G., Lin, X. & Du, R.R. Lowtemperature conductivity of weakly interacting quantum spin Hall edges in strainedlayer InAs/GaInSb. Phys. Rev. B 96, 241406 (2017).
Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).
Qi, X.L. & Zhang, S.C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).
König, M. et al. Quantum spin Hall insulator state in HgTe quantum wells. Science 318, 766–770 (2007).
Du, L. et al. Tuning edge states in strainedlayer InAs/GaInSb quantum spin Hall insulators. Phys. Rev. Lett. 119, 056803 (2017).
Du, L. et al. Coulomb drag in topological wires separated by an air gap. Nat. Electron. 4, 573–578 (2021).
Yang, Y. et al. Timereversalsymmetrybroken quantum spin Hall effect. Phys. Rev. Lett. 107, 066602 (2011).
Pikulin, D. I. & Hyart, T. Interplay of exciton condensation and the quantum spin Hall effect in InAs/GaSb bilayers. Phys. Rev. Lett. 112, 176403 (2014).
Xue, F. & MacDonald, A. H. Timereversal symmetrybreaking nematic insulators near quantum spin Hall phase transitions. Phys. Rev. Lett. 120, 186802 (2018).
Vaöyrynen, J. I., Goldstein, M. & Glazman, L. I. Helical edge resistance introduced by charge puddles. Phys. Rev. Lett. 110, 216402 (2013).
Eisenstein, J. P. & MacDonald, A. H. Bose–Einstein condensation of excitons in bilayer electron systems. Nature 432, 691–694 (2004).
Ma, L. et al. Strongly correlated excitonic insulator in atomic double layers. Nature 598, 585–589 (2021).
Liu, X. et al. Crossover between strongly coupled and weakly coupled exciton superfluids. Science 375, 205–209 (2022).
Jain, J. K. Compositefermion approach for the fractional quantum Hall effect. Phys. Rev. Lett. 63, 199–202 (1989).
Halperin, B. I., Lee, P. A. & Read, N. Theory of the halffilled Landau level. Phys. Rev. B 47, 7312–7343 (1993).
Gopalakrishnan, S., Lamacraft, A. & Goldbart, P. M. Universal phase structure of dilute Bose gases with Rashba spinorbit coupling. Phys. Rev. A 84, 061604(R) (2011).
Wu, X. et al. Electrically tuning manybody states in a Coulombcoupled InAs/InGaSb double layer. Phys. Rev. B 100, 165309 (2019).
Knez, I., Du, R.R. & Sullivan, G. Evidence for helical edge modes in inverted InAs/GaSb quantum wells. Phys. Rev. Lett. 107, 136603 (2011).
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. DMR1157490 and the State of Florida.
Author information
Authors and Affiliations
Contributions
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. cowrote the manuscript with input from other authors. All authors discussed the results. R.R.D. provided overall coordination of the whole project.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature thanks Dmitry Efimkin and the other, anonymous, reviewers(s) for their contribution to the peer review of this work.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Extended data figures and tables
Extended Data Fig. 1 Extraction of energy gap Δ under 0 T.
(a). Bulk conductance of the Corbino device σ_{c} as a function of V_{f} under 0 T at different temperatures. (b) and (c). Arrhenius plot of the conductance σ_{c} at V_{f} from −1.8 V to −2.4 V, and from −2.5 V to −3.2 V, respectively. The data can be fit by σ_{c} ∝ exp (−Δ/2k_{b}T) to obtain Δ. Dashed lines are guides to the eye. For V_{f}≥−2.1 V or V_{f}≤−2.9 V, the σ_{c} exhibits nonactivated temperature dependence, which is also plotted in panel (b) for comparison. Note that according to the ETO theory, as the eh density imbalance increases, the gap that opens around the Fermi level would decrease towards zero, which could describe the data in V_{f}≥−2.1 V or V_{f}≤−2.9 V.
Extended Data Fig. 2 Estimation of carrier densities.
(a). B_{⊥}/eR_{xy} versus V_{f} in a 50 μm × 50 μm Hallbar at 300 mK under B_{⊥}= 1 T. Inset shows a schematic of the asymmetric Hallbar, and the dashed box region is covered by the front gate. The yellow dashed line is obtained from the integration of measured capacitance over V_{f}, representing the netcarrier density n_{e} − n_{p} (see ref. ^{12}). In the linear regime of B_{⊥}/eR_{xy}, n_{p} is negligible with n_{e} − n_{p} ≈ n_{e}, and the yellow line represents n_{e} (blue arrow). B_{⊥}/eR_{xy} deviates from the yellow line due to the increased n_{p} in the twocarrier transport. In this regime, V_{f} would tune both n_{e} and n_{p}. Since the geometry capacitance remains constant and dominates the measured capacitance, n_{e} − n_{p} would still change linearly with V_{f} and follow the yellow