Dynamical signatures and control of time-reversal breaking in twisted nodal superconductors

TL;DR

This work develops a phenomenological, dynamical framework for time-reversal symmetry breaking at twisted nodal-superconductor interfaces. By modeling the interface with a two-harmonic Josephson free energy and employing the RCSJ dynamics, it identifies a soft Josephson plasmon that softens at the TRSB transition and governs linear impedance, while nonlinear driving unveils second-harmonic generation as a necessary and sufficient TRSB signature. Under strong AC driving, the system exhibits dynamical phase transitions between symmetry-broken and symmetric states, including reentrant behavior, characterized by the presence or absence of a finite second-harmonic voltage and tunable by drive amplitude and frequency. The paper provides concrete experimental predictions for twisted Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$, such as measurable plasmon softening and sizable $V_2$ at accessible GHz–tens-of-GHz frequencies, offering practical routes to probe and control TRSB in these interfaces and potentially beyond.

Abstract

Recent observations of time-reversal breaking superconductivity at twisted cuprate interfaces motivate the development of new approaches to better characterize this emergent phenomenon. Here we study the dynamical properties of the order parameters at the twisted unconventional superconductor interfaces. We reveal the emergence of a soft collective mode (Josephson plasmon) at the time-reversal breaking transition, which can be tuned by temperature, twist angle or magnetic field. Furthermore, nonlinear dynamical responses contain direct signatures of both the transition and the broken symmetry itself. In particular, we show that the generation of a second harmonic voltage under alternating current driving is a necessary and sufficient signature of time-reversal symmetry breaking. Finally, we demonstrate that strong nonlinear driving induces dynamical phase transitions between phases with and without spontaneous symmetry breaking, introducing a tool for their out-of-equilibrium control. We discuss the signatures of our predictions in AC current-driven experiments on twisted Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ interfaces.

Figures (13)

• Figure 1: Phase diagram of the interface for varied temperature and twist angle with the panels demonstrating the qualitative behavior of the collective order parameter oscillations (Sec. \ref{['Dynamical Behavior']}). A soft mode appears at the phase transition point (Sec. \ref{['softmode']}) and the system generates second harmonic response in the TRSB phase (Sec. \ref{['sec: reentrance']}).
• Figure 2: Normalized Josephson plasma frequency $\bar{\omega}_{pl}$, Eq. \ref{['eq:omegapl']}, and critical current $\bar{I_c}$ as a function of: a) twist angle $\theta$ and b) temperature (normalized to the critical one), obtained from linearized Eq. \ref{['Normalized RCSJ model']} in the $\beta_c\to \infty$ limit. We take $\bar{J}_{c1}^{\textcolor{black}{T=0}}=10$, giving $\theta_{c\textcolor{black}{,T=0}}^{+}=39.2^{\circ}$. For, a) $T=0$, and for b) $\theta =43^\circ$ and $T_{TRSB}=0.65T_c$ where $T_{TRSB}$ indicate the onset of time reversal breaking. Blue filling indicate symmetric $\phi_0 = 0$ phase whereas green corresponds to the TRSB phase.
• Figure 3: Normalized Josephson plasma frequency $\bar{\omega}_{pl}$, Eq. \ref{['eq:omegapl']}, and critical current $\bar{I_c}$ as a function of in-plane magnetic flux $\bar{\Phi}$ through the system (see Eq. \ref{['eq:Fmag']}) for $\bar{J}_{c1}^{\textcolor{black}{T=0}}=10$and $T=0$. The yellow filling indicates that the system exhibit $C_{2y}$ symmetry and red indicates a symmetry-broken state. a) For $\theta= 35^\circ$, the system is not in TRSB state at zero flux, and $\bar{\omega}_{pl}$ vanishes only together with the critical current. b) For $\theta =43^\circ$, $\bar{\omega}_{pl}$ vanishes at the field-driven phase transition, where $\bar{I}_c$ is finite.
• Figure 4: Josephson plasma frequency $\bar{\omega}_{pl}$ vs. $\theta$ with a finite explicit time-reversal breaking $\bar{J}_m=0.01$, varied applied DC current bias $\bar{I}$, and $\bar{J}^{\textcolor{black}{T=0}}_{c1}=10$at $T=0$. We find that for $\bar{I}=\bar{I}_m=0.01$ the complete plasmon softening is recovered.
• Figure 5: Plot of the absolute values of the linear impedance $|\bar{Z}_1(\bar{\omega}\textcolor{black}{,\theta,T})|$ for a) different twist angles at $T=0$ and b) varying temperatures at $\theta=43^\circ$, where $T/T_c=0.65$ is the TRSB transition temperature. We've taken $\bar{J}^{T=0}_{c1}=10$, yielding $\theta_{c\textcolor{black}{,T=0}}^+=39.2^\circ$, and $\beta_c^{T=0}=20$. In both cases, the peak in the linear impedance shifts towards $\bar{\omega}= 0$ close to the transition indicating the presence of a soft mode.