The project is supported from the programme POLONEZ BIS by the National Science Center and European Union (Grant No. 2022/45/P/ST3/04237).

Classical and quantum quasi-one dimensional (q1D) hard core systems are different in their characteristic physical scales and mathematical descriptions. However, both systems have similar geometry and hard core repulsion so that their studies can be mutually complementing. The main objective of the proposed project is to analytically study classical q1D hard sphere (HS) systems bearing in mind their possible implications for ultracold quantum gases. The motivation is that, on the one hand, incorporation of a pairwise interaction between quantum atoms beyond the delta functional form is a formidable and currently unresolved problem, while, on the other hand, this has become possible in q1D systems due to the recently found exact solution for a q1D systems of hard disks. This solution was obtained for noninteracting hard disks (HDs), but it shows the way to solvable models with interaction between next neighbors (NN) in a wider class of q1D (q2D or q3D) channels where disks are replaced by HSs. We will develop analytically solvable models of interacting classical HSs in two narrow q1D channels, solve these models, describe the orientaional and translational ordering, and establish ensuing phase diagrams. We will consider a plane q1D channel with HDs with a NN interaction (plane q2D system) and channel with a square cross-section with HSs (square q3D system) with and without interaction. The last geometry is particularly interesting. At high densities, the spheres’ centers tend to form a crystalline zigzag which, in a square channel, will be lying in the plane passing through its diagonal. But there are two equivalent diagonals and the system must spontaneously choose one of them. As a result, the symmetry would break and a phase transition liquid-crystalline zigzag is expected to occur. It is known that phase transitions in 1D systems with short range interaction do not exist. Our main hypotheses is that, in some q1D systems, they can exist. The transition type is an interesting question: usually, a spontaneous symmetry breaking is related to a second order continuous transition, but a crystal-liquid transformation is usually related to a discontinuous first order transition. The pair correlation functions in solvable models are expected to be derived analytically . We will especially focus on interactions that can result in dropletlike structures alike the observed in the quantum gases near their Bose condensation. The classical results can help to construct advanced mean field approaches to quantum droplets. The task 1 is to develop solvable models of interacting HDs in the simplest q1D channel of length L and width D where N disks’ centers stay in a 2D plane (plane q1D system). In the thermodynamic limit we will find an analytical expression for the partition function (PF) in the canonical NLT ensemble, equation of state, pair correlation functions, and search for a phase transition liquid state-crystalline zigzag. The task 2 is to develop and solve analytically a q1D model of interacting HSs in a square channel. The canonical PF, equation of state, and pair distribution functions will be studied analytically and numerically in collaboration with Prof. Andij Trokhymchuk and Prof. Taras Bryk from the Institute for Condensed Matter Physics (ICMP) of the Academy of Sciences of Ukraine. The task 3 is to find inhomogeneous states of q1D systems. To include the medium range 1/R 3 potential pertinent to dipolar Bose-Einstein condensates, we will extend the model with a NN interaction to a next-next neighbor (NNN) interaction by using the so-called transfer matrix method (TMM) proposed by Kofke and Post. The ultimate question in task 3 is how to esablish the physical connection of our classical results with the mean field Gross-Pitaevskii type approach to the quantum gases.

Developing and solving a q1D model of interacting hard disks.

Finding the partition function of a classical hard spheres system in a narrow channel with the square cross section. Establishing the role of spontaneous symmetry breaking.

Developing the two-liquid hydrodynamic approach to a cold quantum Bose gas. Incorporation of the slow hydrodynamic and fast locally stationary inner mode in the coarse-grained Schrödinger equation at T = 0.

Consider a quantum analog of hard spheres system in a narrow channel with the square cross section – a supersolid state made of ensemble of quantum droplets. Establishing the role of spontaneous symmetry breaking.