Advances in Nonperturbative Approaches: Methods, Algorithms, and Applications

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Background

Nonperturbative approaches become increasingly important in various fields of physics and other disciplines, including, but not limited to, equilibrium and non-equilibrium dynamics, quantum mechanics and particle physics, nanoscience and nanotechnology, quantum gravity, and complex systems. The importance of nonperturbative effects in many systems, e.g. spin glasses, is a well-established fact. Such effects often appear in critical phenomena, where the renormalization group (RG) is the most extensively used general method. Hence, it is essential to develop nonperturbative approaches, such as functional renormalization, Monte Carlo (MC), and Monte Carlo RG (MCRG) methods. Examples of well-known models in this area include the Wetterich and the Polchinski equations, known as nonperturbative or exact RG equations. However, these equations can be solved only approximately. Therefore, nonperturbative simulation methods, like various MC and MCRG algorithms, play a crucial role in studying the systems of interacting particles and critical phenomena, as they provide a solid basis for comparing and verifying of results. More generally, developing nonperturbative methods and computational algorithms is very important in the description of various complex systems. In particular, the nonequilibrium Green's function (NEGF) method is a well-known nonperturbative approach widely used in condensed matter physics and nanoscience.

In this Research Topic, we focus mainly but not exclusively on the nonperturbative RG and Monte Carlo methods and algorithms. The nonperturbative RG approach traditionally relies on the derivative expansion, which is a small momentum expansion. There are also distinct schemes like the Blaizot-Mendez-Wschebor (BMW) scheme, which preserves the full momentum dependence. It would be important to refine and extend these and other advanced nonperturbative RG schemes with potential applications to equilibrium and nonequilibrium systems, as well as to problems of particle physics and quantum gravity. The Monte Carlo method is a powerful tool of simulation with a large variety of different applications. In particular, sophisticated cluster algorithms, as well as MCRG algorithms and algorithms with inverse MCRG transformations have been developed for efficient simulations near the critical point. Another relevant direction is the development of flat-histogram Monte Carlo algorithms, which are particularly suited for simulations of systems with a complex energy landscape and metastable states. It is essential to further develop and apply these and other nonperturbative methods and computational algorithms, including MCRG and MC simulation algorithms and the NEGF method, to cover as far as possible a wider range of nonperturbative phenomena and provide a solid basis for comparison and verification of results.


The list of specific themes within this Research Topic includes, but is not limited to:

- Statistical mechanics, approximate methods, and applications that drive the development of nonperturbative approaches;
- Recent advances in formulating and solving the nonperturbative RG equations for classical and quantum models. Extensions beyond the scalar field model and applications to nonequilibrium systems and critical dynamics are particularly welcome;
- Optimization and further development of MC and MCRG simulation algorithms. New results with improved MCRG estimations of the correction-to-scaling exponents in the 3D Ising model and other models are particularly welcome;
- Advances, new developments, and recent results with the application of nonperturbative simulation methods and algorithms, such as MC and parallel MC algorithms, simulations with inverse MCRG transformations, nonperturbative approaches to quantum gravity, to dynamics of open systems and complex systems, NEGF calculations, etc.

Original Research papers and Reviews, covering recent advances in these fields, are welcome.

Keywords: nonperturbative dynamics, exact renormalization group equations, nonperturbative renormalization group, Wetterich equation, quantum and statistical field theories, particle physics, Monte Carlo methods, Monte Carlo renormalization group, inverse MCRG transformations, strongly interacting systems, critical phenomena, lattice models, critical exponents, nonequilibrium Green's functions, quantum gravity; computational algorithms, complex systems

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