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

This paper proposes that the dynamical properties of twisted nodal superconductors, specifically the emergence of a soft Josephson plasmon mode and the generation of a second harmonic voltage under AC driving, provide distinct signatures and control mechanisms for characterizing and manipulating time-reversal symmetry breaking at their interfaces.

The Gist

Imagine you have two sheets of a special, super-conducting material (like a super-highway for electricity with zero resistance). If you stack them perfectly on top of each other, they act normally. But, if you twist one sheet slightly relative to the other—like twisting two pieces of paper at a specific angle (around 45 degrees)—something magical and strange happens. The material suddenly breaks a fundamental rule of physics called "time-reversal symmetry."

In simple terms, time-reversal symmetry is like watching a movie of the material's behavior. If the movie looks exactly the same whether you play it forward or backward, the symmetry is intact. If the movie looks different when played backward, the symmetry is broken. This paper explores how to detect this "broken time" and how to control it using electricity.

Here is a breakdown of the paper's main discoveries using everyday analogies:

1. The "Soft" Spot: The Warning Sign

The authors discovered that right at the moment this "broken time" state appears, the material develops a soft collective mode.

• The Analogy: Imagine a playground swing. Normally, if you push it, it swings back and forth at a steady, fast rhythm. But, imagine the swing is attached to a very loose, floppy spring. If you push it, it moves very slowly and sluggishly.
• The Science: As the material approaches the point where it breaks time-reversal symmetry, its natural "swinging" frequency (called a Josephson plasmon) slows down and almost stops. It becomes "soft."
• Why it matters: This slowing down is a clear warning sign that the transition is happening. The paper suggests you can tune this "softness" by changing the temperature, the twist angle, or even applying a magnetic field. It's like tuning a radio to find the exact station where the signal changes.

2. The "Echo" Test: The Second Harmonic

The most exciting finding is a new way to prove that time-reversal symmetry is broken. The authors propose a test using alternating current (AC), which is electricity that flows back and forth like a tide.

• The Analogy: Imagine you are pushing a child on a swing.
  • Normal State (Symmetry Intact): If the swing is perfectly balanced, every time you push forward, it goes forward; every time you pull back, it goes back. The motion matches your push perfectly. If you push at a frequency of 1 push per second, the swing moves at 1 push per second.
  • Broken State (Time-Reversal Broken): Now, imagine the swing is slightly "stuck" or biased to one side. When you push it forward, it flies high. When you pull it back, it barely moves. The motion is lopsided. Because of this lopsidedness, the swing actually creates a "double beat." For every one push you give, the swing creates a distinct "echo" or secondary movement at twice the speed (2 pushes per second).
• The Science: The paper claims that if you drive the twisted superconductor with an AC current, and you measure the voltage, you will see a second harmonic (a signal at double the frequency).
  • No second harmonic? Time symmetry is likely intact.
  • Second harmonic present? Time symmetry is definitely broken.
  • The authors state this is a "necessary and sufficient" test, meaning it's a perfect, foolproof indicator, unlike other tests (like the "diode effect") which can sometimes give false positives.

3. The "Tug-of-War": Controlling the State

The paper also shows that if you push the system hard enough with this alternating current, you can actually force the material to switch states.

• The Analogy: Imagine a ball sitting in a valley with two dips (a "W" shape). The ball can rest in the left dip or the right dip. This represents the two possible "broken time" states.
  • Gentle Push: If you wiggle the ground gently, the ball stays in its dip, just shaking a little.
  • Hard Push: If you shake the ground violently, the ball might get enough energy to jump out of one dip, roll over the hill, and start bouncing back and forth between both dips.
• The Science: When the AC current is strong enough, the material is forced out of its "broken time" state and into a "symmetric" state where it bounces between the two possibilities so fast that, on average, it looks balanced again.
• The Result: This creates a dynamical phase transition. You can use the strength of the electrical current to turn the "broken time" property on and off, effectively controlling the material's quantum state in real-time.

4. Real-World Application

The authors specifically looked at a material called Bi2Sr2CaCu2O8+x (a type of high-temperature superconductor). They calculated that these effects (the slowing down of the swing and the generation of the double-frequency echo) should be observable in real experiments with current technology.

Summary

In short, this paper provides a new "toolkit" for scientists studying twisted superconductors:

1. Look for the slowdown: If the material's natural vibration slows down to a crawl, it's about to break time symmetry.
2. Listen for the echo: If you drive it with AC current and hear a "double beat" (second harmonic), time symmetry is definitely broken.
3. Turn the knob: You can use strong electrical currents to force the material to switch between these states, giving scientists a way to control these exotic quantum properties.

