Renormalization Group Flow Equations Research Articles

Robust marginal Fermi liquid in birefringent semimetals

We investigate the interplay of Coulomb interactions and correlated disorder in pseudospin-3/2 semimetals, which exhibit birefringent spectra in the absence of interactions. Coulomb interactions drive the system to a marginal Fermi liquid, both for the two-dimensional (2d) and three-dimensional (3d) cases. Short-ranged correlated disorder in 2d, or a power-law correlated disorder 3d, has the same engineering dimension as the Coulomb term, in a renormalization group (RG) sense. In order to analyze the combined effects of these two kinds of interactions, we apply a dimensional regularization scheme and derive the RG flow equations. The results show that the marginal Fermi liquid phase is robust against disorder.

Quantum discontinuity fixed point and renormalization group flow of the Sachdev-Ye-Kitaev model

We determine the global renormalization group (RG) flow of the Sachdev-Ye-Kitaev (SYK) model. From a controlled truncation of the infinite hierarchy of the exact functional RG flow equations, we identify several fixed points. Apart from a stable fixed point, associated with the celebrated non-Fermi liquid state of the model, we find another stable fixed point related to an integer-valence state. These stable fixed points are separated by a discontinuity fixed point with one relevant direction, describing a quantum first-order transition. Most notably, the fermionic spectrum continues to be quantum critical even at the discontinuity fixed point. This rules out a description of the transition in terms of a local effective Ising variable as is established for classical transitions. We propose an entangled quantum state at phase coexistence as a possible physical origin of this critical behavior.

(No) phase transition in tensorial group field theory

Continuum spacetime is expected to emerge via phase transition in discrete approaches to quantum gravity. A promising example is tensorial group field theory but its phase diagram remains an open issue. The results of recent attempts in terms of the functional renormalization group method remain inconclusive since they are restricted to truncations of low order. We overcome this barrier with a local-potential approximation for U(1) tensor fields at arbitrary rank r focusing on a specific class of so-called cyclic-melonic interactions. Projecting onto constant field configurations, we obtain the full set of renormalization-group flow equations. At large cut-offs we find equivalence with r−1 dimensional O(N) scalar field theory in the large-N limit, modified by a tensor-specific, relatively large anomalous dimension. However, on small length scales there is equivalence with the corresponding scalar field theory with vanishing dimension and, thus, no phase transition. This is confirmed by numerical analysis of the full non-autonomous equations where we always find symmetry restoration. The essential reason for this effect is isolated zero modes. This result should therefore be true for tensor field theories on any compact domain and including any tensor-invariant interactions. Thus, group field theories with non-compact degrees of freedom will be necessary to describe a phase transition to continuum spacetime.

Transasymptotics and hydrodynamization of the Fokker-Planck equation for gluons

We investigate the non-linear transport processes and hydrodynamization of a system of gluons undergoing longitudinal boost-invariant expansion. The dynamics is described within the framework of the Boltzmann equation in the small-angle approximation. The kinetic equations for a suitable set of moments of the one-particle distribution function are derived. By investigating the stability and asymptotic resurgent properties of this dynamical system, we demonstrate, that its solutions exhibit a rather different behavior for large (UV) and small (IR) effective Knudsen numbers. Close to the forward attractor in the IR regime the constitutive relations of each moment can be written as a multiparameter transseries. This resummation scheme allows us to extend the definition of a transport coefficient to the non-equilibrium regime naturally. Each transport coefficient is renormalized by the non-perturbative contributions of the non-hydrodynamic modes. The Knudsen number dependence of the transport coefficient is governed by the corresponding renormalization group flow equation. An interesting feature of the Yang-Mills plasma in this regime is that it exhibits transient non-Newtonian behavior while hydrodynamizing. In the UV regime the solution for the moments can be written as a power-law asymptotic series with a finite radius of convergence. We show that radius of convergence of the UV perturbative expansion grows linearly as a function of the shear viscosity to entropy density ratio. Finally, we compare the universal properties in the pullback and forward attracting regions to other kinetic models including the relaxation time approximation and the effective kinetic Arnold-Moore-Yaffe (AMY) theory.

RG flows of integrable \u03c3-models and the twist function

In the study of integrable non-linear σ-models which are assemblies and/or deformations of principal chiral models and/or WZW models, a rational function called the twist function plays a central rôle. For a large class of such models, we show that they are one-loop renormalizable, and that the renormalization group flow equations can be written directly in terms of the twist function in a remarkably simple way. The resulting equation appears to have a universal character when the integrable model is characterized by a twist function.

