Arithmetical finite-size corrections in one-dimensional quantum systems

N. Sh. Izmailian, Chin‐Kun Hu

N. M. Bogoliubov, Anatoli G. Izergin, Nicolai Reshetikhin

Barbara Lulek

The kinematics of the one-dimensional finite Heisenberg magnetic ring with impurities is discussed in the light of the general recipe of Weyl (1952). The cyclic group generating the ring, and the group of all its automorphisms, play the roles of the obvious and hidden symmetry respectively. The hidden symmetry group imposes a property of the distribution of quantum states of a translationally invariant ensemble of magnets over the finite Brillouin zone. Inhomogeneities of the distribution arise from irregular orbits of the action of the cyclic group on the set of all magnetic configurations of the ensemble. Size-dependent effects, like rarefied bands, are classified using the arithmetic structure of integers, (prime numbers, socles and arithmetic exponents), which is more appropriate than the linear order in the ring of integers.

Elías Castellanos

We analyze the corrections caused by finite size effects upon the ground state properties of a homogeneous one-dimensional (1D) Bose–Einstein condensate. We assume from the very beginning that the Bogoliubov’s formalism is valid and consequently, we show that in order to obtain a well-defined ground state properties, finite size effects of the system must be taken into account. Indeed, the formalism described in the present paper allows to recover the usual properties related to the ground state of a homogeneous 1D Bose–Einstein condensate but corrected by finite size effects of the system. Finally, this scenario allows us to analyze the sensitivity of the system when the Bogoliubov’s regime is valid and when finite size effects are present. These facts open the possibility to apply these ideas to more realistic scenarios, e.g. low-dimensional trapped Bose–Einstein condensates.

Georg Junker

Jan Philip Solovej