Renormalization Group Flow Equations Research Articles

Higher-derivative four-dimensional sine-Gordon model

The phase structure of a higher derivative sine-Gordon model in four dimensions is analysed. It is shown that the inclusion of a relevant two-derivative term in the action significantly modifies some of the results obtained by neglecting this operator, and the final picture is substantially different from the one describing the phase diagram associated with the two-dimensional Berezinskii-Kosterlitz–Thouless (BKT) transition. The study is carried out with the help of the Renormalization Group (RG) flow equations, determined for a set of three parameters, and numerically solved both for a truncated series expansion approximation, and for the complete set of equations. In both cases, a continuous line of fixed points, terminating at a particular point presenting universal properties, is found, together with a manifold that separates two phases, roughly characterized by the sign of the coupling z˜k associated with this newly included operator. While the phase corresponding to z˜k>0 shows some pathologies, the one with z˜k<0 has a well-behaved infrared limit, where the system reduces to a Gaussian-like model. We also briefly comment about the possibility that our model could capture some of the qualitative features of the ultraviolet (UV) critical manifold of conformally reduced gravity.

Time evolution and thermal renormalization group flow in cosmology

Time-evolution of the Universe as described by the Friedmann equation can be coupled to equations of motion of matter fields. Quantum effects may be incorporated to improve these classical equations of motion by the renormalization group (RG) running of their couplings. Since temporal and thermal evolutions are linked to each other, astrophysical and cosmological treatments based on zero-temperature RG methods require the extension to finite-temperatures. We propose and explore a modification of the usual finite-temperature RG approach by relating the temperature parameter to the running RG scale as T≡kT=τk (in natural units), where kT is acting as a running cutoff for thermal fluctuations and the momentum k can be used for the quantum fluctuations. In this approach, the temperature of the expanding universe is related to the dimensionless quantity τ (and not to kT). We show that by this choice dimensionless RG flow equations have no explicit k-dependence, as it is convenient. We also discuss how this modified thermal RG is used to handle high-energy divergences of the RG running of the cosmological constant and to “solve the triviality” of the ϕ4 model by a thermal phase transition in terms of τ in d=4 Euclidean dimensions.

First-order formalism for β functions in bosonic sigma models from supersymmetry breaking

We consider the renormalization group flow equation for the two-dimensional sigma models with the Kähler target space. The first-order formulation allows us to treat perturbations in these models as current-current deformations. We demonstrate, however, that the conventional first-order formalism misses certain anomalies in the measure, and should be amended. We reconcile beta functions obtained within the conformal perturbation theory for the current-current deformations with traditional “geometric” results obtained in the background field methods, in this way resolving the peculiarities pointed out in O. Gamayun [Peculiarities of beta functions in sigma models, ]. The result is achieved by the supersymmetric completion of the first-order sigma model. Published by the American Physical Society 2024

Original and modified non-perturbative renormalization group equations of the BMW scheme at the arbitrary order of truncation

We consider the non-perturbative renormalization group (RG) equations, obtained as approximations of the exact Wetterich RG flow equation within the Blaizot–Mendez–Wschebor (BMW) truncation scheme. For the first time, we derive explicit RG flow equations for the scalar model at the arbitrary order of truncation. Moreover, we consider original, as well as modified, approximations, used to obtain a set of closed equations. We compare these equations at the s = 2 order of truncation with those recently derived in J. Phys. A: Math. Theor. 53, 415002 (2020) within a new truncation scheme and find a striking similarity. Namely, the first-order equations of the latter scheme, those of the original BMW scheme, and those of the modified BMW scheme (at s = 2) differ only in one term. We solved these equations by a recently proposed and tested method of semi-analytic approximations. Thus, the critical exponents η, ν, and ω were evaluated, recovering also the known results of the original BMW scheme. In addition, we estimated the subleading correction-to-scaling exponent ω2 for the three equations considered. To the best of our knowledge, this exponent has not yet been extracted from the Wetterich equation beyond the local potential (the zeroth order) approximation. Our current estimate for the 3D Ising model is ω2 = 2.02 (40), where the error bars include the expected truncation error in the BMW scheme.

Exact flow equation for the divergence functional

An exact functional renormalization group flow equation is derived for the divergence functional which is a generalization of the Kullback-Leibler divergence to quantum field theories in the Euclidean domain. It compares distributions with different sources and field expectation values. The renormalization group flow for a regularized version of this functional connects two limits: one where the functional is known in terms of the microscopic action or probability distribution, and the other where all fluctuations are taken into account. In the latter limit one can obtain full correlation functions from functional derivatives of the divergence functional. The flow equation provides a possibility to determine this functional non-perturbatively.

