Spin structures of the ground states of four body bound systems with spin 3 cold atoms

We consider the case that four spin-3 atoms are confined in an optical trap. The temperature is so low that the spatial degrees of freedom have been frozen. Exact numerical and analytical solutions for the spin-states have been both obtained. Two kinds of phase-diagrams for the ground states (g.s.) have been plotted. In general, the eigen-states with the total-spin S (a good quantum number) can be expanded in terms of a few basis-states \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{S,i}$$\end{document}fS,i. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{f_{S,i}}^{\lambda }$$\end{document}PfS,iλ be the probability of a pair of spins coupled to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda =0, 2, 4$$\end{document}λ=0,2,4, and 6 in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{S,i}$$\end{document}fS,i state. Obviously, when the strength \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_{\lambda }$$\end{document}gλ of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}λ-channel is more negative, the basis-state with the largest \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{f_{S,i}}^{\lambda }$$\end{document}PfS,iλ would be more preferred by the g.s.. When two strengths are more negative, the two basis-states with the two largest probabilities would be more important components. Thus, based on the probabilities, the spin-structures (described via the basis-states) can be understood. Furthermore, all the details in the phase-diagrams, say, the critical points of transition, can also be explained. Note that, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{S,i}$$\end{document}fS,i, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{f_{S,i}}^{\lambda }$$\end{document}PfS,iλ is completely determined by symmetry. Thus, symmetry plays a very important role in determining the spin-structure of the g.s..

g is the weighted strength which is a product of the strength and the integral φ 4 dr . The latter embodies the effect of spatial profile. The dipole-dipole (d-d) coupling between a pair of atoms is relatively weak (for 52 Cr as an example, the strength of the d-d coupling c dd = 0.004g 6 ), therefore is neglected. In fact, the calculation in 21 demonstrates that the g.s. of 52 Cr does not seem to depend on the d-d coupling. An important feature of H spin is the conservation of the total spin S and its Z-component M . Thus the eigen-energies and eigen-states of H spin are denoted as E SM and ψ SM (the subscript M might be neglected).

Spin-structures based on the pairs
After the diagonalization of H spin , the parameter space can be divided into zones according to S, and the phase diagram thereby can be plotted. To reduce the complexity, we use three 2-dimensional subspaces to replace the 4-dimensional parameter space as shown in Fig. 1. In each of these subspaces g 4 and g 6 are variable, while g 0 and g 2 are fixed. There are three possible cases (1) g 0 < g 2 , (2) g 0 ≃ g 2 , and (3) g 0 > g 2 . Note that the spin-structures will neither be changed when all the {g } are shifted with the same value, nor when the unit for {g } is changed. For case (1), let the set {g } be shifted so that (g 0 + g 2 )/2 = 0 , then a unit is adopted so that g 0 = −0.5 and g 2 = 0.5 (Fig. 1a). For case (2), as an approximation, we assume g 0 = g 2 . Then, {g } is shifted so that g 0 = g 2 = 0 (Fig. 1b). For case (3), similarly, we have g 0 = 0.5 and g 2 = −0.5 (Fig. 1c). For all the three cases, the ranges of g 4 and g 6 are from −1 to +1 . In the qualitative sense, the feature of a 4-dimensional diagram can be roughly illustrated via these three 2-dimensional diagrams.
To understand better the underlying physics, in addition to numerical solutions, we look for analytical solutions. Let be a basis-state,where S is an operator for symmetrization and normalization, χ(i) is the spin-state of the i-th particle, particles 1 and 2 (3 and 4) are coupled to a ( b ), a and b should be even and coupled to S. Note thatϕ S; a b has not yet been symmetrized, but φ (λaλ b )S is. When S is fixed while a and b are variable, the set {φ (λaλ b )S } can also be used as (non-orthogonal) basis-states for ψ SM . It turns out that, for N = 4, the multiplicity of every ψ SM is very small ( ≤ 3 ). Thus H spin can be analytically diagonalized. Examples are given below.
By recoupling the spins, we have where  The eigen-energy E 0 is the root of a two-dimensional homogeneous linear equation, Making use of Table 1, the eigen-energy of the lower S = 0 states is where The normalized spin-state of the lower S = 0 state is , and a 6 = xa 4 . Whereas for S = 3 , 5, 7, 9, 10 and 12 states, all of them have multiplicity one, thus the eigen-state is just , where a and b are arbitrary even numbers adapted to S. For example, when S = 7 we choose a = 6 and b = 4 , then we have where C 7;6,4;2,6 = .6362 , C 7;6,4;4,6 = .3086 , C 7;6,4;6,2 = −C 7;6,4;2,6 , C 7;6,4;6,4 = −C 7;6,4;4,6 , other wise The eigen-energies of other S-states with multiplicity can be similarly obtained.It is emphasized that, when other a and b are chosen, both ψ 7M and E S=7 remain the same. These states are strictly determined by symmetry.
Note that, when a g is more negative than the others, two spins in the g.s. will prefer to be coupled to and form a [ ]-pair. Let �( a b ) S |gs� be a shortened label for the overlap φ (λaλ b )S |Ψ S(gs) . We found that, at the point A (where g 4 = −1 and g 6 = 1 ) marked in Fig. 1a-c, �(4, 4) 0 |gs� = 0.9883 , 0.9996, and 0.9794, respectively. It implies that the g.s. is essentially composed of two [4]-pairs (due to the very negative g 4 ), and they are further coupled to zero, namely, they are lying opposite to each other (due to the very positive g 6 ). Besides, at C, B and A marked in Fig.1c (where g 6 = 0 , 0.5, and 1), �(4, 4) 0 |gs� =0.9703, 0.9765, and 0.9883, respectively. It implies that, when g 6 increases from 0, the structure (4,4) 0 will be more dominant.

