Generalized Quadratic Majorana Bilinears

• Generalized quadratic Majorana bilinears are operator structures that extend traditional iγ_iγ_j couplings, providing effective descriptions of pairing, symmetry, and topological features in diverse fermionic systems.
• They include various forms such as antisymmetric quadratic constructs, local particle-hole bilinears, and reservoir-assisted cotunneling operators, each derived via distinct methodologies and symmetry arguments.
• Beyond quadraticity, these bilinears highlight the persistence of robust algebraic features, notably the role of fermion parity, even when the effective mode ceases to be strictly quadratic.

Generalized quadratic Majorana bilineals are operator structures that extend the canonical Majorana bilinear $i\gamma_i\gamma_j$ beyond the simplest free-fermion setting while preserving its role in pairing, symmetry, topology, transport, or qubit encoding. In the literature considered here, the topic includes antisymmetric quadratic forms in Majorana operators, local particle-hole bilinears of Bogoliubov–de Gennes amplitudes, complete symmetry classifications of bilinears in quadratic-band-touching theories, reservoir-assisted cotunneling bilinears on Majorana islands, non-Hermitian number-conserving bilinears in periodically driven superconductors, and bosonic analogues formulated through quadratic Markovian dynamics (Lee et al., 2013, Sedlmayr et al., 2015, Herbut et al., 23 Apr 2026, Tian, 7 May 2026, Bomantara, 2023, Flynn et al., 2021). A recurrent conclusion is that quadratic bilinears remain the canonical effective couplings, but the algebraic phenomena usually associated with them can persist even when the exact conserved mode is no longer quadratic.

1. Canonical bilinears and the Majorana algebra

The standard starting point is the decomposition of ordinary fermion operators $a_j^\dagger,a_j$ into Majorana operators

$$\gamma_{2j-1}=a_j+a_j^\dagger,\qquad \gamma_{2j}=\frac{a_j-a_j^\dagger}{i},$$

with

$$\gamma_i^\dagger=\gamma_i,\qquad \{\gamma_i,\gamma_j\}=2\delta_{ij},\qquad \gamma_i^2=1.$$

These relations define the Clifford algebra underlying all later constructions (Lee et al., 2013).

Within this algebra, quadratic Majorana Hamiltonians are sums of bilinears $i\gamma_i\gamma_j$. The basic examples are Kitaev’s wire Hamiltonians

$$H_0=-i\sum_{j=1}^{N}\gamma_{2j-1}\gamma_{2j},\qquad H_1=-i\sum_{j=1}^{N-1}\gamma_{2j}\gamma_{2j+1},$$

and, more generally, a quadratic Majorana Hamiltonian may be written as

$H=\frac{i}{2}\sum_{ij}A_{ij}\gamma_i\gamma_j,$

with $A_{ij}$ real antisymmetric (Lee et al., 2013).

These bilinears have the familiar physical interpretation. In $H_0$, the term

$-i\gamma_{2j-1}\gamma_{2j}=2a_j^\dagger a_j-1$

is the occupation number shifted by $a_j^\dagger,a_j$0. In $a_j^\dagger,a_j$1, the bilinears mix neighboring Majoranas from different sites and generate both normal hopping and pairing terms in the underlying electron variables. In topological wires, exponentially localized end operators $a_j^\dagger,a_j$2 satisfy $a_j^\dagger,a_j$3, and the leading coupling between the ends is itself a quadratic Majorana bilinear,

$a_j^\dagger,a_j$4

At this level, quadratic bilinears pair Majoranas into ordinary fermions and split degeneracies (Lee et al., 2013).

A common misconception is that an odd number of Majorana generators automatically guarantees a doubled spectrum. The junction analysis of three Majoranas $a_j^\dagger,a_j$5 shows that the most general quadratic Hamiltonian

$a_j^\dagger,a_j$6

does not by itself produce intrinsic doubling when represented only through the algebra $a_j^\dagger,a_j$7; the missing ingredient is fermion parity (Lee et al., 2013).

2. Parity, odd products, and the nonlinear generalization of bilinears

The decisive extension beyond quadraticity is the inclusion of the parity operator

$a_j^\dagger,a_j$8

which obeys

$a_j^\dagger,a_j$9

for parity-conserving Hamiltonians. Once parity is included, the trijunction admits the nonlinear emergent operator

$$\gamma_{2j-1}=a_j+a_j^\dagger,\qquad \gamma_{2j}=\frac{a_j-a_j^\dagger}{i},$$0

with

$$\gamma_{2j-1}=a_j+a_j^\dagger,\qquad \gamma_{2j}=\frac{a_j-a_j^\dagger}{i},$$1

Because $$\gamma_{2j-1}=a_j+a_j^\dagger,\qquad \gamma_{2j}=\frac{a_j-a_j^\dagger}{i},$$2 commutes with the Hamiltonian but anticommutes with parity, it maps every eigenstate to another state of the same energy and opposite parity; the resulting doubling is therefore a whole-spectrum statement rather than only a ground-state statement (Lee et al., 2013).

