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Register a new userThe nonperturbative functional renormalization group and its applications
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The renormalization group plays an essential role in many areas of physics, both conceptually and as a practical tool to determine the long-distance low-energy properties of many systems on the one hand and on the other hand search for viable ultraviolet completions in fundamental physics. It provides us with a natural framework to study theoretical models where degrees of freedom are correlated over long distances and that may exhibit very distinct behavior on different energy scales. The nonperturbative functional renormalization-group (FRG) approach is a modern implementation of Wilsons RG, which allows one to set up nonperturbative approximation schemes that go beyond the standard perturbative RG approaches. The FRG is based on an exact functional flow equation of a coarse-grained effective action (or Gibbs free energy in the language of statistical mechanics). We review the main approximation schemes that are commonly used to solve this flow equation and discuss applications in equilibrium and out-of-equilibrium statistical physics, quantum many-particle systems, high-energy physics and quantum gravity.
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We study the quantum sine-Gordon model within a nonperturbative functional renormalization-group approach (FRG). This approach is benchmarked by comparing our findings for the soliton and lightest breather (soliton-antisoliton bound state) masses to exact results. We then examine the validity of the Lukyanov-Zamolodchikov conjecture for the expectation value $langle e^{frac{i}{2}nbetavarphi}rangle$ of the exponential fields in the massive phase ($n$ is integer and $2pi/beta$ denotes the periodicity of the potential in the sine-Gordon model). We find that the minimum of the relative and absolute disagreements between the FRG results and the conjecture is smaller than 0.01.
We investigate finite lattice approximations to the Wilson Renormalization Group in models of unconstrained spins. We discuss first the properties of the Renormalization Group Transformation (RGT) that control the accuracy of this type of approximations and explain different methods and techniques to practically identify them. We also discuss how to determine the anomalous dimension of the field. We apply our considerations to a linear sigma model in two dimensions in the domain of attraction of the Ising Fixed Point using a Bell-Wilson RGT. We are able to identify optimal RGTs which allow accurate computations of quantities such as critical exponents, fixed point couplings and eigenvectors with modest statistics. We finally discuss the advantages and limitations of this type of approach.
Using the nonperturbative renormalization group, we study the existence of bound states in the symmetry-broken phase of the scalar $phi^4$ theory in all dimensions between two and four and as a function of the temperature. The accurate description of the momentum dependence of the two-point function, required to get the spectrum of the theory, is provided by means of the Blaizot--Mendez-Galain--Wschebor approximation scheme. We confirm the existence of a bound state in dimension three, with a mass within 1% of previous Monte-Carlo and numerical diagonalization values.
We reexamine the two-dimensional linear O(2) model ($varphi^4$ theory) in the framework of the nonperturbative renormalization-group. From the flow equations obtained in the derivative expansion to second order and with optimization of the infrared regulator, we find a transition between a high-temperature (disordered) phase and a low-temperature phase displaying a line of fixed points and algebraic order. We obtain a picture in agreement with the standard theory of the Kosterlitz-Thouless (KT) transition and reproduce the universal features of the transition. In particular, we find the anomalous dimension $eta(Tkt)simeq 0.24$ and the stiffness jump $rho_s(Tkt^-)simeq 0.64$ at the transition temperature $Tkt$, in very good agreement with the exact results $eta(Tkt)=1/4$ and $rho_s(Tkt^-)=2/pi$, as well as an essential singularity of the correlation length in the high-temperature phase as $Tto Tkt$.
These notes provide a concise introduction to important applications of the renormalization group (RG) in statistical physics. After reviewing the scaling approach and Ginzburg-Landau theory for critical phenomena, Wilsons momentum shell RG method is presented, and the critical exponents for the scalar Phi^4 model are determined to first order in an eps expansion about d_c = 4. Subsequently, the technically more versatile field-theoretic formulation of the perturbational RG for static critical phenomena is described. It is explained how the emergence of scale invariance connects UV divergences to IR singularities, and the RG equation is employed to compute the critical exponents for the O(n)-symmetric Landau-Ginzburg-Wilson theory. The second part is devoted to field theory representations of non-linear stochastic dynamical systems, and the application of RG tools to critical dynamics. Dynamic critical phenomena in systems near equilibrium are efficiently captured through Langevin equations, and their mapping onto the Janssen-De Dominicis response functional, exemplified by the purely relaxational models with non-conserved (model A) / conserved order parameter (model B). The Langevin description and scaling exponents for isotropic ferromagnets (model J) and for driven diffusive non-equilibrium systems are also discussed. Finally, an outlook is presented to scale-invariant phenomena and non-equilibrium phase transitions in interacting particle systems. It is shown how the stochastic master equation associated with chemical reactions or population dynamics models can be mapped onto imaginary-time, non-Hermitian `quantum mechanics. In the continuum limit, this Doi-Peliti Hamiltonian is represented through a coherent-state path integral, which allows an RG analysis of diffusion-limited annihilation processes and phase transitions from active to inactive, absorbing states.
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