Poor Man's Majorana Modes

• Poor Man's Majorana Modes are zero-energy states emerging in quantum dots via balanced elastic cotunneling and crossed Andreev reflection.
• The models demonstrate nonlocal parity encoding with measurable diagnostics like Majorana polarization despite lacking topological protection.
• Various engineering routes—including interacting three-dot devices, YSR states, and cavity embedding—provide practical platforms for tuning and stabilizing PMMs.

Searching arXiv for recent and foundational papers on poor man's Majorana modes. Poor man’s Majorana modes (PMMs) are Majorana-like zero-energy states that arise most commonly in minimal Kitaev-chain realizations built from quantum dots coupled through superconductivity. In the canonical construction, elastic cotunneling (ECT) provides an effective hopping term and crossed Andreev reflection (CAR) provides an effective pairing term; when these amplitudes are balanced at a finely tuned sweet spot, one Hermitian zero mode can localize on each end dot. PMMs reproduce several hallmark properties of Majorana bound states—spatial separation, self-conjugation, charge neutrality, parity encoding, and in idealized protocols even Majorana-like fusion or braiding logic—but they do not derive their existence from a bulk topological phase and therefore lack topological protection (Leijnse et al., 2012, Cayao, 2024, Tsintzis et al., 2023).

1. Foundational construction in double quantum dots

The foundational proposal considers two single-level quantum dots coupled through a common $s$-wave superconductor. In that geometry, the effective Hamiltonian is

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

with $t$ the effective interdot tunneling amplitude mediated by the superconductor and $\Delta$ the effective crossed Andreev pairing amplitude. The dots are rendered effectively spinless by strong Zeeman splitting, while non-collinear magnetic fields on the two dots tune the relative weights of tunneling and CAR according to

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

where $\varphi$ is the angle between the two field directions (Leijnse et al., 2012).

At the canonical sweet spot,

$\varepsilon_1=\varepsilon_2=0,\qquad t=\pm \Delta,$

the BdG spectrum contains two zero modes and two finite-energy modes. The zero-mode operators can be written explicitly as

$$\gamma_1=\frac{d_1+d_1^\dagger}{\sqrt 2},\qquad \gamma_2=\frac{i(d_2-d_2^\dagger)}{\sqrt 2},$$

so that one Majorana operator resides entirely on each dot. This is the basis for the expression “poor man’s Majorana bound states”: the modes are not topologically protected, but otherwise share the properties of MBS formed in topological superconductors (Leijnse et al., 2012).

The same construction also yields a nonlocal Dirac fermion,

$f=\frac{\gamma_1-i\gamma_2}{2},$

whose occupation defines a parity qubit. In the ideal limit, the even- and odd-parity states are locally indistinguishable on a single dot, with $\langle n_1\rangle=\langle n_2\rangle=1/2$, while joint charge fluctuations can distinguish them. This nonlocal encoding was one of the original motivations for PMMs as a minimal platform for parity-qubit physics (Leijnse et al., 2012).

2. Minimal Kitaev-chain interpretation and microscopic generalizations

Within the now standard Kitaev-chain interpretation, PMMs arise because a minimal two-site device can emulate the pairing–hopping structure of the Kitaev model. ECT acts as intersite hopping, CAR acts as intersite pairing, and the sweet spot is the two-site analogue of the Kitaev condition that hopping and pairing be equal. In more realistic devices this structure is often implemented not by a single effective superconductor alone, but by a three-site Hamiltonian in which the left and right quantum dots are coupled through a proximitized middle element, such as an Andreev bound state or a superconducting dot (Tsintzis et al., 2022).

