Critical Casimir amplitudes for n-component φ models with O(n)-symmetry breaking quadratic boundary terms

Euclidean n-component φ theories whose Hamiltonians areO(n) symmetric except for quadratic symmetry breaking boundary terms are studied in the film geometry R×[0, L]. The boundary terms imply the Robin boundary conditions ∂nφα = c̊ (j) α φα at the boundary planes Bj=1 at z = 0 and Bj=2 at z = L. Particular attention is paid to the cases in which mj of the n variables c̊ (j) α associated with plane Bj take the special value c̊mj -sp corresponding to critical enhancement while the remaining ones are larger and hence subcritically enhanced. Under these conditions, the semi-infinite system with boundary plane Bj has a multicritical surface-bulk point, called mjspecial, at which an O(mj) symmetric critical surface phase coexists with the O(n) symmetric bulk phase, provided d is sufficiently large. The L-dependent part of the reduced free energy per cross-section area behaves asymptotically as ∆C/L d−1 as L → ∞ at the bulk critical point. The Casimir amplitudes ∆C are determined for small ǫ = 4− d in the general case where mc,c components φα are critically enhanced at both boundary planes, mc,D +mD,c components are enhanced at one plane but satisfy asymptotic Dirichlet boundary conditions at the respective other, and the remaining mD,D components satisfy asymptotic Dirichlet boundary conditions at both Bj . Whenever mc,c > 0, the corresponding small-ǫ expansions involve, besides integer powers of ǫ, also fractional powers ǫ with k ≥ 3 modulo powers of logarithms. Results to order ǫ are given for general values of mc,c, mc,D + mD,c, and mD,D. These are used to estimate the Casimir amplitudes ∆C of the three-dimensional Heisenberg systems with surface spin anisotropies for the cases with (mc,c,mc,D +mD,c) = (1, 0), (0, 1), and (1, 1).