Finite-size scaling behavior in trapped systems

Understand

Numerical transfer-matrix methods are applied to two-dimensional Ising spin systems, in presence of a confining magnetic field which varies with distance $|{\vec x}|$ to a "trap center", proportionally to $(|{\vec x}|/\ell)^p$, $p>0$.

• On a strip geometry, the competition between the "trap size" $\ell$ and the strip width, $L$, is analysed in the context of a generalized finite-size scaling {\em ansatz}.
• In the low-field regime $\ell \gg L$, 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 $\ell \lesssim L$, correlation-length scaling data broadly confirms an existing picture of $p$-dependent characteristic exponents.