Euclidean $n$-component $\phi^4$ theories whose Hamiltonians are O(n) symmetric except for quadratic symmetry breaking boundary terms are studied in films of thickness $L$. The boundary terms imply the Robin boundary conditions $\partial_n\phi_\alpha =\mathring{ c}^{ (j)}_\alpha \phi_\alpha $ at the boundary planes $\mathfrak{ B}_{ j=1,2}$ at $z=0$ and $z=L$. Particular attention is paid to the cases in which $m_j$ of the $n$ variables $\mathring{ c}^{ (j)}_\alpha$ take the special value $\mathring{ c}_{ m_j\text{ -sp}}$ corresponding to critical enhancement while the remaining ones are subcritically enhanced. Under these conditions, the semi-infinite system bounded by $\mathfrak{ B}_j$ has a multicritical point, called $m_j$-special, at which an $O(m_j)$ 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 area behaves as $\Delta_C/L^{ d-1}$ as $L\to\infty$ at the bulk critical point. The Casimir amplitudes $\Delta_C$ are determined for small $\epsilon=4-d$ in the general case where $m_{ c,c}$ components $\phi_\alpha$ are critically enhanced at both boundary planes, $m_{ c,D} + m_{ D,c}$ components are enhanced at one plane but satisfy asymptotic Dirichlet boundary conditions at the respective other, and the remaining $m_{ D,D}$ components satisfy asymptotic Dirichlet boundary conditions at both $\mathfrak{ B}_j$. Whenever $m_{ c,c}>0$, these expansions involve integer and fractional powers $\epsilon^{ k/2}$ with $k\ge 3$ (mod logarithms). Results to $O(\epsilon^{ 3/2})$ for general values of $m_{ c,c}$, $m_{ c,D}+m_{ D,c}$, and $m_{ D,D}$ are used to estimate the $\Delta_C$ of 3D Heisenberg systems with surface spin anisotropies when $(m_{ c,c}, m_{ c,D}+ m_{ D,c}) = (1,0)$, $(0,1)$, and $(1,1)$.
Categories: cond-mat.stat-mech cond-mat.other hep-th