High-order corrections to inflationary perturbation spectra in quantum gravity

We compute the inflationary perturbation spectra and the quantity r+8nT to the next-to-next-to-leading log order in quantum gravity with purely virtual particles (which means the theory R+R2+C2 with the fakeon prescription/projection for C2). The spectra are functions of the inflationary running coupling α (1/k) and satisfy the cosmic renormalization-group flow equations, which determine the tilts and the running coefficients. The tensor fluctuations receive contributions from the spin-2 fakeon χ μν at every order of the expansion in powers of α ∼ 1/115. The dependence of the scalar spectrum on the χ μν mass mχ, on the other hand, starts from the α2 corrections, which are handled perturbatively in the ratio mϕ/mχ, where mϕ is the inflaton mass. The predictions have theoretical errors ranging from α4∼10−8 to α3 ∼ 10−6. Nontrivial issues concerning the fakeon projection at higher orders are addressed.

Topological phase transitions in four dimensions

We show that four-dimensional systems may exhibit a topological phase transition analogous to the well-known Berezinskii-Kosterlitz-Thouless vortex unbinding transition in two-dimensional systems. We study a suitable generalization of the sine-Gordon model in four dimensions and the renormalization group flow equation of its couplings, showing that the critical value of the frequency is the square of the corresponding value in 2D. The value of the anomalous dimension at the critical point is determined (η=1/32) and a conjecture for the universal jump of the superfluid stiffness (4/π2) presented.

Operator evolution from the similarity renormalization group and the Magnus expansion

The Magnus expansion is an efficient alternative to solving similarity renormalization group (SRG) flow equations with high-order, memory-intensive ordinary differential equation solvers. The numerical simplifications it offers for operator evolution are particularly valuable for in-medium SRG calculations, though challenges remain for difficult problems involving intruder states. Here we test the Magnus approach in an analogous but more accessible situation, which is the free-space SRG treatment of the spurious bound-states arising from a leading-order chiral effective field theory (EFT) potential with very high cutoffs. We show that the Magnus expansion passes these tests and then use the investigations as a springboard to address various aspects of operator evolution that have renewed relevance in the context of the scale and scheme dependence of nuclear processes. These aspects include SRG operator flow with band- versus block-diagonal generators, universality for chiral EFT Hamiltonians and associated operators with different regularization schemes, and the impact of factorization arising from scale separation. Implications for short-range correlations physics and the possibilities for reconciling high- and low-resolution treatments of nuclear structure and reactions are discussed.

A new method of solution of the Wetterich equation and its applications

The known approximation schemes for the solution of the Wetterich exact renormalization group (RG) equation are critically reconsidered, and a new truncation scheme is proposed. In particular, the equations of the derivative expansion up to the ∂2 order for a scalar model are derived in a suitable form, clarifying the role of the off-diagonal terms in the matrix of functional derivatives. The natural domain of validity of the derivative expansion appears to be limited to small values of q/k in the calculation of the critical two-point correlation function, depending on the wave-vector magnitude q and the infrared cut-off scale k. The new approximation scheme has the advantage to be valid for any q/k, and, therefore, it can be auspicious in many current and potential applications of the celebrated Wetterich equation and similar models. Contrary to the derivative expansion, derivatives are not truncated at a finite order in the new scheme. The RG flow equations in the first approximation of this new scheme are derived and approximately solved as an example. It is shown that the derivative expansion up to the ∂2 order is just the small-q approximation of our new equations at the first order of truncation.

Infrared behavior of Weyl gravity: Functional renormalization group approach

Starting from an ultraviolet fixed point, we study the infrared behavior of quantum Weyl gravity in terms of a functional renormalization group (RG) flow equation. To do so, we employ two classes of Bach-flat backgrounds, namely maximally symmetric spacetimes and Ricci-flat backgrounds in the improved one-loop scheme. We show that in the absence of matter fields and with a topological term included, the effective action exhibits dynamical breaking of scale symmetry. In particular, it is shown that apart from a genuine IR fixed point that is reached at a zero-value of the running scale, the RG flow also exhibits bouncing behavior in the IR regime. We demonstrate that both $\beta_C$ and $\beta_E$ reach the RG turning point (almost) simultaneously at the same finite energy scale, irrespectively of the chosen background. The IR fixed point itself is found to be IR-stable in the space of the considered couplings. Ensuing scaling dimensions of both operators are also computed. Salient issues, including the connection of the observed bouncing RG flow behavior with holography and prospective implications in early Universe cosmology, are also briefly discussed.

