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arXiv 1601.04760hep-th
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具有Robin边界条件的对数极小模型

Logarithmic Minimal Models with Robin Boundary Conditions

Jean-Emile Bourgine, Paul A. Pearce, Elena Tartaglia

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摘要：

我们考虑一般的对数极小模型${\cal LM}(p,p')$，$p,p'$互质，在一条$N$列的$(r,s)$罗宾边界条件引入皮尔斯，Rasmussen和Tipunin。相关的共形边界条件由Kac标签$r\in{\Bbb Z}$和$s\in{\Bbb N}$标记。Robin真空边界条件，记作$(r,s\!-\!\frac{1}{2})=(0,\mbox{$ \textstyle \frac{1}{2} $})$，是Neumann和Dirichlet边界条件的线性组合。利用融合方法，通过作用于带有$(r,s)$型缝的Robin真空边界，构造了一般的$(r,s)$ Robin边界条件，该缝由宽度为$w$列的$r$型缝和宽度为$d=s-1$列的$s$型缝组成. $r$型接缝允许一个任意的边界场，我们将其固定为特殊值$ξ=-\tfracλ{2}$，其中$λ=\frac{(p'-p)π}{2p'}$是交叉参数。$s$型边界将$d$缺陷引入到块中。我们认为相关的量子哈密顿和计算分析的$(r,s)$罗宾边界条件的边界自由能。使用有限尺寸的校正和序列外推到系统尺寸$N+w+d\le 26$，边界算子的共形谱可以通过数值对角化的哈密顿算子来获得。固定$r\ne 0$的$N$的奇偶性，并限制到基态序列$w=\big\lfloor\frac{|r|p'}{p}\big\rfloor$，$r\in{\Bbb Z}$与逆$r=(-1)^{N+w+d}\big\lceil \frac{p w}{p'}\big\rceil$，我们发现共形权重取值$Δ^{p,p'}_{r,s-\frac12}$，其中$Δ^{p,p'}_{r,s}$由通常的Kac公式给出。因此，$(r,s)$ Robin边界条件与Kac标签$s-\mbox{$ \textstyle \frac{1}{2} $}$的具有半整数值的标度算子共轭。

英文摘要：

We consider general logarithmic minimal models ${\cal LM}(p,p')$, with $p,p'$ coprime, on a strip of $N$ columns with the $(r,s)$ Robin boundary conditions introduced by Pearce, Rasmussen and Tipunin. The associated conformal boundary conditions are labelled by the Kac labels $r\in{\Bbb Z}$ and $s\in{\Bbb N}$. The Robin vacuum boundary condition, labelled by $(r,s\!-\!\frac{1}{2})=(0,\mbox{$\textstyle \frac{1}{2}$})$, is given as a linear combination of Neumann and Dirichlet boundary conditions. The general $(r,s)$ Robin boundary conditions are constructed, using fusion, by acting on the Robin vacuum boundary with an $(r,s)$-type seam consisting of an $r$-type seam of width $w$ columns and an $s$-type seam of width $d=s-1$ columns. The $r$-type seam admits an arbitrary boundary field which we fix to the special value $ξ=-\tfracλ{2}$ where $λ=\frac{(p'-p)π}{2p'}$ is the crossing parameter. The $s$-type boundary introduces $d$ defects into the bulk. We consider the associated quantum Hamiltonians and calculate analytically the boundary free energies of the $(r,s)$ Robin boundary conditions. Using finite-size corrections and sequence extrapolation out to system sizes $N+w+d\le 26$, the conformal spectrum of boundary operators is accessible by numerical diagonalization of the Hamiltonians. Fixing the parity of $N$ for $r\ne 0$ and restricting to the ground state sequences $w=\big\lfloor\frac{|r|p'}{p}\big\rfloor$, $r\in{\Bbb Z}$ with the inverse $r=(-1)^{N+w+d}\big\lceil \frac{p w}{p'}\big\rceil$, we find that the conformal weights take the values $Δ^{p,p'}_{r,s-\frac12}$ where $Δ^{p,p'}_{r,s}$ is given by the usual Kac formula. The $(r,s)$ Robin boundary conditions are thus conjugate to scaling operators with half-integer values for the Kac label $s-\mbox{$\textstyle \frac{1}{2}$}$.

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