# Advances in Nonperturbative Approaches: Methods, Algorithms, and Applications

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Nonperturbative approaches become increasingly important in various fields of physics, such as equilibrium and non-equilibrium phase transitions, quantum gravity, nanoscience, etc. 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 important to develop nonperturbative approaches, such as the functional renormalization, Monte Carlo (MC) and Monte Carlo RG (MCRG) methods. The Wetterich equation and the Polchinski equation are known as the nonperturbative or exact RG equations, which lie in the basis of the functional RG approach. This general approach is widely used in quantum and statistical field theories, including asymptotically safe quantum gravity. The nonequilibrium Green’s function (NEGF) method is another nonperturbative approach widely used in physics, including nanoscience.

In this Research Topic, we focus mainly but not exclusively on the nonperturbative RG and MCRG methods and algorithms. Although the Wetterich equation is an exact RG equation, it cannot be solved exactly. Closed approximate equations are obtained from it applying certain truncation schemes. Traditionally, the derivative expansion is used, which relies on the small momentum approximation. This method is relatively well developed for the scalar field models. There are also distinct schemes like the Blaizot-Mendez-Wschebor (BMW) scheme and a recent scheme proposed in J. Phys. A: Math. Theor. 53, 415002 (2020), which preserve the full momentum dependence. It would be important to extend these schemes beyond the first order of truncation. Generally, there is a room for new advances in the application of the nonperturbative RG approach to a variety of models, not limited to the scalar field model. It is also very important to further develop and apply other nonperturbative methods, 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:
- Recent advances in the formulation and solution of the nonperturbative RG equations for classical and quantum models. Extensions beyond the scalar field model are particularly welcome;
- Advances in the development and solution of the truncated nonperturbative RG equations, which preserve the full momentum dependence;
- Optimization of the MCRG simulation algorithm and improved estimation of the critical exponents. It particularly refers to the correction-to-scaling exponent of the 3D Ising model, where the earlier MCRG results are somewhat surprising and uncertain;
- Advances, new developments and recent results with application of nonperturbative simulation methods and algorithms, such as MC and parallel MC algorithms, simulations with inverse MCRG transformations, NEGF calculations, etc.

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

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

Important Note: All contributions to this Research Topic must be within the scope of the section and journal to which they are submitted, as defined in their mission statements. Frontiers reserves the right to guide an out-of-scope manuscript to a more suitable section or journal at any stage of peer review.

Nonperturbative approaches become increasingly important in various fields of physics, such as equilibrium and non-equilibrium phase transitions, quantum gravity, nanoscience, etc. 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 important to develop nonperturbative approaches, such as the functional renormalization, Monte Carlo (MC) and Monte Carlo RG (MCRG) methods. The Wetterich equation and the Polchinski equation are known as the nonperturbative or exact RG equations, which lie in the basis of the functional RG approach. This general approach is widely used in quantum and statistical field theories, including asymptotically safe quantum gravity. The nonequilibrium Green’s function (NEGF) method is another nonperturbative approach widely used in physics, including nanoscience.

In this Research Topic, we focus mainly but not exclusively on the nonperturbative RG and MCRG methods and algorithms. Although the Wetterich equation is an exact RG equation, it cannot be solved exactly. Closed approximate equations are obtained from it applying certain truncation schemes. Traditionally, the derivative expansion is used, which relies on the small momentum approximation. This method is relatively well developed for the scalar field models. There are also distinct schemes like the Blaizot-Mendez-Wschebor (BMW) scheme and a recent scheme proposed in J. Phys. A: Math. Theor. 53, 415002 (2020), which preserve the full momentum dependence. It would be important to extend these schemes beyond the first order of truncation. Generally, there is a room for new advances in the application of the nonperturbative RG approach to a variety of models, not limited to the scalar field model. It is also very important to further develop and apply other nonperturbative methods, 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:
- Recent advances in the formulation and solution of the nonperturbative RG equations for classical and quantum models. Extensions beyond the scalar field model are particularly welcome;
- Advances in the development and solution of the truncated nonperturbative RG equations, which preserve the full momentum dependence;
- Optimization of the MCRG simulation algorithm and improved estimation of the critical exponents. It particularly refers to the correction-to-scaling exponent of the 3D Ising model, where the earlier MCRG results are somewhat surprising and uncertain;
- Advances, new developments and recent results with application of nonperturbative simulation methods and algorithms, such as MC and parallel MC algorithms, simulations with inverse MCRG transformations, NEGF calculations, etc.

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

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

Important Note: All contributions to this Research Topic must be within the scope of the section and journal to which they are submitted, as defined in their mission statements. Frontiers reserves the right to guide an out-of-scope manuscript to a more suitable section or journal at any stage of peer review.

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