Bose-哈伯德模型中的陷获曲面和突现曲面空间
Trapped surfaces and emergent curved space in the Bose-Hubbard model
摘要：
作为涌现几何和引力的自旋系统模拟，在以前的工作中引入了动力学晶格上的玻色-哈伯德模型。在晶格中具有高连通性区域的图形被确定为包含捕获表面的时空几何形状的候选类似物。我们对这些系统进行了详细的研究，并明确表明，高度连通的子图陷阱的问题。我们通过在晶格上的物质没有反反应的限制下求解模型来做到这一点，并且对于我们的问题来说，具有某些对称性的状态是自然的。我们发现，在这种情况下，该问题减少到一个一维的哈伯德模型上的一个格子与可变的顶点度和多个边缘之间的相同的两个顶点。此外，我们得到了一个（离散）微分方程的粒子的概率密度的演化是封闭的经典制度。这是一个波动方程，其中顶点度与概率的局部传播速度有关。这使得对粒子概率密度的解释类似于模拟引力系统：在这个模拟系统中的物质看到一个弯曲的时空。我们通过数值模拟验证了我们的分析结果。最后，我们分析了依赖本地化的一个渐进的，而不是突然的，高度连通区域的边界上的顶点度的下降，并发现，该问题是本地化的区域内和周围。
英文摘要：
A Bose-Hubbard model on a dynamical lattice was introduced in previous work as a spin system analogue of emergent geometry and gravity. Graphs with regions of high connectivity in the lattice were identified as candidate analogues of spacetime geometries that contain trapped surfaces. We carry out a detailed study of these systems and show explicitly that the highly connected subgraphs trap matter. We do this by solving the model in the limit of no back-reaction of the matter on the lattice, and for states with certain symmetries that are natural for our problem. We find that in this case the problem reduces to a one-dimensional Hubbard model on a lattice with variable vertex degree and multiple edges between the same two vertices. In addition, we obtain a (discrete) differential equation for the evolution of the probability density of particles which is closed in the classical regime. This is a wave equation in which the vertex degree is related to the local speed of propagation of probability. This allows an interpretation of the probability density of particles similar to that in analogue gravity systems: matter inside this analogue system sees a curved spacetime. We verify our analytic results by numerical simulations. Finally, we analyze the dependence of localization on a gradual, rather than abrupt, fall-off of the vertex degree on the boundary of the highly connected region and find that matter is localized in and around that region.
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