Monte Carlo renormalization of hard sphere polymer chains in two to five dimensions

A renormalization group for polymer chains with hard-core interaction is considered, where a chain ofN ₀ links of lengthl ₀ and hard-core diameterh ₀ is mapped onto a chain ofN ₁=N ₀/s links of lengthl ₁ and hard-core diameterh ₁. The lengthl ₁ is defined in terms of suitable interior distances of the original chain, andh ₁ is found from the condition that the end-to-end distance is left invariant. This renormalization group procedure is carried through by various Monte-Carlo methods (simple sampling is found advantageous for short enough chains or high dimensionalities, while dynamic methods involving kinkjumps or reptation are used else). Particular attention is paid to investigate systematic errors of the method by checking the dependence of the results on bothN ₀ ands. It is found that for dimensionalitiesd=2, 3 only the nontrivial fixed-point is stable, where upon iteration the ratio _k =h _k /l _k tends to nonzero fixed-point value ^*, while ford=4,5 the method converges to the gaussian fixed point with ^*=0. Taking both statistical and systematic errors into account, we estimate the exponentv asv=0.74±0.01 (d=2) andv =0.59±0.01 (d=3). The results are consistent with the expected crossover exponents =1/2 (d=3) and =1 (d=2), respectively.