Generalized Devil's staircase and RG flows

We discuss a two-parameter renormalization group (RG) flow when parameters are organized in a single complex variable, τ, with modular properties. Throughout the work we consider a special limit when the imaginary part of τ characterizing the disorder strength tends to zero. We argue that generalized Riemann-Thomae (gRT) function and the corresponding generalized Devil's staircase emerge naturally in a variety of physical models providing a universal behavior. In 1D we study the Anderson-like probe hopping in a weakly disordered lattice, recognize the origin of the gRT function in the spectral density of the probe and formulate specific RG procedure which gets mapped onto the discrete flow in the fundamental domain of the modular group SL(2,Z). In 2D we consider the generalization of the phyllotaxis crystal model proposed by L. Levitov and suggest the explicit form of the effective potential for the probe particle propagating in the symmetric and asymmetric 2D lattice of defects. Analyzing the structure of RG flow equations in the vicinity of saddle points we claim emergence of BKT-like transitions at Imτ→0. We show that the RG-like dynamics in the fundamental domain of SL(2,Z) for asymmetric lattices asymptotically approaches the “Silver ratio”. For a Hubbard model of particles on a ring interacting via long-ranged potentials we investigate the dependence of the ground state energy on the potential and demonstrate by combining numerical and analytical tools the emergence of the generalized Devil's staircase. Also we conjecture a bridge between a Hubbard model and a phyllotaxis.

On the renormalization of non-polynomial field theories

A class of scalar models with non-polynomial interaction, which naturally admits an analytical resummation of the series of tadpole diagrams is studied in perturbation theory. In particular, we focus on a model containing only one renormalizable coupling that appear as a multiplicative coefficient of the squared field. A renormalization group analysis of the Green functions of the model shows that these are only approximated solutions of the flow equations, with errors proportional to powers of the coupling, therefore smaller in the region of weak coupling. The final output of the perturbative analysis is that the renormalized model is non-interacting with finite mass and vanishing vertices or, in an effective theory limited by an ultraviolet cut-off, the vertices are suppressed by powers of the inverse cut-off. The relation with some non-polynomial interactions derived long ago, as solutions of the linearized functional renormalization group flow equations, is also discussed.

Hierarchical proliferation of higher-rank symmetry defects in fractonic superfluids

Symmetry defects, e.g., vortices in conventional superfluids, play a critical role in a complete description of symmetry breaking phases. In this paper, we develop the theory of symmetry defects in fractonic superfluids, i.e., spontaneously higher-rank symmetry (HRS) breaking phases initially proposed by Yuan et al. [Phys. Rev. Res. 2, 023267 (2020)] and by Chen et al. [Phys. Rev. Res. 3, 013226 (2021)]. By Noether's theorem, HRS is associated with the conservation law of higher moments, e.g., dipoles, quadrupoles, and angular moments. We establish finite-temperature phase diagrams by identifying a series of topological phase transitions via the renormalization group flow equations and Debye-H\"uckel approximation. Accordingly, a series of Kosterlitz-Thouless topological transitions are found to occur successively at different temperatures, which are triggered by proliferation of defects, defect bound states, and so on. Such a hierarchical proliferation brings rich phase structures. Meanwhile, a screening effect from sufficiently high density of defect bound states leads to instability and collapse of the intermediate temperature phases, which further enriches the phase diagrams. For concreteness, we consider a fractonic superfluid in which ``angular moments'' are conserved. We then present the general theory, in which other types of HRS can be analyzed in a similar manner. Further directions are present at the end of the paper.

Renormalization group flow to effective quantum mechanics in the IR in an emergent dual holographic description for spontaneous chiral symmetry breaking

Implementing the Wilsonian renormalization group (RG) transformation in a nonperturbative way, we construct an effective holographic dual description with an emergent extradimension identified with an RG scale. Taking the large$-N$ limit, we obtain an equation of motion of an order-parameter field, here the chiral condensate for our explicit demonstration. In particular, an intertwined structure manifests between the first-order RG flow equations of renormalized coupling functions and the second-order differential equation of the order-parameter field, thus referred to as a nonperturbative RG-improved mean-field theory. Assuming translational symmetry as a vacuum state, we solve these nonlinear coupled mean-field equations based on a matching method between UV- and IR-regional solutions. As a result, we find an RG flow from a weakly-coupled chiral-symmetric UV fixed point to a strongly-correlated chiral-symmetry broken IR fixed point, where the renormalized velocity of Dirac fermions vanishes most rapidly and effective quantum mechanics appears at IR. Furthermore, we translate these RG flows of coupling functions into those of emergent metric tensors and extract out geometrical properties of the emergent holographic spacetime constructed from the UV- and IR-regional solutions. Surprisingly, we obtain the volume law of entanglement entropy in the Ryu-Takayanagi formula, which implies appearance of a black hole type solution in the limit of infinite cutoff even at zero temperature. We critically discuss our field theoretic interpretation for this solution in terms of potentially gapless multi-particle excitation spectra.

