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arxiv: 2606.04608 · v1 · pith:Z6QOMYL6new · submitted 2026-06-03 · ❄️ cond-mat.str-el · cond-mat.mes-hall

Correlated States in Quantum Dot Clusters Coupled to a Common Superconductor

Martin \v{Z}onda , Jakub R\k{e}kas , Tobi\'a\v{s} Pol\'a\v{c}ek , Jana Kodrlov\'a , Vladislav Pokorn\'y , Martin Fri\'ak This is my paper

Pith reviewed 2026-06-28 04:38 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.mes-hall
keywords quantum dot clusterssuperconducting nanostructurescanonical transformationneural quantum statesHeisenberg modelsinglet-doublet transitionstriplet ground statescorrelated regimes
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The pith

A canonical transformation maps quantum dot clusters on a superconductor to a particle-number-conserving model whose three interaction regimes are accessible to neural quantum states.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors apply a canonical transformation to an effective model of regular quantum dot clusters coupled to a common superconductor, converting the problem into a particle-number-conserving form. This step lets them combine exact diagonalization, density-matrix renormalization group, and fermionic neural-network variational Monte Carlo to locate three regimes: a trivial superconducting singlet phase, a strongly correlated regime that reduces to an effective Heisenberg model, and an intermediate critical regime. One-dimensional chains in the intermediate regime undergo a sequence of singlet-doublet transitions and become gapless in the thermodynamic limit, while two-dimensional clusters instead stabilize triplet ground states. The calculations demonstrate that standard neural quantum states suffice for these superconducting nanostructures.

Core claim

By applying a canonical transformation the system is rewritten in a particle-number-conserving representation that is directly treatable by fermionic neural quantum states. Exact, DMRG, and variational Monte Carlo calculations then identify three regimes: a trivial superconducting singlet phase, a strongly correlated regime connected to an effective Heisenberg model, and a critical intermediate regime. In one dimension the intermediate regime shows singlet-doublet transitions and becomes gapless for large systems even at finite Coulomb interaction; in two dimensions the same regime supports robust triplet ground states.

What carries the argument

The canonical transformation that converts the original model into a particle-number-conserving representation, allowing direct application of standard fermionic neural-network variational Monte Carlo.

If this is right

  • The superconducting gap closes at a high-symmetry point that marks crossings between singlet ground states of different character in finite non-interacting systems.
  • One-dimensional systems in the intermediate regime undergo singlet-doublet transitions and become gapless in the thermodynamic limit even with finite Coulomb interaction.
  • Two-dimensional clusters in the intermediate regime support robust triplet ground states.
  • Standard fermionic neural quantum states provide an efficient route to correlated superconducting nanostructures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same mapping and neural-state approach can be applied to larger clusters that remain out of reach for exact methods.
  • Geometry dependence between one and two dimensions implies that device layout can be used to select between gapless and gapped correlated phases.
  • The reduction to an effective Heisenberg model in the strongly correlated regime opens the use of established spin-model techniques for predicting properties of these hybrid systems.

Load-bearing premise

The canonical transformation that produces the particle-number-conserving representation remains valid and faithful once finite Coulomb interactions are present.

What would settle it

Numerical or experimental detection of whether the superconducting gap closes exactly at the high-symmetry point and whether one-dimensional chains exhibit the predicted sequence of singlet-doublet transitions that close the gap in the thermodynamic limit.

Figures

Figures reproduced from arXiv: 2606.04608 by Jakub R\k{e}kas, Jana Kodrlov\'a, Martin Fri\'ak, Martin \v{Z}onda, Tobi\'a\v{s} Pol\'a\v{c}ek, Vladislav Pokorn\'y.

Figure 1
Figure 1. Figure 1: Examples of the band structure and the corresponding single-particle den [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Evolution of the BdG single-particle spectrum for a finite noninteracting [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Local z-moment S (2) z (a1) and mean total electron occupation 〈N〉 (a2) in the ζ-ϵ plane within the HSC regime for a noninteracting chain of length L = 9. Panels (b1) and (b2) show the corresponding quantities away from HSC at ϵ = 0, where the evolution is smooth up to special points ζd (L,n) at t = 0. Panels (c1) and (c2) illustrate the spatial distribution of the local moment and particle occupation for … view at source ↗
Figure 4
Figure 4. Figure 4: Phase diagrams in the ζ–U plane for a one-dimensional chain with an even (first and second row) and odd (third and fourth row) number of sites, ϵ = 0 and t = 0. (a) and (c): Total spin of the many-body ground state. (b) and (d): Average double occupancy D/L. Dashed black lines in (a1) are analytical phase boundaries calculated in App. C. results obtained with the Lanczos method, whereas the remaining curve… view at source ↗
Figure 5
Figure 5. Figure 5: Evolution of the ground-state total spin (first row) and double occupancy [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Finite-size scaling of the entanglement entropy (first row), charge gap [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Phase diagrams in the ζ–U plane for various two-dimensional systems with geometries illustrated in the left corners of particular columns, at trivial HSC points ϵ = 0, t = 0 (a1)-(a4) and away of this point (a5). (a) First row shows the total spin of the many-body ground state. (b) Second row average double occupancy D/L. Blue crosses in panel (a3) indicate the positions of the benchmark calculations liste… view at source ↗
Figure 8
Figure 8. Figure 8: Detail of the Phase diagrams in the ζ–U plane in the vicinity of the critical point (ζ = 0,U = 2Γ ) for various two-dimensional systems, at trivial HSC ϵ = 0, t = 0 (a)-(c) and away from it ϵ = 0, t = 0.02Γ (d). First row: Total spin of the many-body ground state. Second row: average double occupancy D/L. regions. Similarly to the one-dimensional case, and as discussed in detail in App. D, the low-ζ regime… view at source ↗
Figure 9
Figure 9. Figure 9: Evolution of the ground-state total spin (first row), double occupancy per [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Analysis of the second regime at the HSC point ( [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of DMRG results (red) with the effective Heisenberg-model [PITH_FULL_IMAGE:figures/full_fig_p030_11.png] view at source ↗
read the original abstract

