🔬 mesoscale physics

Exact fermionic matrix elements from the Bethe-ansatz solution of the Richardson-Gaudin model

This paper introduces an effective Hamiltonian description for mesoscopic superconducting systems based on the Richardson-Gaudin model, featuring a novel compact determinant formula for fermionic matrix elements that enables the direct calculation of single-electron tunneling and other observables in the canonical ensemble.

Original author: Peter D. Johannsen, Valerii K. Kozin, Jelena Klinovaja, Daniel Loss

Published 2026-11-03

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CC BY 4.0

Original author: Peter D. Johannsen, Valerii K. Kozin, Jelena Klinovaja, Daniel Loss

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of superconductors, electrons do not behave as solitary travelers. Instead, they pair up, forming a coordinated dance that allows them to flow without resistance. This phenomenon, known as superconductivity, is usually described by theories that work best when dealing with vast numbers of particles, where individual quirks are smoothed out by the sheer scale of the crowd. However, when scientists shrink these superconducting materials down to the size of a tiny grain, containing only a few dozen electrons, the old rules begin to falter. In these mesoscopic systems, the discrete nature of the electrons matters; the fact that there is an odd or even number of them can change the entire behavior of the material. This is a realm where the smooth, average descriptions of the past give way to a jagged, particle-by-particle reality, and understanding this transition is crucial for building the next generation of quantum computers, which rely on controlling these very small, very sensitive systems.

For decades, physicists have had a powerful mathematical tool to describe these tiny superconducting grains, a method called the Bethe ansatz. This approach allows researchers to find the exact energy states of the grain, treating every electron with perfect precision. However, a significant gap remained in this picture. While scientists could calculate the energy of the grain itself, they could not easily calculate what happens when a single electron tries to jump from the grain into a neighboring component, such as a quantum dot. This tunneling process is the bridge that connects the isolated grain to the rest of the world, yet the mathematical machinery used to describe the grain's internal states had no way to handle this single-electron exchange. Without a way to calculate these specific jumps, the exact solutions remained isolated, unable to describe the hybrid devices that are actually built in laboratories.

In a new study, researchers have bridged this gap by developing a compact mathematical formula that describes exactly how single electrons move between these states. They focused on a system where an ultrasmall superconducting grain is coupled to a single quantum dot, a setup that mimics the hybrid structures used in modern quantum devices. The team derived a new way to calculate the probability of an electron tunneling from the grain to the dot, or vice versa, using only the fundamental variables that define the grain's energy states. This breakthrough allows them to project the complex, exact solution of the grain onto a smaller, manageable set of states that still capture the essential physics. By doing so, they created an effective description of the entire system that retains the precision of the exact solution while remaining simple enough to analyze.

The researchers applied this new method to a model known as the superconducting Anderson impurity model, which describes a quantum dot attached to a superconducting grain. They found that the system exhibits a rich variety of behaviors depending on how the electrons are charged and how strongly they interact. When the charging energy is low, the system behaves like a superconductor, favoring an even number of electrons on the grain. However, as the charging energy increases, the system begins to behave more like a double quantum dot, where the electrons are more localized. The study revealed that the competition between the superconducting pairing force and the repulsive charging force creates a complex landscape of energy states. In particular, the team observed "subgap states," which are energy levels that exist within the superconducting gap, a region where no other states should exist. These states can switch between being a spin singlet, where the electrons pair up, and a spin doublet, where an unpaired electron remains, depending on the specific conditions of the system.

Beyond simply mapping out the energy levels, the researchers used their new formula to calculate the optical conductivity of the system. This measurement reveals how the system responds to light or an oscillating electric field, but with a crucial difference: it looks at excitations that keep the total number of electrons constant. This is distinct from the usual measurements that involve adding or removing electrons. The results showed that the optical response is highly sensitive to the parity of the system, meaning whether the total number of electrons is even or odd. The team found that as they tuned the interaction strength, the optical conductivity could change abruptly, signaling a shift in the ground state of the system. These transitions are invisible to standard measurements that allow the electron number to fluctuate, highlighting the unique insights gained by working within a fixed-particle-number framework.