Competition in the formation of pairs
From the above section we know that, when a g is more negative than the others, the [ ]-pairs will be important. The relative orientation of the spins of pairs depends on g 6 and will be changed from being anti-parallel to parallel. It is expected that, when g and g ′ are both more negative, there would be a competition between the [ ]-and [ ′ ]-pairs. To clarify, we introduce another kind of phase diagrams as shown in Fig. 2. In Fig. 2a  There is a critical point p 8→4 located at g 4 = g 2 = −0.5 , at which S transits from 8 to 4. Afterward, when g 4 (g 2 ) increases (decreases) further so that g 2 < g 4 , the g.s. is dominated by (2,2) 4 (say, �(2, 2) 4 |gs� g 4 =−0.25 = 0.967 ). Thus the negative g 6 = −0.5 is not sufficient to form the [6]-pairs, but sufficient to bring the spins of the two [4]-pairs or the two [2]-pairs to be parallel.
The case with g 6 = 0.5 (solid line in orange) is similar to the case with g 6 = 0 , except p 0→2 = −0.163 , and the balance point B 2 = −0.274 . Thus, both p 0→2 and B 2 shift to the right.
Furthermore, when g 6 increases, the critical point also shifts to the right. This is due to a similar reason that the appearance of the [6]-pairs in (2,2) 0 is less probable than in (2,2) 2 .
In Fig. 2b both g 0 and g 4 are negative. Figure 2b is comparable with Fig. 2a   The label of the component ( a , b ) S is marked above the horizontal lines, each is for a given g 6 marked at the left end of the line. For 2a, g 0 = 0 , the abscissa is for g 4 , and g 2 = −1 − g 4 . For 2b, g 2 = 0 , the abscissa is also for g 4 , and g 0 = −1 − g 4 . For 2c, g 4 = 0 , the abscissa is for g 2 , and g 0 = −1 − g 2 . www.nature.com/scientificreports/ due to the negative g 6 , either the two [4]-pairs or the two [6]-pairs are parallel to each other. This leads to the transition of S as 8 → 12 → 0 when g 4 increases ( g 0 decreases). (4) The shift of the balance point to the right appears again (i.e., B 1 < B 2 < B 3 ). Note that C 0;0,0;6,6 = 0.5245 .
Thus the appearance of the [6]-pairs in (4,4) 0 is also much less probable than in (0,0) 0 . This causes the shift as before.
In Fig. 2c both g 0 and g 2 are negative. When g 6 = −1 , the g.s. is fully polarized as before. Otherwise, the g.s. is essentially composed of (2, 2) b and (0, 0) 0 (where b = 4 , 2, and 0). When g 6 = −0.5 we see a chain of transitions: S=4 → 2 → 0 → 12 → 0.When g 0 , g 2 and g 6 are all close to −0.5 , there is a small segment in bold black line where (6,6) 12 emerges (similar to the case in Fig. 2b). When g 6 = 0, 0.5, and 0.8 (dotted line), we see the transition of S = 2 → 0 . Where the critical point shifts to the left when g 6 increases. It implies that the appearance of the [6]-pairs in (2,2) 0 is less probable than in (2,2) 2 . Whereas the balance point shifts to the right when g 6 increases. It implies that the appearance of the [6]-pairs in (2,2) 0 is less probable than in (0,0) 0 .

Final remarks
The spin-structures of N = 4 condensates have been studied, both numerical and analytical solutions have been obtained. Thereby two kinds of phase-diagrams for the g.s. have been plotted and explained. From dynamical aspect, the [ ]-pairs would be important constituents when g is more negative. However, the probability of the appearance of a [ ]-pair in a particular component ( a b ) S is determined by symmetry. Thus the structure of the g.s. depends not only on the strengths but also on the symmetry constraint. We have calculated the probabilities P ( a b ) S for finding out the important components. The importance is further confirmed by the calculation of the amplitudes �( a b ) S |� S(gs) � . Obviously, for cold few-body systems, the very small multiplicity of a state is a remarkable feature, thereby the states are tightly (or even completely) constrained by symmetry.
When two or more g are negative and close to each other, there is competition between various [ ]-pairs and the most important pair is thereby determined. Note that the magnitude of S depends on the relative orientation of the pair-spins a and b (if they are nonzero), while the orientation is determined by the strengths. In particular, the sign of g 6 is crucial which determines whether the two pair-spins are parallel or anti-parallel. Thus the variation of {g } will cause the change of the most important pair and the relative orientation of the pair-spins. This leads to the shift of the balance point and the critical point. The chain of transitions is thereby explained.
The approach of this paper can be generalized to systems with a larger N. When N is larger, if g is more negative, the [ ]-pairs would also be more important in the g.s. There would also be competitions among various [ ]-pairs. The study of the probability P where is an assumed basis-state would also be helpful for finding out the important component(s) and their alternation. In particular, some very stable spin-structures found in fewbody systems could be building blocks for large N systems. This is a point to be clarified.