应用多尺度纠缠重整化方法研究阻挫量子磁性及关联电子体系

The main purpose of this research project is to use Multiscale entanglement renormalization ansatz (MERA) to study complex quantum systems. We will focus on two groups of theoretical models in two or higher dimensions: frustrated quantum spin models and correlated electron models. In both cases, the well established quantum Monte Carlo method fails to provide meaningful results due to the notorious sign problem. An unbaised universal numerical method, which captures quantum entanglement more effecively and treats resonably large system, is still much awaited. The MERA method we use is based on the Matrix/Tensor Product State (MPS/TPS) Ansatz, which utilizes matrix/tensor product to construct wave functions that are capable of describing quantum entanglement in the system. The MPS/TPS ansatz has been shown to be effective, especially in low dimensions (Density Matrix Renormalization Group method in 1D). The MERA method combines the TPS Ansatz (which is a natural extension of MPS in 2D) with idea of Renormalization. It focus on long range entanglement that persists in large system scales, by introducing a special kind of tensors called "disentangler", which remove/reduce short range entanglement. Compared with other TPS based methods, the MERA method has the following advantages: it is capable to deal with relatively large systems; it is an exact variational method without approximation, so that it provides upper limit for the ground state energy; it is quite flexible when dealing with boundary conditions; it is capable of keeping track of entanglement evolvement by changing system scale. The first group of models we study- - frustrated quantum spin systems are host to many exciting physics phenomena in quantum regime, including the spin liquid, quantum phase transitions, and entanglement. We will study the Heisenberg model on the checkerboard lattice, which is a 2D projection of the 3D Pyrochlore lattice, as well as other exciting frustrated quantum spin models. The second group focus on correlated elctron models. Fermion statistics may lead to non-locality in the system and then hinder the applications of conventional numerical methods. Following recent development in TPS, we will develop a new numerical algroithm GMERA which integrate Grassmann algebra into MERA method. Technically, Grassmann numbers will be attached to virtual bonds between tensors, and these number will be integrated during contraction of tensors. A successful integration requires local exchange of Grassmann number, which provides the fermionic sign. Hence it is sufficient to recover the fermionic statistics through local operations. By employing the new GMERA method, we will study the t-J model and the Hubbard model on different lattices, as well as other theoretical models that may shed light on the mechanism of high Tc superconductivity.

本课题应用基于张量矩阵乘积态理论的多尺度纠缠重整化方法对复杂量子体系进行数值理论研究。课题研究的第一个方向是阻挫的量子磁性体系。目前较为成熟的计算方法（如量子蒙特卡洛）在处理阻挫体系时面临种种困难，而张量矩阵乘积法是新近发展起来的可以成功处理二维及更高维阻挫体系的新型数值计算方法。我们所采用的多层纠缠重整化方法将张量矩阵理论与重整化概念相结合，与其它方法相比，具有能处理更大体系，善于描述长程量子纠缠，属于严格变分法等优点。我们将对二维checkerboard晶格上的海森堡模型等含有新奇量子物理现象的若干阻挫量子体系进行细致研究。课题研究的第二个方向是将Grassmann代数引入多层纠缠重整化方法，以处理诸如t-J模型、Hubbard模型等关联电子理论模型。这方面研究的重点在于探讨关联电子体系中各种有序态的产生和相互竞争，以及伴随的量子相变，如t-J模型中超导电性与反铁磁性的产生及竞争关系。

本项目利用多尺度纠缠重整化（MERA）对量子阻挫自旋体系和强关联电子体系进行了严格的数值研究。在这类体系中不仅强关联相互作用至关重要，使得基于平均场近似的传统方法无法准确的描述体系的基态性质；而且带有符号问题，导致发展成熟的量子蒙特卡洛类方法的应用也面临种种困难和局限。我们使用的多尺度纠缠重整化方法基于张量矩阵乘积态的思想，是近十年来迅速发展起来的研究强关联体系的强有效的数值方法之一。这类方法能有效的描述体系中的长程量子纠缠，并严格的模拟强关联量子体系的基态性质。.我们的研究的最主要工作是对正方晶格上的t－J模型，这一与铜氧化物高温超导原理密切相关的基础理论模型进行了深入仔细的数值模拟。对于t-J模型的模型的解析研究大多是是基于平均场假设，传统的数值方法在处理该模型时面临各种困难与局限，因此到目前为止并没有严格的数值模拟结果证明有限体系会自发对称破缺进入超导状态。.为了研究强关联电子模型，我们将MERA方法与场论中常用到的Grassmann代数相结合，发展出了GMERA算法，并进行了benchmark计算。然后我们将这一方法用于t-J模型的研究。我们发现，在小体系中，由于有限尺寸效应存在着许多能量极为相近的物象的竞争，如自旋密度波(SDW)，电荷密度波(CDW)等等，体系无法发生自发对称破缺直接进入超导态。在t-J模型基础上加入配对相互作用的过程中，我们验证了D波形式的配对是体系的最优解，且此时体系中形成的D波超导态在很大的参数区间是体系的基态。更为重要的是，通过对更大尺寸体系的计算我们发现，体系增大后，上述D波超导态在在未添加配对相互作用时，在中等掺杂（0.22～0.3）区间仍能量最优，是体系的基态。因此我们的结论是对于t-J模型，随着体系不断增大，有限尺寸带来的各种竞争相能量变差，而D波超导态将成为体系的真正基态。我们的计算是第一次严格的模拟有限体系的t-J模型并对D波超导给出强有力的支持。.我们的工作还包括对量子阻挫自旋体系的研究，如描述铱氧碱金属的Kitaev模型等。在研究中我们发现了新奇的物象与物理。

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