The study also explored what happens when the system is placed in a magnetic flux loop, a setup where the grain and dot are connected at two points, forming a ring. In this configuration, the electrons can travel along two different paths, and the magnetic field passing through the ring influences how these paths interfere with each other. The researchers found that the energy of the system oscillates as the magnetic flux changes, a phenomenon known as the Aharonov-Bohm effect. However, the pattern of these oscillations depends critically on whether the total number of electrons is even or odd. For an odd number of electrons, the energy minimum shifts to a fractional value of the magnetic flux, whereas for an even number, it stays at integer values. This difference creates a distinct signature in the electrical current flowing through the device, which can be tuned by adjusting the system's parameters. The ability to predict these flux-tunable spectra and the associated current patterns provides a new tool for designing quantum devices that can be controlled with magnetic fields.

The significance of this work lies in its ability to connect the abstract world of exact mathematical solutions with the concrete reality of experimental devices. By providing a way to calculate the matrix elements for single-electron tunneling, the researchers have unlocked the ability to use the exact Bethe-ansatz solution for hybrid systems that were previously too complex to solve. This approach is not just a theoretical exercise; it offers a practical method for simulating the behavior of mesoscopic superconducting devices with a level of detail that was previously unattainable. The method is efficient enough to handle the necessary calculations, allowing for the exploration of parameter spaces that would be impossible with other techniques. It captures the subtle interplay between superconductivity, charging effects, and local magnetic moments, providing a clear picture of how these forces compete and cooperate in the nanoscale world.

Ultimately, this research demonstrates that integrability, a property usually reserved for idealized, isolated systems, can be leveraged to understand real-world devices where integrability is broken by external connections. The new framework allows scientists to treat the superconducting grain as a precise, exact component within a larger, more complex circuit. This opens the door to studying more elaborate setups, such as systems with multiple quantum dots or networks of superconducting grains, while maintaining a microscopic, number-conserving description. The findings suggest that the rich physics of these tiny systems, including the even-odd effects and the formation of subgap states, can be harnessed and controlled with greater precision. As the field of quantum computing continues to evolve, the ability to accurately model these hybrid systems will be essential for designing devices that can reliably manipulate quantum information. The work stands as a testament to the power of combining exact mathematical solutions with practical physical insights to unravel the complexities of the quantum world.

Technical Summary: Exact Fermionic Matrix Elements from the Bethe-ansatz Solution of the Richardson–Gaudin Model

Problem Statement
Mesoscopic superconducting systems, such as ultrasmall superconducting grains coupled to quantum dots, exhibit rich physics including the competition between superconducting pairing and local-moment formation, fermionic parity effects, and Andreev bound states. While the reduced Bardeen–Cooper–Schrieffer (BCS) Hamiltonian describing the superconducting grain is integrable via the Richardson–Gaudin (RG) Bethe-ansatz solution, typical experimental devices are hybrid structures where single-electron tunneling breaks this integrability. Furthermore, in the regime of ultrasmall grains where the superconducting gap is comparable to the single-particle level spacing, the grand-canonical mean-field BCS theory becomes inadequate. A rigorous description requires the canonical ensemble (fixed particle number), where charge-conserving methods like Density Matrix Renormalization Group (DMRG) are often used but can be computationally expensive for extensive parameter sweeps. A key challenge has been the lack of exact, finite-size expressions for the matrix elements of single-electron creation and annihilation operators between arbitrary RG eigenstates, which are necessary to project the full non-integrable Hamiltonian onto a basis of exact grain eigenstates.

Methodology
The authors introduce an effective Hamiltonian description for hybrid grain-dot systems by projecting the full Hamiltonian onto a truncated basis of exact Richardson–Gaudin eigenstates. The core of their approach relies on the integrability of the reduced BCS Hamiltonian, which allows the grain's eigenstates to be parametrized by rapidities or, more advantageously for numerical stability, by Eigenvalue-Based Variables (EBVs).