This construction generalizes to any odd number $$\gamma_{2j-1}=a_j+a_j^\dagger,\qquad \gamma_{2j}=\frac{a_j-a_j^\dagger}{i},$$3 of Majorana operators at a junction: $$\gamma_{2j-1}=a_j+a_j^\dagger,\qquad \gamma_{2j}=\frac{a_j-a_j^\dagger}{i},$$4 For odd $$\gamma_{2j-1}=a_j+a_j^\dagger,\qquad \gamma_{2j}=\frac{a_j-a_j^\dagger}{i},$$5, the operator is Hermitian, squares to one, commutes with parity-conserving Hamiltonians built from these modes, and anticommutes with $$\gamma_{2j-1}=a_j+a_j^\dagger,\qquad \gamma_{2j}=\frac{a_j-a_j^\dagger}{i},$$6. For even $$\gamma_{2j-1}=a_j+a_j^\dagger,\qquad \gamma_{2j}=\frac{a_j-a_j^\dagger}{i},$$7, the parity relation changes from anticommutator to commutator, so the doubling argument fails (Lee et al., 2013).

The significance for generalized quadratic Majorana bilinears is structural. Ordinary bilinears remain the natural effective couplings, but the exact algebraic object controlling robustness at odd junctions is an odd Clifford product rather than a bilinear. In the quadratic trijunction case there is also a linear zero mode,

$$\gamma_{2j-1}=a_j+a_j^\dagger,\qquad \gamma_{2j}=\frac{a_j-a_j^\dagger}{i},$$8

but it is not independent: $$\gamma_{2j-1}=a_j+a_j^\dagger,\qquad \gamma_{2j}=\frac{a_j-a_j^\dagger}{i},$$9 When quartic or higher-order terms are included, such a linear $$\gamma_i^\dagger=\gamma_i,\qquad \{\gamma_i,\gamma_j\}=2\delta_{ij},\qquad \gamma_i^2=1.$$0 generally ceases to exist, whereas $$\gamma_i^\dagger=\gamma_i,\qquad \{\gamma_i,\gamma_j\}=2\delta_{ij},\qquad \gamma_i^2=1.$$1 survives. This suggests that the robust content of Majorana bilinear physics is not exhausted by quadraticity itself, but by the parity-based algebra that can persist after quadraticity is lost (Lee et al., 2013).

The same logic extends to larger interacting systems. If additional nearby ordinary fermionic modes are decomposed into Majoranas and added to the junction sector, the product of all Majoranas, with the appropriate power of $$\gamma_i^\dagger=\gamma_i,\qquad \{\gamma_i,\gamma_j\}=2\delta_{ij},\qquad \gamma_i^2=1.$$2, yields a generalized $$\gamma_i^\dagger=\gamma_i,\qquad \{\gamma_i,\gamma_j\}=2\delta_{ij},\qquad \gamma_i^2=1.$$3 that still satisfies the analogues of $$\gamma_i^\dagger=\gamma_i,\qquad \{\gamma_i,\gamma_j\}=2\delta_{ij},\qquad \gamma_i^2=1.$$4, $$\gamma_i^\dagger=\gamma_i,\qquad \{\gamma_i,\gamma_j\}=2\delta_{ij},\qquad \gamma_i^2=1.$$5, and $$\gamma_i^\dagger=\gamma_i,\qquad \{\gamma_i,\gamma_j\}=2\delta_{ij},\qquad \gamma_i^2=1.$$6. The exact conserved emergent Majorana may then become highly nonlinear in both effective Majorana operators and the underlying electron creation and destruction operators (Lee et al., 2013).

3. Local particle-hole bilinears and generalized Majorana polarization

A distinct quadratic generalization appears in Bogoliubov–de Gennes systems with particle-hole symmetry. The generalized Majorana polarization is defined from the local particle-hole self-overlap

$$\gamma_i^\dagger=\gamma_i,\qquad \{\gamma_i,\gamma_j\}=2\delta_{ij},\qquad \gamma_i^2=1.$$7

and the integrated criterion for a localization region $$\gamma_i^\dagger=\gamma_i,\qquad \{\gamma_i,\gamma_j\}=2\delta_{ij},\qquad \gamma_i^2=1.$$8 is

$$\gamma_i^\dagger=\gamma_i,\qquad \{\gamma_i,\gamma_j\}=2\delta_{ij},\qquad \gamma_i^2=1.$$9

The paper states that this is a necessary and sufficient criterion to test whether a state is a Majorana or not (Sedlmayr et al., 2015).

