论文标题
（3+1）D拓扑订单的差距
Gapped boundaries of (3+1)d topological orders
论文作者
论文摘要
给定（3+1）d拓扑顺序的边界（TO），可以在其上堆叠一个脱钩（2+1）d，以获得另一个边界理论。一个人应该将这两个边界视为“不同”？自然的选择是没有。应定义（3+1）d的不同类别的间隙边界，这些脱钩（2+1）d tos。但这足够了吗？ 我们检查了将A（3+1）D与其他（2+1）d TOS或分形系统耦合的可能性，从而导致距离边界的更多可能性。通常，在接触边界时，散装点状激发在添加的（2+1）d相中成为激发，而大块中的弦状激发可能会在边界上结束，但端点在（2+1）d相中的其他激发装饰。对于（3+1）d的间隙边界的“班级”的良好定义，我们也选择使不同的调味料弄清楚。 我们通过类似字符串的激发来表征一类散布的边界，无论它们的端点是什么。一个具体的示例是（3+1）d骨旋转的代码。使用群体的共同体学和类别理论，先前已经找到了三个间隙边界：粗糙的边界，平滑边界和扭曲的平滑边界。我们可以构建超出这些边界的更多差距，它们自然而然地属于两个类，与$ m $ srin的相对应在边界上结束。根据此分类，先前发现的三个边界分为{rugh}，{平滑，扭曲的平滑}。对于以有限的$ g $为特征的（3+1）d，不同的类对应于$ g $的不同子组。我们从各个角度说明了物理图片，包括耦合层构造，沃克 - 瓦模型和现场理论。
Given a gapped boundary of a (3+1)d topological order (TO), one can stack on it a decoupled (2+1)d TO to get another boundary theory. Should one view these two boundaries as "different"? A natural choice would be no. Different classes of gapped boundaries of (3+1)d TO should be defined modulo these decoupled (2+1)d TOs. But is this enough? We examine the possibility of coupling the boundary of a (3+1)d TO to additional (2+1)d TOs or fractonic systems, which leads to even more possibilities for gapped boundaries. Typically, the bulk point-like excitations, when touching the boundary, become excitations in the added (2+1)d phase, while the string-like excitations in the bulk may end on the boundary but with endpoints dressed by some other excitations in the (2+1)d phase. For a good definition of "class" for gapped boundaries of (3+1)d TO, we choose to quotient out the different dressings as well. We characterize a class of gapped boundaries by the string-like excitations that can end on the boundary, whatever their endpoints are. A concrete example is the (3+1)d bosonic toric code. Using group cohomology and category theory, three gapped boundaries have been found previously: rough boundary, smooth boundary and twisted smooth boundary. We can construct many more gapped boundaries beyond these, which all naturally fall into two classes corresponding to whether the $m$-string can or cannot end on the boundary. According to this classification, the previously found three boundaries are grouped as {rough}, {smooth, twisted smooth}. For a (3+1)d TO characterized by a finite group $G$, different classes correspond to different subgroups of $G$. We illustrate the physical picture from various perspectives including coupled layer construction, Walker-Wang model and field theory.
