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.
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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.
Extended data figures and tables
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.
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(ακ0T2, βT5) 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.
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.
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).
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.