Piotr Szulim | Searcharxiv

arXiv2022

Atoms in a spin dependent optical potential: ground state topology and magnetization

We investigate a Bose-Einstein condensate of $F= 1$ $^{87}$Rb atoms in a 2D spin-dependent optical lattice generated by intersecting laser beams with a superposition of polarizations. For $^{87}$Rb the effective interaction of an atom with the electromagnetic field contains a scalar and a vector (called as fictitious magnetic field, $B_{fic}$) potentials. The Rb atoms behave as a quantum rotor (QR) with angular momentum given by the sum of the atomic rotational motion angular momentum and the hyperfine spin. The ground state of the QR is affected upon applying an external magnetic field, $B_{ext}$, perpendicular to the plane of QR motion and a sudden change of its topology occurs as the ratio $B_{ext}/B_{fic}$ exceeds critical value. It is shown that the change of topology of the QR ground state is a result of combined action of Zeeman and Einstein-de Haas effects. The first transfers atoms to the largest hyperfine component to polarize the sample along the field as the external magnetic field is increased. The second sweeps spin to rotational angular momentum, modifying the kinetic energy of the atoms.

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arXiv2019

Integrals in left coideal subalgebras and group-like projections

We develop a theory of right group-like projections in Hopf algebras linking them with the theory of left coideal subalgebras with two sided counital integrals. Every right group-like projection is associated with a left coideal subalgebra, maximal among the ones containing the given group-like projection as an integral, and we show that that subalgebra is finite dimensional. We observe that in a semisimple Hopf algebra $H$ every left coideal subalgebra has an integral and we prove a 1-1 correspondence between right group-like projections and left coideal subalgebras of $H$. We provide a number of equivalent conditions for a right group-like projections to be left group-like projection and prove a 1-1 correspondence between semisimple left coideal subalgebras preserved by the squared antipode and two sided group-like projections. We also classify left coideal subalgebras in Taft Hopf algebras $H_{n^2}$ over a field $\mathbb{k}$, showing that the automorphism group splits them into - a class of cardinality $|\mathbb{k}|-1$ of semisimple ones which correspond to right group-like projections which are not two sided; - finitely many semisimple singletons, each corresponding to two sided group-like projection; the number of those singletons for $H_{n^2}$ is equal to the number of divisors of $n$; - finitely many singletons, each non-semisimple and admitting no right group-like projection; the number of those singletons for $H_{n^2}$ is equal to the number of divisors of $n$. In particular we answer the question of Landstad and Van Daele showing that there do exist right group-like projections which are not left group-like projections.

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