Tensor network methods for lattice gauge theories

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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.