Poor Man’s Majoranas in Minimal Superconductors

• Poor Man’s Majoranas are engineered zero-energy modes in double quantum dot setups that emulate two-site Kitaev chains through balanced elastic cotunneling and crossed Andreev reflection.
• They exhibit hallmark Majorana signatures—zero-bias anomalies, spatial separation, charge neutrality, and non-local parity—but do not benefit from bulk topological protection.
• Their identification relies on spectral diagnostics like parity splitting, charge differences, and Majorana polarization, with tunability via magnetic fields, coupling strengths, or cavity-induced screening.

Poor Man's Majoranas are Majorana-like zero-energy modes engineered in minimal, non-topological superconducting nanostructures, most prominently two quantum dots coupled through a superconductor so as to emulate a two-site Kitaev chain. At the sweet spot, the induced elastic cotunneling and crossed Andreev reflection amplitudes balance, leaving one self-adjoint zero mode on each outer dot and encoding a non-local fermion-parity degree of freedom. They reproduce several hallmark signatures of Majorana bound states—zero-bias anomalies, spatial separation, charge neutrality, and non-local parity structure—yet they lack the bulk topological protection of Majorana bound states in long topological superconductors (Leijnse et al., 2012, Sanches et al., 5 Sep 2025).

1. Conceptual origin and relation to topological Majorana modes

The original formulation considered a double quantum dot connected via a common superconducting lead, with one relevant spin-polarized level on each dot. The superconductor mediates two processes: normal electron tunneling between dots with amplitude $t$, and crossed Andreev reflection with amplitude $\Delta$. In that setup the local magnetic fields form an angle $\varphi$, and the effective amplitudes obey

$$t=t_0\cos(\varphi/2), \qquad \Delta=\Delta_0\sin(\varphi/2),$$

so rotation of the relative field direction tunes the system toward the Majorana-like regime $t\approx \Delta$ (Leijnse et al., 2012).

Subsequent work reformulated this device as the minimal Kitaev-chain implementation: two grounded, effectively spinless quantum dots with a balanced hopping and pairing sector at

$t=\Delta, \qquad \mu_L=\mu_R=0.$

In this language, the modes are “Majorana-like” because they arise as spatially separated zero modes at the ends of a minimal chain, but they are “poor man’s” because the construction is truncated and engineered rather than rooted in a bulk topological phase and bulk-boundary correspondence (Sanches et al., 5 Sep 2025).

The distinction from topological Majorana bound states is not merely terminological. True Majorana bound states in a long Kitaev chain arise from a bulk topological phase protected by a bulk gap. Poor Man’s Majoranas instead occur only at fine-tuned parameter values in a few-site system. A symmetry-based formulation sharpens this contrast: a genuine Majorana “character” requires a local unitary operator $\mathcal V$ such that $\{H,\mathcal V\}=0$ and $\mathcal V^2=1$, which is available in BDI or DIII with mirror symmetry, but not generically in class D. In the absence of such a global operator, zero modes may remain fine-tuned, emergent only at low energy, or weakly protected, which is precisely the conceptual niche occupied by poor man’s states (Sedlmayr et al., 2015).

2. Minimal two-site realization and parity-qubit structure

The canonical effective Hamiltonian of the double-dot realization is

$$H=\varepsilon_1 n_1+\varepsilon_2 n_2+t\,d_1^\dagger d_2+\Delta\,d_1^\dagger d_2^\dagger+\text{h.c.},$$

with $\Delta$0. In Nambu basis, zero modes appear when

$\Delta$1

At this sweet spot the zero-energy eigenvectors correspond to the Hermitian operators

$\Delta$2

while the remaining two eigenstates sit at energies $\Delta$3 (Leijnse et al., 2012).

This decomposition is the origin of the minimal-chain Majorana picture. Rewriting the fermions in terms of Majorana operators produces a dimerized structure in which one Majorana on each outer end remains effectively unpaired when $\Delta$4, whereas the inner Majoranas hybridize into a gapped dimer. In the Green’s-function formulation, the left-dot spectral function at the sweet spot has the characteristic zero-energy weight

$\Delta$5

which is why the PMM is often described as a half-fermion or half-electron excitation in the density of states (Sanches et al., 5 Sep 2025).