We study an effective model of regular quantum dot clusters coupled to a common superconductor. By applying a canonical transformation, we map the system onto a particle-number-conserving representation, making it directly accessible to standard fermionic neural-network quantum-state variational Monte Carlo methods. We show that the superconducting gap closes at a particular high-symmetry point, which, in finite non-interacting systems, corresponds to crossings between singlet ground states of different character. Combining exact methods, density matrix renormalization group, and neural quantum-state variational Monte Carlo calculations, we identify three distinct interacting regimes: a trivial superconducting singlet phase, a strongly correlated regime connected to an effective Heisenberg model, and a critical intermediate regime with qualitatively different behavior in one and two dimensions. In one-dimensional systems, the intermediate regime exhibits a sequence of singlet-doublet transitions and becomes gapless in the thermodynamic limit even for finite Coulomb interaction. In two-dimensional clusters, we find robust triplet ground states. Furthermore, our results demonstrate that relatively standard fermionic neural quantum states provide an efficient approach for correlated superconducting nanostructures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript studies an effective model of regular quantum dot clusters coupled to a common superconductor. A canonical transformation is used to map the system to a particle-number-conserving representation, enabling application of exact methods, DMRG, and fermionic neural quantum-state variational Monte Carlo. The superconducting gap is shown to close at a high-symmetry point. Three interacting regimes are identified: a trivial superconducting singlet phase, a strongly correlated regime connected to an effective Heisenberg model, and a critical intermediate regime. In 1D the intermediate regime exhibits singlet-doublet transitions and becomes gapless in the thermodynamic limit even at finite U; in 2D, robust triplet ground states appear. The work also demonstrates the efficiency of standard fermionic NQS for correlated superconducting nanostructures.

Significance. If the mapping is exact, the paper provides a concrete route to apply NQS-VMC to superconducting nanostructures and reports dimension-dependent critical behavior of potential interest to the cond-mat.str-el community. The multi-method approach (exact + DMRG + NQS) is a positive feature when convergence details are supplied.

major comments (2)
  1. [Section describing the canonical transformation (likely §II or §III)] The canonical transformation that produces the particle-number-conserving Hamiltonian is load-bearing for every subsequent claim. The manuscript must explicitly demonstrate (or derive the conditions under which) the generator of the transformation commutes with the finite-U Coulomb terms; otherwise additional interaction-induced terms appear and the spectrum of the mapped model is not guaranteed to match the original. This issue is not resolved by the abstract statement that the mapping is applied to the effective model.
  2. [Numerical results and figures (e.g., §IV and associated figures showing gap vs. parameters)] Claims of three distinct regimes and the thermodynamic-limit statements in 1D rest on numerical identification of gap closings, singlet-doublet crossings, and ground-state character. The manuscript does not supply sufficient information on DMRG bond-dimension convergence, NQS optimization stability, or finite-size scaling to rule out that the reported distinctions are artifacts of accessible system sizes.
minor comments (2)
  1. [Model definition section] Notation for the transformed operators and the effective Heisenberg coupling should be introduced with an explicit equation reference to avoid ambiguity when comparing to the original model.
  2. [Figure captions] Figure captions would benefit from explicit statements of the system sizes and method parameters used for each data set.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major point below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [Section describing the canonical transformation (likely §II or §III)] The canonical transformation that produces the particle-number-conserving Hamiltonian is load-bearing for every subsequent claim. The manuscript must explicitly demonstrate (or derive the conditions under which) the generator of the transformation commutes with the finite-U Coulomb terms; otherwise additional interaction-induced terms appear and the spectrum of the mapped model is not guaranteed to match the original. This issue is not resolved by the abstract statement that the mapping is applied to the effective model.

    Authors: We agree that an explicit demonstration is required. The transformation is performed on the effective model prior to including the finite-U terms, and the generator commutes with the Coulomb interaction by construction in this setup. In the revised manuscript we will add a dedicated appendix deriving the commutation relations explicitly, confirming that no additional interaction-induced terms are generated and that the spectrum of the mapped model matches the original. revision: yes

  2. Referee: [Numerical results and figures (e.g., §IV and associated figures showing gap vs. parameters)] Claims of three distinct regimes and the thermodynamic-limit statements in 1D rest on numerical identification of gap closings, singlet-doublet crossings, and ground-state character. The manuscript does not supply sufficient information on DMRG bond-dimension convergence, NQS optimization stability, or finite-size scaling to rule out that the reported distinctions are artifacts of accessible system sizes.

    Authors: We acknowledge that additional convergence and scaling details are needed to support the regime identifications and thermodynamic-limit claims. In the revised version we will expand the numerical sections and supplementary material with explicit DMRG bond-dimension convergence data, NQS optimization stability checks (including multiple random seeds and learning-rate schedules), and finite-size scaling analysis for the gap closings and singlet-doublet transitions. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation applies a standard canonical transformation to reach a number-conserving form, followed by independent numerical methods (exact diagonalization, DMRG, NQS-VMC) whose outputs are not forced by the transformation itself or by any fitted parameters. No self-citations are load-bearing for the central claims, no predictions reduce to inputs by construction, and the three-regime identification follows from direct computation on the mapped Hamiltonian without definitional equivalence or ansatz smuggling. The paper is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are stated. The canonical transformation itself is treated as a standard tool whose validity for the interacting case is assumed without further derivation visible here.

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Reference graph

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