Hyperspherical asymptotics of a system of four charged particles

We present a detailed analysis of the charged four-body system in hyperspherical coordinates in the large hyperradial limit. In powers of $R^{-1}$ for any masses and charges, the adiabatic Hamiltonian is expanded to third order in the dimer-dimer limit and to first order in the particle-trimer limit.Comment: 10 pages, 0 figure

Tunable high-temperature thermodynamics of weakly-interacting dipolar gases

We consider dilute gases of dipolar bosons or fermions in the high-temperature limit in a spherically symmetric harmonic trapping potential. We examine the system using a virial expansion up to second order in the fugacity. Using the Born approximation and assuming purely dipolar interactions, we find that the second-order virial coefficient for both bosons and fermions depends quadratically on the dipole length and is negative at high temperature, indicating that to lowest order in the dipole-dipole interactions the dipolar single-component quantum gases are repulsive. If the $s$-wave scattering length for the bosonic system is tunable and its absolute value is made small, then the $s$-wave interactions dominate and the dipolar as behaves like a weakly-interacting Bose gas with isotropic $s$-wave interactions. If the generalized scattering lengths for the fermionic system are tunable, then the dipole length can enter linearly in the virial equation of state, enhancing the dipole-dipole effects in the thermodynamic observables.Comment: 9 pages, 6 figure

Thermodynamics of the two-component Fermi gas with unequal masses at unitarity

We consider mass-imbalanced two-component Fermi gases for which the unequal-mass atoms interact via a zero-range model potential with a diverging s-wave scattering length $a_s$, i.e., with $1/a_s=0$. The high temperature thermodynamics of the harmonically trapped and homogeneous systems are examined using a virial expansion approach up to third order in the fugacity. We find that the universal part of the third-order virial coefficient associated with two light atoms and one heavy atom is negative, while that associated with two heavy and one light atom changes sign from negative to positive as the mass ratio $\kappa$ increases, and diverges when Efimov physics sets in at $\kappa=13.61$. By examining the Helmholtz free energy, we find that the equilibrium polarization of the trapped and homogeneous systems is 0 for $\kappa=1$, but finite for $\kappa \ne 1$ (with a majority of heavy particles). Compared to the equilibrium polarization of the non-interacting system, the equilibrium polarization at unitarity is increased for the trapped system and decreased for the homogeneous system. We find that unequal-mass Fermi gases are stable for all polarizations.Comment: 14+2 pages, 14 figure

Extension of the correlated Gaussian hyperspherical method to more particles and dimensions

The solution of the hyperangular Schr\"odinger equation for few-body systems using a basis of explicitly correlated Gaussians remains numerically challenging. This is in part due to the number of basis functions needed as the system size grows, but also due to the fact that the number of numerical integrations increases with the number of hyperangular degrees of freedom. This paper shows that the latter challenge is no more. Using a delta function to fix the hyperradius $R$, all matrix element calculations are reduced to a single numerical integration regardless of system size $n$ or number of dimensions $d$. In the special case of $d$ an even number, the matrix elements of the noninteracting system are fully analytical. We demonstrate the use of the new matrix elements for the 3-, 4-, and 5-body electron-positron systems with zero total angular momentum $L$, positive parity $\pi$, and varied spins $S_+$ and $S_-$.Comment: 8 pages, 4 figure

Adiabatic hyperspherical analysis of realistic nuclear potentials

Using the hyperspherical adiabatic method with the realistic nuclear potentials Argonne V14, Argonne V18, and Argonne V18 with the Urbana IX three-body potential, we calculate the adiabatic potentials and the triton bound state energies. We find that a discrete variable representation with the slow variable discretization method along the hyperradial degree of freedom results in energies consistent with the literature. However, using a Laguerre basis results in missing energy, even when extrapolated to an infinite number of basis functions and channels. We do not include the isospin $T=3/2$ contribution in our analysis.Comment: 9 pages, 3 figures, 1 tabl