In the spinless case, with local Nambu spinor $i\gamma_i\gamma_j$0 and particle-hole operator $i\gamma_i\gamma_j$1, the local generalized Majorana polarization is

$i\gamma_i\gamma_j$2

In the spinful basis

$i\gamma_i\gamma_j$3

with $i\gamma_i\gamma_j$4, it becomes

$i\gamma_i\gamma_j$5

These are explicitly quadratic bilinears in local particle and hole amplitudes, and the paper identifies them as a universal measure of the same-spin particle-hole overlap (Sedlmayr et al., 2015).

This formulation shifts attention away from density and toward off-diagonal particle-hole coherence. A true Majorana requires not only electron-hole mixing but coherent spatial alignment of the local complex bilinears. The local quantity is represented as a complex two-component pseudo-spin-like vector; aligned vectors correspond to a true Majorana, whereas spatially rotating or oscillating vectors correspond to quasi-Majorana or non-Majorana states. The paper emphasizes that low energy alone is insufficient: a state may have large local Majorana polarization yet fail the integrated criterion because the local directions are misaligned (Sedlmayr et al., 2015).

The generalization relative to the earlier Majorana polarization of Sticlet et al. is symmetry-based. The older definition applied only to a subset of models, essentially BDI or “chiral orthogonal” settings with specific spin structure. The generalized definition is built directly from the particle-hole operator and is therefore applicable to arbitrary quadratic BdG Hamiltonians with particle-hole mixing, including D-, BDI-, and DIII-type systems, spinless and spinful cases, and disordered or noncollinear-spin settings (Sedlmayr et al., 2015).

4. Complete symmetry classification of quadratic bilinears in two-dimensional QBT systems

A different generalization concerns the classification of all quadratic fermion bilinears in two-dimensional quadratic-band-touching Hamiltonians even under momentum reversal. For a theory of $i\gamma_i\gamma_j$6 flavors of two-component complex fermions,

$i\gamma_i\gamma_j$7

with $i\gamma_i\gamma_j$8, the emergent internal symmetry is $i\gamma_i\gamma_j$9, not the orthogonal group familiar from Dirac/Majorana systems. A central identity is

$$H_0=-i\sum_{j=1}^{N}\gamma_{2j-1}\gamma_{2j},\qquad H_1=-i\sum_{j=1}^{N-1}\gamma_{2j}\gamma_{2j+1},$$0

(Herbut et al., 23 Apr 2026).

After rewriting the complex field in a Majorana basis, the paper classifies all bilinears $$H_0=-i\sum_{j=1}^{N}\gamma_{2j-1}\gamma_{2j},\qquad H_1=-i\sum_{j=1}^{N-1}\gamma_{2j}\gamma_{2j+1},$$1 with $$H_0=-i\sum_{j=1}^{N}\gamma_{2j-1}\gamma_{2j},\qquad H_1=-i\sum_{j=1}^{N-1}\gamma_{2j}\gamma_{2j+1},$$2 as irreducible representations of $$H_0=-i\sum_{j=1}^{N}\gamma_{2j-1}\gamma_{2j},\qquad H_1=-i\sum_{j=1}^{N-1}\gamma_{2j}\gamma_{2j+1},$$3. The symmetry generators are

$$H_0=-i\sum_{j=1}^{N}\gamma_{2j-1}\gamma_{2j},\qquad H_1=-i\sum_{j=1}^{N-1}\gamma_{2j}\gamma_{2j+1},$$4

with total number

$$H_0=-i\sum_{j=1}^{N}\gamma_{2j-1}\gamma_{2j},\qquad H_1=-i\sum_{j=1}^{N-1}\gamma_{2j}\gamma_{2j+1},$$5

matching the dimension of $$H_0=-i\sum_{j=1}^{N}\gamma_{2j-1}\gamma_{2j},\qquad H_1=-i\sum_{j=1}^{N-1}\gamma_{2j}\gamma_{2j+1},$$6. These bilinears form the adjoint representation (Herbut et al., 23 Apr 2026).

