Renormalization Flow Equations Research Articles

A ribbon graph derivation of the algebra of functional renormalization for random multi-matrices with multi-trace interactions

We focus on functional renormalization for ensembles of several (say nge 1) random matrices, whose potentials include multi-traces, to wit, the probability measure contains factors of the form exp [-mathrm {Tr}(V_1)times cdots times mathrm {Tr}(V_k)] for certain noncommutative polynomials V_1,ldots ,V_kin {mathbb {C}}_{langle n rangle } in the n matrices. This article shows how the “algebra of functional renormalization”—that is, the structure that makes the renormalization flow equation computable—is derived from ribbon graphs, only by requiring the one-loop structure that such equation (due to Wetterich) is expected to have. Whenever it is possible to compute the renormalization flow in terms of mathrm U(N)-invariants, the structure gained is the matrix algebra M_n( mathcal {A}_{n,N}, star ) with entries in mathcal {A}_{n,N}=({mathbb {C}}_{langle n rangle } otimes {mathbb {C}}_{langle n rangle } )oplus ( {mathbb {C}}_{langle n rangle } boxtimes {mathbb {C}}_{langle n rangle }), being {mathbb {C}}_{langle n rangle } the free algebra generated by the n Hermitian matrices of size N (the flowing random variables) with multiplication of homogeneous elements in mathcal {A}_{n,N} given, for each P,Q,U,Win {mathbb {C}}_{langle n rangle }, by (U⊗W)⋆(P⊗Q)=PU⊗WQ,(U⊠W)⋆(P⊗Q)=U⊠PWQ,(U⊗W)⋆(P⊠Q)=WPU⊠Q,(U⊠W)⋆(P⊠Q)=Tr(WP)U⊠Q,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} (U \\otimes W) \\star ( P\\otimes Q)&= PU \\otimes WQ \\,, \\\\ (U\\boxtimes W) \\star ( P\\otimes Q)&=U \\boxtimes PWQ \\,, \\\\ (U \\otimes W) \\star ( P\\boxtimes Q)&= WPU \\boxtimes Q \\,, \\\\ (U\\boxtimes W) \\star ( P\\boxtimes Q)&= \\mathrm {Tr} (WP) U\\boxtimes Q \\,, \\end{aligned}$$\\end{document}which, together with the condition (lambda U) boxtimes W = Uboxtimes (lambda W) for each complex lambda , fully define the symbol boxtimes .

Fundamental scale invariance

We propose fundamental scale invariance as a new theoretical principle beyond renormalizability. Quantum field theories with fundamental scale invariance admit a scale-free formulation of the functional integral and effective action in terms of scale invariant fields. They correspond to exact scaling solutions of functional renormalization flow equations. Such theories are highly predictive since all relevant parameters for deviations from the exact scaling solution vanish. Realistic particle physics and quantum gravity are compatible with this setting. The non-linear restrictions for scaling solutions can explain properties as an asymptotically vanishing cosmological constant or dynamical dark energy that would seem to need fine tuning of parameters from a perturbative viewpoint. As an example we discuss a pregeometry based on a diffeomorphism invariant Yang-Mills theory. It is a candidate for an ultraviolet completion of quantum gravity with a well behaved graviton propagator at short distances.

Examining instabilities due to driven scalars in AdS

We extend the study of the non-linear perturbative theory of weakly turbulent energy cascades in AdSd+1 to include solutions of driven systems, i.e. those with time-dependent sources on the AdS boundary. This necessitates the activation of non-normalizable modes in the linear solution for the massive bulk scalar field, which couple to the metric and normalizable scalar modes. We determine analytic expressions for secular terms in the renormalization flow equations mass values {m}_{BF}^2<{m}^2le 0 , and for various driving functions. Finally, we numerically evaluate these sources for d = 4 and discuss what role these driven solutions play in the perturbative stability of AdS.

Breaking of scale invariance in the time dependence of correlation functions in isotropic and homogeneous turbulence

In this paper, we present theoretical results on the statistical properties of stationary, homogeneous, and isotropic turbulence in incompressible flows in three dimensions. Within the framework of the non-perturbative renormalization group, we derive a closed renormalization flow equation for a generic n-point correlation (and response) function for large wave-numbers with respect to the inverse integral scale. The closure is obtained from a controlled expansion and relies on extended symmetries of the Navier-Stokes field theory. It yields the exact leading behavior of the flow equation at large wave-numbers |p→i| and for arbitrary time differences ti in the stationary state. Furthermore, we obtain the form of the general solution of the corresponding fixed point equation, which yields the analytical form of the leading wave-number and time dependence of n-point correlation functions, for large wave-numbers and both for small ti and in the limit ti → ∞. At small ti, the leading contribution at large wave-numbers is logarithmically equivalent to −α(εL)2/3|∑tip→i|2, where α is a non-universal constant, L is the integral scale, and ε is the mean energy injection rate. For the 2-point function, the (tp)2 dependence is known to originate from the sweeping effect. The derived formula embodies the generalization of the effect of sweeping to n-point correlation functions. At large wave-numbers and large ti, we show that the ti2 dependence in the leading order contribution crosses over to a |ti| dependence. The expression of the correlation functions in this regime was not derived before, even for the 2-point function. Both predictions can be tested in direct numerical simulations and in experiments.

