Critical Casimir amplitudes for $n$-component ${\varphi }^4$ models with O(n)-symmetry breaking quadratic boundary terms

Abstract

Euclidean $n$-component ${\varphi }^4$ 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 ${\partial }_n{\varphi }_{\alpha }={\mathring{c}}_{\alpha }^{(j)}{\varphi }_{\alpha }$ 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 ${\mathring{c}}_{\alpha }^{(j)}$ 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 ${\varphi }_{\alpha }$ 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 $(1,1)$.

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Cited by in corpus (3)

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