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arXiv:cond-mat/0409664 · Physical Review Letters

Efficient Simulation of Resistively Shunted Josephson Junctions

Philipp Werner, Matthias Troyer

Published Aug 1, 2005• 39 citations
arXivHTMLPDF

Abstract

We present a cluster algorithm for resistively shunted Josephson junctions or similar physical systems, which dramatically improves sampling efficiency, and apply it to the superconductor-to-metal transition in a single junction. Measuring the temperature dependence of the zero bias resistance, we confirm that the critical point does not depend on the strength of the Josephson coupling. However, we find that the correlation exponents vary continuously along the phase boundary, indicating that the Schmid-Bulgadaev transition is a line of fixed points.

DOI
10.1103/physrevlett.95.060201

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We present a cluster algorithm for resistively shunted Josephson junctions or similar physical systems, which dramatically improves sampling efficiency, and apply it to the superconductor-to-metal transition in a single junction. Measuring the temperature dependence of the zero bias resistance, we confirm that the critical point does not depend on the strength of the Josephson coupling. However, we find that the correlation exponents vary continuously along the phase boundary, indicating that the Schmid-Bulgadaev transition is a line of fixed points.
Using a new cluster Monte Carlo algorithm, we study the phase diagram and critical properties of an interacting pair of resistively shunted Josephson junctions. This system models tunnelling between two electrodes through a small superconducting grain, and is described by a double sine-Gordon model. In accordance with theoretical predictions, we observe three different phases and crossover effects arising from an intermediate coupling fixed point. On the superconductor-to-metal phase boundary, the observed critical behaviour is within error-bars the same as in a single junction, with identical values of the critical resistance and a correlation function exponent which depends only on the strength of the Josephson coupling. We explain these critical properties on the basis of a renormalization group (RG) calculation. In addition, we propose an alternative new mean-field theory for this transition, which correctly predicts the location of the phase boundary at intermediate Josephson coupling strength.
Using resistivity-shunted-junction dynamics, we model bond dilutions in Josephson junction arrays. The current-voltage characteristics are obtained by large-scale numerical simulations. By dynamic scaling analysis to the finite-temperature phase transition, we provide clear evidence for the existence of non-Kosterlitz–Thouless-type phase transition, as described by Kosterlitz et al. [J. Phys. C 6, 1181 (1973); J. Phys. C 7, 1046 (1974)] in a wide range of disorder strengths. Critical temperature and exponents of the phase transitions are estimated, which are further confirmed by the finite-size scaling analysis at the transition temperature.
Dynamical simulations and scaling arguments are used to study the current-voltage $(I\ensuremath{-}V)$ characteristics of a two-dimensional model of resistively shunted Josephson-junction arrays in presence of percolative disorder, at zero external field. Two different limits of the Josephson-coupling concentration ppp are considered, where pc{p}_{c}pc​ is the percolation threshold. For p>pcp>{p}_{c}p>pc​ and zero temperature, the $I\ensuremath{-}V$ curves show power-law behavior above a disorder-dependent critical current. The power-law behavior and critical exponents are consistent with a simple scaling analysis. At pc{p}_{c}pc​ and finite temperature T,T,T, the results show the scaling behavior of a T=0T=0T=0 superconducting transition. The resistance is linear but vanishes for decreasing TTT with an apparent exponential behavior. Crossover to nonlinearity appears at currents proportional to ${T}^{1+{\ensuremath{\nu}}_{T}},$ with a thermal-correlation length exponent ${\ensuremath{\nu}}_{T}$ consistent with the corresponding value for the diluted XY\mathrm{XY}XY model at pc.{p}_{c}.pc​.
A Josephson junction subject to an ohmic dissipative environment undergoes a zero-temperature quantum phase transition called Schmid-Bulgadaev transition form an insulating state to a superconducting state upon increasing the impedance of the environment above the resistance quantum. Superconducting qubits based on Josephson junction chains represent a system where it is possible to dynamically explore both sides of this transition by probing the scattering of individual plasma modes of the junction chain on a SQUID at the boundary which was achieved in recent experiments. Here, we provide a theoretical analysis of the scattering in the system paying special attention to the non-perturbative superconducting regime. We find that in this regime, a specific diagrammatic resummation must be performed in order to correctly account for fluctuation renormalization of the SQUID's potential. We find analytic expressions for the modes' frequency shifts and quality factors which together provide a full description of scattering. We argue that the non-perturbative nature of the superconducting phase manifests itself through a power-law dependence of the frequency shifts on the SQUID flux and through non-monotonous dependence of the quality factors on the mode frequency. Our predictions may be readily verified in modern low-temperature experiments.
Schmid transition was introduced first as a superconductor-insulator transition in the zero-frequency response of a shunted Josephson junction in equilibrium at zero temperature. As it is typical for a quantum impurity problem, at finite frequencies the transition is broadened to a crossover. Modern attempts to find Schmid transition rely on finite-frequency measurements of a quantum circuit. We predict the frequency dependence of the admittance and reflection phase shift for a high-impedance transmission line terminated by a Josephson junction for a wide variety of devices, from a charge qubit to a transmon. Our results identify the circuit parameters allowing for the universal scaling of the responses with frequency, thus helping to identify the Schmid transition from the finite-frequency measurements.
The nonequilibrium phase transition of two-dimensional fully frustrated Josephson junction arrays driven by external currents is studied in the framework of the short-time dynamic scaling approach. Within the resistively shunted junction dynamics, besides the critical temperature Tc{T}_{c}Tc​, the static and dynamic critical exponents $2\ensuremath{\beta}∕\ensuremath{\nu}$, $\ensuremath{\nu}$, and zzz are obtained. The resulting values of Tc{T}_{c}Tc​ are consistent with previous steady-state simulations and decrease with the increase of currents. It is found that the values of the exponent $\ensuremath{\nu}$ are not sensitive to external currents, while the values of $2\ensuremath{\beta}∕\ensuremath{\nu}$ and zzz are strongly current dependent.