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Description
The Schmid transition represents a fundamental insulator-to-superconductor phase transition occurring in the resistively shunted Josephson junction (RSJJ). The system is modeled as a parallel circuit with a non-linear inductor (Josephson energy $E_J$), a capacitor (charging energy $E_C = 4e^2/2C$), and a shunt resistor $R_S$. Initially, Schmid and Bulgadaev [1,2] predicted this transition at $\alpha = R_q/R_S = 1$, with $R_q = h/4e^2$ being the resistance quantum. Nevertheless, recent theoretical [3,4] and experimental [5] studies have challenged its existence, reviving the debate on its transport implications. In this work, we clarify the transition's nature by introducing a distinct binary order parameter $S$. We establish that it falls into the Berezinskii-Kosterlitz-Thouless (BKT) universality class [6,7], evidenced by the logarithmic decay of correlation functions near the critical point. Employing linear response theory, we show the transition's hallmark is the weight of the delta function $D_Q$ at $\omega = 0$ in the resistive response function $\Psi_{\text{reg},Q}(\omega)$ [6]. Remarkably, it lacks a low-frequency Drude peak, demonstrating that dissipation is avoided even in the insulating phase. An out-of-equilibrium analysis under small injected currents further confirms this non-dissipative dynamics.
References
[1] A. Schmid, "Diffusion and Localization in a Dissipative Quantum System", Phys. Rev. Lett. 51, 1506 (1983).
[2] S. Bulgadaev, "Phase diagram of a dissipative quantum system", ZhETF Pisma Redaktsiiu 39, 315 (1984).
[3] K. Masuki, H. Sudo, M. Oshikawa, and Y. Ashida, "Absence versus Presence of Dissipative Quantum Phase Transition in Josephson Junctions", Phys. Rev. Lett. 129, 087001 (2022).
[4] C. Altimiras, D. Esteve, C¸ . Girit, H. Le Sueur, and P. Joyez, "Absence of a dissipative quantum phase transition in Josephson junctions: Theory" arXiv:2312.14754, (2023).
[5] A. Murani, N. Bourlet, H. le Sueur, F. Portier, C. Altimiras, D. Esteve, H. Grabert, J. Stockburger, J. Ankerhold, and P. Joyez, "Absence of a dissipative quantum phase transition in Josephson junction", Phys. Rev. X 11, 018002 (2021).
[6] F. G. Capone, A. de Candia, V. Cataudella, N. Nagaosa, C. A. Perroni, and G. De Filippis, "A Clue on Small-Capacitance Josephson Junction: What to Expect from Cooper Pair Ideal Conductor and Ohmic Resistor in Parallel?", arXiv:2504.00258, (2025).
[7] F. G. Capone, A. de Candia, V. Cataudella, R. Fazio, N. Nagaosa, C. A. Perroni, and G. De Filippis "Quantum Brownian Motion: proving that the Schmid transition belongs to the Berezinskii-Kosterlitz-Thouless universality class", arXiv:2603.16227, (2025).