共形流形上的拓扑变换
Topological Transition on the Conformal Manifold
摘要：
尽管在空位相的研究方面取得了巨大的成功，但对无空位相及其转变的全面理解仍处于发展之中。本文研究了具有费米子自由度的（1+1）$d$临界相空间中的一个普遍现象，该临界相空间由连续的共形场论族（CFT）描述，又称共形场论.共形流形沿着共形流形上的一维轨迹，可以有一个过渡点，通过沿着轨迹逐点堆叠可逆费米子拓扑序（IFTO），两侧的费米子CFTs不同。在共形流形上的每一点上，有序和无序算子都有幂律两点函数，但它们的临界指数在过渡点处相互交叉，在过渡点处叠加IFTO使费米子CFT保持不变。我们称这种在费米子共形流形上的连续跃迁为拓扑跃迁。通过测量费米子宇称，IFTO堆叠成为相应玻色子CFTs之间的Kramers-Wannier对偶。IFTO堆积和Kramers-Wannier对偶性都是由（2+1）$d$ $\mathbb{Z}_2$拓扑序的电磁对偶性引起的。我们提供了几个拓扑转变的例子，包括熟悉的无自旋费米子的Luttinger模型（即$c=1$无质量狄拉克费米子与Thirring相互作用），和一类新的$c=2$例子描述$U(1)\times SU(2)$保护的无隙相。
英文摘要：
Despite great successes in the study of gapped phases, a comprehensive understanding of the gapless phases and their transitions is still under developments. In this paper, we study a general phenomenon in the space of (1+1)$d$ critical phases with fermionic degrees of freedom described by a continuous family of conformal field theories (CFT), a.k.a. the conformal manifold. Along a one-dimensional locus on the conformal manifold, there can be a transition point, across which the fermionic CFTs on the two sides differ by stacking an invertible fermionic topological order (IFTO), point-by-point along the locus. At every point on the conformal manifold, the order and disorder operators have power-law two-point functions, but their critical exponents cross over with each other at the transition point, where stacking the IFTO leaves the fermionic CFT unchanged. We call this continuous transition on the fermionic conformal manifold a topological transition. By gauging the fermion parity, the IFTO stacking becomes a Kramers-Wannier duality between the corresponding bosonic CFTs. Both the IFTO stacking and the Kramers-Wannier duality are induced by the electromagnetic duality of the (2+1)$d$ $\mathbb{Z}_2$ topological order. We provide several examples of topological transitions, including the familiar Luttinger model of spinless fermions (i.e. the $c=1$ massless Dirac fermion with the Thirring interaction), and a new class of $c=2$ examples describing $U(1)\times SU(2)$-protected gapless phases.
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