Tunneling in multi-site mesoscopic quantum Hall circuits

Transport properties of single- and two-site mesoscopic quantum Hall (QH) circuits at high transparencies can be described in terms of the lowest-order backscattering processes, enabling a mapping to the boundary sine-Gordon model. We show that this description breaks down in circuits with four or more sites, where higher-order backscattering processes become relevant and qualitatively modify the low-energy physics, while remaining exactly marginal in three-site geometries. Focusing on the four-site circuit, we derive an effective low-energy theory that captures the resulting interaction-driven physics and reveal the emergence of unique quantum-critical points. In the vicinity of these critical points, we obtain universal conductance and scaling behavior and establish the robustness of the associated non-Fermi liquid physics. We further introduce tunneling in multichannel multi-site QH circuits and propose a promising route for realizing diverse quantum-critical phenomena. We show that a boundary sine-Gordon description can be restored in multichannel multi-site QH circuits by appropriately looping selected edge channels, a procedure that is experimentally feasible. Finally, we analyze the non-equilibrium heating effects relevant to transport measurements in QH circuits. Altogether, our results establish multi-site QH circuits as a versatile and highly controllable platform for simulating interaction-driven quantum critical phenomena.