Renormalization group, secular term resummation and AdS (in)stability

We revisit the issues of non-linear AdS stability, its relation to growing (secular) terms in naive perturbation theory around the AdS background, and the need and possible strategies for resumming such terms. To this end, we review a powerful and elegant resummation method, which is mathematically identical to the standard renormalization group treatment of ultraviolet divergences in perturbative quantum field theory. We apply this method to non-linear gravitational perturbation theory in the AdS background at first non-trivial order and display the detailed structure of the emerging renormalization flow equations. We prove, in particular, that a majority of secular terms (and the corresponding terms in the renormalization flow equations) that could be present on general grounds given the spectrum of frequencies of linear AdS perturbations, do not in fact arise.

Schwinger-Dyson renormalization group

We use the Schwinger-Dyson equations as a starting point to derive renormalization flow equations. We show that Katanin's scheme arises as a simple truncation of these equations. We then give the full renormalization group equations up to third order in the irreducible vertex. Furthermore, we show that to the fifth order, there exists a functional of the self-energy and the irreducible four-point vertex whose saddle point is the solution of Schwinger-Dyson equations.

Functional renormalization group for few-nucleon systems: SU(4) symmetry and its breaking

We apply the functional renormalization group to few-nucleon systems. Our starting point is a local effective action that includes three- and four-nucleon interactions, expressed in terms of nucleon and two-nucleon boson fields. The evolution of the coupling constants in this action is described by a renormalization group flow. We derive these flow equations both in the limit of exact Wigner SU(4) symmetry and in the realistic case of broken symmetry. In the symmetric limit we find that the renormalization flow equations decouple and can be combined into two sets, one of which matches the known results for bosons, and the other result matches the one for a single flavor of spin $1/2$ fermions. The equations show universal features in the unitary limit, which is obtained when the two-body scattering length tends to infinity. We calculate the spin-quartet neutron-deuteron scattering length and the deuteron-deuteron scattering lengths in the spin-singlet and spin-quintet channels.

Functional renormalization flow and dynamical chiral symmetry breaking of QCD

The dependence of function renormalization group equation on regulators is investigated. A parameter is introduced to control the suppression of regulators. Functional renormalization group equations will become regulator-independent if this newly introduced parameter is sent to infinity in the end of calculation. One-loop renormalization flow equations of QCD are derived. The novelty is that both the coupling running equation and the mass running equation are mass-dependent. Different flow patterns are explored. A mechanism for non-occurrence of dynamical chiral symmetry breaking is arrived at. The existence of a conformal window is also discussed in the language of renormalization flow.

Non-equilibrium scaling properties of a double quantum dot system: Comparison between perturbative renormalization group and flow equation approach

Since the experimental realization of Kondo physics in quantum dots, its far-from-equilibrium properties have generated considerable theoretical interest. This is due to the interesting interplay of non-equilibrium physics and correlation effects in this model, which has now been analyzed using several new theoretical methods that generalize renormalization techniques to non-equilibrium situations. While very good agreement between these methods has been found for the spin-1/2 Kondo model, it is desirable to have a better understanding of their applicability for more complicated impurity models. In this paper the differences and commons between two such approaches, namely the flow equation method out of equilibrium and the frequency-dependent poor man's scaling approach are presented for the non-equilibrium double quantum dot system. This will turn out to be a particularly suitable testing ground while being experimentally interesting in its own right. An outlook is given on the quantum critical behavior of the double quantum dot system and its accessibility with the two methods.

Theory of Smeared Quantum Phase Transitions

We present an analytical strong-disorder renormalization group theory of the quantum phase transition in the dissipative random transverse-field Ising chain. For Ohmic dissipation, we solve the renormalization flow equations analytically, yielding asymptotically exact results for the low-temperature properties of the system. We find that the interplay between quantum fluctuations and Ohmic dissipation destroys the quantum critical point by smearing. We also determine the phase diagram and the behavior of observables in the vicinity of the smeared quantum phase transition.

