Critical Casimir amplitudes for n-component ϕ4 models with O(n)-symmetry breaking quadratic boundary terms
Abstract
Euclidean n-component ϕ theories whose Hamiltonians are O(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 ∂ϕ=c˚α(j)ϕ at the boundary planes B at z=0 and B at z=L. Particular attention is paid to the cases in which m of the n variables c˚α(j) associated with plane B take the special value c-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 B has a multicritical surface-bulk point, called m-special, at which an O(m) 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 Δ/L as L→∞ at the bulk critical point. The Casimir amplitudes Δ are determined for small ɛ=4-d in the general case where m components ϕ are critically enhanced at both boundary planes, m+m components are enhanced at one plane but satisfy asymptotic Dirichlet boundary conditions at the respective other, and the remaining m components satisfy asymptotic Dirichlet boundary conditions at both B. Whenever m>0, the corresponding small-ɛ expansions involve, besides integer powers of ɛ, also fractional powers ɛ with k⩾3 modulo powers of logarithms. Results to order ɛ are given for general values of m, m+m, and m. These are used to estimate the Casimir amplitudes Δ of the three-dimensional Heisenberg systems with surface spin anisotropies for the cases with (m,m+m)=(1,0), (0,1), and (1,1).
- Publication:
-
Nuclear Physics B
- Pub Date:
- December 2009
- DOI:
- 10.1016/j.nuclphysb.2009.07.010
- arXiv:
- arXiv:0905.3113
- Bibcode:
- 2009NuPhB.822..517D
- Keywords:
-
- 05.70.Jk;
- 68.35.Rh;
- 11.10.Gh;
- 11.10.Kk;
- 75.30.Gw;
- 64.60.Kw;
- Critical point phenomena;
- Phase transitions and critical phenomena;
- Renormalization;
- Field theories in dimensions other than four;
- Magnetic anisotropy;
- Multicritical points;
- Condensed Matter - Statistical Mechanics;
- Condensed Matter - Other Condensed Matter;
- High Energy Physics - Theory
- E-Print:
- Latex source file with 5 eps files