Majorana Zero Modes in Quantum Systems

• Majorana zero modes are self-conjugate, zero-energy quasiparticles localized at topological defects in superconducting systems, defined by particle–hole symmetry.
• They appear in various physical platforms, including Kitaev chains, semiconductor–superconductor heterostructures, and chiral p-wave superconductors, with clear implications for quantum computing.
• Robustness from topological invariants and non-Abelian exchange statistics makes them key candidates for fault-tolerant quantum computation through braiding and fusion operations.

Majorana zero modes (MZMs) are self-conjugate, spatially localized, zero-energy excitations that emerge at topological defects or boundaries in superconducting systems with particle-hole (charge-conjugation) symmetry. Their presence is deeply connected to the topological properties of the system; they often signal the existence of protected quantum information that is robust against local perturbations. MZMs are realized in a variety of physical platforms, including chiral p-wave superconductors, semiconductor–superconductor heterostructures, spin-orbit-coupled materials, two-dimensional electron gases, and engineered systems such as magnetic domain walls and graphene/superconductor interfaces.

1. Fundamental Properties and Formal Definition

A Majorana zero mode is a quasiparticle excitation at zero energy satisfying the self-conjugacy condition

$\gamma = \gamma^\dagger$

where $\gamma$ is the operator representing the mode. This operator anticommutes with itself, $\{\gamma, \gamma\} = 2$, and transforms the many-body ground state between even and odd fermionic parity sectors.

In model Hamiltonians such as the Kitaev chain, MZMs appear at the ends of a one-dimensional topological superconductor when the chemical potential lies within the bulk gap. In the Bogoliubov-de Gennes (BdG) formalism, the existence of such zero modes is guaranteed by the presence of particle–hole symmetry: $$\mathcal{C}^{-1} \mathcal{H} \mathcal{C} = -\mathcal{H}$$ where $\mathcal{C}$ is the charge-conjugation operator. This symmetry enforces a symmetric energy spectrum about zero, allowing for robust zero-energy solutions localized at defects or boundaries.

The non-Abelian exchange statistics of MZMs form the basis for topological quantum computation, where braiding operations manipulate the qubit space in a way that is insensitive to local disturbances.

2. Topological Invariants and Counting Formulas

The presence and number of MZMs are determined by topological invariants related to the system's dimensionality and symmetry class. For systems with charge-conjugation symmetry, the number (or parity) of zero modes bound to a defect can be computed from a conserved quasiparticle charge $Q$ via the relation: $N = -2Q$ where $N$ is the total number of unoccupied zero modes. The local charge density is given in terms of the full Green function $\mathcal{G}(\omega)$ and a reference Green function $\mathcal{G}_0(\omega)$ for the bulk: $$\rho(x) = \int_\gamma \frac{d\omega}{2\pi} \langle x | \mathrm{tr}_R [\mathcal{G}(\omega) - \mathcal{G}_0(\omega)] | x \rangle$$ where the integration contour $\gamma$ encircles all negative-energy poles except for $\omega=0$.

A gradient expansion of the Green function (adiabatic approximation) allows one to express $Q$ as a spatial integral over topological invariants, such as Chern numbers. Notably:

• In $d$-dimensions, for Dirac-type BdG Hamiltonians, the $d$th Chern number governs the zero mode count.
• In one dimension, the induced charge can be written as an integral over $(\omega, p, \theta)$ (frequency, momentum, and order parameter angle), yielding the first Chern number.
• In two dimensions, the induced charge after compactifying space and order-parameter space is given by the second Chern number.

For example, in 2D systems: $$Q_\mathrm{adia} = -\frac{i (2\pi)^2}{60} \int_K \epsilon_{\nu_1 \ldots \nu_5} \mathrm{tr}_R \left[ \mathcal{G} \frac{\partial \mathcal{G}^{-1}}{\partial K_{\nu_1}} \ldots \mathcal{G} \frac{\partial \mathcal{G}^{-1}}{\partial K_{\nu_5}} \right]$$ with $K_\nu = (\omega, p_x, p_y, \theta_1, \theta_2)$ forming a five-vector over frequency, momenta, and order parameter variables (Santos et al., 2010).

