We investigate the self-organization of strongly interacting particles confined in 1D and 2D. We consider hardcore bosons in spinless Hubbard lattice models with short range interactions. We show that, many-body orders with topological characteristics emerge, at different energy bands separated by large gaps. These topological orders manifest in the way the particles organize in real space to form states with different energy. Each of these states contains topological defects/condensations whose Euler characteristic can be used as a topological number to categorize states belonging to the same energy band. We provide analytical formulas for this topological number and the full energy spectrum of the system for both sparcly and densely filled systems. Furthermore, we discuss the connection with the Gauss-Bonnet theorem of differential geometry, by using the curvature generated in real space by the particle structures. Our result is a demonstration of how topological orders can arise in many-body systems with simple underlying rules, without considering the spin or long-range microscopic interactions.
In the framework of adiabatic approximation the energy spectrum and wave functions of two impenetrable particles in an infinitely deep potential well are considered for two cases of approximation of the effective confining potential of the “slow” subsystem. In case of the quadraticterm approximation the obtained energy spectrum is equidistant. The probability distribution in the range of “fast” particle has a symmetric shape while that in the range of the “slow” particle is asymmetric and the peak of localization of the system in its ground state is shifted towards the “fast” particle. In the first excited state the center of the probability distribution of the “slow” particle is shifted towards the impenetrable wall.
The density fluctuation in a Bose gas interacting via a strong short-range repulsive potential and a weak long-range attractive potential is studied by the methods of quantum-field theory. The density-density correlation function is generated by taking the functional derivative of the single-particle matrix Green's function
G1(1,2) with respect to an auxiliary scalar field. For
G1(1,2), we use the Hartree approximation for the attractive potential and the Hartree-Fock approximation for the short-range repulsive potential. We show that the density fluctuation spectrum is identical to the Beliaev approximation for the single-particle excitation spectrum.