Minimal Kitaev Chains in Quantum Dot Architectures

• Minimal Kitaev chains are few-site realizations of the spinless p-wave Kitaev model, engineered in mesoscopic devices using quantum dots with tunable hopping and pairing.
• They host near-zero-energy end states known as poor man’s Majoranas, which mimic topological Majorana bound states through fine tuning and controlled experimental protocols.
• Experimental realizations in two- and three-site chains reveal distinct sweet spot conditions, nuanced conductance signatures, and enhanced finite-size protection in three-dot architectures.

Searching arXiv for papers on minimal Kitaev chains and closely related few-site quantum-dot Kitaev-chain realizations. {"query":"all:(\"minimal Kitaev chain\" OR \"poor man's Majorana\" OR \"three-site Kitaev chain\" quantum dots superconductor)", "max_results": 10} Minimal Kitaev chains are few-site realizations of the spinless $p$-wave Kitaev model in which the essential ingredients of the long-chain problem—onsite energies, nearest-neighbor hopping, and nearest-neighbor pairing—are engineered in highly tunable mesoscopic devices. In the literature, the term usually denotes the two-site limit, most commonly realized by two quantum dots coupled through a superconducting element, although the three-site chain has emerged as the smallest extension that exhibits qualitatively stronger Majorana-like protection (Dvir et al., 2022, Bordin et al., 2024). Their central interest lies in the appearance of near-zero-energy end states known as poor man’s Majoranas (PMMs): finite-size, fine-tuned analogs of Majorana bound states that reproduce nonlocal parity structure and end localization without constituting a thermodynamic topological phase (Luethi et al., 2024).

1. Definition and effective Hamiltonians

In its standard form, a minimal Kitaev chain is the two-site truncation of the spinless Kitaev model. For two spin-polarized dot levels $L$ and $R$, the effective Hamiltonian used in experiment and theory is

$$H_{\mathrm{eff}} = \varepsilon_L c_L^\dagger c_L + \varepsilon_R c_R^\dagger c_R + t\, c_L^\dagger c_R + t\, c_R^\dagger c_L + \Delta\, c_L^\dagger c_R^\dagger + \Delta\, c_R c_L ,$$

where $t$ is the effective hopping and $\Delta$ the effective pairing. In quantum-dot realizations, $t$ is generated by elastic cotunneling (ECT), while $\Delta$ is generated by crossed Andreev reflection (CAR); the two dots therefore play the role of the two lattice sites of the Kitaev chain (Dvir et al., 2022).

The same structure can also emerge without a superconducting coupler that directly mediates CAR. In a double quantum dot where each dot is only locally proximitized by its own bulk superconductor, interdot tunneling plus local pairing generate an effective anomalous intersite term in the low-energy sector. In the “Kitaev limit” $V_z,\Delta\gg t,t_{\mathrm{so}}$, projecting onto one low-energy Bogoliubov mode per dot yields a two-site Kitaev Hamiltonian with effective parameters $t_K$ and $L$0, and the superconducting phase difference $L$1 becomes the tuning knob that balances hopping against pairing (Samuelson et al., 2023).

For three sites, the effective model acquires a second bond and a middle site,

$L$2

This is the finite-size spinless Kitaev-chain description realized experimentally with three quantum dots separated by two hybrid semiconductor-superconductor segments, with ECT engineering $L$3 and CAR engineering $L$4 (Bordin et al., 2024).

2. Sweet spots, poor man’s Majoranas, and finite-size Majorana criteria

The defining two-site “sweet spot” is the fine-tuned condition

$L$5

or, in the effective notation of the locally proximitized double-dot proposal,

$L$6

At this point the two-site chain hosts a pair of zero-energy PMMs, one on each site. These states are Majorana-like in that they generate nearly degenerate even- and odd-parity ground states, vanish in local charge at the ideal point, and can be spatially separated across the two ends; they are not topologically protected, because the degeneracy exists only at a finely tuned point or narrow region of parameter space (Dvir et al., 2022, Samuelson et al., 2023).

Realistic microscopic models replace the exact sweet spot by a threshold region. In the three-dot microscopic model consisting of two outer normal dots and a proximitized middle dot, the Hamiltonian conserves total fermion parity, so the low-energy sector is naturally analyzed through the parity splitting $L$7, the end charge differences $L$8, the Majorana polarization $L$9, and the excitation gap $R$0. Exact conditions $R$1, $R$2, $R$3, and $R$4 were not found generically; instead the relevant PMM regime is a “region of threshold” in which these diagnostics are approximately satisfied (Luethi et al., 2024).

