有限维Hilbert空间上的不可缺边理论
Ungappable edge theories with finite dimensional Hilbert spaces
摘要：
我们构造了一类新的边理论，用于一类具有$K = \begin{pmatrix} k_1 & 0 \\ 0 & - k_2 \end{pmatrix}$形式的$K$ -矩阵的费米子阿贝尔拓扑相，其中$k_1, k_2 > 0$是奇数。我们的边缘理论是值得注意的两个原因：（i）他们有有限维希尔伯特空间（有限大小的系统）和（ii）根据$k_1, k_2$的值，一些边缘理论描述的边界，不能被任何局部相互作用。这种不可应用的边界的最简单的例子出现在$(k_1, k_2) = (1, 3)$中，它是由$ν= 2/3$的双稳态实现的。我们得到我们的边缘理论开始与标准的手征玻色子边缘理论，由两个反向传播的手征玻色子模式，然后引入一个阵列的点状杂质散射。我们解决了这个杂质模型完全在无限杂质散射的限制，我们表明，能谱包括一个带隙的声子谱连同基态简并指数标度与杂质的数量。这个基态子空间形成了我们的边缘理论的希尔伯特空间。我们相信，类似的边缘理论可以为任何阿贝尔拓扑相位消失的热霍尔系数，$κ_H = 0$。
英文摘要：
We construct a new class of edge theories for a family of fermionic Abelian topological phases with $K$-matrices of the form $K = \begin{pmatrix} k_1 & 0 \\ 0 & - k_2 \end{pmatrix}$, where $k_1, k_2 > 0$ are odd integers. Our edge theories are notable for two reasons: (i) they have finite dimensional Hilbert spaces (for finite sized systems) and (ii) depending on the values of $k_1, k_2$, some of the edge theories describe boundaries that cannot be gapped by any local interaction. The simplest example of such an ungappable boundary occurs for $(k_1, k_2) = (1, 3)$, which is realized by the $ν= 2/3$ FQH state. We derive our edge theories by starting with the standard chiral boson edge theory, consisting of two counterpropagating chiral boson modes, and then introducing an array of pointlike impurity scatterers. We solve this impurity model exactly in the limit of infinite impurity scattering, and we show that the energy spectrum consists of a gapped phonon spectrum together with a ground state degeneracy that scales exponentially with the number of impurities. This ground state subspace forms the Hilbert space for our edge theory. We believe that similar edge theories can be constructed for any Abelian topological phase with vanishing thermal Hall coefficient, $κ_H = 0$.
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