Majorana Zero Modes in Topological Superfluids

Updated 13 August 2025

Majorana zero modes are self-conjugate quasiparticles localized at defects, exhibiting non-Abelian braiding and robust zero-energy behavior.
They form via Hamiltonian-induced and dissipative pairing in systems like chiral p-wave superfluids and spin-orbit coupled wires.
Their unique properties underpin quantum computing innovations and deepen our understanding of topological phases in condensed matter.

Majorana zero modes (MZMs) in topological fermionic superfluids are localized, self-conjugate quasiparticle excitations appearing at topological defects or boundaries in paired states of fermions. These excitations are central to both fundamental studies of topological phases and applied pursuits such as fault-tolerant topological quantum computation. Their properties, existence conditions, and physical manifestations depend on the microscopic details of the superfluid, the dimensionality, and the mechanism—Hamiltonian or dissipative—by which pairing is induced.

1. Basic Concepts and Defining Properties

Majorana zero modes are solutions to the Bogoliubov–de Gennes (BdG) equations (or appropriate non-equilibrium analogues) that are self-adjoint quasiparticles: γ = γ†, satisfying canonical anticommutation relations. In 2D or 1D topological superfluids (e.g., chiral p-wave, s-wave with spin–orbit coupling and Zeeman field, or number-conserving ladders), MZMs emerge at topological defects, such as vortex cores (in 2D) or boundaries/domain walls (in 1D/2D), and are protected by global invariants such as the Chern number, winding number, or more generally, by symmetry and topology.

MZMs are fundamentally non-local: two MZMs spatially separated (e.g., at the ends of a wire) combine to form a conventional fermionic mode, but individually, each Majorana mode represents “half” a fermion—this notion is formalized via the quantum dimension (d = √2) (Ye et al., 2015).

Key qualitative features:

Localized spatial profile, typically exponential decay from a defect or boundary.
Protected zero energy (or zero damping, in dissipative settings), up to exponentially small splittings.
Obey non-Abelian exchange (braiding) statistics when multiple modes are present—relevant for fault-tolerant quantum computation.
Robustness to moderate disorder and certain perturbations, due to the topological or symmetry-protected nature of the host phase.

2. Mechanisms of Formation in Fermionic Superfluids

Hamiltonian-Induced Pairing

In chiral p-wave (e.g., p_x + ip_y) superfluids, Cooper pairing occurs with finite angular momentum, breaking time-reversal symmetry. In 2D, vortices host MZMs at their cores (Marienko et al., 2012, Silaev, 2013, Silaev et al., 2014).
In spin-orbit coupled s-wave superfluids with a substantial Zeeman field, the system is mapped to an effective p-wave model. The condition for a topological phase supporting MZMs is h² > Δ² + μ^2, where h is the Zeeman field, Δ the pairing gap, and μ the chemical potential (Ruhman et al., 2014, Toikka, 2019).
In 1D wires or ladders, with attractive interactions and suitable spin-orbit coupling (or engineered correlated hoppings), MZMs localize at the system ends or domain walls (Chua et al., 2020, Tausendpfund et al., 2022).

Dissipative and Floquet Engineering

Dissipation-induced pairing, as in the Lindblad formalism with tailored Lindblad operators combining s- and p-wave components, can stabilize p-wave paired steady states and create MZMs. In the model of Bardyn et al., the dissipative vortex supports a single MZM tied to non-equilibrium topological features absent from purely Hamiltonian systems (Bardyn et al., 2012).
Floquet (periodically driven) systems facilitate “Floquet MZMs,” including those with quasienergy ε = 0 and π/T at topological defects, even when bulk invariants vanish (“anomalous” bulk-boundary correspondence) (Yang, 2014).

Number Conserving and Synthetic Dimension Approaches

Particle-number conserving setups realize MZMs via correlated pair hopping and engineered parity symmetry (e.g., using Aharonov–Bohm cages in ladder geometries) (Tausendpfund et al., 2022).
Synthetic dimension schemes, realized through multicomponent atomic gases with Raman-induced couplings, support robust MZMs when the effective 2D lattice (physical + synthetic) has odd Chern number and the BdG Hamiltonian possesses chiral symmetry (Yan et al., 2015).

3. Bulk-Defect Correspondence and Topological Invariants

The appearance of MZMs is tied to nontrivial bulk topology:

In 2D chiral p-wave superfluids, the first Chern number C = (1/2π) ∫ F(k) d^2k is nonzero, enforcing the existence of zero modes at boundaries or vortices (Silaev et al., 2014).
In time-reversal-invariant p-wave phases (e.g., ^3He-B), symmetry-protected invariants (with anti-unitary symmetry) guarantee Dirac-like surface states; dimensional reduction links these to planar or lower-dimensional systems.
For polar phases with nodal lines, the winding number defined as N_K = (1/(4πi)) tr{K ∮_C dl H⁻¹∇_lH} protects 2D flat bands of zero modes on surfaces.
In dissipative contexts, the relevant invariant can switch from the Chern number (which may vanish) to a winding number in a mapped 1D system, as for the dissipative vortex (Bardyn et al., 2012).

