One-Dimensional Traps, Two-Body Interactions, Few-Body Symmetries: II. $N$ Particles

Abstract

This is the second in a pair of articles that classify the configuration space and kinematic symmetry groups for $N$ identical particles in one-dimensional traps experiencing Galilean-invariant two-body interactions. These symmetries explain degeneracies in the few-body spectrum and demonstrate how tuning the trap shape and the particle interactions can manipulate these degeneracies. The additional symmetries that emerge in the non-interacting limit and in the unitary limit of an infinitely strong contact interaction are sufficient to algebraically solve for the spectrum and degeneracy in terms of the one-particle observables. Symmetry also determines the degree to which the algebraic expressions for energy level shifts by weak interactions or nearly-unitary interactions are universal, i.e.\ independent of trap shape and details of the interaction. Identical fermions and bosons with and without spin are considered. This article analyzes the symmetries of $N$ particles in asymmetric, symmetric, and harmonic traps; the prequel article treats the one, two and three particle cases.

v2 46 pages, 2 figures, 2 tables. Revised notation, added commentary and references, and made corrections. Some of this materially originally appeared in version 1 of arXiv:1501.00215, but that article has been split into two pieces, and this is the second piece

References in corpus (11)

• Exact Solution of Strongly Interacting Quasi-One-Dimensional Spinor Bose Gases
• Exact solution for infinitely strongly interacting Fermi gases in tight waveguides
• Soluble Models of Strongly Interacting Ultracold Gas Mixtures in Tight Waveguides
• Strongly Interacting Quantum Gases in One-Dimensional Traps
• Distinguishability, degeneracy and correlations in three harmonically trapped bosons in one-dimension
• Symmetries of Three Harmonically-Trapped Particles in One Dimension
• One-dimensional Fermi gas with a single impurity in a harmonic trap: Perturbative description of the upper branch
• A variational approach to repulsively interacting three-fermion systems in a one-dimensional harmonic trap
• Ground-state properties of interacting two-component Bose gases in a hard-wall trap
• Sum rules for spin-$1/2$ quantum gases in states with well-defined spins: spin-independent interactions and spin-dependent external fields
• Entanglement of one-magnon Schur-Weyl states

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• Impenetrable Mass-Imbalanced Particles in One-Dimensional Harmonic Traps
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• Integrable families of hard-core particles with unequal masses in a one-dimensional harmonic trap
• Ultracold few fermionic atoms in needle-shaped double wells: spin chains and resonating spin clusters from microscopic Hamiltonians emulated via antiferromagnetic Heisenberg and t-J models
• Strongly correlated one-dimensional Bose-Fermi quantum mixtures: symmetry and correlations
• Topological Exchange Statistics in One Dimension
• Infinite Barriers and Symmetries for a Few Trapped Particles in One Dimension
• Probing the edge between integrability and quantum chaos in interacting few-atom systems
• Exact ordering of energy levels for one-dimensional interacting Fermi gases with $SU(n)$ symmetry
• Entanglement of an Impurity in a Few-Body One-Dimensional Ideal Bose System
• Entanglement production by statistical operators
• Computation of local exchange coefficients in strongly interacting one-dimensional few-body systems: local density approximation and exact results
• Identical Wells, Symmetry Breaking, and the Near-Unitary Limit
• Experimentally accessible invariants encoded in interparticle correlations of harmonically trapped ultra-cold few-fermion mixtures
• The Goldilocks model of separable, zero-range, few-body interactions in one-dimensional harmonic traps
• Connecting scrambling and work statistics for short-range interactions in the harmonic oscillator
• Long-lived states with well-defined spins in spin-$1/2$ homogeneous Bose gases
• Beyond braid statistics: Constructing a lattice model for anyons with exchange statistics intrinsic to one dimension
• Ground-State Magnetization in Mixtures of a Few Ultra-Cold Fermions in One-Dimensional Traps
• Phase transitions in definite total spin states of two-component Fermi gases
• Coincidence Structures and Hard-Core Few-Body Interactions
• Sum rules for spin-$1/2$ quantum gases in states with well-defined spins: II. Spin-dependent two-body interactions
• Impact of dark states on the stationary properties of quantum particles with off-centered interactions in one dimension
• Necklace Ansatz for strongly repulsive spin mixtures on a ring
• Symmetry oscillations sensitivity to SU(2)-symmetry breaking in quantum mixtures