Majorana flat bands create competing superconducting states

Researchers at Uppsala University have discovered that a standard uniform superconducting state is impossible at zero temperature when Majorana flat bands are present. Their self-consistent calculations reveal two competing superconducting phases, a “pair density wave” with amplitude modulations and a “phase crystal” with phase modulations, both localized to the edges of the topological superconductor. These phases emerge because they lower the system’s free energy by gapping out the Majorana flat band, a process primarily controlled by the chemical potential, and establish the ubiquity of emergent nonuniform superconducting phases in these materials.

Majorana Flat Bands Drive Competing Superconducting Phases

The team’s self-consistent calculations further demonstrate the existence of a second competing phase, a “phase crystal,” distinguished by edge-localized phase modulations in the superconducting order parameter. Unlike the pair density wave, which exhibits amplitude modulations at the edges, the phase crystal relies on alterations in the phase of the superconducting order parameter to achieve stability. The fate of these two phases is intricately linked to “winding numbers,” topological invariants that are directly tunable through manipulation of the chemical potential.

The stability of these nonuniform superconducting phases extends to finite temperatures, though with notable differences from previous studies of Majorana flat bands. The pair density wave persists up to approximately 80% of the bulk superconducting transition temperature, while the phase crystal requires even lower temperatures to emerge. A broad intermediate region exists where both amplitude and phase modulations are comparable, creating a complex interplay between the two competing states.

Above these transition temperatures, the uniform superconducting state, with its characteristic Majorana flat bands, reappears, indicating a temperature-dependent shift in the system’s ground state. This work establishes the prevalence of these emergent nonuniform superconducting phases and their temperature-dependent behavior in topological superconductors, offering new avenues for controlling and manipulating these exotic states of matter.

Topological Superconductivity via Magnetic Adatom Chains

Calculations reveal that the stability of nonuniform superconducting phases depends heavily on the magnetic field and chemical potential, as demonstrated by a ground-state phase diagram constructed from discrete data points. The work expands understanding beyond previous studies of Majorana flat band stability by examining finite temperature effects, which is an important step toward realizing practical applications. Atomic spin chains with a helical magnetic texture deposited on a s-wave superconducting substrate is the platform for these investigations, using recent advances in scanning tunneling microscopy that enable control of superconducting sub-gap states.

Arrays of magnetic adatoms create multiple spin-polarized sub-gap states that hybridize, potentially leading to topologically non-trivial superconducting phases supporting flat bands of Majorana edge states. A tight-binding model for coplanar spin spirals on a 2D s-wave superconductor forms the basis for self-consistent calculations of the superconducting order parameter, used to identify potential ground states.

The spiral pitch vector, specifically the x-component, is a key parameter influencing the number and stability of Majorana states, while a finite component in the y-direction is found to destabilize Majorana flat bands along that direction. Self-consistent calculations are performed numerically on a grid of lattice sites to solve the BdG Hamiltonian, an iterative scheme used to compute the superconducting order parameter.

Self-Consistent Calculations Reveal Nonuniform Order Parameters

The team employed the Barzilai-Borwein method to accelerate convergence during these calculations, ensuring robust results across multiple initial conditions for the superconducting order parameter. Initial guesses for the pair density wave phase used edge-localized amplitude modulation with minimal phase variation, while the phase crystal’s starting configuration featured uniform amplitude and a specific phase ansatz. The researchers clarify that these modulations are not merely artifacts of the calculation, but represent distinct, emergent phases driven by the need to gap out the Majorana flat band and minimize the system’s overall energy.

Pair Density Wave: Edge-Localized Amplitude Modulation

These gaps form as winding numbers, dictated primarily by the chemical potential, control the hybridization of Majorana states. The pair density wave facilitates scattering between Majorana states with opposite windings, which is important for hybridizing these states and fully gapping the Majorana flat band. Specifically, amplitude modulations with a dominant wave vector allow efficient coupling between wires of opposite windings. Scattering is essential because Majorana states on the same edge but with opposing windings can otherwise remain unpaired, leaving a zero-energy state.

The pair density wave acts as an edge-localized perturbation, breaking lattice translation symmetry in a way that enables scattering between opposite winding numbers, effectively eliminating all possible Majorana edge states. These calculations demonstrate that the emergence of these nonuniform superconducting phases is a physical consequence of the interplay between topology and superconductivity.

Phase Crystal: Spontaneous Phase Gradients and Currents

Spontaneous currents arise within the superconducting phase crystal due to a unique interplay between edge states and winding numbers, a phenomenon observed in recent calculations of topological superconductors. These currents break both translational and chiral symmetries, driven by a finite superfluid momentum resulting from spatially varying phase gradients. The emergence of this phase crystal hinges on the ability of these gradients to shift Majorana states away from zero energy, minimizing the overall free energy of the system.

The phase crystal distinguishes itself from a competing state, the pair density wave, through its modulation characteristics; while the pair density wave relies on amplitude modulations to hybridize Majorana states with differing winding numbers, the phase crystal primarily uses phase modulations accompanied by smaller amplitude variations. Calculations reveal that a phase crystal emerges when amplitude modulations are insufficient to fully hybridize the Majorana flat bands, a condition dictated by a mismatch in the winding numbers. The ability of the phase crystal to remain robust even with strong correlations, disorder, and finite-temperature effects is notable, as it aligns with experimental observations.

Winding Numbers Tune Competition Between Superconducting States

Specifically, the emergence of either a “pair density wave” or a “phase crystal”, both nonuniform superconducting phases, is governed by the proportion of these winding numbers, which can be tuned by adjusting the chemical potential. This tunability offers a potential route to controlling these unusual states, which arise from the presence of flat bands within the superconducting gap.

Temperature Dependence of Pair Density Wave and Phase Crystal

The order parameter then transitions into a “phase crystal” as the ability of amplitude modulations to hybridize all Majorana states diminishes. A broad intermediate region connects these two phases, exhibiting comparable modulations in both amplitude and phase, and breaking both translational and chiral symmetries. However, this region is significantly suppressed as temperature increases, indicating a sensitivity to thermal excitations.

This difference in thermal stability is linked to the phase crystal’s emergence in a parameter regime where both the number of Majorana states and the magnitude of the superconducting gap are relatively small. Calculations reveal that the pair density wave transition temperature decreases from 80% to 10% of $T_c$ as the chemical potential increases, mirroring the observed weakening of amplitude variations at zero temperature.

In contrast, the phase crystal transition temperature remains approximately 10% of $T_c$ throughout the phase diagram. These findings establish a clear temperature dependence for these states in topological superconductors, suggesting potential avenues for their control and manipulation.

Chemical Potential Controls Phase Diagram Transitions

The emergence of both pair density wave and phase crystal states is directly linked to the chemical potential, according to new calculations detailing the behavior of topological superconductors. At zero temperature, the system consistently defaults to a pair density wave phase, characterized by edge-localized amplitude modulations, rather than the uniform superconducting state. Increasing the chemical potential then drives a transition to a phase crystal, distinguished by dominant phase modulations along the edges with accompanying, though smaller, amplitude modulations.

This shift occurs as the number of Majorana states decreases, requiring less phase variation to move them away from zero energy. The resulting zero-temperature phase diagram maps these two states across ranges of chemical potential and magnetic field, revealing a clear delineation between regions dominated by amplitude versus phase modulation.

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