3. Physical Realizations and Model Systems

Kitaev Chain and Quantum Wires

The prototypical model for MZMs is the Kitaev chain—a one-dimensional spinless $p$-wave superconductor. The zero mode condition is $2t > |\mu|$, where $t$ is the hopping amplitude and $\mu$ is the chemical potential (Chen et al., 2014). MZMs are robust at phase boundaries and decay exponentially from the chain ends.

Spin-orbit coupled quantum wires proximitized by superconductors realize the Kitaev paradigm, with experimental demonstrations using InSb or InAs nanowires. Notably, Majorana zero modes persist even when the superconductor provides only algebraic (power-law) superconducting correlations, rather than true long-range order (Fidkowski et al., 2011). In these cases, bosonization approaches show that the system retains an energy gap and the edge mode degeneracy, though backscattering or quantum phase slips may introduce power-law-suppressed degeneracy splitting.

Chiral p-wave and Magnetic Domain Wall Models

In chiral $p$-wave superconductors, vortices in the order parameter bind Majorana zero modes due to the topological structure of the pairing field. The number of MZMs associated with a defect is again determined by a topological invariant: the winding or Chern number of the configuration (Santos et al., 2010). An explicit example involves using a one-dimensional magnetic domain wall proximity-coupled to an $s$-wave superconductor, in which a spin-dependent gauge transformation reveals effective spin-orbit and Zeeman terms. The BdG Hamiltonian falls into class D with a $\mathbb{Z}_2$ invariant,

$$\mathcal{Q} = \mathrm{sign}\left( \Delta_0^2 + {\mu'}^2 - V_\mathrm{ex}^2 \right)$$

governing the topological transition (Rameshti et al., 2016). The spatial structure and localization length of the MZM are set by the relevant gap scales and system-specific velocities.

Two-Dimensional Electron Gases, Graphene, and Silicene

In engineered 2DEG platforms with antidot and kink structures, topologically protected MZMs can be formed without the fragility to multiband effects that plague 1D wires; such setups utilize mass inversion domain walls and Rashba spin-orbit coupling (Cheng et al., 2019). In graphene, the interplay between zero Landau level physics, magnetic order (canted antiferromagnetism), and proximity-induced superconductivity leads to topological superconductivity and localized MZMs at the ends of proximitized junctions (San-Jose et al., 2015).

In silicene, oppositely aligned local magnetizations in the two sub-lattices generate zero modes that show sub-Planck-scale interference in phase space, and their coupling to external spins reveals robust quantum coherence features (Varma et al., 2018).

Metallic Surface States and Hybrid Designs

Superconducting gold (Au) surface states, gapped via strong Rashba spin-orbit coupling and Zeeman fields from EuS overlayer, also exhibit spatially separated MZMs evidenced by zero-bias conductance peaks in STM spectroscopy (Manna et al., 2019). These metallic platforms offer advantages such as large energy scales, clean patterning, and straightforward integration into complex qubit architectures.

4. Robustness, Manipulation, and Quantum Computation

MZMs gain their protection from the non-trivial topological invariants of the underlying systems and from the exponential suppression of hybridization at large separation (e.g., mode splitting $\sim e^{-L/\xi}$) (Albrecht et al., 2016). This property underlies their proposed use in fault-tolerant quantum information processing, where quantum information is stored nonlocally in pairs of MZMs and manipulated via braiding operations that realize non-Abelian unitary transformations in Hilbert space (Sarma et al., 2015).