Convergent scattering series solution of the inhomogeneous Helmholtz equation via renormalization group and homotopy continuation approaches

The inhomogeneous Helmholtz equation can be represented by an equivalent integral equation of the Lippmann-Schwinger type. Since it is very costly to solve the Lippmann-Schwinger equation exactly via matrix inversion, researchers often try to use the popular Born series, which represents a physics based iterative solution. However, the popular Born series is only guaranteed to converge if the contrast volume is sufficiently small. We here use renormalization group (RG) and homotopy continuation (HC) approaches to derive novel scattering series solutions of the Lippmann-Schwinger equation which are guaranteed to converge for arbitrary large contrast volumes (perturbations). Our RG approach is based on the use of an auxillary set of scale-dependent scattering potentials and Green's operators which gradually evolve toward the real physical scattering potential and the physical Green's operator, respectively. We show that the auxillary Green's operators satisfies the renormalization group law and we derive the corresponding renormalization group flow equation by using familiar “scaling” arguments. The HC approach comes from pure mathematics and is based on the deformation of a relatively simple reference equation (initial condition) into a more complicated target equation (the Lippmann-Schwinger equation for the physical scattering potential), via a parameter that varies continuously between 0 and 1. Our HC approach is more general than the RG approach since it involves a convergence control operator, but the two approaches are nevertheless closely connected. We show that the RG flow and HC equations can be implemented very efficiently using the Fast Fourier Transform (FFT) algorithm. Also, we present a very efficient iterative scheme based on a combination of the zeroth-order deformation equation with a floating initial. We illustrate our results with a numerical example related with seismic wavefield modeling in strongly scattering media.

Fluctuating vector mesons in analytically continued functional RG flow equations

In this work we study contributions due to vector and axial-vector meson fluctuations to their in-medium spectral functions in an effective low-energy theory inspired by the gauged linear sigma model. In particular, we show how to describe these fluctuations in the effective theory by massive (axial-)vector fields in agreement with the known structure of analogous single-particle or resonance contributions to the corresponding conserved currents. The vector and axial-vector meson spectral functions are then computed by numerically solving the analytically continued functional renormalization group flow equations for their retarded two-point functions at finite temperature and density in the effective theory. We identify the new contributions that arise due to the (axial-)vector meson fluctuations, and assess their influence on possible signatures of a QCD critical end point and the restoration of chiral symmetry in thermal dilepton spectra.

BRST in the exact renormalization group

AbstractWe show, explicitly within perturbation theory, that the quantum master equation and the Wilsonian renormalization group flow equation can be combined such that for the continuum effective action, quantum BRST invariance is not broken by the presence of an effective ultraviolet cutoff $\Lambda$, despite the fact that the structure demands quantum corrections that naïvely break the gauge invariance, such as a mass term for a non-Abelian gauge field. Exploiting the derivative expansion, BRST cohomological methods fix the solution up to choice of renormalization conditions, without inputting the form of the classical, or bare, interactions. Legendre transformation results in an equivalent description in terms of solving the modified Slavnov–Taylor identities and the flow of the Legendre effective action under an infrared cutoff $\Lambda$ (i.e. effective average action). The flow generates a canonical transformation that automatically solves the Slavnov–Taylor identities for the wavefunction renormalization constants. We confirm this structure in detail at tree level and one loop. Under flow of $\Lambda$, the standard results are obtained for the beta function, anomalous dimension, and physical amplitudes, up to the choice of the renormalization scheme.

Holographic Renormalization Group Flows

We briefly review recent applications of holographic renormalization group flow equations in a hot and dense quark-gluon plasma (QGP). We especially focus on presenting a few examples typically used in the holographic description of the QGP. We study the influence of the chemical potential on the holographic β-function in these examples.