For the interacting three-dot platform, the microscopic Hamiltonian contains outer-dot chemical potentials, Zeeman splitting, Coulomb repulsion, spin-conserving tunneling, spin-flip tunneling from SOI, and induced pairing on the central dot. In the perturbative regime, integrating out the proximitized middle dot generates effective ECT and CAR couplings between the outer dots. A key feature of this construction is that the center-dot gate $$H=\varepsilon_1 n_1+\varepsilon_2 n_2+t\,d_1^\dagger d_2+\Delta\, d_1^\dagger d_2^\dagger+\mathrm{h.c.},$$0 controls the relative strength of CAR and ECT, so a PMM sweet spot can be reached without tuning the spin-orbit angle or spin-polarization direction; a fixed SOI is sufficient (Tsintzis et al., 2022).

Analytical sweet-spot conditions have also been generalized beyond the original infinite-Zeeman limit. For a spinful two-dot model with finite Zeeman energy and $$H=\varepsilon_1 n_1+\varepsilon_2 n_2+t\,d_1^\dagger d_2+\Delta\, d_1^\dagger d_2^\dagger+\mathrm{h.c.},$$1, exact conditions can be derived for perfect PMMs, including relations among local Andreev reflection, CAR, ECT, and the Zeeman-shifted onsite energies. In that idealized spinful setting, perfect PMMs still require exact zero-energy degeneracy, exact charge neutrality, unit Majorana polarization, and a finite excitation gap. However, when the same couplings are generated microscopically through superconducting bulk states or an Andreev bound state, those analytical conditions can only be approximated rather than satisfied exactly (Luethi et al., 2024).

This separation between the effective Kitaev description and microscopic realizations is central to the PMM literature. The effective model admits exact zero modes at a fine-tuned point; microscopic models generally deform that point into a narrow region supporting only approximate, near-zero-energy end states (Tsintzis et al., 2022, Luethi et al., 2024).

3. Diagnostics, quality measures, and operational meaning

Because PMMs are intrinsically fine-tuned, the literature places unusual emphasis on diagnostics of “Majorana quality.” A widely used set of measures compares the lowest even- and odd-parity sectors. In one common formulation,

$$H=\varepsilon_1 n_1+\varepsilon_2 n_2+t\,d_1^\dagger d_2+\Delta\, d_1^\dagger d_2^\dagger+\mathrm{h.c.},$$2

$$H=\varepsilon_1 n_1+\varepsilon_2 n_2+t\,d_1^\dagger d_2+\Delta\, d_1^\dagger d_2^\dagger+\mathrm{h.c.},$$3

and the Majorana polarization on site $$H=\varepsilon_1 n_1+\varepsilon_2 n_2+t\,d_1^\dagger d_2+\Delta\, d_1^\dagger d_2^\dagger+\mathrm{h.c.},$$4 is

$$H=\varepsilon_1 n_1+\varepsilon_2 n_2+t\,d_1^\dagger d_2+\Delta\, d_1^\dagger d_2^\dagger+\mathrm{h.c.},$$5

The excitation gap is

$$H=\varepsilon_1 n_1+\varepsilon_2 n_2+t\,d_1^\dagger d_2+\Delta\, d_1^\dagger d_2^\dagger+\mathrm{h.c.},$$6

A perfect PMM requires $$H=\varepsilon_1 n_1+\varepsilon_2 n_2+t\,d_1^\dagger d_2+\Delta\, d_1^\dagger d_2^\dagger+\mathrm{h.c.},$$7, $$H=\varepsilon_1 n_1+\varepsilon_2 n_2+t\,d_1^\dagger d_2+\Delta\, d_1^\dagger d_2^\dagger+\mathrm{h.c.},$$8, $$H=\varepsilon_1 n_1+\varepsilon_2 n_2+t\,d_1^\dagger d_2+\Delta\, d_1^\dagger d_2^\dagger+\mathrm{h.c.},$$9, and $t$0 (Luethi et al., 2024).

Realistic microscopic models usually replace this exact criterion by threshold inequalities. The literature refers to the resulting domains as a threshold region (TR) or region of threshold (ROT), depending on the paper. In that language, PMMs become “imperfect PMMs”: near-zero-energy states with small but nonzero parity splitting, small but nonzero charge difference, high but subunit Majorana polarization, and a finite gap (Luethi et al., 2024, Luethi et al., 2024).