4. Experimental Realizations and Detection Schemes

Ultracold atomic gases: Realization of MZMs via Kohn–Luttinger induced pairing in two-species fermion mixtures (density-imbalanced), imprinted vortices, or optical lattice-based synthetic dimensions (Marienko et al., 2012, Yan et al., 2015, Bardyn et al., 2012).
Semiconductor heterostructures: Proximity effect-induced topological superconductivity in nanowires or quantum dot-coupled structures, tested via charge pumping or zero-bias tunneling spectroscopy (Zhang et al., 2023).
Thermal and transport measurements in superconductors: The transition to a phase with gapless Majorana Bloch bands in vortex lattices directly modifies the transverse thermal conductivity, providing an experimental signature (Silaev, 2013).
Detection protocols: Localized RF spectroscopy or two-photon Raman pulses in cold atom setups; measurement of revival of end-to-end correlations in ladders; entanglement spectrum degeneracy via DMRG for SPT phases; time-of-flight imaging or noise correlations in cold atom systems (Yan et al., 2015, Tausendpfund et al., 2022, Xu et al., 2014).

5. Interactions, Symmetry Protection, and Robustness

Protection of MZMs can be governed by discrete symmetries, e.g., mirror symmetry in spinful superconductors allows even integer quantum vortices to support robust MZMs (mirror Majorana modes), circumventing the need for half quantum vortices (Sato et al., 2013).
Emergent Z_2 or higher symmetries (such as BDI class chiral symmetry in synthetic dimension systems) can stabilize multiple MZMs, with classification possibly reduced to Z_8 by interactions (Yan et al., 2015).
In one-dimensional systems with fractionalized excitations, spin–charge separation enables the coexistence of Majorana and fractional fermion edge modes protected by independent symmetries (Zhang et al., 2016).
Particle-number-conserving treatments clarify the conditions for topological protection and the operational meaning of MZMs in the presence of superselection rules and parity switches (Ortiz et al., 2016, Lin et al., 2018).
Dynamical robustness: Collisions of soliton-hosted MZMs exhibit effective repulsive interactions resulting from wavefunction overlap, preventing fusion into bulk states and ensuring elastic behavior—a positive sign for qubit network stability (Wang et al., 2023).

6. Braiding, Fusion, and Quantum Computation Implications

MZMs exhibit non-Abelian braiding statistics: Exchanging two MZMs (e.g., bound to vortices or defects) transforms the state space via a non-commuting operator, crucial for implementing topological quantum gates (Sato et al., 2013, Toikka, 2019).
The nontrivial fusion rules (e.g., γ × γ = I + Ψ) can be probed experimentally using engineered fusion loops involving MZMs and auxiliary fermionic modes; quantized charge pumping serves as a robust signature distinguishing MZMs from trivial Andreev bound states (Zhang et al., 2023).
Braiding and measurement protocols in both Hamiltonian and dissipative (engineered dissipation) topological superfluids are being explored; corrections due to particle number conservation, finite size, and condensate deformation are under active investigation (Lin et al., 2018).
The quantum dimension (d = √2) of each MZM and the structure of their Hilbert space underpin the non-local encoding of quantum information in topological qubits, offering protection from local noise and decoherence (Ye et al., 2015).

7. Topological Phase Transitions and Limitations

Topological-to-trivial transitions correspond to closing of the bulk excitation gap and a change in the relevant topological invariant (Chern number, winding number, etc.), at which point MZMs disappear (Silaev et al., 2014, Bardyn et al., 2012). In dissipative or Floquet settings, such transitions can be non-equilibrium in nature, with bulk invariants discontinuously changing at isolated critical points. Quantum phase transitions may involve, for example, divergence of the MZM localization length and abrupt shifts in steady-state properties.

Limitations include:

Sensitivity to perturbations breaking the protecting symmetry (e.g., mirror symmetry, particle number parity).
The necessity of precise control in engineered dissipation schemes.
Finite-size-induced splitting of MZMs, scaling as e^{–L/ξ}.
Fundamental constraints on coherent manipulation of MZMs arising from particle number superselection, particularly for encoded qubits in number-conserving systems (Ortiz et al., 2016, Lin et al., 2018).

Majorana zero modes in topological fermionic superfluids thus represent a nexus of topological order, symmetry, and non-Abelian quantum statistics, realized through a range of mechanisms—Hamiltonian and dissipative—and manifesting in a variety of physical systems. Their ongoing paper continues to advance understanding in condensed matter physics, ultracold atoms, and quantum information science.