Nonuniform superconducting states from Majorana flat bands

This paper demonstrates that zero-energy Majorana flat bands in topological superconductors drive the spontaneous emergence of nonuniform superconducting states, specifically pair density waves and phase crystals, which lower the system's free energy by gapping out the flat band and exhibit distinct temperature-dependent stability regimes.

The Gist

Imagine a superconductor as a perfectly synchronized dance floor where electrons move in perfect pairs, gliding without any friction. Usually, this dance floor is uniform; everyone is doing the same steps in the same rhythm everywhere.

However, this paper explores a special, tricky kind of superconductor that hosts "ghosts" on its edges. These ghosts are called Majorana flat bands. Think of them as a row of invisible, zero-energy dancers standing perfectly still right at the edge of the floor. Because they have zero energy and there are so many of them (a "flat band"), they make the system unstable, like a tower of cards that is ready to collapse. The system desperately wants to get rid of this instability to save energy.

The researchers asked: How does the superconductor fix this? They found that instead of staying uniform, the dance floor spontaneously rearranges itself into two distinct, non-uniform patterns to "kick out" these zero-energy ghosts.

Here are the two ways the system reorganizes, explained through analogies:

1. The Pair Density Wave (The "Staggered Step")

In this state, the superconductor decides to change the strength of the dance pairs along the edge, but keeps the rhythm (phase) the same.

• The Analogy: Imagine the dancers on the edge suddenly start doing a "staggered step." One dancer holds hands tightly, the next holds loosely, the next tightly, and so on. It's like a zipper or a bumpy road.
• What it does: This "bumpy" pattern breaks the perfect symmetry of the edge. By doing this, it forces the stationary "ghost" dancers to mix with each other and move off the zero-energy spot. They gain a little energy and disappear from the dangerous zero-energy state.
• When it happens: This happens when the chemical conditions (like the number of electrons) are set to a specific range. It's the system's first line of defense.

2. The Phase Crystal (The "Twisting Spiral")

In this state, the strength of the dance pairs stays mostly the same, but the rhythm (phase) starts to twist and turn along the edge.

• The Analogy: Imagine the dancers on the edge are all holding hands, but they start twisting their bodies in a wave. One faces forward, the next faces slightly right, the next more right, creating a spiral or a crystal-like pattern. This twisting creates tiny, spontaneous currents (like little whirlpools) flowing along the edge.
• What it does: This twisting breaks a different kind of symmetry. It also forces the "ghost" dancers to mix and gain energy, but it does so by changing the direction of the dance rather than the strength of the grip.
• When it happens: This happens when the conditions change (specifically, when the chemical potential increases) and the "staggered step" (Pair Density Wave) isn't strong enough to clear out all the ghosts. The system switches to this twisting mode to finish the job.

The "Middle Ground"

Between these two distinct states, there is a large "intermediate zone."

• The Analogy: Think of this as a dance floor where the dancers are doing both the staggered step and the twisting spiral at the same time. It's a messy mix of changing grip strength and changing rhythm.
• The Finding: At absolute zero temperature, this messy middle ground is very common. The system is willing to do a bit of both to ensure all the zero-energy ghosts are removed.

The Temperature Effect

The paper also looked at what happens when you heat the system up (add thermal energy).

• The Analogy: Imagine the dance floor getting crowded with random, jittery people (heat).
• The Result:
  • The "Staggered Step" (Pair Density Wave) is tough. It survives even when the room gets quite hot (up to 80% of the temperature where the superconductivity breaks down completely).
  • The "Twisting Spiral" (Phase Crystal) is fragile. It only survives in a very cold room. As soon as it gets a little warmer, the twisting stops, and the system reverts to a uniform state with the ghosts back on the edge.
  • The "Messy Middle" disappears almost entirely when the temperature rises.

The Big Picture

The main takeaway is that topology dictates the dance. The "ghosts" (Majorana states) are protected by the mathematical rules of the system (topology). To get rid of them and lower the system's energy, the superconductor must break its own uniformity.

The researchers found that the system doesn't just randomly choose a pattern; it chooses the specific pattern (Staggered Step vs. Twisting Spiral) based on the "winding numbers" (a topological count of how the electrons are arranged). If the count is balanced one way, it does the Staggered Step. If it's unbalanced, it does the Twisting Spiral.

In short: Majorana flat bands are so unstable that they force the superconductor to become a complex, non-uniform patterned state to survive, and the specific pattern depends on the system's topological rules and temperature.

Technical Summary

Technical Summary: Nonuniform Superconducting States from Majorana Flat Bands

Problem Statement
The paper investigates the thermodynamic stability of superconducting ground states in systems hosting Majorana flat bands. While zero-energy flat bands within a superconducting gap are topologically protected, their extensive ground-state degeneracy renders them thermodynamically unstable against interactions. In conventional $d$-wave superconductors, this instability leads to nonuniform ordered phases, such as the superconducting phase crystal, which shift zero-energy states to finite energies to lower the system's free energy. However, it remains unclear how the topological protection of Majorana states in weak topological superconductors influences this competition. Specifically, the authors address whether the topological nature of Majorana flat bands hinders or dictates the emergence of nonuniform superconducting phases, such as pair density waves (PDW) and phase crystals, and how these phases behave at finite temperatures.

