Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Observation of half-integer thermal Hall conductance

An Author Correction to this article was published on 14 August 2018

This article has been updated


Topological states of matter are characterized by topological invariants, which are physical quantities whose values are quantized and do not depend on the details of the system (such as its shape, size and impurities). Of these quantities, the easiest to probe is the electrical Hall conductance, and fractional values (in units of e2/h, where e is the electronic charge and h is the Planck constant) of this quantity attest to topologically ordered states, which carry quasiparticles with fractional charge and anyonic statistics. Another topological invariant is the thermal Hall conductance, which is harder to measure. For the quantized thermal Hall conductance, a fractional value in units of κ0 (κ0 = π2kB2/(3h), where kB is the Boltzmann constant) proves that the state of matter is non-Abelian. Such non-Abelian states lead to ground-state degeneracy and perform topological unitary transformations when braided, which can be useful for topological quantum computation. Here we report measurements of the thermal Hall conductance of several quantum Hall states in the first excited Landau level and find that the thermal Hall conductance of the 5/2 state is compatible with a half-integer value of 2.5κ0, demonstrating its non-Abelian nature.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Get just this article for as long as you need it


Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Device configuration and Hall data.
Fig. 2: Heat flow of the two outermost edge modes at bulk filling ν = 5/2.
Fig. 3: Normalized heat conductance at bulk fillings ν = 7/3 and ν = 8/3.
Fig. 4: Summary of the normalized thermal conductance coefficient results for ν = 5/2.
Fig. 5: Possible orders predicted for the ν = 5/2 state.

Change history

  • 14 August 2018

    In this Article, the publication details for references 33, 34 and 40 have been corrected online.


  1. Nayak, C. et al. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008).

    Article  ADS  MathSciNet  MATH  CAS  Google Scholar 

  2. Willett, R. et al. Observation of an even-denominator quantum number in the fractional quantum Hall effect. Phys. Rev. Lett. 59, 1776–1779 (1987).

    Article  ADS  PubMed  CAS  Google Scholar 

  3. Moore, G. et al. Nonabelions in the fractional quantum Hall effect. Nucl. Phys. B 360, 362–396 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  4. Greiter, M. et al. Paired Hall state at half filling. Phys. Rev. Lett. 66, 3205–3208 (1991).

    Article  ADS  PubMed  CAS  Google Scholar 

  5. Read, N. et al. Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect. Phys. Rev. B 61, 10267–10297 (2000).

    Article  ADS  CAS  Google Scholar 

  6. Dolev, M. et al. Observation of quarter of an electron charge at the ν = 5/2 quantum Hall state. Nature 452, 829–834 (2008).

    Article  ADS  PubMed  CAS  Google Scholar 

  7. Radu, I. P. et al. Quasi-particle properties from tunneling in the ν = 5/2 fractional quantum Hall state. Science 320, 899–902 (2008).

    Article  ADS  PubMed  CAS  Google Scholar 

  8. Bid, A. et al. Observation of neutral modes in the fractional quantum Hall regime. Nature 466, 585–590 (2010).

    Article  ADS  PubMed  CAS  Google Scholar 

  9. Morf, R. H. Transition from quantum Hall to compressible states in the second Landau level: new light on the ν = 5/2 enigma. Phys. Rev. Lett. 80, 1505–1508 (1998).

    Article  ADS  CAS  Google Scholar 

  10. Storni, M. et al. Fractional quantum Hall state at ν = 5/2 and the Moore–Read Pfaffian. Phys. Rev. Lett. 104, 076803 (2010).

    Article  ADS  PubMed  CAS  Google Scholar 

  11. Rezayi, E. H. Landau level mixing and the ground state of the ν = 5/2 quantum Hall effect. Phys. Rev. Lett. 119, 026801 (2017).