This finite-size character is also why the two-site Majoranas are described as “weak Majoranas” in interacting formulations: in the interacting two-site chain, the sweet-spot operators connect the degenerate ground states but not the full spectrum. A plausible implication is that minimal Kitaev chains are best regarded as controlled descendants of the Kitaev model that isolate its low-energy algebraic structure, rather than as direct realizations of bulk topological superconductivity (Samuelson et al., 2023).

3. Experimental realizations and spectroscopic diagnostics

The first direct realization used an InSb nanowire in an $R$5-QD-$R$6-QD-$R$7 geometry, with two spin-polarized quantum dots strongly coupled by both ECT and CAR through a grounded superconducting hybrid segment. The crucial experimental signatures were the opposite avoided-crossing orientations associated with ECT- and CAR-dominated regimes, the sign change of the nonlocal conductance, and the collapse of the avoided crossing at the sweet spot $R$8, where zero-bias peaks and side peaks at approximately $R$9 were observed (Dvir et al., 2022).

Because zero-energy crossings alone do not establish spatially separated Majorana structure, one proposal added a probe dot coupled to one end of the minimal chain. In the spinless toy model, sweeping the additional dot distinguishes a good sweet spot from an overlapping or trivial zero mode: a persistent zero-energy line indicates well-separated Majoranas, a “diamond” pattern signals that the extra dot couples to both Majoranas because the far mode leaks to the probed end, and a “bow-tie” pattern signals intrinsic splitting with relatively good localization (Souto et al., 2023). A later phase-controlled experiment implemented this idea in two- and three-site chains and found that, at the sweet spot, tuning the additional dot produced no resolvable splitting within the linewidth, while deliberate detuning generated the predicted splitting patterns (Bordin et al., 18 Apr 2025).

Transport is not the only probe. RF gate reflectometry on a two-dot hybrid device resolved charge stability diagrams, distinguished ECT from CAR by the orientation of the avoided crossings and the reflectometry response, and remained operational even when both normal leads were pinched off. In that closed regime, the observed quantum-capacitance signal was interpreted as evidence of parity switching between even and odd ground states, making gate sensing a noninvasive probe of interdot coupling and parity dynamics in isolated minimal Kitaev-chain devices (Zhang et al., 8 Aug 2025).

4. Three-site chains and the onset of enhanced protection

The three-site chain is the smallest extension that improves the protection properties of the two-site PMM system. Experimentally and analytically, the three-site sweet spot is

$$H_{\mathrm{eff}} = \varepsilon_L c_L^\dagger c_L + \varepsilon_R c_R^\dagger c_R + t\, c_L^\dagger c_R + t\, c_R^\dagger c_L + \Delta\, c_L^\dagger c_R^\dagger + \Delta\, c_R c_L ,$$0

At this point, the energy splitting due to homogeneous detuning scales as

$$H_{\mathrm{eff}} = \varepsilon_L c_L^\dagger c_L + \varepsilon_R c_R^\dagger c_R + t\, c_L^\dagger c_R + t\, c_R^\dagger c_L + \Delta\, c_L^\dagger c_R^\dagger + \Delta\, c_R c_L ,$$1

so for equal detuning it is cubic rather than quadratic. By contrast, in the two-site chain

$$H_{\mathrm{eff}} = \varepsilon_L c_L^\dagger c_L + \varepsilon_R c_R^\dagger c_R + t\, c_L^\dagger c_R + t\, c_R^\dagger c_L + \Delta\, c_L^\dagger c_R^\dagger + \Delta\, c_R c_L ,$$2

which is only quadratic. The three-site device therefore realizes the smallest chain in which no single-parameter perturbation alone can directly couple and split the two edge Majoranas (Bordin et al., 2024).

The three-site problem is nonetheless not unique. A systematic theoretical analysis identified three distinct sweet-spot classes: genuine 3-site sweet spots with well-localized MBSs at the ends, effective 2-site sweet spots where the middle site acts as a barrier, and sweet spots with delocalized MBSs that overlap in the middle. These three cases differ in localization, excitation gap, and robustness. The genuine 3-site case is the most stable; the effective 2-site case inherits the quadratic-splitting logic of the two-site chain; the delocalized case can have a larger gap but is linearly sensitive to middle-site detuning because the two MBSs overlap there (Dourado et al., 26 Feb 2025).

