Finite-size scaling behavior in trapped systems

Numerical transfer-matrix methods are applied to two-dimensional Ising spin systems, in the presence of a confining magnetic field which varies with distance |x| to a "trap center," proportionally to (|x|/ℓ)p, where p>0 . On a strip geometry, the competition between the "trap size" ℓ and the strip width L is analyzed in the context of a generalized finite-size scaling ansatz. In the low-field regime ℓ >> L, we use conformal-invariance concepts in conjunction with a linear-response approach to derive the appropriate (p-dependent) limit of the theory, which agrees very well with numerical results for magnetization profiles. For high fields ℓ ≲ L, correlation-length scaling data broadly confirm an existing picture of p-dependent characteristic exponents. Standard spin-1/2 and spin-1 Ising systems are considered, as well as the Blume-Capel model.