We introduce a versatile and scalable framework for constructing effective models of superconducting (SC) nanostructures described by the generalized SC Anderson impurity model with multiple quantum dots and leads. Our chain expansion (ChE) method maps each SC lead onto a finite tight-binding chain with parameters obtained from Pad\'e approximants of the tunneling self-energy. We provide an explicit algorithm for the general case as well as simple analytical expressions for the chain parameters in the wide-band and infinite-chain limits. This mapping preserves low-energy physics while enabling efficient simulations: short chains are tractable using exact diagonalization, and longer ones are handled with density matrix renormalization group methods. The approach remains reliable and computationally efficient across diverse geometries, both in and out of equilibrium. We use ChE to map the ground-state phase diagrams of double, triple, and quadruple quantum dots coupled to a single SC lead. While half-filled symmetric systems show similar overall diagrams, the particular phases differ substantially with the dot number. Here, large parameter regions are entirely missed by the widely used zero-bandwidth approximation but are captured by ChE. Away from half-filling, additional dots markedly increase diagram complexity, producing a rich variety of stable phases. These results demonstrate ChE as a fast, accurate, and systematically improvable tool for exploring complex SC nanostructures.
We introduce a versatile and scalable framework for constructing effective models of superconducting (SC) nanostructures described by the generalized SC Anderson impurity model with multiple quantum dots and leads. Our chain expansion (ChE) method maps each SC lead onto a finite tight-binding chain with parameters obtained from Pad'e approximants of the tunneling self-energy. We provide an explicit algorithm for the general case as well as simple analytical expressions for the chain parameters in the wide-band and infinite-chain limits. This mapping preserves low-energy physics while enabling efficient simulations: short chains are tractable using exact diagonalization, and longer ones are handled with density matrix renormalization group methods. The approach remains reliable and computationally efficient across diverse geometries, both in and out of equilibrium. We use ChE to map the ground-state phase diagrams of double, triple, and quadruple quantum dots coupled to a single SC lead. While half-filled symmetric systems show similar overall diagrams, the particular phases differ substantially with the dot number. Here, large parameter regions are entirely missed by the widely used zero-bandwidth approximation but are captured by ChE. Away from half-filling, additional dots markedly increase diagram complexity, producing a rich variety of stable phases. These results demonstrate ChE as a fast, accurate, and systematically improvable tool for exploring complex SC nanostructures.
Systems of quantum dots coupled to superconducting (SC) leads designed for practical applications are becoming increasingly complicated, ren- dering traditional analytical and numerical methods insufficient. Therefore, we tested new Neural Network Quantum States (NNQS) methods based on various network architectures and on variational Monte Carlo. To simplify their appli- cation, a new Chain Expansion method developed in our research group, which constructs effective models, was used. We applied these methods to systems of one to three quantum dots coupled to one to three SC leads, benchmarking them against conventional techniques and analyzing their phase diagrams. While at least one NNQS variant typically yielded correct predictions, no single method performed consistently well across all studied configurations. Thus, we conclude that current NNQS methods still face significant limitations when systematically studying complex hybrid su- perconducting nanostructures. This thesis also introduces a focused analytical study of effective models for fundamental quantum dot configurations. 1
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, enabling an efficient treatment with standard fermionic neural-network quantum-state variational Monte Carlo methods. We show that the excitation gap can close under a particular high-symmetry condition, which, in finite non-interacting systems, corresponds to crossings between singlet ground states of different character. At this high-symmetry condition the model can be mapped to a standard Hubbard model in grand-canonical ensemble with fixed chemical potential away from half-filling. 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 related 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. These results connect the physics of superconducting nanostructures with correlated lattice models and show that standard fermionic neural quantum states can capture the relevant interacting regimes.
