Tensor Networks and their use for Lattice Gauge Theories (1810.12838v2)

Abstract: Tensor Network States are ans\"atze for the efficient description of quantum many-body systems. Their success for one dimensional problems, together with the fact that they do not suffer from the sign problem and can address the simulation of real time evolution, have turned them into one of the most promising techniques to study strongly correlated systems. In the realm of Lattice Gauge Theories they can offer an alternative to standard lattice Monte Carlo calculations, which are suited for static properties and regimes where no sign problem appears. The application of Tensor Networks to this kind of problems is a young but rapidly evolving research field. This paper reviews some of the recent progress in this area, and how, using one dimensional models as testbench, some fundamental milestones have been reached that may pave the way to more ambitious goals.

Tensor Networks and Applications in Lattice Gauge Theories

The paper offers a comprehensive overview of the application of Tensor Network States (TNS) in studying Lattice Gauge Theories (LGTs), with a particular focus on one-dimensional models. It presents TNS as an efficient method for describing quantum many-body systems, showing advantages over traditional Monte Carlo simulations, especially in scenarios involving the sign problem and real-time evolution. TNS, including Matrix Product States (MPS), have been recognized for their capability to represent these systems and address computational challenges inherent in LGTs.

Key Insights and Numerical Results

The paper begins with a detailed introduction to the mathematical foundation of TNS, emphasizing the role of entanglement and the area law in developing effective ansätze for physical states. It identifies several tensor structures, such as Matrix Product States (MPS), Projected Entanglement Pair States (PEPS), and MERA, explaining their relevance in capturing entanglement patterns.

In the context of lattice gauge theories, TNS enables overcoming limitations of classical simulations, concentrating on the ability to work without a sign problem. Notable results include high precision calculations using TNS, surpassing traditional methods and achieving highly accurate characterizations of particle spectra and physical properties.

Numerical Findings:

1. Spectral Properties: The paper reports precise mass gap calculations for the Schwinger model, demonstrating accurate determination of binding energies with MPS. The contrasting methodologies—open boundary MPS versus gauge invariant uniform MPS—provide valuable computational insights.
2. Chiral Condensate and Entropy: Through TNS, the paper explores the chiral condensate measurements, examining both massless and massive fermion scenarios. The paper further explores entanglement entropies and their dependence on different gauge theories, contributing to understanding central properties like the critical behavior.
3. Thermal and Finite Density Studies: TNS facilitate probing temperature-dependent phenomena, like the restoration of chiral symmetry at high temperatures of the Schwinger model. Moreover, finite density analyses showcase the resolution of the sign problem in complex phase scenarios not accessible by prior numerical techniques.
4. Real-time Evolution: One of the striking capabilities of TNS is simulating non-equilibrium dynamics, providing a window into real-time phenomena like Schwinger pair production with impressive fidelity.

Implications and Future Directions

The insights gained from applying TNS to one-dimensional LGTs create a compelling case for their extension to higher-dimensional models, promising to enhance the scope of lattice gauge theory simulations significantly. Although current studies remain limited to lower-dimensional cases, the paper highlights the theoretical groundwork and recent advances fostering this progression.

The possibility of utilizing TNS for a path integral representation indicates a further frontier in lattice gauge computations. Parallel developments in quantum simulations using ultracold atoms underscore the interdisciplinary potential of these methodologies, proposing experimental pathways alongside computational advances.

Overall, the paper provides a firm grounding in both theoretical and practical aspects of TNS applications in LGT. As researchers continue refining TNS algorithms and adapting them to diverse LGT challenges, this work remains a pivotal reference point for both theoretical exploration and practical implementations in quantum field theory simulations.