Direct test of defect-mediated laser-induced melting theory for two-dimensional solids

We investigate by direct numerical solution of appropriate renormalization flow equations the validity of a recent dislocation unbinding theory for laser-induced freezing and melting in two dimensions. The bare elastic moduli and dislocation fugacities are obtained for three different two-dimensional systems namely, the hard disk, inverse 12th power, and Derjaguin-Landau-Verwey-Overbeek potentials. A restricted Monte Carlo simulation sampling only configurations without dislocations is used to obtain these quantities. These are then used as inputs to the flow equations. Numerical solution of the flow equations then yields the phase diagrams. We conclude that (a) the flow equations need to be correct at least up to third order in defect fugacity to reproduce meaningful results, (b) there is excellent quantitative agreement between our results and earlier conventional Monte Carlo simulations for the hard disk system, and (c) while the qualitative form of the phase diagram is reproduced for systems with soft potentials there is some quantitative discrepancy which we explain.

Applications of field-theoretic renormalization group methods to reaction–diffusion problems

We review the application of field-theoretic renormalization group (RG) methods to the study of fluctuations in reaction-diffusion problems. We first investigate the physical origin of universality in these systems, before comparing RG methods to other available analytic techniques, including exact solutions and Smoluchowski-type approximations. Starting from the microscopic reaction-diffusion master equation, we then pedagogically detail the mapping to a field theory for the single-species reaction k A -> l A (l < k). We employ this particularly simple but non-trivial system to introduce the field-theoretic RG tools, including the diagrammatic perturbation expansion, renormalization, and Callan-Symanzik RG flow equation. We demonstrate how these techniques permit the calculation of universal quantities such as density decay exponents and amplitudes via perturbative eps = d_c - d expansions with respect to the upper critical dimension d_c. With these basics established, we then provide an overview of more sophisticated applications to multiple species reactions, disorder effects, L'evy flights, persistence problems, and the influence of spatial boundaries. We also analyze field-theoretic approaches to nonequilibrium phase transitions separating active from absorbing states. We focus particularly on the generic directed percolation universality class, as well as on the most prominent exception to this class: even-offspring branching and annihilating random walks. Finally, we summarize the state of the field and present our perspective on outstanding problems for the future.

Bond-disordered spin systems: Theory and application to doped high-Tccompounds

We examine the stability of magnetic order in a classical Heisenberg model with quenched random exchange couplings. This system represents the spin degrees of freedom in high-${T}_{c}$ compounds with immobile dopants. Starting from a replica representation of the nonlinear-$\ensuremath{\sigma}$ model, we perform a renormalization-group analysis. The importance of cumulants of the disorder distribution to arbitrarily high orders necessitates a functional renormalization scheme. From the renormalization flow equations we determine the magnetic correlation length numerically as a function of the impurity concentration and temperature. From our analysis it follows that two-dimensional layers can be magnetically ordered for arbitrarily strong but sufficiently diluted defects. We further consider the dimensional crossover in a stack of weakly coupled layers. The resulting phase diagram is compared with experimental data for ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4}.$

Flow equations forN point functions and bound states

We discuss the exact renormalization group or flow equation for the effective action and its decomposition into one particle irreducible N point functions. With the help of a truncated flow equation for the four point function we study the bound state problem for scalar fields. A combination of analytic and numerical methods is proposed, which is applied to the Wick-Cutkosky model and a QCD-motivated interaction. We present results for the bound state masses and the Bethe-Salpeter wave function. (Figs. 1-4 attached as separate uuencoded post-script files.)

Helmholtz free energy of finite spin systems near criticality.

We present a quantitative comparison between 4 − e expansion and Monte Carlo estimates of critical finite-size properties: Specifically we consider the Helmholtz free energy (where the overall order parameter rather than its conjugate field is kept fixed) at the bulk critical temperature for a cube with periodic boundary conditions within a 4 − e expansion to one-loop order. An estimate for the complete asymptotic scaling function obtained from renormalization flow equations as well as systematic e-expansion estimates for several amplitude ratios agree well with corresponding Monte Carlo results in three dimensions. The nonequivalence of thermodynamic ensembles for critical finite-size properties is discussed in some detail.

Renormalization of the gas-liquid transition

Abstract Renormalization flow equations are constructed for fluids. The structure of the equations is an explicit hierarchy involving the thermodynamic potential, the two-particle potential, three-particle potential, etc. The transformation of the pressure, density and correlation functions under this renormalization transformation are also deduced explicitly. The conditions on the weights in the renormalization procedure are discussed.

Exact differential renormalization group equations for Ising models on square lattices

The differential real space renormalization theory of Hilhorst et al. is applied to Ising models on square lattices with nearest-neighbour interactions only. The renormalization flow equations for these two interaction parameters contain two auxiliary parameters; these parameters have to be determined by solving two additional equations, one partial differential equation and one ordinary equation. The necessity of introducing these additional parameters is explained by arguing that for the present formulation of differential real space renormalization theory in d dimensions, at least d + 1 parameters are required. The concept of local fixed point is introduced; this fixed point can be determined by solving algebraic equations. The linearized flow around it describes local properties of the system and is therefore related to the critical properties of homogeneous Ising systems. We study temperature-like perturbations around the local fixed point and find a unique eigenvalue yT = 1, in agreement with the known exact result.