The corresponding many-body structure is naturally expressed in the occupation basis $\Delta$6. At $\Delta$7 and $\Delta$8, the eigenstates form Bell-like even- and odd-parity pairs,

$\Delta$9

together with the orthogonal $\varphi$0 and $\varphi$1. The non-local fermion

$\varphi$2

encodes a parity qubit whose logical states differ by the occupation of this non-local fermion. At the sweet spot, local measurements on one dot cannot distinguish even from odd parity because $\varphi$3 for both parity states; only joint observables involving both dots resolve the parity sector (Leijnse et al., 2012).

An alternative gate-tunable implementation replaces the direct superconducting link by a central proximitized dot coupled to two interacting outer dots. In that three-dot architecture, second-order processes through the central dot generate effective elastic cotunneling and crossed Andreev reflection, and gate control of the central level $\varphi$4 provides a direct tuning knob for the CAR/ECT balance and thus for the sweet spots with one Majorana-like mode on each outer dot (Tsintzis et al., 2022).

3. Diagnostics: spectral criteria, Green’s functions, and pairing symmetry

Later literature formalized PMM identification using several diagnostics. In microscopic models, the relevant quantities are the parity splitting

$\varphi$5

the charge difference on dot $\varphi$6,

$\varphi$7

the Majorana polarization

$\varphi$8

and the excitation gap

$\varphi$9

An ideal sweet spot satisfies $$t=t_0\cos(\varphi/2), \qquad \Delta=\Delta_0\sin(\varphi/2),$$0, $$t=t_0\cos(\varphi/2), \qquad \Delta=\Delta_0\sin(\varphi/2),$$1, $$t=t_0\cos(\varphi/2), \qquad \Delta=\Delta_0\sin(\varphi/2),$$2, and $$t=t_0\cos(\varphi/2), \qquad \Delta=\Delta_0\sin(\varphi/2),$$3 (Luethi et al., 2024).

In realistic models, however, exact sweet spots are typically unattainable. This has led to threshold-based definitions of “imperfect PMMs,” where one demands small $$t=t_0\cos(\varphi/2), \qquad \Delta=\Delta_0\sin(\varphi/2),$$4, small $$t=t_0\cos(\varphi/2), \qquad \Delta=\Delta_0\sin(\varphi/2),$$5, large $$t=t_0\cos(\varphi/2), \qquad \Delta=\Delta_0\sin(\varphi/2),$$6, and finite $$t=t_0\cos(\varphi/2), \qquad \Delta=\Delta_0\sin(\varphi/2),$$7 within chosen tolerances. That formulation captures the experimental and numerical fact that realistic PMMs are usually near-zero-energy states that are highly, but not perfectly, localized (Luethi et al., 2024).

The cleanest spectral analysis uses retarded Green’s functions. For superconducting hybrid systems, both normal and anomalous sectors are essential because the PMM resides in an electron-hole-coherent environment. The spectral functions are extracted from

$$t=t_0\cos(\varphi/2), \qquad \Delta=\Delta_0\sin(\varphi/2),$$8

and the Majorana spectral function from

$$t=t_0\cos(\varphi/2), \qquad \Delta=\Delta_0\sin(\varphi/2),$$9

Within this framework, PMM side peaks and zero-bias anomalies are understood as interference features involving electron propagation, hole propagation, and local Andreev processes rather than as bare eigenvalue counting alone (Sanches et al., 5 Sep 2025).

The superconducting correlations induced by PMMs are also highly structured. In a two-site Kitaev chain with onsite energies $t\approx \Delta$0, hopping $t\approx \Delta$1, and induced $t\approx \Delta$2-wave pairing $t\approx \Delta$3, the anomalous Green’s functions support local odd-frequency triplet even-site even-index pairing (OTEE), nonlocal odd-frequency OTEE pairing, and nonlocal even-frequency triplet odd-site even-index pairing (ETOE). At the PMM sweet spot $t\approx \Delta$4 and $t\approx \Delta$5, the odd-frequency pair amplitudes acquire a $t\approx \Delta$6 divergence around zero frequency. The authors interpret this divergence as reflecting intrinsic Majorana nonlocality, but without any relation to topology (Cayao, 2024).

4. Perfect, imperfect, true, and false PMMs

The modern literature distinguishes several categories of PMM-like states.