Occupation numbers of the harmonically trapped few-boson system

We consider a harmonically trapped dilute $N$-boson system described by a low-energy Hamiltonian with pairwise interactions. We determine the condensate fraction, defined in terms of the largest occupation number, of the weakly-interacting $N$-boson system ($N \ge 2$) by employing a perturbative treatment within the framework of second quantization. The one-body density matrix and the corresponding occupation numbers are compared with those obtained by solving the two-body problem with zero-range interactions exactly. Our expressions are also compared with high precision {\em{ab initio}} calculations for Bose gases with $N=2-4$ that interact through finite-range two-body model potentials. Non-universal corrections are identified to enter at subleading order, confirming that different low-energy Hamiltonians, constructed to yield the same energy, may yield different occupation numbers. Lastly, we consider the strongly-interacting three-boson system under spherically symmetric harmonic confinement and determine its occupation numbers as a function of the three-body "Efimov parameter".Comment: 16 pages, 7 figure

Three s-wave interacting fermions under anisotropic harmonic confinement: Dimensional crossover of energetics and virial coefficients

We present essentially exact solutions of the Schroedinger equation for three fermions in two different spin states with zero-range s-wave interactions under harmonic confinement. Our approach covers spherically symmetric, strictly two-dimensional, strictly one-dimensional, cigar-shaped, and pancake-shaped traps. In particular, we discuss the transition from quasi-one-dimensional to strictly one-dimensional and from quasi-two-dimensional to strictly two-dimensional geometries. We determine and interpret the eigenenergies of the system as a function of the trap geometry and the strength of the zero-range interactions. The eigenenergies are used to investigate the dependence of the second- and third-order virial coefficients, which play an important role in the virial expansion of the thermodynamic potential, on the geometry of the trap. We show that the second- and third-order virial coefficients for anisotropic confinement geometries are, for experimentally relevant temperatures, very well approximated by those for the spherically symmetric confinement for all s-wave scattering lengths.Comment: 13 figures (multiple subfigures

Low-Energy Scattering Properties of Ground-State and Excited-State Positronium Collisions

Low-energy elastic and inelastic scattering in the Ps(1$s$)-Ps(2$s$) channel is treated in a four-body hyperspherical coordinate calculation. Adiabatic potentials are calculated for triplet-triplet, singlet-singlet, and singlet-triplet spin symmetries in the spin representation of coupled electrons and coupled positrons, with total angular momentum $L=0$ and parity equal to $+1$. The s-wave scattering lengths for the asymptotic Ps(1$s$)-Ps(2$s$) channel are calculated for each spin configuration. Results obtained for the s-wave scattering lengths are $a_{\mathrm{TT}}=$~$7.3(2)a_0-i0.02(1)a_0$, $a_{\mathrm{SS}}=$~$13.2(2)a_0-i0.9(2)a_0$, and $a_{\mathrm{ST}}=$~$9.7(2)a_0$ for each spin configuration. Spin recoupling is implemented to extract the scattering lengths for collisions of Ps in different spin configurations through properly symmetrized unitary transformations. Calculations of experimentally relevant scattering lengths and cross-sections are carried-out for Ps atoms initially prepared in different uncoupled spin states

Scattering properties of the 2e−2e+2e^-2e^+ polyelectronic system

We study the $2e^-2e^+$ equal-mass charge-neutral four-body system in the adiabatic hyperspherical framework. The lowest few adiabatic potentials are calculated for zero orbital angular momentum, positive parity, and charge conjugation symmetries. Propagating the R-matrix, the low-energy $s$-wave scattering lengths of the singlet-singlet and triplet-triplet spin configurations are calculated. Lastly, we calculate the S-matrix for energies above the ionic threshold to estimate the transition rates between the single ionic fragmentation channel and the lowest few dimer-dimer fragmentation channels.Comment: 8 pages, 5 figure

Hyperspherical theory of the quantum Hall effect: the role of exceptional degeneracy

By separating the Schr\"odinger equation for $N$ noninteracting spin-polarized fermions in two-dimensional hyperspherical coordinates, we demonstrate that fractional quantum Hall (FQH) states emerge naturally from degeneracy patterns of the antisymmetric free-particle eigenfunctions. In the presence of Coulomb interactions, the FQH states split off from a degenerate manifold and become observable as distinct quantized energy eigenstates with an energy gap. This alternative classification scheme is based on an approximate separability of the interacting $N$-fermion Schr\"odinger equation in the hyperradial coordinate, which sheds light on the emergence of Laughlin states as well as other FQH states. An approximate good collective quantum number, the grand angular momentum $K$ from $K$-harmonic few-body theory, is shown to correlate with known FQH states at many filling factors observed experimentally.Comment: 15 pages, 10 figures, 3 table