⚛️ quantum physics

Efficient determination of eigenenergies and eigenstates of $N$ ( $N=3$ --$4$) identical 1D bosons and fermions under external harmonic confinement

This paper presents a new numerical approach for efficiently determining the energy spectra and eigenstates of small one-dimensional systems of identical bosons and fermions with zero-range interactions under external harmonic confinement.

Original authors: J. D. Norris, D. Blume

Published 2026-09-01

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CC BY 4.0

Original authors: J. D. Norris, D. Blume

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 authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of quantum physics, particles like atoms do not behave like the solid objects we see in daily life. Instead, they exist as waves of probability, and when many of them are crowded together, they can act in unison, creating strange new states of matter. For decades, scientists have been fascinated by what happens when these particles are squeezed into very narrow spaces, such as a thin tube where they can only move back and forth in a single line. In these tight, one-dimensional corridors, the rules of interaction change dramatically. The particles can no longer easily slip past one another; instead, they are forced to interact in ways that create powerful, collective behaviors. Understanding how just a few of these particles behave in such a confined space is crucial. It serves as a bridge, helping physicists understand the transition from the simple behavior of a few atoms to the complex, chaotic behavior of a vast cloud of gas.

A team of researchers at the University of Oklahoma has developed a new and highly efficient way to solve the mathematical puzzles that describe these tiny systems. They focused on groups of three or four identical atoms, either all bosons or all fermions, trapped inside a smooth, bowl-shaped energy field created by lasers. In this setup, the atoms are held in place but are free to move back and forth along a single line. The challenge has always been that when these atoms get close to each other, their interactions become incredibly difficult to calculate, especially when the atoms are fermions, a type of particle that naturally avoids occupying the same space. The researchers created a new computational method that allows them to determine the exact energy levels and the specific arrangement of these atoms with high precision, covering a wide range of interaction strengths from weak to very strong.

The core of their work involves solving a complex equation that predicts how the atoms will arrange themselves and what energy they will possess. For groups of three or four bosons, which are particles that like to clump together, this calculation has been done before, but the researchers applied their new method to verify it with greater speed and stability. More significantly, they tackled the much harder problem of three identical fermions. Because fermions have a strict rule that prevents them from being in the same place at the same time, their mathematical description requires a special kind of interaction that involves how the particles' waves change as they approach each other. Previous attempts to solve this directly were often unstable or required indirect tricks to work. The new method, however, handles these tricky interactions directly and stably, proving that the underlying mathematics works perfectly even for these complex, multi-particle systems.

The researchers tested their approach by calculating the energy spectra for these small groups of atoms. An energy spectrum is essentially a map of all the possible energy states the system can occupy. They found that for three fermions, their direct calculation produced results that matched perfectly with a famous theoretical prediction known as the Bose-Fermi mapping. This mapping suggests that under certain conditions, a system of strongly interacting fermions behaves exactly like a system of weakly interacting bosons. By showing that their direct calculation for fermions aligns with the known results for bosons, the team confirmed that their new method is accurate and reliable. They also extended this work to four bosons, a system for which no complete energy spectrum had been calculated before using this specific type of direct approach.

The results revealed a rich variety of behaviors depending on how strongly the atoms attract or repel each other. When the atoms repel each other, they behave like a gas, spreading out to fill the available space. When they attract, they can form bound groups, sticking together like tiny clusters. For the four-boson system, the researchers identified distinct families of states, ranging from loose collections of four separate atoms to tightly bound groups of four, as well as intermediate states where two pairs of atoms are bound together or where a group of three is bound to a single atom. The calculations showed that as the attraction becomes infinitely strong, the energy of these bound groups drops to specific values that match the known binding energies of atoms in free space, confirming that the method captures the essential physics of how these particles stick together.

One of the most important aspects of this work is the stability of the calculations. The researchers found that while their method works smoothly for bosons, the fermion calculations are much more sensitive to numerical errors. To overcome this, they used a specialized computer technique that keeps track of numbers with extreme precision, far beyond what standard computers usually do. This allowed them to see that the mathematical expressions for the fermion interactions are well-defined and do not break down, even when dealing with the complex derivatives required to describe them. This stability is a significant step forward, as it opens the door for studying more complex systems without relying on indirect workarounds.

The paper concludes that this new approach is not only accurate but also efficient enough to be run on a standard laptop, taking only a few minutes to a few hours to generate the results. This efficiency suggests that the method can be extended to even larger systems, such as five particles, or to more complex mixtures of different types of atoms. By providing a reliable way to calculate the behavior of these few-particle systems, the work offers a solid foundation for understanding the quantum world. It helps bridge the gap between the simple physics of two particles and the complex physics of many, offering a clearer view of how quantum matter behaves when squeezed into the tightest of spaces.

