Topologically non-trivial gap function and topology-induced time-reversal symmetry breaking in a superconductor with singular dynamical interaction

This paper demonstrates that adding a repulsive Hubbard-type interaction with a finite cutoff to a model with singular dynamical interactions can stabilize a topologically non-trivial superconducting state, which emerges from the topologically trivial ground state via an intermediate phase characterized by topology-induced time-reversal symmetry breaking.

The Gist

Imagine a crowded dance floor where electrons are the dancers. In most materials, these dancers move in a predictable, orderly way (like a Fermi liquid). But in some exotic materials, the music is so chaotic that the dancers lose their rhythm, creating a "non-Fermi liquid" state.

Physicists have long known that in this chaotic state, electrons can still pair up to become a superconductor (dancing in perfect unison with zero resistance). However, there's a catch: usually, the "easiest" way for them to pair up is a boring, topologically simple dance.

This paper, by Yue Yu and Andrey Chubukov, discovers a way to force the electrons to perform a complex, topologically twisted dance instead. Even better, they find that to switch from the simple dance to the complex one, the electrons must pass through a mysterious "twilight zone" where they break the rules of time itself.

Here is the breakdown using everyday analogies:

1. The Two Types of Dances (The Gap Functions)

In the world of superconductors, the "gap function" is the choreography the electrons follow.

• The Simple Dance (Topologically Trivial): Imagine a group of dancers moving in a perfect circle. It's smooth, stable, and has no knots. This is the standard state that usually wins because it requires the least energy.
• The Twisted Dance (Topologically Non-Trivial): Imagine the dancers are performing a complex routine where their paths cross and twist, creating "knots" or "vortices" in the choreography. In physics terms, these are points where the phase of the wave function spins around. This state is usually unstable and loses to the simple dance.

The Problem: Scientists wanted to know: Can we make the "Twisted Dance" the winner?

2. The New Ingredient: The "Repulsive Wall"

The authors added a new rule to the dance floor: a repulsive force (like a wall that pushes dancers apart) that only exists up to a certain speed (energy cutoff).

• Think of the original chaotic music as a soft, fuzzy blanket that encourages pairing.
• The new repulsive force is like a stiff, high-speed barrier.

The Discovery: When they tuned the strength of this "wall" and the speed limit, they found a sweet spot. In this specific zone, the "Twisted Dance" becomes the most energy-efficient option! The electrons prefer the complex, knotted choreography over the simple circle.

3. The "Twilight Zone": Breaking Time

Here is the most magical part. You cannot simply morph a simple circle into a twisted knot without tearing the fabric of the dance.

• The Analogy: Imagine trying to turn a plain rubber band (simple) into a figure-eight knot (twisted). You can't just stretch it; you have to cut it, twist it, and rejoin it.
• The Physics: To get from the "Simple Dance" to the "Twisted Dance," the system must pass through an intermediate phase where Time-Reversal Symmetry (TRS) is broken.

What does "Breaking Time" mean?
In a normal superconductor, if you played the dance movie backward, it would look exactly the same (symmetry).
In this "Twilight Zone," the dance looks different going forward than it does backward. It's like a clock that suddenly starts running backward for a moment. The electrons create tiny, spontaneous current loops that act like tiny magnets, effectively creating a magnetic field that distinguishes "past" from "future."

This phase is topologically protected. It exists only because the system is forced to bridge the gap between two very different types of choreography. It's a necessary bridge.

4. The Map of the Dance Floor

The paper draws a map (a phase diagram) showing where these states exist:

• Zone A (Simple): Low repulsion. The electrons do the simple circle dance.
• Zone B (Twisted): High repulsion. The electrons do the complex knotted dance.
• Zone C (The Twilight Zone): A narrow strip between A and B. Here, the electrons are confused. They are doing a mix of both dances simultaneously, creating a complex, time-breaking state.