    Article  ADS  PubMed  Google Scholar 

  12. Levin, M. et al. Particle-hole symmetry and the Pfaffian state. Phys. Rev. Lett. 99, 236806 (2007).

    Article  ADS  PubMed  CAS  Google Scholar 

  13. Lee, S. S. et al. Particle–hole symmetry and the ν = 5/2 quantum Hall state. Phys. Rev. Lett. 99, 236807 (2007).

    Article  ADS  PubMed  CAS  Google Scholar 

  14. Wen, X. G. Non-Abelian statistics in the fractional quantum Hall states. Phys. Rev. Lett. 66, 802–805 (1991).

    Article  ADS  MathSciNet  PubMed  MATH  CAS  Google Scholar 

  15. Halperin, B. I. Theory of the quantized Hall conductance. Helv. Phys. Acta 56, 75–102 (1983).

    CAS  Google Scholar 

  16. Yang, G. et al. Influence of device geometry on tunneling in ν = 5/2 quantum Hall liquid. Phys. Rev. B 88, 085317 (2013).

    Article  ADS  CAS  Google Scholar 

  17. Yang, G. et al. Experimental constraints and a possible quantum Hall state at ν = 5/2. Phys. Rev. B 90, 161306 (2014).

    Article  ADS  CAS  Google Scholar 

  18. Son, D. T. Is the composite fermion a Dirac particle? Phys. Rev. X 5, 031027 (2015).

    Google Scholar 

  19. Zucker, P. T. et al. Stabilization of the particle–hole Pfaffian order by Landau-level mixing and impurities that break particle–hole symmetry. Phys. Rev. Lett. 117, 096802 (2016).

    Article  ADS  PubMed  CAS  Google Scholar 

  20. Fidkowski, L. et al. Non-Abelian topological order on the surface of a 3D topological superconductor from an exactly solved model. Phys. Rev. X 3, 041016 (2013).

    Google Scholar 

  21. Bonderson, P. et al. A time-reversal invariant topological phase at the surface of a 3D topological insulator. J. Stat. Mech. 2013, P09016 (2013).

    Article  MathSciNet  CAS  Google Scholar 

  22. Kane, C. L. et al. Pairing in Luttinger liquids and quantum Hall states. Phys. Rev. X 7, 031009 (2017).

    Google Scholar 

  23. Schwab, K. et al. Measurement of the quantum of thermal conductance. Nature 404, 974–977 (2000).

    Article  ADS  PubMed  CAS  Google Scholar 

  24. Meschke, M. et al. Single-mode heat conduction by photons. Nature 444, 187–190 (2006).

    Article  ADS  PubMed  CAS  Google Scholar 

  25. Jezouin, S. et al. Quantum limit of heat flow across a single electronic channel. Science 342, 601–604 (2013).

    Article  ADS  MathSciNet  PubMed  MATH  CAS  Google Scholar 

  26. Banerjee, M. et al. Observed quantization of anionic heat flow. Nature 545, 75–79 (2017).

    Article  ADS  PubMed  CAS  Google Scholar 

  27. Wen, X. G. Quantum Field Theory of Many-body Systems: From the Origin of Sound to an Origin of Light and Electrons (Oxford Univ. Press, Oxford, 2004).

  28. Wellstood, F. C. et al. Hot-electron effects in metals. Phys. Rev. B 49, 5942–5955 (1994).

    Article  ADS  CAS  Google Scholar 

  29. Umansky, V. et al. in Molecular Beam Epitaxy: From Research to Mass Production (ed. Henini, M.) 121–137 (Elsevier, Amsterdam, 2013).

  30. Mooney, P. M. Deep donor levels (DX centers) in III–V semiconductors. J. Appl. Phys. 67, R1–R26 (1990).

    Article  ADS  CAS  Google Scholar 

  31. Dolev, M. et al. Characterizing neutral modes of fractional states in the second Landau level. Phys. Rev. Lett. 107, 036805 (2011).

    Article  ADS  PubMed  CAS  Google Scholar 

  32. Jain, J. K. Composite Fermions (Cambridge Univ. Press, Cambridge, 2007).

  33. Read, N. & Rezayi, E. Beyond paired quantum Hall states: parafermions and incompressible states in the first excited Landau level. Phys. Rev. B 59, 8084–8092 (1999).