EBV Formulation: The authors utilize conserved charges $\hat{Q}_m$ whose eigenvalues $\Lambda_m$ (the EBVs) satisfy a set of coupled equations (Eq. 5) that are numerically better behaved than the standard Richardson equations. These EBVs fully determine the energy and wave functions of the system.
State Selection: To manage the exponential growth of the Hilbert space, the authors employ a low-energy projection. They define an effective excitation degree $k_{\text{eff}}$ that accounts for both pair excitations and blocked levels (unpaired electrons). The Hamiltonian is projected onto the subspace where $k_{\text{eff}} \leq 1$ , reducing the basis size from exponential to quadratic in the system size while retaining the essential low-energy physics.
Exact Matrix Elements: The central technical achievement is the derivation of a compact, exact determinant formula for the matrix elements of fermionic creation ( $\hat{c}^\dagger$ ) and annihilation ( $\hat{c}$ ) operators between arbitrary RG eigenstates (Eq. 14). Unlike previous results limited to overlaps or specific form factors, this formula expresses these matrix elements entirely in terms of EBVs, allowing for the direct calculation of single-electron tunneling terms between sectors differing by one electron.
Effective Hamiltonian Construction: Using these matrix elements, the authors construct projected effective Hamiltonians for the superconducting Anderson impurity model (grain coupled to a dot) and a flux-tunable loop geometry. These effective Hamiltonians are then diagonalized to obtain spectra and response functions.

Key Contributions

Determinant Formula for Fermionic Operators: The paper provides the first exact, finite-size, eigenvalue-based determinant expression for matrix elements of single-fermion operators between arbitrary RG states. This bridges the gap between integrable models and experimentally relevant hybrid devices.
Canonical Ensemble Framework: The method operates naturally in the canonical ensemble, preserving particle-number conservation and capturing even-odd effects and parity-dependent physics that are often lost in grand-canonical treatments.
Computational Efficiency: By combining exact grain eigenstates with a low-energy projection and closed-form matrix elements, the method allows for the inexpensive calculation of projected spectra, response functions, and dense parameter sweeps, complementing direct numerical methods like DMRG.

Results
The authors apply their framework to two specific models:

Superconducting Anderson Impurity Model:
- Ground State Analysis: They analyze the competition between superconductivity and local-moment formation for a singly occupied dot. They demonstrate that for weak charging, the system forms a spin singlet where the dot spin is screened by an unpaired grain electron, while strong charging favors local-moment formation.
- Charge-Stability Diagrams: The method reproduces complex charge-stability diagrams showing transitions between superconducting (even-grain-occupation favored) and double-quantum-dot-like behaviors depending on the charging energy $V$ .
- Subgap States: The approach captures subgap states (analogous to Yu-Shiba-Rusinov states) and tracks their evolution through zero energy, marking ground-state parity crossings.
- Optical Conductivity: The authors calculate the fixed-particle-number optical conductivity. They show that the optical response distinguishes between neutral excitations and single-particle addition/removal spectra. Notably, they observe sharp discontinuities in the optical conductivity at parity-changing ground-state transitions, where new subgap transitions become optically active.
Flux-Tunable Grain-Dot Loop:
- The model is extended to a grain coupled to a dot via two spatially separated contacts, introducing Aharonov-Bohm interference.
- The spectrum exhibits sensitivity to the magnetic flux $\Phi$ . The authors observe a transition in the dominant periodicity of the energy spectrum from the single-electron flux quantum $\Phi_e = h/e$ to the superconducting flux quantum $\Phi_{sc} = h/2e$ as the tunneling strength increases.
- They calculate persistent currents and identify 0– $\pi$ transitions (changes in the phase of the ground state) driven by gate voltages and flux, with current-flux relations showing discontinuities at parity-switching points.

Significance and Claims
The paper claims to provide a "proof of principle" for bridging the gap between the mathematical structure of integrable models and the description of experimentally relevant mesoscopic devices. By deriving exact matrix elements for single-electron operators, the authors enable the coupling of an integrable superconducting grain to degrees of freedom that break integrability (such as a quantum dot) while retaining the exact solution for the grain.

The authors emphasize that their method naturally handles the canonical ensemble, capturing even-odd effects and subgap excitations without the need for grand-canonical approximations. They state that the optical conductivity and flux-tunable spectra calculated for grain-dot systems in the canonical ensemble have not been previously computed using this approach. Furthermore, the method is presented as a route to embed exact superconducting grains as single sites in tensor-network calculations (e.g., DMRG), allowing for number-conserving descriptions of proximity-induced pairing in larger systems. The work does not propose new experimental setups but rather offers a theoretical tool to analyze existing hybrid device physics with higher precision and in regimes where standard mean-field theories fail.

Technical Summary: Exact Fermionic Matrix Elements from the Bethe-ansatz Solution of the Richardson–Gaudin Model

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