Ground states and dynamical properties of a dipolar Bose–Einstein condensate are analyzed based on the Gross–Pitaevskii–Poisson system (GPPS) and its dimension reduction models under an anisotropic confining potential. We begin with the three-dimensional (3D) GPPS and review its quasi-two-dimensional (2D) approximate equations when the trap is strongly confined in the z-direction and quasi-one-dimensional (1D) approximate equations when the trap is strongly confined in the x- and y-directions. In fact, in the quasi-2D equations, a fractional Poisson equation with the operator $(-\Delta)^{1/2}$ is involved which brings significant difficulties into the analysis. Existence and uniqueness as well as nonexistence of the ground state under different parameter regimes are established for the quasi-2D and quasi-1D equations. Well-posedness of the Cauchy problem for both types of equations and finite time blow-up in two dimensions are analyzed. Finally, we rigorously prove the convergence with linear convergence rate for the solutions of the 3D GPPS and its quasi-2D and quasi-1D approximate equations in the weak interaction regime.Ground states and dynamical properties of a dipolar Bose–Einstein condensate are analyzed based on the Gross–Pitaevskii–Poisson system (GPPS) and its dimension reduction models under an anisotropic confining potential. We begin with the three-dimensional (3D) GPPS and review its quasi-two-dimensional (2D) approximate equations when the trap is strongly confined in the z-direction and quasi-one-dimensional (1D) approximate equations when the trap is strongly confined in the x- and y-directions. In fact, in the quasi-2D equations, a fractional Poisson equation with the operator $(-\Delta)^{1/2}$ is involved which brings significant difficulties into the analysis. Existence and uniqueness as well as nonexistence of the ground state under different parameter regimes are established for the quasi-2D and quasi-1D equations. Well-posedness of the Cauchy problem for both types of equations and finite time blow-up in two dimensions are analyzed. Finally, we rigorously prove the convergence with linear convergence rate for the solutions of the 3D GPPS and its quasi-2D and quasi-1D approximate equations in the weak interaction regime.

arXiv:1212.4945SIAM Journal on Mathematical AnalysisJan 1, 201234 citations

Ground-State and Dynamical Properties of Two-Dimensional Systems

Dmitrii Michail Kachintsev

Nov 1, 19950 citations

Quasi-one-dimensional systems

Douglas J. Scalapino

Springer eBooksOct 29, 20051 citation

Incommensurability enabled quasi-fractal order in 1D narrow-band moiré systems

Miguel Gonçalves, Bruno Amorim, Flavio Elias Riche +2 more

Abstract We demonstrate that quasiperiodicity can radically change the ground state properties of 1D moiré systems with respect to their periodic counterparts. By studying an illustrative example we show that while narrow bands play a significant role in enhancing interactions both for commensurate and incommensurate structures, only quasiperiodicity is able to extend the ordered phase down to an infinitesimal interaction strength. In this regime, the quasiperiodic-enabled state has contributions from infinitely many wave vectors. This quasi-factal regime cannot be stabilized in the commensurate case even in the presence of a narrow band. These findings suggest that quasiperiodicity may be a critical factor in stabilizing non-trivial ordered phases in interacting moiré structures and signal out multifractal non-interacting phases, recently found in 2D incommensurate moiré systems, as particularly promising parent states.Abstract We demonstrate that quasiperiodicity can radically change the ground state properties of 1D moiré systems with respect to their periodic counterparts. By studying an illustrative example we show that while narrow bands play a significant role in enhancing interactions both for commensurate and incommensurate structures, only quasiperiodicity is able to extend the ordered phase down to an infinitesimal interaction strength. In this regime, the quasiperiodic-enabled state has contributions from infinitely many wave vectors. This quasi-factal regime cannot be stabilized in the commensurate case even in the presence of a narrow band. These findings suggest that quasiperiodicity may be a critical factor in stabilizing non-trivial ordered phases in interacting moiré structures and signal out multifractal non-interacting phases, recently found in 2D incommensurate moiré systems, as particularly promising parent states.

arXiv:2305.03800Nature PhysicsOct 11, 202425 citations

Ground states in one-dimensional electron systems

W. Hansen, D. Schmerek, Chr. Steinebach

Physics-UspekhiFeb 28, 19987 citations

(-\Delta)^{1/2}

is involved which brings significant difficulties into the analysis. Existence and uniqueness as well as nonexistence of the ground state under different parameter regimes are established for the quasi-2D and quasi-1D equations. Well-posedness of the Cauchy problem for both types of equations and finite time blow-up in two dimensions are analyzed. Finally, we rigorously prove the convergence with linear convergence rate for the solutions of the 3D GPPS and its quasi-2D and quasi-1D approximate equations in the weak interaction regime.