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arxiv: 2506.23636 · v2 · submitted 2025-06-30 · ❄️ cond-mat.mes-hall · cond-mat.str-el

Quantum phase transition in a double quantum dot Josephson junction driven by electron-electron interactions

Cong Li , Yiyan Wang , Bing Dong This is my paper

Pith reviewed 2026-05-19 07:53 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.str-el
keywords quantum phase transitiondouble quantum dotJosephson junctionelectron-electron interactionnon-local magnetizationferromagnetic-antiferromagnetic transitionmagnetic field control
0
0 comments X

The pith

Electron interactions in two quantum dots coupled to superconducting leads produce a sequence of distinct phase transitions tunable by interaction strengths and magnetic field.

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

The work models a Josephson junction formed by two quantum dots, one coupled to superconducting leads, and uses exact diagonalization on a simplified BCS description to map out how on-site repulsion and inter-dot coupling alter the ground state. An initial transition appears when the interaction on the second dot is increased while the first remains non-interacting; further tuning of the first dot's interaction and then the coupling between dots each trigger additional changes. Parallel magnetic field is shown to switch the system reversibly between ferromagnetic and antiferromagnetic order, while weak field applied to only one dot generates magnetization on the other dot whose direction can be flipped by changing the second dot's interaction.

Core claim

Exact diagonalization of the surrogate BCS model reveals that the double quantum dot Josephson junction undergoes three successive interaction-driven phase transitions: the first when U2 is varied at U1 = 0, the second when U1 is subsequently turned on, and the third when inter-dot coupling is adjusted. Parallel magnetic field induces reversible ferromagnetic-antiferromagnetic transitions, and weak field on QD1 produces non-local magnetization whose orientation is controlled by the value of U2 on the second dot.

What carries the argument

Surrogate BCS model with discrete energy levels solved by exact diagonalization, which computes ground-state properties including magnetization and phase boundaries as functions of interaction parameters and magnetic field.

If this is right

  • Varying the interaction strength on QD2 alone induces a phase transition even when QD1 remains non-interacting.
  • Subsequent adjustment of the interaction on QD1 produces a second distinct phase transition.
  • Modulation of the inter-dot coupling strength triggers a third phase transition.
  • Application of a parallel magnetic field drives reversible transitions between ferromagnetic and antiferromagnetic states under appropriate conditions.
  • Weak magnetic field applied only to QD1 generates non-local magnetization whose orientation can be reversed by changing the interaction strength U2 on QD2.

Where Pith is reading between the lines

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

  • The non-local magnetization effect implies that interaction tuning on one dot could remotely control spin properties on a distant dot without direct field application to the second site.
  • The sequence of interaction-driven transitions suggests a route to switch between ordered states using gate voltages that control charging energies rather than external magnets.
  • Similar interaction tuning in extended arrays of quantum dots might produce richer phase diagrams with additional controllable transitions.

Load-bearing premise

The discrete-level BCS surrogate model accurately represents the low-energy physics of the actual double quantum dot Josephson junction without introducing artifacts from level discretization or the mean-field treatment of the leads.

What would settle it

Fabrication and measurement of a real double quantum dot Josephson junction showing that the critical values of U1, U2, or magnetic field at which magnetization or current-phase relation changes do not match the locations predicted by the discrete-level calculations.

Figures

Figures reproduced from arXiv: 2506.23636 by Bing Dong, Cong Li, Yiyan Wang.

Figure 1
Figure 1. Figure 1: (Color online) Schematic diagram of a QD connected [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: (Colour online) System properties analysis: (a) en [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (Colour online) System properties analysis: (a) en [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (Colour online) (a) The interdot spin correlation [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (Colour online) The entropy with temperature is 0.005, [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: (Colour online) (a) ⟨S⃗1·S⃗2⟩ as a function of U1 for various external magnetic fields. (b) The spin correlation as a function of external parallel magnetic field B = −B1 = −B2 with U1 = 10∆. (c) IJ /I0 as a function of external parallel magnetic field when phase difference between two SC leads is 0.3π. The interaction of QD2 is U2 = 10∆. The other parameters are given the same as in [PITH_FULL_IMAGE:figu… view at source ↗
Figure 9
Figure 9. Figure 9: (Color online) Schematic diagram of first-order elec [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: (Colour online) (a) The magnetization intensity of [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
read the original abstract

In this work, we employ a surrogate BCS model with discrete energy levels to investigate a hybrid system comprising two quantum dots (QD1 and QD2), where QD1 is tunnel-coupled to two superconducting leads. Through exact diagonalization of this system, we obtain numerically exact solutions that enable rigorous computation of key physical quantities. Our analysis reveals a rich phase diagram featuring multiple controllable phase transitions mediated by quantum dot interactions. Specifically, the system first undergoes an initial phase transition when tuning QD2's interaction strength while maintaining QD1 in the non-interacting regime. Subsequent adjustment of QD1's interaction induces a secondary phase transition, followed by a third transition arising from inter-dot coupling modulation. Furthermore, we demonstrate that parallel magnetic field application can drive reversible ferromagnetic-antiferromagnetic phase transitions under specific parameter conditions. Finally, we report the emergence of non-local magnetization phenomena when subjecting QD1 to weak magnetic fields. And our results demonstrate that the orientation of nonlocal magnetization can be precisely manipulated through systematic adjustment of the on-site interaction strength $U_2$ in QD2.

