From One to Two Dimensions in the Weak Coupling Limit

H. Schweitzer, G. Czycholl

DAVID M. VAN DER WALT, R. M. Quick, MANUEL DE LLANO

Analytic expressions for the Bardeen–Cooper–Schrieffer (BCS) gap of a many-electron system with the BCS model interaction are obtained in one, two, and three dimensions in the weak coupling limit, but for arbitrary interaction width ν≡ℏωD/EF, 0<ν<∞. Here ℏωD is the maximum energy of a force-mediating boson, and EF is the Fermi energy (which is fixed by the electronic density). Our results address both phononic (ν≪1) as well as nonphononic (e.g., exciton, magnon, plasmon, etc.) pairing mechanisms where the mediating boson energies are not small compared with EF, provided weak electron–boson coupling prevails. The essential singularity in coupling, sometimes erroneously attributed to the two-dimensional character of the BCS model interaction with ν≪1, is shown to appear in one, two, and three dimensions before the limit ν→0 is taken.

Luigi Accardi, Robert Alicki, A. Frigerio +1 more

Luigi Accardi, A. Frigerio, Yun Gang Lu

James K. Freericks

The local approximation is applied to the spinless Falicov-Kimball model in one and two dimensions. The local approximation may be viewed as a zeroth-order expansion in 1/d that is exact in the weak-coupling limit. The local approximation rapidly becomes inaccurate as a function of interaction strength in one dimension, but is accurate in two dimensions. This result indicates that the 1/d expansion converges rapidly for the Falicov-Kimball model in d\ensuremath{\ge}2 and that exact solutions of many-body problems in infinite dimensions are quantitatively relevant for physical dimensions.

Tomohiro Tanaka, Yusuke Nishida

Transport coefficients are typically divergent for quantum integrable systems in one dimension, such as a Bose gas with a two-body contact interaction. However, when a one-dimensional system is realized by confining bosons into a tight matter waveguide, an effective three-body interaction inevitably arises as leading perturbation to break the integrability. This fact motivates us to study the thermal conductivity of a Bose gas in one dimension with both two-body and three-body interactions. In particular, we evaluate the Kubo formula exactly to the lowest order in perturbation by summing up all contributions that are naively higher orders in perturbation but become comparable in the zero-frequency limit due to the pinch singularity. Consequently, a self-consistent equation for a vertex function is derived, showing that the thermal conductivity in quasi-one-dimension is dominated by the three-body interaction rather than the two-body interaction. Furthermore, the resulting thermal conductivity in the weak-coupling limit proves to be identical to that computed based on the quantum Boltzmann equation and its temperature dependence is numerically determined.