    Article  ADS  CAS  Google Scholar 

  34. Bonderson, P. & Slingerland, J. K. Fractional quantum Hall hierarchy and the second Landau level. Phys. Rev. B 78, 125323 (2008).

    Article  ADS  CAS  Google Scholar 

  35. Dolev, M. et al. Dependence of the tunneling quasiparticle charge determined via shot noise measurements on the tunneling barrier and energetics. Phys. Rev. B 81, 161303 (2010).

    Article  ADS  CAS  Google Scholar 

  36. Kane, C. L. et al. Quantized thermal transport in the fractional quantum Hall effect. Phys. Rev. B 55, 15832–15837 (1997).

    Article  ADS  CAS  Google Scholar 

  37. Mross, D. F. et al. Theory of disorder-induced half-integer thermal Hall conductance. Preprint at (2017).

  38. Wang, C., Vishwanath, A. & Halperin, B. I. Topological order from disorder and the quantized Hall thermal metal: possible applications to the ν = 5/2 state. Preprint at (2017).

  39. Lian, B. et al. Theory of disordered ν = 5/2 quantum thermal Hall state: emergent symmetry and phase diagram. Phys. Rev. B 97, 165124 (2018).

    Article  ADS  Google Scholar 

  40. Simon, S. H. On the interpretation of thermal conductance of the ν = 5/2 edge. Phys. Rev. B 97, 121406 (2018).

    Article  ADS  Google Scholar 

  41. Willett, R. L. The quantum Hall effect at 5/2 filling factor. Rep. Prog. Phys. 76, 076501 (2013).

    Article  ADS  PubMed  CAS  Google Scholar 

  42. Samani, M. et al. Low-temperature illumination and annealing of ultrahigh quality quantum wells. Phys. Rev. B 90, 121405 (2014).

    Article  ADS  CAS  Google Scholar 

  43. Rössler, C. et al. Gating of high-mobility two-dimensional electron gases in GaAs/AlGaAs heterostructures. New J. Phys. 12, 043007 (2010).

    Article  ADS  CAS  Google Scholar 

  44. Slobodeniuk, A. O. et al. Equilibration of quantum Hall edge states by an Ohmic contact. Phys. Rev. B 88, 165307 (2013).

    Article  ADS  CAS  Google Scholar 

  45. Dahlem, F. Cryogenic scanning force microscopy of quantum Hall samples: adiabatic transport originating in anisotropic depletion at contact interfaces. Phys. Rev. B 82, 121305 (2010).

    Article  ADS  CAS  Google Scholar 

  46. Gelfand, B. Y. et al. Edge electrostatics and a mesa-etched sample and edge-state-to-bulk scattering rate in the fractional quantum Hall regime. Phys. Rev. B 49, 1862–1866 (1994).

    Article  ADS  CAS  Google Scholar 

Download references


We acknowledge B. Halperin and S. Simon for discussions. M.B. acknowledges the help and advice of Y. Gross regarding fabrication processes and R. Bhattacharyya for help with the cold amplifiers and Y. C. Chung and H. K. Choi for their help with the dilution refrigerator. M.H. acknowledges the continuous support of the Sub-Micron Center staff, and in particular Y. Rotblat, without whom this work would not be possible. M.H. acknowledges the support of the European Research Council under the European Community’s Seventh Framework Program (FP7/2007-2013)/ERC under grant agreement number 339070, the partial support of the Minerva foundation under grant number 711752, the Israeli Science Foundation ISF under grant number 459/16 and, together with V.U., the German Israeli Foundation (GIF) under grant number I-1241-303.10/2014. A.S and Y.O. acknowledge support from the European Research Council under the European Union’s Seventh Framework Program (FP7/2007-2013)/ERC Project MUNATOP, the DFG (CRC/Transregi 183, EI 519/7-1) and the Israel Science Foundation. Y.O. acknowledges the Binational Science Foundation (BSF). D.E.F.’s research was supported in part by the National Science Foundation under grant number DMR-1607451.