Category	Defining criterion	Long-chain interpretation
Perfect PMM	$t\approx \Delta$7, $t\approx \Delta$8, $t\approx \Delta$9, $t=\Delta, \qquad \mu_L=\mu_R=0.$0	Idealized effective-model sweet spot
Imperfect PMM	Threshold-based near-zero, highly localized state	Practical classification in realistic models
True PMM	Imperfect PMM that evolves into a topological MBS in the long-chain limit	Long-chain invariant is topological
False PMM	Imperfect PMM that evolves into a trivial localized low-energy state	Not connected to topological MBSs

A central result of the long-chain analyses is that not all PMMs observed in minimal devices are related to topological states. In microscopic Kitaev-chain extensions, some PMMs evolve into a topological phase in the long-chain limit, but others evolve into trivial highly localized low-energy states. The latter are termed false PMMs. The common assumption that a good PMM in a minimal chain will necessarily become a true MBS when the chain is lengthened is therefore not always correct (Luethi et al., 2024).

The microscopic origin of many false PMMs can be traced to zero-energy states that already exist in the absence of superconductivity. In two-site artificial Kitaev chains, the corresponding normal double-dot subsystem obeys the analytic zero-mode condition

$t=\Delta, \qquad \mu_L=\mu_R=0.$1

and for the branch $t=\Delta, \qquad \mu_L=\mu_R=0.$2 the zero-energy eigenvector becomes strongly localized on the leftmost dot when $t=\Delta, \qquad \mu_L=\mu_R=0.$3. Such a state can survive the addition of superconductivity and extra sites, thereby masquerading as a PMM in a short chain while remaining topologically trivial in the long-chain limit (Luethi et al., 9 Apr 2025).

This classification problem has an immediate experimental consequence. Transport signatures ordinarily associated with PMMs—crossings and anticrossings in zero-bias conductance, sign changes in nonlocal conductance, and qualitatively similar finite-energy conductance maps—occur for both parameter sets that evolve into true topological MBSs and parameter sets that evolve into trivial false PMMs. There is therefore no clear conductance signature, neither zero-bias nor finite-energy, local nor nonlocal, that reliably separates PMMs that evolve into topological MBSs from PMMs that evolve into trivial localized states (Luethi et al., 2024).

A further conceptual complication is that threshold-based classification is partly arbitrary. Realistic models mediated either by superconducting bulk states or by a single Andreev bound state do not satisfy the analytic sweet-spot conditions exactly, so the existence of PMMs becomes dependent on chosen thresholds for $t=\Delta, \qquad \mu_L=\mu_R=0.$4, $t=\Delta, \qquad \mu_L=\mu_R=0.$5, $t=\Delta, \qquad \mu_L=\mu_R=0.$6, and $t=\Delta, \qquad \mu_L=\mu_R=0.$7. This has motivated a stricter vocabulary in which “sweet spot” is reserved for exact ideal conditions and realistic short-chain states are called imperfect PMMs (Luethi et al., 2024).

5. Perturbations, controllability, and engineered robustness

Because PMMs are not topologically protected, perturbations are not merely detrimental; they are also a spectroscopic resource. One extensively studied perturbation is an Ising-like exchange coupling of the left dot to a localized quantum spin,

$t=\Delta, \qquad \mu_L=\mu_R=0.$8

which in the effective spinless description shifts the left-dot chemical potential by $t=\Delta, \qquad \mu_L=\mu_R=0.$9. This exchange term destabilizes the isolated left-end Majorana and causes spillover: the PMM spectral weight delocalizes from the host dot into neighboring degrees of freedom, while the right-end mode remains comparatively unaffected. The resulting subgap spectrum acquires a statistics-dependent multiplet structure, with $\mathcal V$0 satellite states for half-integer $\mathcal V$1 and $\mathcal V$2 for integer $\mathcal V$3, symmetrically distributed around the zero-bias anomaly. In a multi-terminal environment, however, this spillover can be strongly suppressed; for moderate $\mathcal V$4 in their units, the perturbed PMM remains localized on its host dot while the neighboring Majorana spectral weight stays essentially zero (Sanches et al., 5 Sep 2025).

A closely related formulation describes the same phenomenon as a squeezing of half of the spin-induced fine structure into the zero-frequency PMM. In that picture the zero mode remains pinned at $\mathcal V$5 and does not mix with the explicit split states, so the PMM can be regarded as robust against the quantum spin even though the system remains non-topological (Sanches et al., 2024).

Other control strategies seek to improve the PMM quality directly. A detuned single-mode cavity coupled through a Peierls phase in the hopping renormalizes the chemical potential, hopping, and interaction into $\mathcal V$6, $\mathcal V$7, and $\mathcal V$8. The cavity-modified sweet spot is defined by

$\mathcal V$9

so photon-mediated screening can cancel intrinsic interdot interactions that would otherwise hybridize the Majorana end modes. In that setting the cavity is not only a detector but an active tuning element, and the PMM quality can be inferred from the cavity transmission and the photon propagator (Gómez-León et al., 2024).

