Wigner-Majorana Systems: Topology & Symmetry

• Wigner-Majorana systems are fermionic systems characterized by self-conjugate Majorana modes and robust topological degeneracies arising from symmetry constraints.
• They manifest in condensed matter through defect-bound zero modes, translation-symmetric lattices, and supersymmetric structures, linking theoretical models with experimental signatures.
• They bridge abstract geometric reformulations and practical bosonization or qubit mappings, enhancing our understanding of phase-space dynamics and fractionalization in quantum materials.

Wigner-Majorana (WM) systems, as the term is used across several strands of the literature, are fermionic systems in which Majorana self-conjugacy, topological defects, projective symmetry actions, or real-field/BdG structures impose nontrivial constraints on spectra, state spaces, and observables. In condensed-matter realizations they include defect-bound Majorana zero modes in superconductors, translation-symmetric Majorana lattices with symmetry-enforced degeneracy, and spin/qubit reformulations of fermionic models; in more formal settings they include geometric descriptions via orthogonal complex structures and phase-space constructions related to fermionic Wigner functionals (Jackiw, 2014, Chamon et al., 2010, Hsieh et al., 2016, Calderón-García et al., 2017, Roux, 2022). A recurring theme is that Majorana degrees of freedom are not merely neutral fermionic excitations but algebraic objects whose self-conjugacy, parity structure, and topology force protected degeneracies, constrain admissible quantizations, and reorganize the relation between fermions, bosons, and symmetries.

1. Defects, zero modes, and fractionalization

A foundational WM mechanism is the appearance of zero-energy bound states in Dirac-like systems with topological defects. In the defect background formulation,

$$[\boldsymbol{\alpha}\cdot \mathbf{p} + \beta\, \varphi(\mathbf{r})]\psi = E\psi ,$$

the mass term is replaced by a spatially varying field $m\to \varphi(\mathbf r)$. For nontrivial $\varphi(\mathbf r)$, the spectrum contains continuum states with $E>0$ and $E<0$, together with one or more normalizable zero-energy modes whose number is controlled by an index theorem. In the Dirac case this produces the classic fractionalization result: the zero mode carries charge $Q=-\frac12$ when empty and $Q=+\frac12$ when filled, as an eigenvalue rather than merely an expectation value. The review connects this mechanism to the SSH chain and to two-dimensional settings such as graphene and the quantum Hall effect (Jackiw, 2014).

The Majorana analogue replaces charged Dirac fields by self-conjugate neutral fields. In the Weyl-basis formulation summarized in the review, the Majorana condition reduces the four-component Dirac equation to the two-component equation

$$\boldsymbol{\sigma}\cdot \mathbf{p}\,\psi + i\sigma^2\,m\,\psi^\ast = i\frac{\partial}{\partial t}\psi .$$

Because $\psi$ mixes with $\psi^\ast$, there is no conserved particle number or charge. The condensed-matter realization highlighted in the review is a two-dimensional superconductor on a topological-insulator surface, with Hamiltonian density

$m\to \varphi(\mathbf r)$0

When $m\to \varphi(\mathbf r)$1 has a vortex profile, a zero-energy bound state localized at the vortex appears (Jackiw, 2014).

The algebraic signature of a single Majorana zero mode is the Hermitian operator $m\to \varphi(\mathbf r)$2 satisfying

$m\to \varphi(\mathbf r)$3

Two realizations are possible. A one-dimensional realization diagonalizes $m\to \varphi(\mathbf r)$4 but violates fermion parity. A two-dimensional realization preserves fermion parity by introducing two degenerate vacua $m\to \varphi(\mathbf r)$5 with

$m\to \varphi(\mathbf r)$6

The review treats the parity-preserving option as the physically natural one for condensed matter. For $m\to \varphi(\mathbf r)$7 vortices, the zero-mode operators form a Clifford algebra,

$m\to \varphi(\mathbf r)$8

and the protected Hilbert-space dimension grows as

$m\to \varphi(\mathbf r)$9

This establishes the basic WM pattern: topological defect $\varphi(\mathbf r)$0 Majorana zero mode $\varphi(\mathbf r)$1 protected degeneracy (Jackiw, 2014).