The mass sector is defined by matrices $$H_0=-i\sum_{j=1}^{N}\gamma_{2j-1}\gamma_{2j},\qquad H_1=-i\sum_{j=1}^{N-1}\gamma_{2j}\gamma_{2j+1},$$7 that anticommute with the QBT kinetic matrices: $$H_0=-i\sum_{j=1}^{N}\gamma_{2j-1}\gamma_{2j},\qquad H_1=-i\sum_{j=1}^{N-1}\gamma_{2j}\gamma_{2j+1},$$8 and the allowed mass matrices are

$$H_0=-i\sum_{j=1}^{N}\gamma_{2j-1}\gamma_{2j},\qquad H_1=-i\sum_{j=1}^{N-1}\gamma_{2j}\gamma_{2j+1},$$9

There are $H=\frac{i}{2}\sum_{ij}A_{ij}\gamma_i\gamma_j,$0 such matrices. One of them,

$H=\frac{i}{2}\sum_{ij}A_{ij}\gamma_i\gamma_j,$1

is a $H=\frac{i}{2}\sum_{ij}A_{ij}\gamma_i\gamma_j,$2 singlet, while the rest form a nontrivial irrep of dimension $H=\frac{i}{2}\sum_{ij}A_{ij}\gamma_i\gamma_j,$3. Two additional families,

$H=\frac{i}{2}\sum_{ij}A_{ij}\gamma_i\gamma_j,$4

yield two nematic representations, each of dimension $H=\frac{i}{2}\sum_{ij}A_{ij}\gamma_i\gamma_j,$5. The paper states that the generators of $H=\frac{i}{2}\sum_{ij}A_{ij}\gamma_i\gamma_j,$6, the masses, and the nematics exhaust the set of possible Majorana bilinears (Herbut et al., 23 Apr 2026).

This classification is notable because it is complete for arbitrary $H=\frac{i}{2}\sum_{ij}A_{ij}\gamma_i\gamma_j,$7. Under reduction to $H=\frac{i}{2}\sum_{ij}A_{ij}\gamma_i\gamma_j,$8, the non-singlet mass representation splits into $H=\frac{i}{2}\sum_{ij}A_{ij}\gamma_i\gamma_j,$9-neutral insulating masses and $A_{ij}$0-charged pairing masses. A plausible implication is that generalized quadratic Majorana bilinears can be organized systematically once the kinetic structure selects the appropriate internal symmetry group, and that the familiar complex-fermion classification reappears as the overlap of orthogonal and symplectic viewpoints (Herbut et al., 23 Apr 2026).

5. Reservoir-assisted bilinears and non-Markovian Majorana transport

In floating Coulomb-blockaded Majorana islands, generalized quadratic bilinears arise as cotunneling operators. The island hosts $A_{ij}$1 Majorana zero modes $A_{ij}$2 with $A_{ij}$3, and the effective cotunneling operators are

$A_{ij}$4

These operators are Hermitian for $A_{ij}$5, parity-even, and satisfy the $A_{ij}$6 commutation relations

$A_{ij}$7

(Tian, 7 May 2026).

The microscopic tunneling Hamiltonian is linear in single Majoranas, but a Schrieffer–Wolff transformation in the cotunneling regime produces second-order reservoir-assisted transitions generated by the bilinears $A_{ij}$8. The effective cotunneling Hamiltonian is

$A_{ij}$9

The elementary low-energy processes are therefore not bare island bilinears alone, but reservoir-assisted bilinear transition operators $H_0$0 (Tian, 7 May 2026).

For structured reservoirs, the channel-resolved kernels are

$H_0$1

while the island state retains only the bilinear-labeled sums

$H_0$2

The reduced island dynamics depends only on these projected kernels: $H_0$3 The paper’s central result is that complete knowledge of the non-Markovian island-state dynamics does not in general determine the thermodynamic transport statistics measured in the leads, because the projection to island kernels loses which reservoir channel supplied or absorbed the electron and which channel carried the energy exchange (Tian, 7 May 2026).

This establishes a further generalization of quadratic Majorana bilinears: the effective operator algebra on the island is bilinear and complete for island-state observables within cotunneling order, yet it is not complete for lead-resolved charge noise, heat noise, or mixed charge-energy correlations. Distinct microscopic channel assignments can generate identical island-state tomography and relaxation while differing in transport statistics (Tian, 7 May 2026).

6. Number-conserving Floquet bilinears and generalized parity operators

In a number-conserving periodically driven $H_0$4-wave superconductor, the generalized Majorana edge operators are

$H_0$5

They are explicitly non-Hermitian, since $H_0$6, but the paper states that they form a set of mutually anticommuting operators and behave as generalized Majorana edge modes in the enlarged Hilbert space of chain plus Cooper-pair reservoir (Bomantara, 2023).

