Renormalization and Universality of Van der Waals forces?

E. Ruiz Arriola, A. Calle Cordón

Published Apr 12, 2010• 3 citations

Abstract

\n Renormalization ideas can proﬁtably be exploited in conjunction with the\n superposition principle of boundary conditions in the description of model\n independent and universal scaling features of the singular and long range Van\n der Waals force between neutral atoms. The dominance of the leading power law is\n highlighted both in the scattering as well as in the bound state problem. The\n role of oﬀ-shell two-body unitarity and causality within the Eﬀective Field\n Theory framework on the light of universality and scaling at low energies is\n analyzed.\n

DOI: 10.1051/epjconf/20100302005/pdf

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Renormalization and Universality of Van der Waals forces?

E. Ruiz Arriola, A. Calle Cordón

Published Apr 12, 2010• 3 citations

Publisher

Abstract

DOI: 10.1051/epjconf/20100302005/pdf

Related papers

Renormalization and Universality of Van der Waals forces

E. Ruiz Arriola, A. Calle Cordón

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At long distances, interactions between neutral ground-state atoms can be described by the van der Waals potential. In the ultracold regime, atom-atom scattering is dominated by

s

-waves phase shifts given by an effective range expansion in terms of the scattering length ${\ensuremath{\alpha}}_{0}$ and the effective range

{r}_{0}

. We show that while the scattering length cannot be predicted for these potentials, the effective range is given by the universal low-energy theorem ${r}_{0}=A+B/{\ensuremath{\alpha}}_{0}+C/{\ensuremath{\alpha}}_{0}^{2}$, where

A

B

, and

C

depend on the dispersion coefficients

{C}_{n}

and the reduced diatom mass. We confront this formula to about 100 determinations of

{r}_{0}

and ${\ensuremath{\alpha}}_{0}$ and show why the result is dominated by the leading dispersion coefficient

{C}_{6}

. Universality and scaling extend much beyond naive dimensional analysis estimates.

We analyze low-energy scattering for arbitrary short-range interactions plus an attractive

1/{r}^{6}

tail. We derive the constraints of causality and unitarity and find that the van der Waals length scale dominates over parameters characterizing the short-distance physics of the interaction. This separation of scales suggests a separate universality class for physics characterizing interactions with an attractive

1/{r}^{6}

tail. We argue that a similar universality class exists for any attractive potential $1/{r}^{\ensuremath{\alpha}}$ for $\ensuremath{\alpha}\ensuremath{\geqslant}2$. We also discuss the extension to multichannel systems near a magnetic Feshbach resonance. We discuss the implications for effective field theory with attractive singular power-law tails.

We present a variational Monte Carlo formulation for the universal equations of state at the van der Waals length scale [B. Gao, J. Phys. B 37, L227 (2004)] for

N

Bose atoms in a trap. The theory illustrates both how such equations of state can be computed exactly, and the existence and the importance of long-range atom-atom correlation under strong confinement. Explicit numerical results are presented for

N=3

and 5, and used to provide a quantitative understanding of the shape-dependent confinement correction that is important for few atoms under strong confinement.