Generalized quantum impurity models—which feature a few localized and strongly correlated degrees of freedom coupled to itinerant conduction electrons—describe diverse physical systems, from magnetic moments in metals to nanoelectronics quantum devices such as quantum dots or single-molecule transistors. Correlated materials can also be understood as self-consistent impurity models through dynamical mean-field theory. Accurate simulation of such models is challenging, especially at low temperatures, due to many-body effects from electronic interactions, resulting in strong renormalization. In particular, the interplay between local impurity complexity and Kondo physics is highly nontrivial. A common approach, which we further develop in this work, is to consider instead a simpler effective impurity model that still captures the low-energy physics of interest. The mapping from a bare to an effective model is typically done perturbatively, but even this can be difficult for complex systems, and the resulting effective model parameters can nevertheless be quite inaccurate. Here we develop a nonperturbative, unsupervised machine learning approach to systematically obtain low-energy effective impurity-type models, based on the renormalization-group framework. The method is shown to be general and flexible, as well as accurate and systematically improvable. We benchmark the method against exact results for the Anderson impurity model, and we provide an outlook for more complex models beyond the reach of existing methods. Published by the American Physical Society 2024
The case of a quantum dot connected to two superconducting leads is studied by using the narrow-band limit to describe the superconducting degrees of freedom. The model provides a simple theoretical framework, almost analytical, to analyze the interplay between the Kondo effect, superconductivity, and finite temperature. In the quantum dot Kondo regime, the model is completely characterized by the ratio $\ensuremath{\Delta}/J$, with $\ensuremath{\Delta}$ the superconducting gap and J an effective antiferromagnetic exchange coupling between the dot and the leads. The model allows us to calculate, at any temperature T, the equilibrium Josephson current through the dot in a very straightforward way as a function of $\ensuremath{\Delta}/J$. The behavior of the current allows us to distinguish the four types of hybrid junctions: 0, ${0}^{\ensuremath{'}},{\ensuremath{\pi}}^{\ensuremath{'}}$, and $\ensuremath{\pi}$. The presence of the 0- and ${0}^{\ensuremath{'}}$-junction configurations are intrinsically linked to the Kondo effect in the quantum dot, while the $\ensuremath{\pi}$- and ${\ensuremath{\pi}}^{\ensuremath{'}}$-junction configurations are driven by the superconductivity in the leads. The Josephson critical current has a non-monotonic behavior with temperature, that may be used for the experimental characterization of the fundamental $0\ensuremath{-}\ensuremath{\pi}$ transition. The model allows us to obtain easily a phase diagram $\ensuremath{\Delta}/J$ vs temperature, from where we can obtain an overall picture on the stability of the different types of junctions. From the explicit analytical expressions for the ground-state, low-energy excitations, free energy, and Josephson current, it is easy to understand the physical nature of the main features of the critical current and the phase diagram. The results, obtained with a minimum of numerical effort, are in a good qualitative agreement with more demanding calculational approaches aimed to solve the full model.
This work focuses on the time evolution of a quantum dot coupled to superconduct- ing leads following sudden parameter changes (quenches). The system is modeled using a superconducting Anderson impurity framework reformulated through chain expansion. We simulate the system dynamics and identify three regimes: full relaxation, persistent oscillations, and oscillations centered around the expected equilibrium value. We show that relaxation strongly depends on the local density of states on the quantum dot corre- sponding to the post-quench system configuration. While relaxation is suppressed within the superconducting gap, it can reappear at specific energies due to Andreev bound states. 1
The Neural-Network Quantum States (NNQS) method is rapidly emerging as a powerful tool for investigating \nquantum many-body physics. By combining variational Monte Carlo techniques with neural network-based \nvariational functions, this approach leverages the remarkable advancements in deep learning achieved in \nrecent years. While substantial progress has been made in simulating magnetic systems on lattices, simple \nmolecules, and even continuous systems, the ab initio simulation of complex strongly correlated electron \nsystems continues to pose significant challenges. With the help of the generalized atomic limit (GAL) – a \nrecently developed model describing a system of quantum dots coupled to a superconducting lead – we \nattempt to show the efficiency and accuracy of NNQS, mainly the Restricted Boltzmann Machine, by comparing \nthe acquired results to other available methods such as exact diagonalization. The simultaneous study of the \nsystem’s properties such as the energy spectrum and quantum phase transitions could bring advancements \nin electronics, sensors or the design of high-quality qubits used in quantum computers.