Braiding and fusion of MZMs have been explored in detail:

• Fusion experiments and simulations, e.g., in Kitaev chains and real magnetic-superconductor platforms, directly probe the Ising anyon fusion rules and can realize qubit readout by charge measurement, especially when the fused fermion is projected into a quantum dot or non-superconducting region for quantized charge detection (Hodge et al., 12 Mar 2025).
• The limitation on braiding speed is fundamentally set by the “speed of light” in the system and the resulting Lorentz-contracted spatial and energetic scales. Moving domain walls that host MZMs experience Lorentz contraction and time dilation effects, with the discrete bound state spectrum dissolving into a continuum at superluminal velocities ($v \ge u$), setting a bound on maximal braiding frequency (Karzig et al., 2013).

Novel proposals include multi-locational MZMs (where a single zero mode’s wave function is distributed over multiple, spatially separated locations in a network such as a three-terminal Josephson junction) (Nagae et al., 2023) and “multiplicative Majorana zero modes” arising as “tensor product” boundary states of two (or more) topological phases, exhibiting enriched internal Hilbert space structure and alternative quantum gate paradigms (Pal et al., 2023).

5. Classification, Recursion Relations, and Analytical Treatments

Analytical solutions for MZMs in extended models—such as generalized Ising chains with long-range interactions or next-nearest neighbor couplings—demonstrate that the zero mode wave function amplitudes $A_j$ satisfy recursion relations: $$-\mu A_{j} + \lambda_1 A_{j+1} + \lambda_2 A_{j+2} = 0$$ with characteristic equation

$\lambda_2 r^2 + \lambda_1 r - \mu = 0$

and roots $q_\pm$. The modulus and complex phase of $q_\pm$ determine the spatial localization and possible oscillatory decay of the MZM. Only regimes with $|q| < 1$ yield normalizable Majorana zero-energy edge states (Pathak et al., 25 Jun 2025). At phase transitions, the modulus reaches unity and the corresponding zero mode ceases to be localized—signaling a topological transition.

This framework connects directly to the explicit construction and classification of MZM solutions, providing necessary and sufficient criteria for the spatial localization and topological character of the zero modes in a broad class of quantum spin chains and fermionic lattice models.

6. Experimental Signatures and Odd-Frequency Pairing

Majorana zero modes manifest in measurable observables:

• Tunneling spectroscopy yields robust zero-bias conductance peaks due to perfect Andreev reflection; in ideal 1D systems, the quantized conductance at $2e^2/h$ or $4e^2/h$ (with spin degeneracy) serves as a distinctive signature (Sarma et al., 2015, Manna et al., 2019).
• Anomalous (odd-frequency, spin-triplet, $s$-wave) pairing emerges at boundaries hosting MZMs, observable as zero-bias anomalies or as persistent zero-energy peaks in the density of states, even in normal metals adjacent to the superconductor (Tanaka et al., 1 Feb 2024).
• The Josephson effect across topological junctions involves a fractional $4\pi$-periodicity in the current-phase relation, a direct consequence of MZM parity crossings at phase difference $\phi = \pi$.

7. Outlook and Future Directions

The theoretical and experimental landscape of Majorana zero modes is defined by the confluence of topological band theory, unconventional quantum statistics, and practical realizations in engineered materials. Precise analytical and numerical tools—gradient expansions, adiabatic approximations, Green function techniques, and advanced device fabrication—allow for the robust identification and manipulation of MZMs, paving the way for their integration into scalable, fault-tolerant quantum computing architectures.

Critical open directions include:

• Achieving unambiguous demonstration of non-Abelian exchange statistics through direct fusion/braiding experiments in both semiconductor and metallic platforms.
• Extending platforms to support multi-locational and “multiplicative” modes, potentially enabling alternative quantum computation schemes.
• Addressing challenges such as quasiparticle poisoning, finite-size effects, and the control of disorder and interactions.
• Further exploration of the relation between zero modes, Chern numbers, odd-frequency pairing, and experimental signatures for definitive identification and utilization of Majorana zero modes as the building blocks of topological quantum devices.