Methodology
The authors model a two-dimensional system consisting of an array of magnetic adatoms with a helical magnetic texture deposited on a conventional $s$-wave superconducting substrate. This system is described by a tight-binding Bogoliubov-de Gennes (BdG) Hamiltonian in the BDI topological class, characterized by chiral symmetry. The model supports a weak topological phase hosting multiple Majorana edge states along the system boundaries.

To determine the ground state, the authors perform self-consistent numerical calculations of the superconducting order parameter $\Delta_r = |\Delta_r|e^{i\theta_r}$. They employ an iterative scheme on a $120 \times 46$ lattice with open boundary conditions in the $x$-direction and periodic conditions in the $y$-direction. To avoid biasing the results toward specific symmetry-breaking patterns, the calculations utilize three distinct initial guesses:

1. Edge-localized, site-to-site staggered amplitude modulation with vanishing phase variation (targeting PDW).
2. Uniform amplitude with a specific ansatz for phase variation (targeting phase crystals).
3. Uniform solutions.

The converged solution with the lowest free energy ($F = \langle H \rangle - TS$) is identified as the ground state. The study maps the phase diagram as a function of chemical potential ($\mu$), magnetic field strength ($B$), and temperature ($T$), analyzing the resulting amplitude ($|\Delta_r|$) and phase ($\theta_r$) modulations, as well as the low-energy eigenvalue spectrum.

Key Contributions and Results

• Emergence of Competing Nonuniform Phases: The study demonstrates that the uniform superconducting state with Majorana flat bands is never the ground state at zero temperature. Instead, the system spontaneously transitions into nonuniform phases to gap out the Majorana flat band. Two primary phases are identified:

  • Pair Density Wave (PDW): Characterized by edge-localized, site-to-site staggered amplitude modulations ($|\Delta_r|$) with a constant phase ($\theta_r$). This state breaks lattice translation symmetry but preserves chiral symmetry. It hybridizes Majorana states with opposite winding numbers ($W = \pm 1$) on the same edge.
  • Phase Crystal: Characterized by strong, edge-localized phase modulations ($\theta_r$) on the scale of the superconducting coherence length, accompanied by weak amplitude modulations. This state breaks both translation and chiral symmetries, driving spontaneous superfluid momentum and loop currents. It hybridizes Majorana states by breaking the chiral symmetry that otherwise protects them.

• Role of Topology and Winding Numbers: The competition between the PDW and the phase crystal is governed by the distribution of winding numbers ($W_+$ and $W_-$) in the $k_y$-resolved band structure, which is tunable via the chemical potential $\mu$.

  • When $W_+$ and $W_-$ are comparable, the PDW is energetically favorable as it efficiently hybridizes opposite winding states via amplitude modulation.
  • When there is a significant imbalance (e.g., $W_+ \gg W_-$), amplitude modulation alone is insufficient to hybridize the remaining states. In this regime, the system forms a phase crystal, which breaks chiral symmetry to hybridize the remaining states.

• Intermediate Region: At zero temperature, a broad intermediate region exists between the PDW and phase crystal phases. In this regime, the order parameter exhibits comparable modulations in both amplitude and phase. This region arises when remnant Majorana states with identical windings cannot be hybridized by pure amplitude modulation, necessitating the addition of phase modulation to further lower the free energy.

• Finite Temperature Behavior: The stability of these phases differs significantly at finite temperatures:

  • The PDW is robust, surviving up to approximately 80% of the bulk superconducting transition temperature ($T_c$) at low chemical potentials, decreasing to ~10% as $\mu$ increases.
  • The Phase Crystal is more fragile, appearing only at lower temperatures (up to ~10% of $T_c$). Its fragility is attributed to the parameter regime where it exists (higher $\mu$, smaller bulk gap), which is more susceptible to thermal excitations.
  • The Intermediate region is strongly suppressed at finite temperatures, persisting only up to ~4–5% of $T_c$.
  • Above the transition temperatures of these nonuniform states, the uniform superconducting state with the Majorana flat band reappears as a self-consistent solution.

Significance
The paper establishes the ubiquity of emergent nonuniform superconducting phases in topological superconductors hosting Majorana flat bands. It clarifies that the topological protection of Majorana states does not prevent symmetry breaking; rather, the underlying topology (specifically the winding numbers) dictates the nature of the emergent nonuniform phase. The findings highlight that Majorana flat bands are extremely unstable against spatial modulations of the superconducting order parameter, whether in amplitude or phase. The study further distinguishes the thermal stability of these phases, showing that amplitude-driven instabilities (PDW) are significantly more robust against thermal fluctuations than phase-driven instabilities (phase crystals). This work provides a comprehensive energy-based argument for the competition between energy minimization and topological protection in weak topological superconductors.