These diagnostics are operational rather than merely formal. A research roadmap for PMM experiments begins with measuring Majorana quality for a single pair, then proceeds to initialization and readout of parity, measurement of quasiparticle poisoning times, coupling of two PMM systems into a qubit, and finally fusion and braiding-like protocols. The same roadmap emphasizes that minimal two-site Kitaev chains can demonstrate nonabelian physics only with appropriately modified protocols and careful attention to platform-specific imperfections (Tsintzis et al., 2023).

Transport and charge sensing play complementary roles in this program. Local zero-bias conductance peaks track parity degeneracy, nonlocal conductance is more sensitive to nonlocality and wave-function structure, and quantum-capacitance-based readout can convert parity differences into charge response away from the sweet spot. At the ideal sweet spot, by contrast, the original double-dot model predicts that local measurements on a single dot cannot distinguish the parity states, because the encoded fermion is nonlocal (Leijnse et al., 2012, Tsintzis et al., 2023).

4. Imperfect, true, and false PMMs

A major development in the field is the distinction between perfect PMMs of the ideal model and the imperfect PMMs of microscopic models. In realistic descriptions where ECT, CAR, and LAR are not independently tunable, exact sweet spots are generally absent. Instead, one finds narrow parameter regions in which the PMM criteria are only approximately satisfied. The resulting classification depends on threshold choices, which introduces some arbitrariness into the concept of a PMM (Luethi et al., 2024).

An even sharper distinction emerges when one asks what happens as a minimal chain is extended toward the long-chain limit. In that setting, not all PMMs are precursors of true topological Majorana bound states. Some evolve into trivial, highly localized low-energy states rather than into topological edge modes. These have been termed false PMMs, while PMMs that evolve into topological MBSs in long chains are termed true PMMs (Luethi et al., 2024, Luethi et al., 9 Apr 2025).

The microscopic explanation advanced for false PMMs is boundary asymmetry. In a minimal chain, the outer normal dots are not equivalent to bulk normal dots in a long alternating chain: the first and last normal dots each couple directly to only one superconducting dot, whereas a bulk normal dot couples to two superconducting dots. This asymmetry allows boundary-induced near-zero states to form even when they are not finite-size remnants of a topological phase (Luethi et al., 2024).

A complementary analytic mechanism has been identified in artificial two- and three-site chains. In two-site systems, many false PMMs can be traced to zero-energy states already present in the absence of superconductivity. In three-site systems, the same mechanism remains operative, but additional false-PMM regimes also appear outside the range predicted by the two-site analysis, suggesting richer false-PMM physics as the chain length increases (Luethi et al., 9 Apr 2025).

The experimental consequence is restrictive. Transport signatures usually associated with PMMs do not cleanly distinguish true from false cases. In particular, the conventional zero-bias and finite-energy conductance patterns can appear in both classes, so those signatures are necessary but not sufficient evidence for topological Majoranas (Luethi et al., 2024). A plausible implication is that robustness of a PMM-like feature and its relevance to long-chain topology need not be correlated: one study finds that the PMM-like states most stable to chemical-potential perturbations and with the largest excitation gaps occur in parameter regions that also have a large ratio of false to true PMMs (Luethi et al., 9 Apr 2025).

5. Engineering routes and stabilization strategies

Several architectures now exist for engineering PMMs or PMM-like states, each modifying the original two-dot recipe by introducing new control knobs.

