论文标题
量子晶格模型中的紧急时空
Emergent Spacetime in Quantum Lattice Models
论文作者
论文摘要
许多量子晶格模型在其连续限制中具有新兴的相对论描述。著名的例子是石墨烯，其连续限制由Minkowski时空上的Dirac方程描述。连续限制不仅为我们提供了一个几何可观测值的字典来描述模型，而且还允许人们求解这些模型，而这些模型原本是分析性棘手的。在本文中，我们研究了对一系列量子晶格模型的这种相对论描述的新颖特征。特别是，我们演示了如何在晶格水平上产生新兴的弯曲空间并识别可观察到的物体，从而揭示了这种新兴行为，从而使一个人能够模拟实验室中的相对论效应。我们首先研究了具有边缘的系统碳纳米管，它使我们能够测试DIRAC方程的有趣特征，该功能允许在系统边缘上提供散装状态。然后，我们研究Kitaev的Honeycomb模型，该模型具有连续性极限，描述了Minkowski时空上的主要旋转器。我们展示了如何在该模型的连续限制中生成非平凡的度量，以及如何观察该度量标准及其相应的曲率在晶格可观察物中的效果，例如Majorana相关器，Majorana Majorana零模式和自旋密度。我们还讨论了晶格缺陷和$ \ Mathbb {z} _2 $量规场在晶格级别如何在连续限制中生成手性量规场，并揭示其绝热等效性。最后，我们讨论了1D XY模型的手性修改，该模型使模型相互作用并引入了非平凡的相图。我们看到，这会在连续限制中产生一个黑洞度量，其中黑洞的内部和外部处于不同的阶段。然后，我们证明，通过淬灭该模型，我们可以模拟鹰辐射。
Many quantum lattice models have an emergent relativistic description in their continuum limit. The celebrated example is graphene, whose continuum limit is described by the Dirac equation on a Minkowski spacetime. Not only does the continuum limit provide us with a dictionary of geometric observables to describe the models with, but it also allows one to solve models that were otherwise analytically intractable. In this thesis, we investigate novel features of this relativistic description for a range of quantum lattice models. In particular, we demonstrate how to generate emergent curved spacetimes and identify observables at the lattice level which reveal this emergent behaviour, allowing one to simulate relativistic effects in the laboratory. We first study carbon nanotubes, a system with an edge, which allows us to test the interesting feature of the Dirac equation that it allows for bulk states with support on the edges of the system. We then study Kitaev's honeycomb model which has a continuum limit describing Majorana spinors on a Minkowski spacetime. We show how to generate a non-trivial metric in the continuum limit of this model and how to observe the effects of this metric and its corresponding curvature in the lattice observables, such as Majorana correlators, Majorana zero modes and the spin densities. We also discuss how lattice defects and $\mathbb{Z}_2$ gauge fields at the lattice level can generate chiral gauge fields in the continuum limit and we reveal their adiabatic equivalence. Finally, we discuss a chiral modification of the 1D XY model which makes the model interacting and introduces a non-trivial phase diagram. We see that this generates a black hole metric in the continuum limit, where the inside and outside of the black hole are in different phases. We then demonstrate that by quenching this model we can simulate Hawking radiation.