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 investigates a double quantum dot Josephson junction using a surrogate BCS Hamiltonian with discrete energy levels for the superconducting leads. Exact diagonalization is employed to compute the phase diagram as a function of on-site interactions U1 and U2, inter-dot coupling, and parallel magnetic field. The central claims are an initial phase transition driven by tuning U2 (with QD1 non-interacting), a secondary transition upon varying U1, a tertiary transition from inter-dot coupling, reversible ferromagnetic-antiferromagnetic transitions under magnetic field, and the emergence of tunable non-local magnetization on QD1 controlled by U2.

Significance. If the discrete-level model faithfully reproduces the low-energy Andreev physics of the continuum limit, the results demonstrate multiple interaction-tunable phase transitions and field-reversible spin configurations in a hybrid superconducting system. The use of exact diagonalization for the finite model is a clear strength, providing numerically exact spectra and magnetization values without mean-field approximations on the dots themselves. This could inform designs for interaction-controlled Andreev qubits or spin filters, provided the discretization artifacts are controlled.

major comments (2)
  1. [Model and Methods] The surrogate BCS model with a finite number of discrete levels for the leads is central to all reported phase transitions and non-local magnetization. The manuscript must specify the number of levels retained and include explicit convergence tests (e.g., phase boundaries vs. level number) to demonstrate that the initial, secondary, and tertiary transitions remain stable in the continuum limit; without this, discretization could shift Andreev-state crossings and alter the reported diagram.
  2. [Results] The identification of the three successive phase transitions and the reversible FM-AFM switching relies on changes in ground-state magnetization or parity. The results section should show the full energy spectrum or order-parameter curves versus U2, U1, and inter-dot coupling (with fixed other parameters) to make the transition points unambiguous and to allow readers to assess their sharpness.
minor comments (2)
  1. [Model] Notation for the tunnel couplings to the two leads and the inter-dot hopping should be defined explicitly in the Hamiltonian equation to avoid ambiguity when comparing to experimental double-QD devices.
  2. [Abstract and Results] The abstract states that non-local magnetization orientation is manipulated by U2, but the corresponding figure or panel should be referenced directly in the text for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments, which will help strengthen the presentation of our results. We address each major comment below.

read point-by-point responses

  1. Referee: [Model and Methods] The surrogate BCS model with a finite number of discrete levels for the leads is central to all reported phase transitions and non-local magnetization. The manuscript must specify the number of levels retained and include explicit convergence tests (e.g., phase boundaries vs. level number) to demonstrate that the initial, secondary, and tertiary transitions remain stable in the continuum limit; without this, discretization could shift Andreev-state crossings and alter the reported diagram.

    Authors: We agree that explicit specification of the discretization and convergence tests are necessary to substantiate the robustness of the reported phase transitions. The original manuscript employed a finite but fixed number of discrete levels in the surrogate BCS Hamiltonian for each lead; however, the precise count and associated convergence data were not included. In the revised manuscript we will state the number of levels retained and add a dedicated subsection (or supplementary figure) that plots the locations of the initial, secondary, and tertiary transition points versus the number of retained levels. This will demonstrate that the phase boundaries remain stable once a sufficient number of levels is included, thereby confirming that the essential low-energy Andreev physics is captured. revision: yes

  2. Referee: [Results] The identification of the three successive phase transitions and the reversible FM-AFM switching relies on changes in ground-state magnetization or parity. The results section should show the full energy spectrum or order-parameter curves versus U2, U1, and inter-dot coupling (with fixed other parameters) to make the transition points unambiguous and to allow readers to assess their sharpness.

    Authors: We appreciate this recommendation to improve the transparency of the transition points. While the manuscript already identifies the transitions through jumps in ground-state magnetization and parity, we concur that additional curves would make the crossings unambiguous. In the revised version we will augment the results section with new figures that display the lowest few eigenenergies and the relevant order parameters (magnetization on each dot and parity) as continuous functions of U2 (at fixed U1=0), of U1 (at fixed U2), and of the inter-dot coupling (at fixed magnetic field). These plots will allow readers to directly inspect the sharpness and character of each transition. revision: yes

Circularity Check

0 steps flagged

No significant circularity: results follow from explicit Hamiltonian diagonalization

full rationale

The paper defines a surrogate BCS Hamiltonian with discrete levels for the leads, performs exact diagonalization to obtain the spectrum, and extracts phase transitions, magnetization, and non-local effects directly from the numerical eigenvalues and eigenvectors. No observable is defined in terms of itself, no parameter is fitted to the output data and then relabeled as a prediction, and no load-bearing step reduces to a self-citation or ansatz smuggled from prior work by the same authors. The derivation chain is therefore self-contained: model construction to numerical solution to reported quantities.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central claims rest on a discretized BCS approximation for the leads and on the assumption that the finite-level model remains representative when interaction strengths are varied over the reported ranges.

free parameters (3)
  • U1 (on-site interaction in QD1)
    Tuned parameter used to induce the secondary phase transition; its specific values are chosen to demonstrate the effect rather than derived from first principles.
  • U2 (on-site interaction in QD2)
    Tuned parameter used to induce the initial transition and to control non-local magnetization orientation.
  • inter-dot coupling strength
    Tuned parameter used to induce the third transition.
axioms (2)
  • domain assumption The superconducting leads can be represented by a surrogate BCS model with a finite number of discrete energy levels.
    This approximation is invoked to enable exact diagonalization of the full hybrid Hamiltonian.
  • domain assumption Electron-electron interactions are captured entirely by local on-site and inter-dot repulsion terms.
    Standard modeling choice for quantum-dot systems; no long-range or higher-order interaction terms are included.

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Reference graph

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