Exact renormalization group for quantum spin systems

We show that the diagrammatic approach to quantum spin systems developed in a seminal work by Vaks, Larkin, and Pikin [Sov. Phys. JETP 26, 188 (1968)] can be embedded in the framework of the functional renormalization group. The crucial insight is that the generating functional of the time-ordered connected spin correlation functions of an arbitrary quantum spin system satisfies an exact renormalization group flow equation which resembles the corresponding flow equation of a system of interacting bosons. The $SU (2)$ spin algebra is implemented via a non-trivial initial condition for the renormalization group flow. Our method is rather general and offers a new non-perturbative approach to quantum spin systems.

Phase structure of the Euclidean three-dimensional O(1) ghost model

We have treated the Euclidean three-dimensional O(1) ghost model with a modified version of the effective average action (EAA) renormalization group (RG) method, developed by us. We call it Fourier–Wetterich RG approach and it is used to investigate the occurrence of a periodic condensate in terms of the functional RG. The modification involves additional terms in the ansatz of the EAA, corresponding to the Fourier-modes of the periodic condensate. The RG flow equations are derived keeping the terms up to the fourth order of the gradient expansion (GE), however the numerical calculations are conducted in the second order (or next-to-leading order, NLO) of the GE. The expansion of the flow equations around the nontrivial minimum of the local potential takes into account properly the vertices induced by the periodic condensate even if the wave function renormalization is set to be field-independent. The numerical analysis reveals several different phases with three multicritical points.

Information loss under coarse graining: A geometric approach

Information theoretic geometry near critical points in classical and quantum systems is well understood for exactly solvable systems. Here we show that renormalization group flow equations can be used to construct the information metric and its associated quantities near criticality, for both classical and quantum systems, in an universal manner. We study this metric in various cases and establish its scaling properties in several generic examples. Scaling relations on the parameter manifold involving scalar quantities are studied, and scaling exponents are identified. The meaning of the scalar curvature and the invariant geodesic distance in information geometry is established and substantiated from a renormalization group perspective.

Renormalization group for the φ^{4} theory with long-range interaction and the critical exponent η of the Ising model.

We calculate the critical exponent η of the D-dimensional Ising model from a simple truncation of the functional renormalization group flow equations for a scalar field theory with long-range interaction. Our approach relies on the smallness of the inverse range of the interaction and on the assumption that the Ginzburg momentum defining the width of the scaling regime in momentum space is larger than the scale where the renormalized interaction crosses over from long range to short range; the numerical value of η can then be estimated by stopping the renormalization group flow at this scale. In three dimensions our result η=0.03651 is in good agreement with recent conformal bootstrap and Monte Carlo calculations. We extend our calculations to fractional dimensions D and obtain the resulting critical exponent η(D) between two and four dimensions. For dimensions 2≤D≤3 our result for η is consistent with previous calculations.

Physical renormalization schemes and asymptotic safety in quantum gravity

The methods of the renormalization group and the $\varepsilon$-expansion are applied to quantum gravity revealing the existence of an asymptotically safe fixed point in spacetime dimensions higher than two. To facilitate this, physical renormalization schemes are exploited where the renormalization group flow equations take a form which is independent of the parameterisation of the physical degrees of freedom (i.e. the gauge fixing condition and the choice of field variables). Instead the flow equation depends on the anomalous dimensions of reference observables. In the presence of spacetime boundaries we find that the required balance between the Einstein-Hilbert action and Gibbons-Hawking-York boundary term is preserved by the beta functions. Exploiting the $\varepsilon$-expansion near two dimensions we consider Einstein gravity coupled to matter. Scheme independence is generically obscured by the loop-expansion due to breaking of two-dimensional Weyl invariance. In schemes which preserve two-dimensional Weyl invariance we avoid the loop expansion and find a unique ultra-violet (UV) fixed point. At this fixed point the anomalous dimensions are large and one must resum all loop orders to obtain the critical exponents. Performing the resummation a set of universal scaling dimensions are found. These scaling dimensions show that only a finite number of matter interactions are relevant. This is a strong indication that quantum gravity is renormalizable.

Wilsonian RG-flow approach to holographic transport with momentum dissipation

We systematically present a new approach for studying the coupled linear transport of holographic systems. In this approach, the set of equations for the linear perturbations can be reduced to a first-order nonlinear ordinary differential equation expressed as the radial (renormalization group) flow equation of the transport matrices. As an important application, we use this approach to compute the dc and ac conductivities of a holographic model with momentum dissipation, which can be easily read off from the nonlinear flow equations. This method also works for transport in the presence of an external magnetic field.