Technical Summary: Efficient Determination of Eigenenergies and Eigenstates of N (N = 3–4) Identical 1D Bosons and Fermions

Problem Statement
This work addresses the challenge of determining the energy spectra and eigenstates for small, strictly one-dimensional (1D) systems of identical bosons and fermions under external harmonic confinement. Specifically, the authors focus on systems containing three or four particles ( $N=3, 4$ ) interacting via additive two-body zero-range potentials. For bosons, the interaction is a standard $\delta$ -function potential ( $V_+$ ), while for fermions, the interaction involves derivative operators ( $V_-$ ) to account for the odd-parity nature of the wavefunction. While the Bose-Fermi mapping theorem allows for the derivation of fermionic spectra from bosonic ones in the Tonks-Girardeau regime, this work seeks to solve the fermionic problem directly without relying on such duality, particularly to validate the mathematical well-definedness of the odd-parity interaction potential for systems with more than two particles.

Methodology
The authors employ the Lippmann-Schwinger equation approach, starting from the time-independent Schrödinger equation for the relative Hamiltonian. By utilizing the Green's function formalism, the problem is reduced to solving eigenvalue equations for the interaction strength $g_{\pm}$ as a function of energy $E$ .

Theoretical Framework:
- The system is transformed from single-particle coordinates to Jacobi relative coordinates and a center-of-mass coordinate. The center-of-mass motion is separated, leaving the relative Hamiltonian to be solved.
- For the three-boson (BBB) and four-boson (BBBB) systems (positive relative parity, $\Pi=+1$ ), and the three-fermion (FFF) system (negative relative parity, $\Pi=-1$ ), the authors derive specific eigenvalue equations.
- The eigenvalue equations take the form of matrix equations where $1/g_{\pm}$ acts as the eigenvalue. The matrix elements involve integrals of the two-body Green's function and harmonic oscillator eigenstates.
Numerical Scheme:
- A key contribution is the development of a computationally efficient and numerically stable scheme to evaluate the integrals appearing in the matrix elements (e.g., $I^{(3B)}_{\ell k}$ , $J^{(3F)}_{\ell k}$ , $K^{(3F)}_{\ell k}$ , and $I^{(4B)}_{stk\ell}$ ).
- The authors utilize recursion relationships for integrals involving products of Hermite polynomials and Gaussian functions. By expressing the Green's function as an infinite sum of harmonic oscillator states, the multidimensional integrals are reduced to sums of one-dimensional integrals.
- A critical technical challenge arises in the FFF system due to the derivative operators in the potential, which introduce discontinuities and require high numerical precision. The authors implement the C++ code using the Boost Multiprecision library (50 decimal digits) for the FFF system to ensure stability, whereas standard double-precision arithmetic suffices for the bosonic systems.
- Convergence is controlled by two cutoffs: the basis set size (maximum quantum numbers $\ell, k, s, t$ ) and the truncation of the infinite sum over the intermediate quantum number $p$ (denoted $p_{max}$ ).

Key Results

BBB System: The energy spectrum for three identical bosons was calculated and found to be numerically robust. The results agree with existing literature. The spectrum shows gas-like behavior for repulsive interactions ( $g_+ > 0$ ) and the emergence of bound states (dimers and trimers) for attractive interactions ( $g_+ < 0$ ).
FFF System: The authors present the first direct calculation of the energy spectrum for three identical fermions in a harmonic trap without invoking the Bose-Fermi mapping. The results confirm that the odd-parity pseudopotential yields mathematically well-defined expressions for $N > 2$ . The FFF spectrum was benchmarked against the BBB spectrum using the Bose-Fermi mapping relation ( $g_+ = -2\hbar^4/(m^2 g_-)$ ), showing essentially perfect overlap, thereby validating the direct fermionic calculation.
BBBB System: This work presents the first calculations of the full energy spectrum for four identical bosons in a harmonic trap. The spectrum categorizes states into various configurations (atom-atom-atom-atom, atom-atom-dimer, dimer-dimer, atom-trimer, and tetramer) based on their behavior as $g_+$ varies. For strong attraction, the lowest energy state corresponds to a bound tetramer, and its energy converges to the free-space binding energy of four bosons.

Significance and Claims
The paper claims several specific contributions:

Direct Fermionic Solution: It demonstrates that the Lippmann-Schwinger equation approach can be successfully applied to identical fermions with derivative-based interactions, yielding stable results. This validates the use of the odd-parity interaction potential for multi-particle systems without conceptual issues.
Novelty in Four-Body Systems: The calculations for the four-boson system are presented as the first of their type, providing a detailed energy spectrum across a broad range of interaction strengths.
Computational Efficiency: The derived numerical scheme, based on recursion relations and pre-calculated energy-independent terms, allows for the determination of 20 or more converged eigenenergies efficiently. The authors note that calculations for the four-particle system can be performed on a laptop within minutes to hours.
Foundation for Future Work: The authors state that this framework complements previous work on few-atom systems and provides a foundation for extending studies to two-dimensional single-component fermions (predicted to support super-Efimov states) and three-dimensional systems. They also suggest the method can be adapted to treat anyons and mixtures of bosons and fermions.

The work does not propose new experimental setups but emphasizes its relevance to state-of-the-art cold-atom experiments where few-atom 1D systems can be prepared. The primary goal is to provide a rigorous theoretical tool for understanding the transition from few- to many-body quantum behaviors in low-dimensional systems.

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