Platform	Key control knob	Representative result
Interacting three-dot QD device	Center-dot gate $t$1	CAR/ECT balance without tuning SOI angle (Tsintzis et al., 2022)
YSR-based two-site chain	ABS detuning and strong QD–SC hybridization	Gap larger than $t$2, reduced charge dispersion (Zatelli et al., 2023)
Cavity-embedded two-site chain	$t$3, $t$4, photon number $t$5	Cavity-controlled sweet spot with $t$6 (Gómez-León et al., 2024)
Double 3TJJ minimal chain	Superconducting phases and QPC coupling	Field-free PMMs with $t$7 in the minimal model (Escribano et al., 24 Jan 2025)
Reservoir-coupled non-Hermitian chain	Lead asymmetry $t$8	Zero-real-energy lines between EPs (Cayao et al., 2024)
Finite-length QD–SC–QD	Superconductor length $t$9	PMM number oscillates between 0 and 2 (Zhang et al., 14 Apr 2026)

In YSR-based devices, proximitized quantum dots host Yu–Shiba–Rusinov states rather than ordinary dot states. Strong hybridization with a shared superconducting segment reduces effective charge and greatly suppresses charge dispersion. The reported PMM sweet spot features a gap of about $\Delta$0, with Extended Data showing a similar value of about $\Delta$1; the main text describes this as three times larger than in the previous QD-only PMM realization. The lever arm drops from $\Delta$2 in the above-gap QD regime to $\Delta$3 at the YSR sweet spot, the curvature of the PMM splitting is about 150 times smaller than ადრე, and the estimated dephasing from QD charge noise is $\Delta$4 MHz rather than $\Delta$5 MHz (Zatelli et al., 2023).

Cavity embedding introduces a different stabilization mechanism. In the large-detuning regime, adiabatic elimination of photons yields renormalized parameters $\Delta$6, $\Delta$7, and $\Delta$8, so the interacting sweet spot becomes

$\Delta$9

In this picture the cavity does not merely shift the original PMM point; it creates a tunable sweet-spot manifold and can screen intrinsic interdot interactions that otherwise hybridize the end Majoranas (Gómez-León et al., 2024).

A distinct route replaces magnetic control by superconducting phase control. In a double three-terminal Josephson junction embedded in a planar semiconductor, each 3TJJ acts as an effective Kitaev-chain site, while a QPC provides inter-site hopping. PMMs emerge near $$t=t_0\cos(\varphi/2), \qquad \Delta=\Delta_0\sin(\varphi/2),$$0 and $$t=t_0\cos(\varphi/2), \qquad \Delta=\Delta_0\sin(\varphi/2),$$1, reaching $$t=t_0\cos(\varphi/2), \qquad \Delta=\Delta_0\sin(\varphi/2),$$2 in the reduced model and $$t=t_0\cos(\varphi/2), \qquad \Delta=\Delta_0\sin(\varphi/2),$$3 in realistic continuum simulations, with a finite minigap of $$t=t_0\cos(\varphi/2), \qquad \Delta=\Delta_0\sin(\varphi/2),$$4 in the minimal model and $$t=t_0\cos(\varphi/2), \qquad \Delta=\Delta_0\sin(\varphi/2),$$5 in the full simulation. Because the proposal is all-electric and field-free, it is explicitly aimed at materials classes, such as Ge-based heterostructures, for which Zeeman-based PMM engineering is less favorable (Escribano et al., 24 Jan 2025).

Two further developments alter the standard notion of the PMM sweet spot. In non-Hermitian minimal chains, coupling to normal reservoirs generates exceptional points that connect stable zero-real-energy lines. These are presented as non-Hermitian generalizations of Hermitian PMMs, broadening the PMM regime from isolated points into finite regions bounded by EP transitions (Cayao et al., 2024). By contrast, a finite-length microscopic treatment of the intermediary superconductor shows that PMMs are highly sensitive to the superconducting length: the number of PMMs oscillates between zero and two with a period set by the Fermi wavelength, while four PMMs appear in the long-SC limit. The same analysis proves that strictly separately localized PMMs at opposite ends of the full hybrid structure do not exist for finite $$t=t_0\cos(\varphi/2), \qquad \Delta=\Delta_0\sin(\varphi/2),$$6; only nearly localized PMMs appear in strong fields (Zhang et al., 14 Apr 2026).

