Pushing the frontier of simulations of two-dimensional multi-orbital fermionic lattice and Heisenberg models by fully exploiting non-abelian symmetries in tensor network states

在费米子晶格模型中，自旋和轨道量子自由度的相互作用导致了低能量下高度复杂的竞争相关相。对于他们的理解，控制良好的非微扰方法是典型的。 在这方面，张量网络状态（TNS）提供了一种非常有前途的新方法，但它对数值要求极高。因此，该项目的主要重点是对称多轨道模型，我们的目标是充分利用所有潜在的阿贝尔以及非阿贝尔对称性，以推进二维（2D）晶格模型的TNS模拟远远超出当前的最先进的。为此，我们将利用最近开发的QSpace张量库，可以处理非阿贝尔对称性的通用基础。对于1D系统，这已经在计算效率的增益方面产生了许多数量级。 在这里，我们将使用QSpace来研究2D中几个范例模型系统的低能相图以及纠缠和拓扑性质。这些包括N轨道费米子系统，如哈伯德和tJ模型与高Tc超导相关，以及衍生的SU（N）海森堡模型与自旋和轨道自由度，表现出非常规的磁性。对这些系统进行更好的计算控制，最终有望对我们理解强关联量子多体系统产生根本性的影响。

The interplay of spin and orbital quantum degrees of freedom in fermionic lattice models leads to highly complex competing correlated phases at low energies. For their understanding well-controlled non-perturbative methods are quintessential. A very promising new approach in that respect is provided by tensor network states (TNS) which nevertheless are extremely demanding numerically. Therefore the main focus of this project is on symmetric multi-orbital models where we aim to fully exploit all underlying abelian as well as non-abelian symmetries in order to advance TNS simulations of two dimensional (2D) lattice models far beyond the current state-of-the-art. To this end, we will utilize the recently developed QSpace tensor library that can deal with non-abelian symmetries on a generic footing. For 1D systems, this has already yielded many orders of magnitude in gain of computational efficiency. Here we will use QSpace to study the low-energy phase diagram as well as entanglement and topological properties of several paradigmatic model systems in 2D. These include N-orbital fermionic systems such as the Hubbard and tJ-model with relevance to high-Tc superconductivity, as well as derived SU(N) Heisenberg models with spin and orbital degrees of freedom that exhibit unconventional magnetism. A better computational control of these systems ultimately promises a fundamental impact on our understanding of strongly-correlated quantum many-body systems.