二维重力的张量网络公式

M. Asaduzzaman, S. Catterall, J. Unmuth-Yockey
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引用次数: 10

摘要

我们展示了如何制定一个晶格规范理论,其朴素连续统极限对应于二维(欧几里得)量子引力,包括一个正的宇宙常数。更准确地说,由此产生的连续统理论对应于引力的一阶形式,其中局部坐标系和自旋连接被视为独立的场。将这个点阵理论转换为张量网络,使我们可以在不遇到符号问题的情况下研究强耦合理论。在二维中,这个张量网络是完全可溶的,我们证明了系统有一系列临界点,这些临界点发生在纯虚耦合中,并且与一阶相变有关。我们通过在具有不同拓扑的格上表述晶格理论,证明了晶格理论在本质上是纯拓扑的,并证明了配分函数仅依赖于三角剖分的欧拉特征。
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Tensor network formulation of two-dimensional gravity
We show how to formulate a lattice gauge theory whose naive continuum limit corresponds to two dimensional (Euclidean) quantum gravity including a positive cosmological constant. More precisely the resultant continuum theory corresponds to gravity in a first order formalism in which the local frame and spin connection are treated as independent fields. Recasting this lattice theory as a tensor network allows us to study the theory at strong coupling without encountering a sign problem. In two dimensions this tensor network is exactly soluble and we show that the system has a series of critical points that occur for pure imaginary coupling and are associated with first order phase transitions. We show evidence that the lattice theory is purely topological in nature by formulating it on lattices with differing topologies and show that the partition function depends only on the Euler character of the triangulation.
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