Quantum Hall transformer in a quantum point contact over the full range of transmission

A recent experiment [Cohen et al., Science 382, 542 (2023)] observed a quantized e²/2h conductance tunneling into a fractional quantum hall edge in the strong coupling limit, which is consistent with the prediction of a quantum Hall transformer. However, details of this description, including renormalizability, remain unresolved. We model this quantum point contact as a quantum wire with a spatially-varying Luttinger parameter and a backscattering impurity. Mapping this model to a spin chain, we use a tensor-network calculation to find the wavepacket scattering and microwave (i.e. finite frequency) conductance at the interface. We then compare the microwave conductance to an analytic solution of a boundary sine-Gordon model of the same model to determine the effective backscattering terms from microscopic parameters of the Luttinger liquid model, which can then be used to predict temperature, frequency and bias dependence of conductance and noise.