2. Majorana quantization, BdG reality, and geometric reformulation

A second WM strand generalizes the Majorana idea from isolated zero modes to the full superconducting field theory. For the surface Dirac theory of a topological insulator proximitized by an $\varphi(\mathbf r)$2-wave superconductor, the equation of motion

$\varphi(\mathbf r)$3

already mixes particle and hole sectors. The central claim is that the Majorana feature is not confined to the zero mode: after a suitable unitary transformation to a Majorana basis, the Bogoliubov-de Gennes Hamiltonian becomes purely imaginary,

$\varphi(\mathbf r)$4

so that

$\varphi(\mathbf r)$5

is a real equation of motion admitting real field solutions. This reproduces, within superconducting BdG theory, the relativistic Majorana construction in which the operator is imaginary and the field is real (Chamon et al., 2010).

The same work argues that this is generic to superconductors, not special to Dirac dispersion. For a general Nambu-space BdG Hamiltonian

$\varphi(\mathbf r)$6

fermionic statistics imply the universal conjugation symmetry

$\varphi(\mathbf r)$7

Positive- and negative-energy modes therefore occur in paired conjugates, and there always exists a unitary transformation sending the conjugation matrix to the identity. A common misconception is that “Majorana quantization” refers only to isolated vortex-bound states; the generic BdG argument shows instead that real-field/Majorana structure is a property of the full superconducting field theory once the appropriate doubled Nambu description is brought to Majorana basis (Chamon et al., 2010).

This structural viewpoint is sharpened by the orthogonal-complex-structure formulation of quadratic fermionic Hamiltonians. In that framework one starts from a real vector space $\varphi(\mathbf r)$8 with positive symmetric form $\varphi(\mathbf r)$9 and an orthogonal complex structure

$E>0$0

The complex Hilbert space is then $E>0$1, with Hermitian form

$E>0$2

and the irreducible Fock representation is $E>0$3. The vacuum is not an arbitrary state but is encoded by the choice of $E>0$4; equivalently, ground states of quadratic fermionic Hamiltonians are parametrized by orthogonal complex structures. This removes the apparent Hilbert-space doubling of BdG/Nambu formalisms and yields the equivalence

$E>0$5

Within the same framework, the $E>0$6 invariant is expressed as

$E>0$7

so topology, vacuum structure, and Majorana splitting are encoded in a single geometric object (Calderón-García et al., 2017).

3. Translation symmetry, projective action, and exact supersymmetry

The sharpest symmetry-enforced WM result concerns one- and two-dimensional arrays of Majorana modes with translation symmetry and an odd number of modes per unit cell. In one dimension, for a ring of Majorana operators $E>0$8 with even $E>0$9, fermion parity is

$E<0$0

and translation by one site acts as

$E<0$1

With periodic boundary conditions and an odd number of Majoranas per unit cell, translation anticommutes with fermion parity,

$E<0$2

This single algebraic fact implies that if $E<0$3 and $E<0$4 with $E<0$5, then $E<0$6 is a distinct eigenstate with the same energy and opposite parity. Hence every energy level is at least doubly degenerate. The paper presents this as a Kramers-like theorem in which translation symmetry replaces time reversal (Hsieh et al., 2016).

The doubling is elevated to an exact $E<0$7 supersymmetry after shifting the spectrum so that all eigenvalues are nonnegative. A fermionic, non-Hermitian supercharge is defined as

$E<0$8

and the relations

$E<0$9

follow from $Q=-\frac12$0. Thus the Hamiltonian is an anticommutator of supercharges, and every state has a partner of opposite fermion parity. Because the construction includes the ground state, the Witten index vanishes. This result is unusually strong in that it requires only translation symmetry, periodic boundary conditions, and an odd Majorana count per unit cell (Hsieh et al., 2016).

For one-dimensional anti-periodic boundary conditions, the same paper proves a different obstruction. The direct parity-translation anticommutation argument no longer doubles every level, but there cannot be both a unique ground state and a finite gap in the thermodynamic limit. The proof doubles the system, maps it to a spin chain through

$Q=-\frac12$1

and identifies an onsite $Q=-\frac12$2 symmetry represented projectively like spin-$Q=-\frac12$3. This yields a Lieb-Schultz-Mattis-type obstruction rather than full spectral doubling (Hsieh et al., 2016).