A different route uses proximitized quantum dots hosting Yu–Shiba–Rusinov states. Strong hybridization with the superconductor yields PMM zero modes with a gap of about $\{H,\mathcal V\}=0$0, a comparable $\{H,\mathcal V\}=0$1 gap in another configuration, and a curvature of the quadratic charge dispersion about a factor of $\{H,\mathcal V\}=0$2 smaller than in earlier non-proximitized devices. The extracted lever arm is reduced from about $\{H,\mathcal V\}=0$3 for an above-gap dot state to about $\{H,\mathcal V\}=0$4 near the Andreev-bound-state charge minimum, and the estimated dephasing time reaches $\{H,\mathcal V\}=0$5 ns, while the adiabatic time scale remains $\{H,\mathcal V\}=0$6 ps (Zatelli et al., 2023).

In interacting three-dot implementations, high Majorana polarization and parity degeneracy need to occur at the same parameter point. Nonlocal conductance,

$\{H,\mathcal V\}=0$7

is then a sharper diagnostic than local spectroscopy: at a genuine sweet spot with good localization, the zero lines of $\{H,\mathcal V\}=0$8 coincide with the parity-degeneracy lines and cross at the sweet spot, whereas low-polarization apparent sweet spots lack that crossing pattern (Tsintzis et al., 2022).

6. Generalizations: Floquet, dressed states, edge modes, Josephson physics, and networks

The PMM concept has been generalized beyond static double-dot dimers. A driven double-quantum-dot system connected by an $\{H,\mathcal V\}=0$9-wave superconductor and subjected to periodically varying electric or magnetic fields can host Floquet poor man’s Majorana fermions. In that setting high-frequency Floquet PMMs depend on the phase difference between the external drives on the two dots, numerical calculations also find many low-frequency Floquet PMMs, and the modes survive nonzero dot energies and interdot interaction, although the emergence frequencies shift (Li et al., 2013).

A microscopic reformulation of the canonical double-dot setup retains the superconducting quasiparticles explicitly instead of integrating them out into effective CAR and EC parameters. This produces dressed Majorana fermions, or “Poor Man’s Majoranons,” that are superpositions of dot and superconductor quasiparticles and satisfy the zero-energy condition

$\mathcal V^2=1$0

Unlike the phenomenological sweet spot, this defines a continuous manifold in parameter space. In the weak-tunneling and strong-Zeeman limit, the dressed state reduces to the conventional phenomenological PMM (Zhang et al., 12 Jun 2025).

The terminology has also expanded to geometries outside artificial Kitaev chains. On the surface of a 3D topological insulator, a single normal–superconductor interface can support a helical Majorana edge mode at the Dirac point through specular Andreev reflection, even without a magnetic insulator. The mode has equal electron and hole weight and can be written in a real Majorana basis, but it coexists with gapless bulk states and therefore lacks the topological protection of the chiral Majorana mode in the Fu–Kane geometry; it is accordingly described as a poor-man’s Majorana edge mode (Beenakker, 2024).

In Rashba superconductors with coupled antiferromagnetic dimers of magnetic adatoms, near-zero-energy Yu–Shiba–Rusinov states can behave as weakly coupled PMM excitations. Their presence generates highly dispersive and phase-asymmetric Andreev-bound-state spectra, a nonreciprocal Josephson current with large diode efficiency, and an equilibrium $\mathcal V^2=1$1-periodic Josephson effect mediated by triplet-induced mixing of YSR states (Kotetes et al., 2024).

Network constructions go further still. A four-dot device coupled via a floating superconducting island defines a “poor man’s Majorana tetron,” whose charging energy yields a two-fold degenerate odd-parity ground state and an effective Anderson impurity model when leads are attached; under suitable tuning, the system can approach a regime featuring the topological Kondo effect (Nitsch et al., 2024). In parallel, braiding studies of three minimal Kitaev chains show that imperfect Majoranas with finite overlap can still exhibit robust nonabelian adiabatic exchange when the unwanted ground-state splitting is compensated by a tunable correction term. The braid unitary becomes overlap-dependent rather than ideal, but it remains nonabelian except in the perfect fermion limit (Nitsch et al., 15 Jul 2025).

Taken together, these developments position Poor Man’s Majoranas not as substitutes for topological Majorana zero modes, but as a distinct research program: minimal, highly tunable, spectroscopically rich Majorana analogues whose lack of topological protection is simultaneously their defining limitation and a source of extraordinary controllability.