5. Why Does This Matter?

• New Physics: It proves that "topology" (the shape of the dance) can force a system to break time symmetry, rather than time symmetry breaking causing the topology. It's the reverse of what we usually see.
• Real-World Application: The authors suggest that by using electric gates (to control the "wall") and pressure (to tune the "music"), experimentalists could actually create this state in a lab.
• Detection: Because this state creates tiny, swirling currents (like a microscopic altermagnet), scientists can detect it using sensitive magnetic probes or by twisting the material (strain) to make the hidden currents visible.

Summary

Imagine you are trying to get from a flat road (Simple Superconductor) to a mountain peak (Twisted Superconductor). Usually, you just drive up. But this paper shows that sometimes, the road is blocked. To get to the peak, you must drive through a parallel dimension (the Time-Breaking Phase) where the laws of physics feel slightly different, just to bridge the gap between the two destinations.

This "parallel dimension" isn't just a glitch; it's a necessary, stable state of matter that emerges purely because of the shape of the electron dance.

Technical Summary

1. Problem Statement

The paper addresses the nature of superconductivity in strongly correlated electron systems exhibiting non-Fermi liquid (NFL) behavior. In these systems (e.g., near quantum critical points, $\nu=1/2$ Landau levels, or quantum dots), pairing is mediated by a singular dynamical interaction (often modeled as a soft boson with a frequency-dependent susceptibility $\chi(\Omega_m) \sim 1/|\Omega_m|^\gamma$).

Key issues identified:

• Multiplicity of Solutions: Unlike conventional BCS theory, which yields a single gap solution, pairing out of an NFL generates a "tower" of topologically distinct gap functions, $\Delta_n$. These solutions are characterized by the number of zeros (dynamical vortices) of the gap function $\Delta(z)$ in the upper half of the complex frequency plane.
• Ground State Selection: In previously studied models (e.g., pure $\gamma$-models), the topologically trivial solution ($n=0$, no vortices on the Matsubara axis) always possesses the lowest energy.
• The Gap: There is a lack of understanding regarding conditions under which a topologically non-trivial solution (with $n > 0$) becomes the ground state and how the system transitions between these distinct topological phases. Specifically, can the topological distinction between solutions induce a new phase of matter, such as one with broken time-reversal symmetry (TRS)?

2. Methodology

The authors employ a combination of analytical reasoning and numerical analysis on a minimal model:

• The Model: A $\gamma$-model describing fermions coupled to a soft boson with dynamical susceptibility $\chi(\Omega_m) = 1/(|\Omega_m|^\gamma + M^\gamma)$, where $M$ is the boson mass.
• Perturbation: A repulsive Hubbard-type interaction $V$ with a hard energy cutoff $E_V$ is added to the model. This mimics experimental control via external gates or screening.
• Equations: The authors solve the Eliashberg gap equation on the Matsubara axis (Eq. 1 in the text) numerically.
  $$\Delta(\omega_m) = \int_0^\infty \dots - V \int_0^{E_V} \dots$$
• Analytical Continuation: To determine the topological structure (number of vortices) in the causal (upper) half-plane, the authors use Padé approximants to analytically continue the gap function from the imaginary Matsubara axis to the real frequency axis ($z = \omega + i0^+$).
• Phase Diagram Construction: They map the phase diagram in the $(E_V, V)$ plane for varying boson masses $M$ relative to a critical mass $M_c$.

3. Key Contributions and Results

A. Selection of Topologically Non-Trivial Ground States

The authors demonstrate that adding a repulsive interaction $V$ with a finite cutoff $E_V$ can stabilize the subleading, topologically non-trivial solution ($\Delta_1$, containing one vortex) as the ground state.

• Mechanism: The repulsive term penalizes the gap function at low frequencies. By tuning $E_V$, the system can be forced into a configuration where the trivial solution $\Delta_0$ (which is roughly constant at low frequencies) is energetically unfavorable compared to $\Delta_1$ (which has a node/vortex structure that minimizes the repulsive energy cost).
• Phase Diagram:
  • $M > M_c$: Only the trivial solution exists in the pure model. With $V$, the system undergoes a series of topological transitions (0 $\to$ 2 $\to$ 1 vortices) as $V$ increases. Crucially, a region of normal state (no superconductivity) emerges between the low-$V$ and high-$V$ superconducting phases.
  • $M = M_c$: The normal state region shrinks to a line ($E_V = E_V^c$). At a critical point $A = (E_V^c, V^*)$, the trivial solution $\Delta_0$ and the infinitesimal non-trivial solution $\Delta_1$ become degenerate.
  • $M < M_c$: The non-trivial solution $\Delta_1$ has a finite magnitude. The normal state vanishes entirely.