At the static solvable point $H_0$7, $H_0$8, and $H_0$9, the Hamiltonian becomes

$-i\gamma_{2j-1}\gamma_{2j}=2a_j^\dagger a_j-1$0

This is the number-conserving analogue of a quadratic Majorana chain, but the natural generalized bilinear is now $-i\gamma_{2j-1}\gamma_{2j}=2a_j^\dagger a_j-1$1 rather than $-i\gamma_{2j-1}\gamma_{2j}=2a_j^\dagger a_j-1$2 (Bomantara, 2023).

The most important bilinears are the generalized parity operators

$-i\gamma_{2j-1}\gamma_{2j}=2a_j^\dagger a_j-1$3

For two generalized Majorana modes, the paper gives

$-i\gamma_{2j-1}\gamma_{2j}=2a_j^\dagger a_j-1$4

With four generalized Majorana modes, bilinears such as

$-i\gamma_{2j-1}\gamma_{2j}=2a_j^\dagger a_j-1$5

anticommute and may be interpreted as effective logical $-i\gamma_{2j-1}\gamma_{2j}=2a_j^\dagger a_j-1$6 and $-i\gamma_{2j-1}\gamma_{2j}=2a_j^\dagger a_j-1$7 operators in the encoded qubit space (Bomantara, 2023).

The Floquet setting introduces generalized Majorana zero modes and generalized Majorana $-i\gamma_{2j-1}\gamma_{2j}=2a_j^\dagger a_j-1$8 modes. If $-i\gamma_{2j-1}\gamma_{2j}=2a_j^\dagger a_j-1$9 is a zero mode and $a_j^\dagger,a_j$00 a $a_j^\dagger,a_j$01 mode, then

$a_j^\dagger,a_j$02

Accordingly, bilinears constructed from two zero modes or two $a_j^\dagger,a_j$03 modes commute with the Floquet operator, whereas bilinears constructed from one zero mode and one $a_j^\dagger,a_j$04 mode anticommute with it. The paper uses this structure to define conserved Floquet parities, $a_j^\dagger,a_j$05-shifting nonlocal operators, qubit encoding, and braiding diagnostics (Bomantara, 2023).

Finite charging energy shifts the zero and $a_j^\dagger,a_j$06 quasienergies and breaks the chiral symmetries that pin them exactly, but the generalized parity bilinears remain approximately conserved in larger systems. This places the bilinears, rather than the constituent non-Hermitian edge operators alone, at the center of the robust qubit structure (Bomantara, 2023).

7. Bosonic analogues, conceptual boundaries, and recurring lessons

A conceptually adjacent development appears in quadratic bosonic Markovian dynamics. Here the fundamental Hermitian operators are bosonic quadratures

$a_j^\dagger,a_j$07

which are Majorana-like only in the sense of being Hermitian linear combinations of annihilation and creation operators; they obey canonical commutation relations rather than a Clifford algebra. The relevant quadratic structure is encoded in the Nambu vector $a_j^\dagger,a_j$08, the non-Hermitian single-particle generator

$a_j^\dagger,a_j$09

and the closed equations

$a_j^\dagger,a_j$10

The paper’s “Majorana bosons” are edge-localized Hermitian linear combinations of quadratures that emerge as approximate conserved zero modes or approximate weak-symmetry generators in a metastable quadratic Lindbladian, with topology formulated through winding of the rapidity spectrum of $a_j^\dagger,a_j$11 in the complex plane (Flynn et al., 2021).

This bosonic case marks an important boundary. The paper does not construct fermionic Majorana bilinears in the strict sense; instead, it develops a bosonic analogue in which edge physics is inherited from quadratic covariance dynamics, non-normality, and pseudospectral bulk-boundary correspondence. The modes are not Clifford Majoranas, and the analogue of Majorana pairing is split by dissipation into a conserved mode on one edge and a symmetry generator on the opposite edge (Flynn et al., 2021).

Across these works, several recurring lessons emerge. First, quadratic bilinears $a_j^\dagger,a_j$12 remain the canonical language for pairing, tunneling, and effective low-energy couplings. Second, the physically relevant generalization can take different forms: local particle-hole bilinears of BdG amplitudes, symmetry-classified antisymmetric matrices, reservoir-assisted cotunneling bilinears, or non-Hermitian number-conserving bilinears $a_j^\dagger,a_j$13. Third, several robust Majorana phenomena often attributed to bilinears alone actually depend on additional algebraic structure, especially fermion parity, locality of particle-hole self-overlap, or symmetry constraints. Finally, the literature shows that when quadraticity is relaxed, the exact surviving object may cease to be a bilinear altogether; nevertheless, the bilinear framework remains the reference point from which these broader many-body constructions are defined (Lee et al., 2013).