In two dimensions, for translation-symmetric arrays with a single Majorana per unit cell and even linear dimensions, the translation operators satisfy

$Q=-\frac12$4

Since both commute with the Hamiltonian, every eigenstate has a partner at the same energy with different translation quantum numbers. The paper explicitly stresses that this two-dimensional degeneracy is not due to supersymmetry in the same way as the one-dimensional case; it follows directly from the anticommutation of the translation operators. The underlying reason is that translations of Majorana modes are represented projectively, with a representative of the form

$Q=-\frac12$5

where $Q=-\frac12$6 is antisymmetric. The extra $Q=-\frac12$7 factor expresses the anomalous, “fractional” character of a single Majorana mode (Hsieh et al., 2016).

4. Spin, qubit, and bosonic reformulations

WM systems admit exact mappings to spin and qubit models that preserve locality more effectively than the one-dimensional Jordan-Wigner chain would suggest. For tree graphs, the Jordan-Wigner transformation can be extended by introducing additional spin-$Q=-\frac12$8 degrees of freedom at junctions. On each chain $Q=-\frac12$9,

$Q=+\frac12$0

with Klein factors $Q=+\frac12$1 built recursively from ancillary junction spins. The essential effect is that fermionic sign information is stored locally at branch points rather than in one global string. Parent-child and sibling couplings remain local in the spin representation, dressed only by the relevant junction-spin components. Majorana operators on a branch become

$Q=+\frac12$2

and the standard braiding unitary

$Q=+\frac12$3

is translated into multi-spin or multi-qubit operations mediated by the ancilla when the exchanged Majoranas lie on different branches. The mapping enlarges the Hilbert space by an extra spin at each inner vertex, but it preserves exact correspondence in the enlarged space (Backens et al., 2018).

A higher-dimensional Jordan-Wigner construction uses auxiliary Majorana fermions rather than ancillary spins. Complex fermions are first decomposed as

$Q=+\frac12$4

and then rewritten in terms of two kinds of auxiliary Majorana partons: $Q=+\frac12$5, which carry internal quantum numbers, and $Q=+\frac12$6, which encode spatial and directional structure. The basic bosonic operator building blocks are

$Q=+\frac12$7

The construction applies to lattices with even coordination number and an arbitrary number of fermion flavors. Square, triangular, and cubic lattices are treated explicitly, with local parity constraints, plaquette constraints, and Wilson-loop constraints organizing the bosonic Hilbert space and fixing global fermion-parity sectors (Li et al., 2021).

These reformulations are significant for two reasons. First, they show that WM physics is compatible with qubit and spin architectures rather than being tied exclusively to microscopic fermionic hardware. Second, the auxiliary structures introduced to preserve fermionic statistics resemble $Q=+\frac12$8 gauge fields, which is why the higher-dimensional construction connects naturally to Wen’s plaquette model, Kitaev-type models, and Ryu’s three-dimensional spin liquid. A plausible implication is that WM systems form a bridge between Majorana zero-mode physics, bosonization, and exactly solvable spin-liquid constructions (Backens et al., 2018, Li et al., 2021).

5. Phase-space and nonstandard Wigner constructions

A separate WM-related direction concerns fermionic phase-space representations. In that setting, Majorana operators are identified as the standard fermionic analogue of bosonic quadratures, but they are judged unsuitable for constructing Wigner functionals “in the bosonic way” because their algebra is structurally opposite to bosonic quadratures: they anticommute with each other and do not anticommute with themselves. To avoid this obstruction, a different set of fermion quadrature operators is introduced through a relative spin transformation between ladder operators,

$Q=+\frac12$9

with $$\boldsymbol{\sigma}\cdot \mathbf{p}\,\psi + i\sigma^2\,m\,\psi^\ast = i\frac{\partial}{\partial t}\psi .$$0 the antisymmetric spin matrix (Roux, 2022).

