Tensor network methods for lattice gauge theories

规范理论的格点表述在我们理解这些理论中起着重要的作用。特别是，它们提供了一个独特的机会，定量研究量子色动力学（QCD），强相互作用的公认理论。格点QCD已经导致了几个重要的结果，例如从第一性原理计算强子谱。然而，某些物理问题并不适合其标准工具，即蒙特卡罗模拟。这类问题的两个例子是非零化学势和实时动力学。因此，有必要寻找新的替代方法，可以帮助解决这些问题。这样的方法是一类方法称为张量网络（TN）。这些非微扰技术有助于理解量子多体问题的几个方面，特别是在凝聚态物理和量子信息的背景下。然而，它们也非常适合格点规范理论的数值模拟。在我们早期的研究中，我们使用最成功的TN技术之一，称为矩阵乘积态（MPS），证明了TN方法应用于格点Schwinger模型（即1+1维量子电动力学）的可行性。拟议项目的主要目标是研究TN方法在格点规范理论中的应用，特别是使它们在QCD中应用成为现实的一个长期目标。具体而言，这涉及使用TN方法来解决上述对标准技术具有挑战性的问题。这些问题妨碍了对强力的一些重要方面的理解，例如从阐明早期宇宙物理学的角度来看。在项目的实施阶段之后，我们将考虑Schwinger模型中的以下物理问题。首先，我们将计算有限温度下的手征凝聚体。然后，我们将MPS方法应用到有限费米子密度的情况。作为Schwinger模型的最终目标，我们计划研究非平衡性质。对于许多被认为是可观的分析预测存在于无质量的情况下，并将允许与TN技术得到的结果进行交叉检查。在某些情况下，例如，具有多于2种风味的模型的情况下，非平凡的物理问题仍然没有解决，并且将被解决。另一个将在项目后期考虑和实现的方向是将研究扩展到更高维度的非阿贝尔规范理论和系统。当然，这两种可能性都是为了使所考虑的模型更接近最终目标--完全量子色动力学。

Lattice formulations of gauge theories play an important role in our understanding of these theories. In particular, they provide a unique opportunity to quantitatively study Quantum Chromodynamics (QCD), the accepted theory of the strong interaction. Lattice QCD has led to several important results, e.g. the computation of the hadron spectrum from first principles. However, some classes of physical problems are not well-suited for its standard tools, i.e. Monte Carlo simulations. Two examples of such problems are non-vanishing chemical potential and real-time dynamics. It is therefore essential to look for new alternative approaches that can help to solve these problems.One such approach is the class of methods called Tensor Networks (TN). These non-perturbative techniques helped to understand several aspects of quantum many-body problems, especially in the context of condensed matter physics and quantum information. However, they are also well-suited for numerical simulations of lattice gauge theories. In our earlier research, we showed the feasibility of application of TN methods for the lattice Schwinger model, i.e. quantum electrodynamics in 1+1 dimensions, using one of the most successful TN techniques, called Matrix Product States (MPS).The main objective of the proposed project is to investigate the application of TN methods to lattice gauge theories, in particular to make realistic their application to QCD as a long-termgoal. Specifically, this concerns the use of TN methods to tackle the above mentioned problems that are challenging to standard techniques. Such problems have obtruded understanding of a few important aspects of the strong force, relevant e.g. from the point of view of elucidating the physics of the early Universe. After the implementation phase of the project, we will consider the following physical problems in the Schwinger model. Firstly, we will calculate the chiral condensate at finite temperature. Then, we will apply the MPS method to the case of finite fermion density. As the final objective in the Schwinger model, we plan the investigation of non-equilibrium properties. For many considered observables analytical predictions exist in the massless case and will allow for cross-checks of the results obtained with the TN techniques. In some cases, e.g. the case of the model with more than 2 flavours, non-trivial physical questions are still unresolved and will be addressed. Another direction that will be considered and realized at a later stage of the project, is to extend the studies to non-Abelian gauge theories and systems in higher dimensions. Both possibilities are intended, naturally, to bring the model under consideration closer to the ultimate aim - full QCD.