Reviewer information

Nature thanks K. Shtengel, S. Simon and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Author information

Authors and Affiliations



M.B. and M.H. designed the experiment, preformed the measurements, did the analysis and guided the experimental work. M.B. fabricated the devices with input from M.H., D.E.F. and Y.O., and A.S. worked on the theoretical aspects. V.U. grew the two-dimensional electron-gas heterostructures. All authors contributed to the write up of the manuscript.

Corresponding author

Correspondence to Moty Heiblum.

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.

Extended data figures and tables

Extended Data Fig. 1 Details of the growth structure.

Schematic of the conduction band in the MBE-grown structures that were studied. The SPSL doping scheme comprises δ–Si doping planes placed in narrow GaAs quantum wells (QW). The thickness of the GaAs and AlAs quantum wells in SPSL is chosen in such a way that the X-band minima of the AlAs layers reside below the Γ-band minimum of the GaAs. Electrons that spill over to the AlAs wells have low mobility and thus do not participate effectively in the conduction process. This structure suffers from substantial added bulk heat conductance. The structure used in our study, with δ–Si doping in low-Al-mole-fraction AlGaAs, did not have a visible bulk thermal conductance.

Extended Data Fig. 2 Longitudinal resistance of the high-mobility SPSL-grown heterostructure.

Longitudinal resistance measured in a Hall bar 100 µm wide and 200 µm long. Fractional filling factors are more pronounced than in the δ–Si-doped structure. Yet, the structure suffers from added thermal conductance in the bulk (see main text).

Extended Data Fig. 3 Thermal noise analysis at ν = 2 in the bulk in the high-mobility SPSL structure.

Dissipated power in the floating reservoir is plotted as a function of Tm for different numbers of open arms, N, with one edge mode allowed to flow in each arm (controlled by the surface gates). Dashed curves show the one-parameter fit of α from ΔP(ακ0T2βT5) for a given β (the value deduced from all the experiments). The apparent total thermal conductance is K = 7.34κ0 at N = 4 instead of K = 4κ0, and K = 5.33κ0 at N = 2 instead of K = 2κ0. The inset shows the dissipated power obtained when subtracting the contributions of a different number of open arms; this cancels the added phonons and bulk contributions, both of which depend only on Tm. The fit line leads to the average thermal conductance per channel gQ = (1.03 ± 0.04)κ0T, which agrees with the expectations. (Errors mentioned here correspond to a confidence level of better than 95%.) This device was not used in the experiments.

Extended Data Fig. 4 Equal branching of current in all arms at ν = 5/2.

Current is sourced from the source, S, and measured in the drain, D, in the same arm (see Fig. 1a). The blue curve shows the reflection coefficient of the current measured in the drain as a function of the pinching of the arm gate. The reflection coefficient value starts from 0.25, when all the arm gates are fully open, and reaches 1.00, when all the current is reflected. The red, green and magenta curves correspond to measurements for the fully open ‘measurement arm’, performed while the other arm gates deplete gradually one by one. Four open arms give a reflection coefficient of r = 0.25, whereas three open arms lead to r = 0.33 and two open arms give r = 0.50. The dotted lines are guides for the eyes indicating equal branching of currents.

Extended Data Fig. 5 Thermal noise analysis at ν = 7/3 and ν = 8/3.

a, b, Standard analysis (see main text), without subtracting the number of participating arms, but using the phonon contribution coefficient β, which fits extremely well in a large range of temperatures and at different filling factors (errors of the fit are 99% confidence levels). The agreement with the expected data is clear. We note the added thermal heat conductance at ν = 8/3 (b; see text).

Extended Data Fig. 6 Upstream neutral modes in ν = 5/2 and ν = 8/3.

a, b, The noise measured at an upstream floating contact connected to the cold amplifier (with respect to ground) is clear evidence of upstream neutral modes. Such upstream noise is not found in particle-like states21.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Banerjee, M., Heiblum, M., Umansky, V. et al. Observation of half-integer thermal Hall conductance. Nature 559, 205–210 (2018).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


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.


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing