Majoranas Beyond Mean Field: Fixed Charge & Many-body States

Mean-field superconducting wavefunctions do not have a definite particle number, while a genuinely isolated device has fixed total charge. Does the Majorana description survive that change? Agarwal and collaborators approach the question through number projection, multiple-wire state spaces, and correlation diagnostics in interacting models. The issue is not whether mean-field theory is useful—it often is—but which of its spectral and information-processing claims remain valid once the physical charge constraint is imposed.

Figure to read

Odd-even excitation gaps versus chain length in number-conserving wire models. Figure 1 in the source paper. Thomas-Markarian, Jaden; Agarwal, Kartiek; Martin, Ivar. Majorana Edge Modes in Isolated Wires, Physical Review Letters 137, 086504 (2026). Figure linked rather than reproduced; the archived manuscript does not provide an explicit open reproduction license.

Compare the topological and trivial parameter choices. The plotted topological splitting scales approximately as an inverse length in this example; do not replace the observed scaling with an assumed exponential.

A zero mode is not automatically a qubit

The standard Majorana picture associates edge operators with nearly degenerate states of different fermion parity. If a single isolated wire is restricted to one exact particle number, those parity sectors are not both freely available as computational states. One must specify a larger system or another accessible degree of freedom before discussing nontrivial operations within a protected subspace. The general topological-computation framework makes the state space and allowed operations indispensable parts of a proposal. 1

A useful thought experiment is to compare spectroscopy with storage. Adding or removing a particle can reveal a low-energy spectral feature even when the fixed-number ground state is unique. That spectral observation is meaningful, but it does not by itself provide two states between which a logical operation can act while conserving the total charge. The distinction prevents an otherwise easy slide from an edge signature to a working isolated qubit.

Number-projected wavefunctions

Sajith, Agarwal, and Martin study Kitaev-type wavefunctions projected onto fixed particle number. Their work finds that important edge-related spectral characteristics survive projection and constructs many-body counterparts of familiar Majorana signatures. The result gives a concrete bridge between the simple mean-field picture and a charge-conserving description. It does not justify assuming that every mean-field degeneracy survives unchanged in every isolated geometry. 2

Projection is also a variational construction. Its quality depends on the Hamiltonian and parameter regime, and a useful projected state need not establish every dynamical property of a physical device. Comparing projected wavefunctions with direct calculations in number-conserving interacting models is therefore an important independent test. The broad Majorana literature supplies the baseline expectations, while the isolated-system problem tests where those expectations need refinement. 3

Multiple wires and braiding-equivalent operations

Martin and Agarwal next consider multiple wires with a conserved total charge. Charge can be redistributed within the combined system even though the total remains fixed. Starting from number-projected states, they identify a many-body state structure analogous to the parity basis used in mean-field treatments and analyze operations equivalent to braiding. The study also discusses limitations to fidelity and protection associated with intrinsic structure and external perturbations. 4

The practical lesson is to identify the actual low-energy manifold before choosing a gate protocol. Interwire tunneling can resolve a large degeneracy, and the resulting splittings and matrix elements determine the useful operating window. A formal braid unitary is only part of the story; initialization, dynamical phases, leakage, and sensitivity to local noise must be evaluated in that same charge-conserving state space.

Seeing an edge in a many-body state

Thomas-Markarian, Agarwal, and Martin develop a correlation-based method for identifying Majorana structure in interacting, number-conserving wires. Their density-matrix-renormalization calculations compare odd-even energy splittings and parity-dependent off-diagonal correlations. The latter reveal spatial information that is not apparent from inspecting a many-body wavefunction as though it were a single-particle orbital. The published work also discusses the greater fragility of short-range interacting cases lacking a bulk excitation gap. 5

Finite-size scaling remains essential. A decreasing splitting and edge-sensitive correlations are complementary evidence, but their scaling law should be measured rather than assumed exponential. Long-range interactions change the comparison with strictly local models. The next research question is whether the same microscopic description can support robust preparation, manipulation, and readout under realistic charging and environmental conditions. This page links Majorana transport to information diagnostics: both require distinguishing an accessible observable from the stronger claim that a useful quantum state space has been certified.

References

1. Nayak, Chetan; Simon, Steven H.; Stern, Ady; Freedman, Michael; Sarma, Sankar Das. Non-Abelian anyons and topological quantum computation. Reviews of Modern Physics 80, 1083-1159 (2008). Open manuscript.
2. Sajith, Rohith; Agarwal, Kartiek; Martin, Ivar. Signatures of Majorana Zero-Modes in an isolated one-dimensional superconductor. Physical Review B 109, 184509 (2024). Open manuscript.
3. Alicea, Jason. New directions in the pursuit of Majorana fermions in solid state systems. Rep. Prog. Phys. 75, 076501 (2012). Open manuscript.
4. Martin, Ivar; Agarwal, Kartiek. Understanding Majorana braiding in superconductors with a fixed total number of particles. Physical Review B 113, 155149 (2026). Open manuscript.
5. Thomas-Markarian, Jaden; Agarwal, Kartiek; Martin, Ivar. Majorana Edge Modes in Isolated Wires. Physical Review Letters 137, 086504 (2026). Open manuscript.

Independent research synthesis. Literature checked 5 September 2026; preprints are identified in the references.