The crucial technical novelty is the fermionic adjoint,

$$\boldsymbol{\sigma}\cdot \mathbf{p}\,\psi + i\sigma^2\,m\,\psi^\ast = i\frac{\partial}{\partial t}\psi .$$1

under which the quadrature operators are fermionic selfadjoint even though they are not Hermitian selfadjoint. Their eigenstates are displaced squeezed vacua, and once the dual space is defined using the fermionic adjoint rather than the ordinary Hermitian adjoint, the quadrature bases become orthogonal and complete: $$\boldsymbol{\sigma}\cdot \mathbf{p}\,\psi + i\sigma^2\,m\,\psi^\ast = i\frac{\partial}{\partial t}\psi .$$2

$$\boldsymbol{\sigma}\cdot \mathbf{p}\,\psi + i\sigma^2\,m\,\psi^\ast = i\frac{\partial}{\partial t}\psi .$$3

The resulting Wigner functional is

$$\boldsymbol{\sigma}\cdot \mathbf{p}\,\psi + i\sigma^2\,m\,\psi^\ast = i\frac{\partial}{\partial t}\psi .$$4

This construction is explicitly presented as an alternative to Majorana-based quadrature schemes rather than a direct restatement of them (Roux, 2022).

The phase-space geometry induced by these quadratures is symplectic rather than merely Grassmann-orthogonal. That point matters conceptually: “Wigner-Majorana” can refer not only to Wigner-style symmetry/topology around Majorana modes, but also to Wigner-functional attempts to place fermionic systems into a phase-space language. The literature here indicates that these are related but not identical enterprises, and that a fermionic Wigner theory may require abandoning Majorana operators as the preferred quadrature variables (Roux, 2022).

6. Observables, dynamical extensions, and interpretive scope

In translation-symmetric Majorana lattices, the symmetry-enforced degeneracy has a direct experimental consequence: a zero-bias peak in tunneling conductance. The relevant matrix element is generated by a local Majorana operator $$\boldsymbol{\sigma}\cdot \mathbf{p}\,\psi + i\sigma^2\,m\,\psi^\ast = i\frac{\partial}{\partial t}\psi .$$5 connecting supersymmetric partner states of opposite parity, and the characteristic scale is

$$\boldsymbol{\sigma}\cdot \mathbf{p}\,\psi + i\sigma^2\,m\,\psi^\ast = i\frac{\partial}{\partial t}\psi .$$6

For the illustrative Hamiltonian

$$\boldsymbol{\sigma}\cdot \mathbf{p}\,\psi + i\sigma^2\,m\,\psi^\ast = i\frac{\partial}{\partial t}\psi .$$7

the free case $$\boldsymbol{\sigma}\cdot \mathbf{p}\,\psi + i\sigma^2\,m\,\psi^\ast = i\frac{\partial}{\partial t}\psi .$$8 gives

$$\boldsymbol{\sigma}\cdot \mathbf{p}\,\psi + i\sigma^2\,m\,\psi^\ast = i\frac{\partial}{\partial t}\psi .$$9

and numerically the matrix element scales roughly as $\psi$0 over a range of $\psi$1. The signal therefore weakens with system size, but the paper states that for experimentally realistic sizes a zero-bias peak should remain visible. It also stresses that this mechanism is special to an odd number of Majoranas per unit cell; for an even number, the translation/supersymmetry mechanism is absent and the zero-bias peak is not generically expected (Hsieh et al., 2016).

A more interpretive extension appears in pilot-wave studies of Majorana spinors. There the Majorana constraint

$\psi$2

leads to a lightlike current,

$\psi$3

so the Bohmian velocity has $\psi$4 despite the presence of a mass parameter. For free states this produces helical trajectories whose diameter is of order the Compton wavelength; under coarse-graining with respect to the Compton wavelength, the motion appears subluminal. Simulations reported in that work show relaxation toward equilibrium for both Dirac and Majorana systems, but more slowly for Majorana systems, and do not conclusively demonstrate survival of quantum non-equilibrium at sub-Compton scales (Colin, 2013).

Taken together, these results delimit the scope of WM systems. In the strict condensed-matter sense they are topological and symmetry-constrained Majorana systems with protected degeneracy, parity structure, and experimentally accessible low-energy signatures. In a broader formal sense they include real-field quantization principles, geometric descriptions of BdG vacua, bosonization schemes using auxiliary Majoranas or junction spins, and even nonstandard phase-space or pilot-wave formulations. This suggests that the unifying content of the WM label is not a single model class but a recurring algebraic structure: self-conjugate fermionic degrees of freedom whose topology or symmetry is represented projectively and whose observables, Hilbert spaces, and effective descriptions differ sharply from those of ordinary complex fermions.