line. The green dashed line represents the V_{f} dependence of n_{e} (blue arrow) in the twocarrier transport regime^{12}. The difference between the green and yellow lines corresponds to n_{p} (red arrow). At V_{f} = −2 V, B_{⊥}/eR_{xy} drops because the topological edge occurs and contributes a large R_{xx} component on the R_{xy} due to the asymmetrical Hallbar geometry^{12}. The edge state persists in a voltage range from −2 V to −3 V (topological EI regime). At V_{f} = −2 V, n_{e} ≈1.1 × 10^{11} cm^{−2} and n_{p} ≈4.5 × 10^{10} cm^{−2}. From −2 V to −2.5 V, the eh 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 n_{e} and n_{p} using a twocarrier 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.
R_{m} is the measured resistance. R_{q} denotes quantized values calculated by the LB formula, that is, h/2e^{2} (Hall bar, circle)^{23}, h/4e^{2} (H bar, uptriangle), h/2e^{2} (two terminal bar, downtriangle)^{23}, and h/4e^{2} (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 contacts^{42} where channels equilibrate over λ; within λ, the helicallike edge transport is ballistic and gives the value of a quantum resistor h/2e^{2}. The upleft 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 lowerleft inset shows the schematic plot of counterpropagating edge channels with a bulk EI state and arrows on the perimeters indicate the helicallike 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, σ_{xx} and σ_{xy}, converted from R_{xx}, R_{xy} measured in a 50 μm × 50 μm Hallbar device C as a function of V_{f} at 20 mK under B_{⊥} up to 18 T. Magnetotransport 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 σ_{xx} peak starts to emerge in the transition regime, and σ_{xy} collapses. The σ_{xx} peaks become more visible for higher B_{⊥}. Meanwhile, σ_{xy} in the topological EI regime increases because the edge channels start to separate under B_{⊥}. At 18 T, σ_{xy} between the v = 1 QH state and topological EI is close and the σ_{xy} collapse is replaced by a bump due to twocarrier transport. The corresponding σ_{xx} 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 Hallbar (\({\sigma }_{{\rm{xx}}}^{{\rm{EI}}}\) < e^{2}/2h at 0 T): the former makes \({\sigma }_{{\rm{xx}}}^{{\rm{EI}}}\) increase towards e^{2}/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 manybody gap, with its edge state formed by a pair of the weaklybound particle (−e) and hole (+e) channels, as demonstrated by the theory of ETO; the robustness of the edge states is protected by a BCSlike bulk gap. Considering the quasiparticlequasihole symmetry near the Fermi level, the excitonic edge state is viewed as equivalent to a helicallike 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 chirallike transport.
Extended Data Fig. 6 Calculation of the ETO edge transport under B_{⊥}.
To calculate the edge resistance, we consider a sixterminal Hall bar with edge separation under B_{⊥} and virtual contacts (Methods). (a) and (b) respectively show the calculated R_{xx} and R_{xy} as functions of B_{⊥} compared with the experimental data. Data R_{xx} in (a) is measured from the Hallbar device C (see Extended Data Fig. 4), and data in (b) is extracted from R_{xy} 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 R_{xy} 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 V_{f} 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 σ_{c} 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 σ_{c} ∝ exp (−Δ/2k_{b}T), 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 (σ_{c} < 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.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author selfarchiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
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/s4158602306065w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s4158602306065w
This article is cited by

Evidence for chiral graviton modes in fractional quantum Hall liquids
Nature (2024)

Evidence for electron–hole crystals in a Mott insulator
Nature Materials (2024)

Unveiling chiral states in the XXZ chain: finitesize scaling probing symmetryenriched c = 1 conformal field theories
Journal of High Energy Physics (2024)

Dual quantum spin Hall insulator by densitytuned correlations in TaIrTe4
Nature (2024)

Z3 and (×Z3)3 symmetry protected topological paramagnets
Journal of High Energy Physics (2023)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.