= ЯДРА ^^
HIGHER-ORDER BRUNNIAN STRUCTURES AND POSSIBLE PHYSICAL REALIZATIONS
© 2014 N. A. Baas1), D. V. Fedorov2), A. S. Jensen2), K. Riisager2), A. G. Volosniev2), N. T. Zinner2)
Received January 29, 2013
We consider few-body bound state systems and provide precise definitions of Borromean and Brunnian systems. The initial concepts are more than a hundred years old and originated in mathematical knot-theory as purely geometric considerations. About thirty years ago they were generalized and applied to the binding of systems in nature. It now appears that recent generalization to higher-order Brunnian structures may potentially be realized as laboratory-made or naturally occurring systems. With the binding energy as measure, we discuss possibilities of physical realization in nuclei, cold atoms, and condensed-matter systems. Appearance is not excluded. However, both the form and the strengths of the interactions must be rather special. The most promising subfields for present searches would be in cold atoms because of external control of effective interactions, or perhaps in condensed-matter systems with non-local interactions. In nuclei, it would only be by sheer luck due to a lack of tunability.
DOI: 10.7868/S0044002714030027
We cordially dedicate this work to Vladimir B. Belyaev on the occasion of his 80th birthday
1. MOTIVATION
Tying knots has from ancient times been a very practical skill to master. Two circular closed strings (rings) can be tied together and only removed from each other after cutting one of them. Such arrangements are called linked geometries. With more than two rings numerous possibilities arise for distinctly different topological structures. If one part cannot be removed from another part without untying or cutting, these parts are called linked and, if removable, called unlinked. The mathematical classification and description of such geometric or topological structures are called knot theory, and it has a hundred year long history.
One of the simplest cases is obviously three rings. They can be arranged so that they are linked, but if any one is cut and removed, the two remaining ones can be removed without further cutting. This is illustrated in Fig. 1. This construction is now often called Borromean rings, since they are the heraldic symbol of the dukes of Borromeo. However, the structure was fascinating already in the Norse mythology, shown as triangles and known as Odins knot, and in various
^Department of Mathematical Sciences, NTNU, Trondheim, Norway.
2)Department of Physics and Astronomy, Aarhus University, Aarhus, Denmark.
disguises especially in religious contexts throughout history. Not surprisingly, psychology (emphasized by for instance Jacques Lacan) could use it as symbols of interlocked triads like past, present, and future, etc.
In science, the Borromean rings have triggered several generalizations. The present paper discusses whether a new family of generalized structures may have counterparts in physics. We shall start by giving a brief review of previous work. Several early papers [1,2] preceeded the first mathematically rigorous treatments [3, 4] in the 1960's. Since then links of multiplicity n, essentially n (deformed) rings in three-dimensional space, are called Brunnian if they form a connected structure where all sybsystems are unlinked.
A link is of type B(n, k) if it consists of n rings in such a way that any subsystem of m rings with m < k is unlinked. For k = 2 the link is called Borromean, for k = n — 1 it is called Brunnian. If n = 3, the two notions coincide. The notion of B(n,k) links was introduced in [5, 6] extending previous notions in [7].
Further generalizations of higher-order links — in particular, higher-order Brunnian links were made in [5] as special case of higher-order structures introduced in [8], expanded in [9]. First-order structures are for example Brunnian links, second-order structures are Brunnian rings of Brunnian rings, etc., see Figs. 1—3. See [5] for more illustrations of higherorder links.
We shall focus on the occurence of related higherorder structures in physics, but should point out
Fig. 1. Illustration of the Borromean rings which are held together only by the presence of all three. If any one of the rings is cut open, the entire structure falls apart.
first that more direct examples may be found in chemistry or biology. Molecular rings with nonstandard topologies have been contemplated for a long time [10] and may in certain cases have different properties (such as optical activity). Biological molecules are known to form long chains of folded structures, often with knots. Borromean rings have been constructed from DNA [11] and even high-order topological structures have been envisaged [6].