B. Topology-Induced Time-Reversal Symmetry Breaking (TRSB)

The most significant finding is the emergence of an intermediate TRSB phase required to connect topologically distinct superconducting states.

• The Topological Obstacle: The transition from the low-$V$ state (0 vortices) to the high-$V$ state (1 vortex) cannot occur continuously while preserving Time-Reversal Symmetry (TRS).
  • In the TRS limit, vortices must enter/exit the causal plane in pairs (at $\pm \omega$) to preserve symmetry.
  • To go from 2 vortices to 1 vortex, one vortex must exit the causal plane at a finite real frequency. This breaks the symmetry between $+\omega$ and $-\omega$.
• The TRSB Phase: To bridge the gap between the 2-vortex and 1-vortex states, the system enters a phase where the gap function becomes complex:
  $$\Delta(z) \approx \cos\theta \Delta_0(z) \pm i \sin\theta \Delta_1(z)$$
  Here, $\theta$ varies from $0$ to $\pi/2$. The imaginary component spontaneously breaks TRS.
• Topology as the Driver: Unlike conventional cases where TRSB leads to topology (e.g., chiral edge modes), here the topological distinction between $\Delta_0$ and $\Delta_1$ forces the system to break TRS. The TRSB phase is topologically protected.

C. Phase Diagram Features

• Critical Point A: At $M=M_c$, the transition is a point where the normal state, the trivial SC, and the non-trivial SC meet.
• TRSB Region: For $M < M_c$, the TRSB phase occupies a finite area in the $(E_V, V)$ plane. The size of this region increases as $M$ decreases (approaching $M=0$).
• Vortex Dynamics: The paper details how vortices move:
  • In the adiabatic limit ($E_V \gg 1$), vortices exit at infinity.
  • In the anti-adiabatic limit ($E_V \ll 1$), vortices exit at the origin.
  • The TRSB phase allows a smooth interpolation where a vortex exits at a finite frequency.

4. Physical Implications and Significance

• Experimental Realizability: The proposed mechanism relies on three experimentally tunable parameters:
  1. Boson Mass ($M$): Tunable via pressure or doping (distance to quantum criticality).
  2. Repulsion Strength ($V$) and Cutoff ($E_V$): Tunable via electrostatic gating and screening in 2D materials or quantum dots.
• Nature of the TRSB State:
  • The TRSB order corresponds to a staggered current loop order (orbital altermagnetism).
  • Since the gap function is single-component, it does not break crystal mirror symmetry globally, resulting in zero net orbital angular momentum ($L_z = 0$). However, it induces local currents.
  • Detection: The local magnetic fields from these currents can be detected by $\mu$SR (muon spin rotation). Global detection of net $L_z$ can be achieved by applying shear strain ($\epsilon_{xy}$) to break mirror symmetry, leading to a measurable Kerr effect.
• Theoretical Impact: This work establishes that topological constraints in the space of gap functions can drive symmetry breaking, a reversal of the usual paradigm where symmetry breaking creates topological phases. It provides a concrete strategy to realize and detect topologically non-trivial superconductivity in NFL systems.

5. Conclusion

Yu and Chubukov demonstrate that in superconductors with singular dynamical interactions, the addition of a repulsive interaction with a cutoff can select a topologically non-trivial ground state. The transition between topologically distinct gap functions (differing by the number of dynamical vortices) necessitates an intermediate phase with spontaneously broken time-reversal symmetry. This TRSB phase is a direct consequence of the topological structure of the gap function, offering a new pathway to discover and characterize exotic superconducting states in quantum critical materials.