A frequent misconception is that a three-site chain is therefore already “topological.” The more careful statement is that it is a stronger finite-chain precursor: it has a middle “bulk” site, higher-order sensitivity to local perturbations, and a richer phase structure, but it remains a short finite chain rather than a thermodynamic topological superconductor (Bordin et al., 2024).

5. Topological ambiguity: true and false PMMs

A central conceptual development is the distinction between PMMs that are adiabatically connected to the topological phase of a long Kitaev chain and PMMs that are not. In the microscopic three-dot model of a minimal chain, some parameter sets that satisfy all standard short-chain PMM diagnostics—small parity splitting, small end-charge response, high Majorana polarization, and finite excitation gap—evolve into a topological long chain with $$H_{\mathrm{eff}} = \varepsilon_L c_L^\dagger c_L + \varepsilon_R c_R^\dagger c_R + t\, c_L^\dagger c_R + t\, c_R^\dagger c_L + \Delta\, c_L^\dagger c_R^\dagger + \Delta\, c_R c_L ,$$3, while others evolve into a trivial phase with $$H_{\mathrm{eff}} = \varepsilon_L c_L^\dagger c_L + \varepsilon_R c_R^\dagger c_R + t\, c_L^\dagger c_R + t\, c_R^\dagger c_L + \Delta\, c_L^\dagger c_R^\dagger + \Delta\, c_R c_L ,$$4 despite retaining zero-energy end-localized states (Luethi et al., 2024).

The latter are “false PMMs.” Their origin is a boundary inequivalence built into the alternating normal-dot/proximitized-dot architecture: bulk normal dots couple to superconducting dots on both sides, whereas the first and last normal dots couple directly to only one superconducting neighbor. This boundary environment allows trivial states to split from the bulk and localize near the ends even in a topologically trivial phase. A plausible implication is that end localization, near-zero parity splitting, and even a finite excitation gap are insufficient by themselves to certify the topological origin of a PMM (Luethi et al., 2024).

This point extends directly to transport. The minimal device can display the experimentally used PMM signature—a crossing at $$H_{\mathrm{eff}} = \varepsilon_L c_L^\dagger c_L + \varepsilon_R c_R^\dagger c_R + t\, c_L^\dagger c_R + t\, c_R^\dagger c_L + \Delta\, c_L^\dagger c_R^\dagger + \Delta\, c_R c_L ,$$5 inside the threshold region that turns into an anticrossing upon detuning $$H_{\mathrm{eff}} = \varepsilon_L c_L^\dagger c_L + \varepsilon_R c_R^\dagger c_R + t\, c_L^\dagger c_R + t\, c_R^\dagger c_L + \Delta\, c_L^\dagger c_R^\dagger + \Delta\, c_R c_L ,$$6, together with a sign change in the nonlocal conductance—both for true PMMs and for false PMMs. There is therefore no clear conductance signature in the minimal Kitaev chain that distinguishes a PMM evolving into a topological long-chain Majorana bound state from one evolving into a trivial boundary-localized state (Luethi et al., 2024).

6. Control, readout, and qubit-oriented architectures

Because minimal Kitaev chains are fine-tuned devices, automated characterization and control have become a distinct subfield. One approach trained a convolutional conditional generative adversarial neural network on simulated differential-conductance data from the two-site Hamiltonian and applied it to experimental conductance maps of a two-dot minimal chain. The model inferred whether measurements were in the ECT- or CAR-dominated regime with an average success probability of $$H_{\mathrm{eff}} = \varepsilon_L c_L^\dagger c_L + \varepsilon_R c_R^\dagger c_R + t\, c_L^\dagger c_R + t\, c_R^\dagger c_L + \Delta\, c_L^\dagger c_R^\dagger + \Delta\, c_R c_L ,$$7, providing a rapid Hamiltonian-learning route to the sweet spot $$H_{\mathrm{eff}} = \varepsilon_L c_L^\dagger c_L + \varepsilon_R c_R^\dagger c_R + t\, c_L^\dagger c_R + t\, c_R^\dagger c_L + \Delta\, c_L^\dagger c_R^\dagger + \Delta\, c_R c_L ,$$8 (Koch et al., 2023).

