Atoms and Molecules in Restricted Geometries at Low Temperatures

The spherical model is solved on a hypercubic lattice in $d$ dimensions, each bond of which is decorated with $l$ spins. The thermodynamic functions and the helicity modulus, analogous to a superfluid density, are calculated. We find that at least two spherical fields are required for the model to exhibit low-temperature properties that can approximate reasonably those of $O(n)$ models. The heuristic prediction that the critical temperature behaves as ${T}_{c}(l)\ensuremath{\sim}{(l+1)}^{\ensuremath{-}1}$ is checked for the model and found to hold quite accurately even for small $l(\ensuremath{\gtrsim}2)$. The helicity modulus and magnetization of the two-constraint spherical model are found to scale approximately with the critical temperature, but the relation between them is more complex than in the undecorated model. This relation is used to check heuristic arguments concerning the helicity modulus at low temperatures. We comment on the relevance to physical systems, in particular, the problem of boson condensation in a restricted geometry.The spherical model is solved on a hypercubic lattice in $d$ dimensions, each bond of which is decorated with $l$ spins. The thermodynamic functions and the helicity modulus, analogous to a superfluid density, are calculated. We find that at least two spherical fields are required for the model to exhibit low-temperature properties that can approximate reasonably those of $O(n)$ models. The heuristic prediction that the critical temperature behaves as ${T}_{c}(l)\ensuremath{\sim}{(l+1)}^{\ensuremath{-}1}$ is checked for the model and found to hold quite accurately even for small $l(\ensuremath{\gtrsim}2)$. The helicity modulus and magnetization of the two-constraint spherical model are found to scale approximately with the critical temperature, but the relation between them is more complex than in the undecorated model. This relation is used to check heuristic arguments concerning the helicity modulus at low temperatures. We comment on the relevance to physical systems, in particular, the problem of boson condensation in a restricted geometry.

Physical review. B, Condensed matterAug 1, 198212 citations

New possibilities for obtaining cold atoms and molecules

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The spherical model is solved on a hypercubic lattice in

d

dimensions, each bond of which is decorated with

l

spins. The thermodynamic functions and the helicity modulus, analogous to a superfluid density, are calculated. We find that at least two spherical fields are required for the model to exhibit low-temperature properties that can approximate reasonably those of

O(n)

models. The heuristic prediction that the critical temperature behaves as ${T}_{c}(l)\ensuremath{\sim}{(l+1)}^{\ensuremath{-}1}$ is checked for the model and found to hold quite accurately even for small $l(\ensuremath{\gtrsim}2)$. The helicity modulus and magnetization of the two-constraint spherical model are found to scale approximately with the critical temperature, but the relation between them is more complex than in the undecorated model. This relation is used to check heuristic arguments concerning the helicity modulus at low temperatures. We comment on the relevance to physical systems, in particular, the problem of boson condensation in a restricted geometry.