Technical Summary

Technical Summary: Dynamical Signatures and Control of Time-Reversal Breaking in Twisted Nodal Superconductors

Problem Statement
Recent experimental observations of time-reversal symmetry breaking (TRSB) superconductivity at twisted interfaces of cuprates, specifically $Bi_2Sr_2CaCu_2O_{8+x}$, have motivated the need for robust methods to characterize this emergent phenomenon. While the Josephson diode effect has been observed, it is a sufficient but not necessary criterion for TRSB, and experimental "memory effects" suggest the presence of additional, potentially explicit symmetry-breaking mechanisms. The field lacks definitive dynamical signatures to distinguish spontaneous TRSB from other effects and to probe the collective modes associated with the phase transition.

Methodology
The authors develop a theoretical framework based on the Resistively and Capacitively Shunted Junction (RCSJ) model applied to the phase difference $\phi$ between two twisted nodal superconductors. The equilibrium free energy is modeled with a two-harmonic Josephson coupling term ($J_{c1}\cos\phi - \frac{J_{c2}}{2}\cos2\phi$), which allows for a continuous phase transition to a TRSB state near a $45^\circ$ twist angle.

The study proceeds in three stages:

1. Linear Response: The authors linearize the equations of motion to identify collective modes (Josephson plasmons) and analyze their behavior near the critical point where the TRSB transition occurs. They also incorporate phenomenological terms to model explicit symmetry breaking (e.g., "memory effects") and external magnetic fields.
2. Perturbative Nonlinear Response: Using a Taylor expansion of the phase variable in powers of the driving current amplitude, the authors derive analytical expressions for higher-order harmonic responses, specifically focusing on the generation of a second-harmonic voltage.
3. Strong Nonlinearity and Numerical Simulation: To address regimes where perturbative expansions fail, the authors numerically solve the full RCSJ equation under strong AC driving. They analyze the resulting non-equilibrium phase diagrams, phase portraits, and time-averaged free energies to identify dynamical phase transitions.

Key Contributions and Results

• Emergence of a Soft Josephson Plasmon: The study reveals that a soft collective mode (Josephson plasmon) emerges at the TRSB transition point. The frequency of this mode, $\omega_{pl}$, vanishes precisely at the transition as the potential energy landscape flattens. Unlike conventional Josephson junctions where the plasma frequency is tied to the critical current, here the critical current remains finite while the plasma frequency softens to zero. This softening serves as a signature of the transition and can be tuned by temperature, twist angle, or in-plane magnetic field.
• Diagnosing Explicit Symmetry Breaking: The authors demonstrate that the presence of explicit TRSB (modeled as a "memory effect" or DC bias) prevents the complete softening of the plasmon mode. However, applying a specific DC current bias can counteract this explicit term, restoring the complete softening of the mode. This provides a spectroscopic method to precisely determine the magnitude of explicit symmetry-breaking terms.
• Second-Harmonic Generation as a Necessary and Sufficient Signature: A central finding is that the generation of a second-harmonic voltage under AC current driving is a necessary and sufficient signature of the TRSB phase (with the exception of the exact $45^\circ$ twist angle where symmetry is restored). In the symmetric phase, the free energy is symmetric about $\phi=0$, forbidding odd-order terms in the expansion required for second-harmonic generation. In the TRSB phase, the broken symmetry allows for a non-zero second-harmonic response. This distinguishes it from the diode effect, which is not a necessary condition.
• Dynamical Phase Transitions: Under strong AC driving, the system undergoes non-equilibrium dynamical phase transitions. As the drive amplitude increases, the system can transition from a symmetry-broken state (characterized by a non-zero time-averaged phase and second-harmonic voltage) to a symmetric state (where the phase oscillates between two minima, averaging to zero and suppressing the second harmonic). This transition is identified as a bifurcation in the system's limit cycles.
• Reentrant Behavior: In the high-frequency and low-frequency limits, the authors identify reentrant dynamical transitions where the second-harmonic response reappears at very high drive amplitudes. This is explained analytically using the Jacobi-Anger expansion and the concept of a time-averaged effective free energy, which can renormalize the system's potential from a double-well to a single-well and back.

Significance and Claims
The paper claims to establish order parameter dynamics as a primary tool for both probing and controlling twisted nodal superconductors. The authors assert that their proposed signatures—specifically the softening of the Josephson plasmon and the presence of second-harmonic voltage—offer robust, mechanism-independent methods to identify TRSB.

The work suggests that these effects are experimentally accessible in twisted $Bi_2Sr_2CaCu_2O_{8+x}$ interfaces. The authors estimate that the soft plasmon mode should be detectable via low-frequency AC impedance measurements, and the second-harmonic voltage (estimated at $\sim 20\mu V$) should be observable using standard lock-in techniques. Furthermore, the demonstration that AC driving can induce sharp dynamical phase transitions introduces a new method for the out-of-equilibrium control of these superconducting phases, potentially achievable with conventional microwave electronics. The phenomenological approach, relying only on the two-harmonic Josephson effect, implies that these results may be applicable to a wide variety of other platforms beyond the specific cuprate system studied.