The overall goal of our research is to understand the interplay of few- and many-body physics by asking when we may apply the formalism designed for many and when we may apply formalism designed for few particles to a given system of interest. Challenges in both pure and applied physics involve a mixture of few- and many-body settings and a unified qualitative and quantitative approach is of great interest and utility.
On the right-hand side we show a sketch that illustrates the situation where bound states of a few particles are embedded in a larger system. This is a very typical case since microscopic systems are very rarely completely isolated neither in laboratory conditions or in technological applications. The situation on the left side shows a bound state with three particles (red spheres) that are bound to each other. This means that it takes a certain energy to extract the particles, i.e. to pull the system apart (just like the Earth-Sun system under the gravitational force). The system has more particles (shown as the blue spheres) but they are on average far away as compared to the distance between the particles that are bound together. On the right side, the situation is different since the average distance between all particles is now similar, irrespective of whether they are in the bound state or not. One may now ask whether the bound state picture makes sense at all. If we assume that the red and blue particles are actually physically the same then the blue or red particles may equally well participate in formation of bound states. If we imagine a large system for which we have (external) control over the density (and thus the average distance between particles) then we may interpolate between the situation on the left and on the right. The question then becomes when the bound state loses its unique identity as a bound state in a big system, or from the other side, when will a big system start to form small bound states as the density becomes lower?
The chart above illustrates some of the striking differences of the few- and many-body environments that we face. One of the most straightforward ways to see this is to look at the dimensionality of a system. Modern experiments and technological applications requires control of physical systems not only in three dimensions but also on two-dimensional planes and one-dimensional lines. If we consider first few-body physics, then quantum mechanics teaches us that to bind particles it is better to be in low dimension. In 3D, it takes a certain amount of attraction between two bodies before they will bind. This is no longer true in 1D and 2D where any attraction, however small, will do! On the other hand, if we are interested in many-body states such as a superconductor, a superfluid or a Bose-Einstein condensate, then 3D is much better in order for the state to remain coherent and for all the constituents to behave in unison. In 1D or 2D its is much harder to get such coherent behavior because fluctuations due to either thermal or quantum fluctuations are more prolific. This implies that having a true coherent state of a large system in a low-dimensional setup is, if not impossible, then extremely challenging.
A famous example from the theory of superconductivity is that of the Cooper pair, a bound pair of electrons that can only achieve binding due to the presence of a so-called Fermi sea of background electrons. The Fermi sea appears because of the Pauli principle which states that only one electron may occupy any one quantum state at any given time. The background electrons will thus fill up a lot of states and thus leave less room for the two remaining electrons that we are trying to bind. What happens is that we are actually trying to bind particle that sit on top of a sphere (the Fermi sea) and this reduces our problem from 3D to effectively only 2D (the 2D surface of the spherical Fermi sea). But in 2D bound states are easier to produce and any small attraction (provided by the ion lattice in the solid-state environment of typical superconductors) will bind the electrons into a Cooper pair.
What remains is still the question of fluctuations since the Fermi sea may have holes in that have to be taken into account. This was taken into account for the two-body Cooper pairs shortly after the theory of superconductivity was introduced in 1957. However, it is generally unknown what happens to bound states and to fluctuations as we proceed to consider three-body systems in the presence of a many-body background such as a Fermi sea as in the example above.
