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arXiv:2112.07885 · Physica Scripta

The band structure of the whole spectrum of an N-body cold system containing atoms with arbitrary integer spin and dominated by singlet pairing force

Y. Z. He, Y M Liu, Zhibing Li, Bao Cheng-guang

Published Mar 3, 2022• 3 citations
arXivHTMLPDF

Abstract

Abstract The spectra of N -boson systems with arbitrary nonzero spin f have been studied. Firstly, only the singlet pairing interaction is considered, a set of exact eigenstates together with the eigenenergies are analytically obtained. The completeness of this set is proved. The analytical expression allows us to see clearly the spin textures of various states different in N and/or f , and to find out the similarity and relationship lying among them. Secondly, the effect of other interactions is evaluated via exact numerical calculations on the systems with a smaller N . Some features and notable phenomena that might emerge in high- f systems, say, the ground band might have extremely high level density, have been discussed.

DOI
10.1088/1402-4896/ac5a89

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Abstract The spectra of N -boson systems with arbitrary nonzero spin f have been studied. Firstly, only the singlet pairing interaction is considered, a set of exact eigenstates together with the eigenenergies are analytically obtained. The completeness of this set is proved. The analytical expression allows us to see clearly the spin textures of various states different in N and/or f , and to find out the similarity and relationship lying among them. Secondly, the effect of other interactions is evaluated via exact numerical calculations on the systems with a smaller N . Some features and notable phenomena that might emerge in high- f systems, say, the ground band might have extremely high level density, have been discussed.
Abstract The spin-textures of trapped medium-body cold systems of spin-4 atoms with the interaction dominated by the attractive λ = 4 channel (the two interacting spins are coupled to 4) have been studied via an exact diagonalization of the spin-dependent Hamiltonian under the single spatial mode approximation. Particle number N (3 ≤ N ≤ 13) is assumed. The eigenstates have been analyzed to find out the details of spin-structures and the inherent excitation modes, and the relation between all of them. The emphasis is to find out how the spectra depend on N. We found that the singlet-triplets (a 3-body subsystem where the three spins are coupled to zero) and various excited triplets are basic constituents and all the eigenstates are composed of them. For low-lying states, only a few particles are involved in the excitation modes, while the other particles would 3-by-3 keep themselves in singlet-triplets.
The spin-textures of bound medium-body systems with spin-[Formula: see text] atoms ([Formula: see text]) have been studied. The Hamiltonian is assumed to be dominated by the two-body interaction favoring parallel spins. The system with particle number [Formula: see text] and [Formula: see text] is first chosen, and the Hamiltonian is exactly diagonalized by using Fock-states as basis-states, thereby all the eigenenergies and eigenstates are obtained and a detailed analysis is made. Then the cases with [Formula: see text] and [Formula: see text] are further studied. Since the total spin S is conserved, the eigenstates having the same S form an S-group. Let the lowest (highest) energy state of an S-group be called a bottom-state (top-state). We found that all the bottom-states are bipartite product states with constituent states describing fully polarized subsystems containing [Formula: see text] and [Formula: see text] ([Formula: see text]) particles, respectively. For two bottom-states different in [Formula: see text], the one with a larger [Formula: see text] is higher. For two having the same [Formula: see text], the one with a smaller S is higher. Whereas all the top-states are found to be essentially a product state of the pairs, in each pair the two spins are coupled to [Formula: see text] if the strength of the [Formula: see text]-channel is more repulsive than the others. For the states belonging to an S-group, the higher one would contain more pieces. As the energy goes up, larger pieces (those containing more than two particles) will disappear.
Abstract When the Hamiltonian of a condensate containing spin- f bosons is dominated by attractive singlet pairing interaction with the strength − ∣G 0∣ and when the temperature T is in the neighborhood of zero, the many-body Schrödinger equation can be exactly solved. Based on the complete and exact spectra the thermodynamic properties of these systems different in particle number N and f have been studied. The variation of the function ∂ C ∂ x against x is firstly studied, where C is the capacity, x ≡ lg(k B T/∣G 0∣), and k B is the Boltzmann constant. It was found that the variation can be divided into four stages. In the very narrow first stage, ∂ C ∂ x appears as a sharp peak. The second stage is broad where ∂ C ∂ x = 0 . The third stage is also very narrow where ∂ C ∂ x appears as a sharp anti-peak. In the fourth stage, ∂ C ∂ x becomes zero again. All the thermodynamic functions: the internal energy, the capacity, and the entropy vary accordingly. How these functions are affected by N, f , and ∣G 0∣ in each stage has been clarified. The spin-thermodynamics would replace the traditional thermodynamics in the neighborhood of zero temperature, where the spin degrees of freedom play an essential role.
We have found a stable spin-structure of N [Formula: see text] 3 system in which three spin-3 atoms are trapped and coupled to total spin S [Formula: see text] 3. We have proved that a pair of this structure is nearly an exact solution for N [Formula: see text] 6 systems in a very broad district in the parameter-space. Comparing with the well-known singlet pairs, this pair is a more promising candidate to serve as a building block for large N systems with spin-3 atoms. This is because the spin-structure of the latter can be modified depending on the interactions to reduce the energy while the singlet pairs can not. In fact, we have proved that, for a specific set of strengths (a point in the parameter-space) the product state based on this pair is an exact solution of the N-body Hamiltonian. Thus, in the neighborhood of this point, the product state will appear as an approximate solution. However, how broad this neighborhood would be remains to be clarified.
We investigate the spin structure of many-fermion systems with a spin-conserving two-body random interaction. We find a strong dominance of spin-0 ground states and considerable correlations between energies and wave functions of low-lying states with different spin, but no indication of pairing. The spectral densities exhibit spin-dependent shapes and widths, and depend on the particle density as well as on the relative strengths of the spin-0 and spin-1 couplings in the two-body random matrix. The spin structure of low-lying states can largely be explained analytically.
We study the ground-state properties of a mixture formed by two spin-1 condensates in the absence of an external magnetic field. As the collisional symmetry between interspecies bosonic atoms is broken, the interspecies coupling interaction ($\ensuremath{\beta}$) and interspecies singlet-pairing interaction ($\ensuremath{\gamma}$) arise. The ground state can be calculated using the angular momentum theory analytically for $\ensuremath{\gamma}=0$. The full quantum approach of exact diagonalization is adopted numerically to consider the more general case as $\ensuremath{\gamma}\ensuremath{\ne}0$. We illustrate the competition between the two interspecies interactions and find that as singlet-pairing interaction dominates (or the total spin vanishes), there are still different types of singlet formations which are well determined by $\ensuremath{\beta}$.