In [5] it was proposed that higher-order links, for example, of Brunnian type may suggest new physical states. The concept of Borromean systems was introduced in microscopic physics [12] based not on geometrical linking (topology), but on binding energy. We shall in a similar manner take over all of the above topological concepts by replacing geometrically (un)linked by energetically (un)bound. Note that since the exact geometrical configurations now are of less direct importance, we can (and will) also consider two-dimensional structure rather than confining ourselves to three spatial dimensions. In the following section we shall give a general discussion of the possibilities for having higher-order Brunnian systems. This is followed by more specific looks at atomic, condensed-matter, and nuclear systems.
2. PHYSICAL REALIZATIONS OF HIGHER-ORDER BRUNNIAN SYSTEMS
In order to realize higher-order Brunnian systems in physical setups, one needs to consider multi-particle dynamics. For concreteness, we will focus here on the case of a second-order Brunnian state built from nine particles. At first we discuss the basic Hamiltonian setup and the interactions that one can expect. We then go on to consider subfields of physics where such a system might be accessible. This includes ultracold atomic gases of both nonpolar and polar single- and two-species gases and
a
Fig. 2. Brunnian rings of first and second order: (a) a first order Brunnian ring, (b) a second order Brunnian ring.
nuclear systems. We also make some comments about potential relevance of higher-order Brunnian structures in traditional condensed-matter and solidstate systems.
Assume that we have three species of particles, a, b, c, which could be distinguished by either internal quantum numbers (f.x. electron spin or hyperfine spin in neutral atoms) or external quantum numbers (f.x. different mass particle). At this point we do not assume anything about the quantum statistics of the different species and they can as such be either fermionic, bosonic, or a mixture of the two. We assume that the system has two-body interaction terms between the same and different species, which we denote Vaa, Vbb, Vcc, Vab, Vbc, Vac with the subscript indicating the particular pair of particles. These interactions are indicated on Fig. 3.
In order to have a second-order Brunnian system of nine particles, we need to have three Borromean
Fig. 3. A possible second-order Brunnian system with nine particles grouped into three different species, a, b, and c. The assumption is that each species holds a Borromean three-body bound state (indicated by the black rings). The intra-species interactions are denoted Vaa, Vbb, and Vcc, while the inter-species are Vab, Vbc, and Vac. In an effective description, we assume that we have three clusters that interact through effective interactions Vab, Vbc, and Vac .
three-body systems, that are three bound three-body subsystems wherein no two-body interaction supports a bound state. This is well known to occur on the weakly-interacting side of a Feshbach resonance in cold atomic gas systems that display the Efimov effect [13]. Numerically, such a state can be produced by using model potentials such as Gaussian or Yukawa-type interactions with interaction ranges that are short compared to the overall size of the Borromean three-body states [14]. In this type of setup we are in a sense working effectively with three Borromean systems and ask when those three will form a bound structure (see Fig. 3). The idea is then that the effective interaction between these three-body Borromean clusters should support a Borromean state, i.e. if you remove one cluster, you break the system into three unbound three-body states where each of the three-body systems forms a Borromean state.
The total potential is
v = £ ^ >
i>j
(1)
the idea above that one should be able to describe the system as a clusterized structure with three Borromean three-body systems, we now restructure the Hamiltonian, H. Following the schematics in Fig. 3, we denote the interactions between each of the clusters as Vab, Vbc , and Vac, which generally depend on the two-body interactions with the lower case subscripts, Vij. For a well-developed cluster structure it would be a good approximation to decompose the total wave function as ty = §ABC, where
(i = A, B, C) satisfies
(Ti + V)h = Ei<fii
(3)
where Ti is the relative kinetic energy operator of the corresponding three-body system, and VA = = Vaa (12) + Vaa (23) + Vaa (13) denotes the potential of all pairs within the three-body cluster containing species a, and likewise for B and C. The "relative" wave function between the three clusters, $ABC, satisfies the equation
(Tabc + Vab + Vbc + Vac )&abc = (4)
= EABC&ABC,
where
TABC = —
H2 2MA
V pia —
K,
H2 2MB
V pb —
ti,
(5)
h2
- ÏCM - 77TT^RC 2Mc rcm
where 1 < i,j < 9 and Vij denotes the interaction of the ith and jth particles which generally depend on the relative distance of the two particles. The Schrodinger equation in turn becomes
H V:=(T + V)^ = Ety, (2)
where ty is the total (nine-particle) wave function, E is the energy, and T is the relative kinetic ener
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