6. Extensions, reinterpretations, and broader usage

The PMM concept has expanded well beyond the static double-dot setting. A periodically driven double quantum dot coupled through an $$t=t_0\cos(\varphi/2), \qquad \Delta=\Delta_0\sin(\varphi/2),$$7-wave superconductor can host Floquet poor man’s Majorana fermions in the high-frequency region, depending on the phase difference between the periodic fields applied to the two dots, while numerical calculations also find many Floquet PMMs in the low-frequency region (Li et al., 2013).

The superconducting correlations associated with PMMs have likewise been analyzed directly. In a two-site Kitaev chain, local pairing is purely odd-frequency, spin-triplet, even-site, even-superconductor-index ($$t=t_0\cos(\varphi/2), \qquad \Delta=\Delta_0\sin(\varphi/2),$$8), while nonlocal pairing contains both $$t=t_0\cos(\varphi/2), \qquad \Delta=\Delta_0\sin(\varphi/2),$$9 and even-frequency triplet odd-site even-superconductor-index ($\varphi$0) components. At the PMM sweet spot $\varphi$1, the odd-frequency components can diverge as $\varphi$2 near $\varphi$3, and this divergence is interpreted as reflecting intrinsic Majorana nonlocality without any relation to topology (Cayao, 2024).

Several works revisit PMMs under exchange coupling to a quantum spin. In this setting, the exchange term acts as an effective chemical potential on one dot, induces a fine structure with $\varphi$4 sublevels, and generates PMM spillover: the zero mode delocalizes across the two ends while part of the fine structure is squeezed into the zero-energy resonance. The claim of “protection” here is explicitly local and conditional rather than topological, because the zero mode remains non-topological even when it stays pinned at $\varphi$5 (Sanches et al., 2024).

PMMs have also been invoked to explain equilibrium Josephson phenomena in topologically trivial systems. In a Rashba superconductor decorated by two coupled AFM dimers of magnetic adatoms, near-zero YSR states behave as weakly coupled PMM excitations. The resulting Andreev spectrum becomes highly dispersive and phase asymmetric, producing a nonreciprocal Josephson current and an equilibrium $\varphi$6-periodic Josephson effect. The diode efficiency is reported to reach around $\varphi$7 in some parameter regimes (Kotetes et al., 2024).

The term has even been generalized to edge-mode physics outside quantum-dot Kitaev chains. On the surface of a 3D topological insulator proximitized by a single superconductor, specular Andreev reflection at an NS interface can bind a helical Majorana edge mode when the normal-side Fermi level is tuned to the Dirac point. The simplified geometry dispenses with the magnetic insulator required in the chiral Fu–Kane construction, but the mode is explicitly described as lacking the topological protection of its chiral counterpart; in that sense it is a “poor man’s Majorana edge mode” (Beenakker, 2024).

A final broadening of scope concerns composite and dressed constructions. The “poor man’s Majorana tetron” replaces a topological tetron by four quantum dots coupled through a floating superconducting island, yielding a two-fold degenerate odd-parity ground state that maps onto an effective Anderson impurity and can approach a regime featuring the topological Kondo effect under suitable tuning (Nitsch et al., 2024). A different microscopic reinterpretation treats the PMM not as a strictly dot-only mode but as a dressed Majorana-like zero mode built from both QD excitations and SC quasi-excitations; in that formulation, the dressed zero-mode condition defines a continuous parameter region rather than an isolated point (Zhang et al., 12 Jun 2025).

Across these variants, one theme remains stable. PMMs are valuable precisely because they isolate Majorana algebra, nonlocality, and spectroscopic phenomenology in experimentally tractable, highly tunable, and often topologically trivial systems. The same feature that makes them accessible—the absence of topological protection—also makes their interpretation delicate: PMM signatures can encode true Majorana-like structure, trivial boundary physics, or platform-specific dressed and driven analogues, and the distinction must be established at the microscopic level rather than inferred from a zero-bias anomaly alone (Luethi et al., 2024, Zhang et al., 12 Jun 2025).