A later cross-platform transfer-learning scheme used theory data, retrained on a two-dimensional electron-gas realization, and then deployed on a nanowire realization of the same two-site effective model. The autonomous loop predicted the ratio $$H_{\mathrm{eff}} = \varepsilon_L c_L^\dagger c_L + \varepsilon_R c_R^\dagger c_R + t\, c_L^\dagger c_R + t\, c_R^\dagger c_L + \Delta\, c_L^\dagger c_R^\dagger + \Delta\, c_R c_L ,$$9 from conductance images and tuned the hybrid gate toward the sweet spot. It converged within $t$0 mV of a verified sweet spot in $t$1 of attempts and within $t$2 mV in $t$3 of cases, typically finding a sweet spot in $t$4 minutes or less (Driel et al., 2024).

Minimal chains have also been embedded in circuit-QED architectures. A proposed Kitaev-transmon or “Kitmon” consists of a Josephson junction between two double-quantum-dot minimal chains. In the deep transmon regime, the $t$5-Josephson effect of the junction produces microwave transitions whose frequencies agree with analytical expressions in terms of the double-dot parameters only, and the microwave response can be used to extract the Majorana polarization in the junction (Pino et al., 2023).

7. Open-system, Josephson, nonequilibrium, and entanglement extensions

Once minimal Kitaev chains are treated as open quantum devices rather than isolated Hamiltonian systems, non-Hermitian effects become intrinsic rather than incidental. For a double quantum dot coupled through a superconductor and also to normal reservoirs, the effective non-Hermitian self-energy broadens the isolated PMM point into zero-real-energy lines bounded by exceptional points. In this setting, non-Hermiticity stabilizes poor man’s Majorana-like zero modes over a much broader parameter range than in the Hermitian problem (Cayao et al., 2024).

The same logic extends to Josephson structures. A Josephson junction formed by two minimal Kitaev chains and coupled to normal reservoirs has a complex Andreev spectrum that hosts phase-controlled second-order exceptional points, exceptional lines, and protected two-dimensional zero-real-energy areas. Depending on how the non-Hermiticity is distributed, these Andreev exceptional points can occur at zero or finite energies and connect stable energy lines protected by non-Hermitian topology (Cayao et al., 22 Jun 2026).

Minimal-chain Josephson networks also exhibit phase phenomena without long-chain analogues. In a device composed of three laterally coupled minimal Kitaev chains, an imbalance $t$6 in the middle chain produces an asymmetric $t$7-periodic Andreev spectrum and a nonlocal Josephson diode effect. The effect requires breaking local time-reversal and local charge-conjugation symmetries, the latter identified as unique to minimal Kitaev chains, and the reported diode efficiencies can exceed $t$8 (Cayao et al., 29 Nov 2025).

Nonequilibrium fluctuations and entanglement dynamics provide complementary perspectives. In a voltage-biased double-dot minimal chain, the differential effective charge $t$9 takes the value $\Delta$0 only in a very narrow vicinity of $\Delta$1, whereas $\Delta$2 over almost the whole sweet-spot region and even in the high-voltage tails, making $\Delta$3 the robust fluctuation signature of PMM-dominated transport (Smirnov, 26 Mar 2026). In closed-system dynamics, two-site chains generate Bell-like states from separable inputs and three-site chains generate GHZ-type states and imperfect W-type states; at the three-site sweet spot, edge PMMMs suppress edge concurrence, while finite detuning restores it, showing that Majorana localization and conventional bipartite entanglement need not coincide (Vimal et al., 23 Jul 2025).

Minimal Kitaev chains therefore occupy a distinctive place between toy models and extended topological matter. They are the smallest systems that realize the algebraic structure of the Kitaev chain, the smallest experimentally tractable platforms for PMMs, and the smallest networks in which one can study the separation between local spectroscopy, nonlocal parity structure, and true topological origin. Their chief limitation is equally clear: short-chain Majorana signatures do not, by themselves, establish topological superconductivity. Their chief strength is control: few-site architectures expose hopping, pairing, phase, boundary conditions, reservoir coupling, and probe geometry at a level of microscopic tunability that is inaccessible in most longer one-dimensional platforms (Luethi et al., 2024).