具有O（n）对称破缺二次边界项的$n$分量$ϕ^4$模型的临界Casimir振幅
Critical Casimir amplitudes for $n$-component $ϕ^4$ models with O(n)-symmetry breaking quadratic boundary terms
摘要：
研究了在$L$厚度的薄膜中的欧几里得$n$ -分量$ϕ^4$理论，其哈密顿量除了二次对称破缺边界项外都是O（n）对称的。边界项意味着在$z=0$和$z=L$的边界平面$\mathfrak{B}_{j=1,2}$处的Robin边界条件$\partial_nϕ_α=\mathring{c}^{(j)}_αϕ_α$。特别注意$n$变量$\mathring{c}^{(j)}_α$的$m_j$取对应于临界增强的特殊值$\mathring{c}_{m_j\text{-sp}}$，而其余的是亚临界增强的情况。在这些条件下，以$\mathfrak{B}_j$为界的半无限系统有一个多临界点，称为$m_j$ -special，在该点上，如果$d$足够大，则$O(m_j)$对称临界表面相与O（n）对称体相共存.单位面积的约化自由能的$L$依赖部分在体临界点处表现为$Δ_C/L^{d-1}$和$L\to\infty$。在一般情况下，$m_{c,c}$分量$ϕ_α$在两个边界平面处都被临界增强，$m_{c,D} + m_{D,c}$分量在一个平面处被增强，但在相应的另一个平面处满足渐近狄利克雷边界条件，并且剩余的$m_{D,D}$分量在两个$\mathfrak{B}_j$处都满足渐近狄利克雷边界条件，确定小$ε=4-d$的卡西米尔振幅$Δ_C$。每当$m_{c,c}>0$时，这些展开涉及整数和分数幂$ε^{k/2}$与$k\ge 3$（模幂）。利用$O(ε^{3/2})$对$m_{c,c}$、$m_{c,D}+m_{D,c}$和$m_{D,D}$的一般值的计算结果，估算了具有表面自旋各向异性的三维海森堡系统在$(m_{c,c}, m_{c,D}+ m_{D,c}) = (1,0)$、$(0,1)$和$(1,1)$时的$Δ_C$。
英文摘要：
Euclidean $n$-component $ϕ^4$ 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 $\partial_nϕ_α=\mathring{c}^{(j)}_αϕ_α$ 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 $\mathring{c}^{(j)}_α$ 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 $Δ_C/L^{d-1}$ as $L\to\infty$ at the bulk critical point. The Casimir amplitudes $Δ_C$ are determined for small $ε=4-d$ in the general case where $m_{c,c}$ components $ϕ_α$ 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 $ε^{k/2}$ with $k\ge 3$ (mod logarithms). Results to $O(ε^{3/2})$ for general values of $m_{c,c}$, $m_{c,D}+m_{D,c}$, and $m_{D,D}$ are used to estimate the $Δ_C$ 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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