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arXiv:2403.06519 · Few-Body Systems

Confinement of N-Body Systems and Non-integer Dimensions

Eric Garrido, A. S. Jensen

Published Apr 6, 2024• 4 citations
arXivHTMLPDF

Abstract

Abstract The squeezing process of a three-dimensional quantum system by use of an external deformed one-body oscillator potential can also be described by the d-method, without external field and where the dimension can take non-integer values. In this work we first generalize both methods to N particles and any transition between dimensions below 3. Once this is done, the use of harmonic oscillator interactions between the particles allows complete analytic solutions of both methods, and a direct comparison between them is possible. Assuming that both methods describe the same process, leading to the same ground state energy and wave function, an analytic equivalence between the methods arises. The equivalence between both methods and the validity of the derived analytic relation between them is first tested for two identical bosons and for squeezing transitions from 3 to 2 and 1 dimensions, as well as from 2 to 1 dimension. We also investigate the symmetric squeezing from 3 to 1 dimensions of a system made of three identical bosons. We have in all the cases found that the derived analytic relations between the two methods work very well. This fact permits to relate both methods also for large squeezing scenarios, where the brute force numerical calculation with the external field is too much demanding from the numerical point of view, especially for systems with more than two particles.

DOI
10.1007/s00601-024-01906-4

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The resources required to solve the general interacting quantum NNN-body problem scale exponentially with NNN, making the solution of this problem very difficult when NNN is large. In a previous series of papers we develop an approach for a fully interacting wave function for a confined system of identical bosons with a general two-body interaction. This method tames the NNN scaling by developing a perturbation series that is order-by-order invariant under a point group isomorphic with SN{S}_{N}SN​. Group theory and graphical techniques are then used to solve for the wave function exactly and analytically at each order, yielding a solution for the general NNN-body problem which scales as N0{N}^{0}N0 at any given order. Recently this formalism has been used to obtain the first-order fully interacting wave function for a system of harmonically confined bosons interacting harmonically. In this paper, we report the application of this NNN-body wave function to a system of NNN fully interacting bosons in three dimensions. We derive an expression for the density profile for a confined system of harmonically interacting bosons. Choosing this simple interaction is not necessary or even advantageous for our method, however this choice allows a direct comparison of our exact results through first order with exact results obtained in an independent solution. Our density profile for the wave function through first order in three dimensions is indistinguishable from the first-order exact result obtained independently and shows strong convergence to the exact result to all orders.
Continuum structures of three short-range interacting particles in a deformed external one-body field are investigated. We use the equivalent ddd method employing noninteger dimension, ddd, in a spherical calculation with a dimension-dependent angular momentum barrier. We focus on dimensions close to the critical dimension, d=dEd={d}_{E}d=dE​, between 2 and 3, defined by zero two-body energies, where the Efimov effect can occur. We design for this dimension region a schematic, long-distance realistic, square-well based, three-body spherical model, which is used to derive analytic expressions for the wave functions, scattering lengths, phase shifts, and elastic-scattering cross sections. The procedure and the results are universal, valid for all short-range potentials, and for large scattering lengths. We discuss the properties and validity of the derived expressions by means of the simplest system of three identical bosons. The derived expressions are particularly useful for very small energies, where full numerical calculations are often not feasible. For energies where the numerical calculations can be performed, a good agreement with the analytic results is found. These model results may be tested by scattering experiments for three particles in an equivalent external deformed oscillator potential. The cross sections all vanish in the zero-energy limit for d<3d<3d<3 with definite ddd-dependent power of energy.
Abstract Quantum mechanical few-body systems in reduced dimensionalities can exhibit many interesting properties such as scale-invariance and universality. Analytical descriptions are often available for integer dimensionality, however, numerical approaches are necessary for addressing dimensional transitions. The Fully-Correlated Gaussian method provides a variational description of the few-body real-space wavefunction. By placing the particles in a harmonic trap, the system can be described at various degrees of anisotropy by squeezing the confinement. Through this approach, configurations of two and three identical bosons as well as heteronuclear (Cs–Cs–Li and K–K–Rb) systems are described during a continuous deformation from three to one dimension. We find that the changes in binding energies between integer dimensional cases exhibit a universal behavior akin to that seen in avoided crossings or Zeldovich rearrangement.
The resources required to solve the general interacting quantum N-body problem scale exponentially with N, making the solution of this problem very difficult when N is large. In a previous series of papers we develop an approach for a fully-interacting wave function with a general two-body interaction which tames the N-scaling by developing a perturbation series that is order-by-order invariant under a point group isomorphic with S_N . Group theory and graphical techniques are then used to solve for the wave function exactly and analytically at each order. Recently this formalism has been used to obtain the first-order, fully-interacting wave function for a system of harmonically-confined bosons interacting harmonically. In this paper, we report the first application of this N-body wave function to a system of N fully-interacting bosons in three dimensions. We determine the density profile for a confined system of harmonically-interacting bosons. Choosing this simple interaction is not necessary or even advantageous for our method, however this choice allows a direct comparison of our exact results through first order with exact results obtained in an independent solution. Our density profile through first-order in three dimensions is indistinguishable from the first-order exact result obtained independently and shows strong convergence to the exact result to all orders.
We consider the energetics and structural properties of a many-particle system in one dimension with pairwise contact interactions confined in a parabolic external potential. To render the problem analytically solvable, we use the harmonic approximation scheme at the level of the Hamiltonian. We investigate the scaling with particle number of the ground-state energies for systems consisting of identical bosons or fermions. We then proceed to focus on bosonic systems and make a detailed comparison with known exact results in the absence of the parabolic external trap for three-body systems. We also consider the thermodynamics of the harmonic model, which turns out to be similar for bosons and fermions due to the lack of degeneracy in one dimension.