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The search for anyons, quasiparticles with fractional charge and exotic exchange statistics, has inspired decades of condensed matter research. Quantum Hall interferometers enable direct observation of the anyon braiding phase via discrete interference phase jumps when the quasiparticle number changes. Here, we observe the universal anyonic braiding phase in both the $\nu = 1/3$ and $4/3$ fractional quantum Hall states by probing three-state random telegraph noise (RTN) in real-time. We find that the observed RTN stems from anyon quasiparticle number $n$ fluctuations and reconstruct three Aharonov-Bohm oscillation signals phase shifted by $2\pi/3$, corresponding to the three possible interference branches from braiding around $n$ (mod 3) anyons. Hence, we fully characterize the anyon exchange statistics using new methods that can readily extend to non-abelian states., Comment: 19 pages, 11 figures

In low-dimensional systems, the combination of reduced dimensionality, strong interactions, and topology has led to a growing number of many-body quantum phenomena. Thermal transport, which is sensitive to all energy-carrying degrees of freedom, provides a discriminating probe of emergent excitations in quantum materials and devices. However, thermal transport measurements in low dimensions are dominated by the phonon contribution of the lattice, requiring an experimental approach to isolate the electronic thermal conductance. Here, we show how the measurement of nonlocal voltage fluctuations in a multiterminal device can reveal the electronic heat transported across a mesoscopic bridge made of low-dimensional materials. By using 2-dimensional graphene as a noise thermometer, we demonstrate quantitative electronic thermal conductance measurements of graphene and carbon nanotubes up to 70 K, achieving a precision of ~1% of the thermal conductance quantum at 5 K. Employing linear and nonlinear thermal transport, we observe signatures of long-range interaction-mediated energy transport in 1-dimensional electron systems, in agreement with a theoretical model. Our versatile nonlocal noise thermometry allows new experiments probing energy transport in emergent states of matter and devices in low dimensions., Comment: Authors' revision. Main text: 11 pages, 4 figures. Supplementary information: 26 pages, 11 figures. Methods and references included. Published in Nature Nanotechnology