Renormalization group and UV completion of cosmological perturbations: Gravitational collapse as a critical phenomenon

Cosmological perturbation theory is known to converge poorly for predicting the spherical collapse and void evolution of collisionless matter. Using the exact parametric solution as a testing ground, we develop two asymptotic methods in spherical symmetry that resolve the gravitational evolution to much higher accuracy than Lagrangian perturbation theory (LPT), which is the current gold standard in the literature. One of the methods selects a stable fixed-point solution of the renormalization-group flow equation, thereby predicting already at the leading order the critical exponent of the phase transition to collapsed structures. The other method completes the truncated LPT series far into the UV regime, by adding a non-analytic term that captures the critical nature of the gravitational collapse. We find that the UV method most accurately resolves the evolution of the nonlinear density as well as its one-point probability distribution function. Similarly accurate predictions are achieved with the renormalization-group method, especially when paired with Pad\'e approximants. Further, our results yield new, very accurate, formulas to relate linear and nonlinear density contrasts. Finally, we chart possible ways on how to adapt our methods to the case of cosmological random field initial conditions.

A Coherent Approach towards Quantum Gravity

This paper typically focuses on the rescaling or equivocally a phase transition from the asymptotic approach of renormalizing the quantum gravity to a more granular approach of the loop quantum gravity (LQG) and then merging it with the Regge calculus for deriving the spin-(2) graviton as the basis of the unified theory. To construct a successful Ultraviolet (UV) completed theory via the fixed-point renormalization group flow equations (FRGE) results in an asymptotic safety approach of the quantum gravity (QG). From the loop-(2) onwards, the higher derivative divergence terms like the higher derivative curvatures, and quadratic divergences with higher derivative scalars make the momentum go to infinity which assaults a problem in renormalizing the QG. If the Einstein-Hilbert (E-H) action, which is the principle of least action is being computed, arising an equation of motion, and a localized path integral (or partition functions) is defined over a curved space, then that action is shown to be associated with the higher order dimension in a more compactified way, resulting in an infinite winding numbers being accompanied through the exponentiality coefficients of the partition integrals in the loop expansions of the second order term onwards, and based on that localization principle, the entire path integral got collapsed to isolated points or granules that if corresponds the aforesaid actions, results in negating the divergences’ with an implied bijections’ and reverse bijections’ of a diffeomorphism of a continuous differentiable functional domains. If those domains are being attributed to the spatial constraints, Hamiltonian constraints, and Master constraints then, through Ashtekar variables, it can be modestly shown that the behavior of quantum origin of asymptotic safety behavior is similar to the LQG granules of spin foam spacetime. Then, we will proceed with the triangulation of the “zoomed in” entangled-points that results in the inclusion of Regge poles via the quantum number (+2,-2,0) as the produced variables of the spin-(2) graviton and spin-(0) dilaton.

Constructing CFTs from AdS flows

We study the renormalization group flow equations for correlation functions of weakly coupled quantum field theories in AdS. Taking the limit where the external points approach the conformal boundary, we obtain a flow of conformally invariant correlation functions. We solve the flow for one- and two-point functions and show that the corrections to the conformal dimensions can be obtained as an integral over the Mellin amplitude of the four-point function. We also derive the flow of the Mellin amplitude for higher n-point functions. We then consider the flows at tree level and one loop (in AdS), and show that one obtains exactly the recursion relations for the corresponding Mellin amplitudes derived earlier by Fitzpatrick et al. [1] at tree level and Yuan [2, 3] at one loop. As an application, we furthermore compute one-loop corrections to the conformal dimensions for some operators in the CFT dual to an O(N) scalar model in AdS.

Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases. II. Entropy production and irreversibility of RG flows

We demonstrate that the reformulation of renormalization group (RG) flow equations as nonlinear heat equations has severe implications on the understanding of RG flows in general. We demonstrate by explicitly constructing an entropy function for a zero-dimensional ${\mathbb{Z}}_{2}$-symmetric model that the dissipative character of generic nonlinear diffusion equations is also hard-coded in the functional RG equation. This renders RG flows manifestly irreversible, revealing the semigroup property of RG transformations on the level of the flow equation itself. Additionally, we argue that the dissipative character of RG flows, its irreversibility and the entropy production during the RG flow may be linked to the existence of a so-called $\mathcal{C}\text{\ensuremath{-}}/\mathcal{A}$-function. In total, this introduces an asymmetry in the so-called RG time---in complete analogy to the thermodynamic arrow of time---and allows for an interpretation of infrared actions as equilibrium solutions of dissipative RG flows equations. The impossibility of resolving microphysics from macrophysics is evident in this framework. Furthermore, we directly link the irreversibility and the entropy production in RG flows to an explicit numerical entropy production, which is manifest in diffusive and non-linear partial differential equations (PDEs) and a standard mathematical tool for the analysis of PDEs. Using exactly solvable zero-dimensional ${\mathbb{Z}}_{2}$-symmetric models, we explicitly compute the (numerical) entropy production related to the total variation nonincreasing property of the PDE during RG flows toward the infrared limit. Finally, we discuss generalizations of our findings and relations to the $\mathcal{C}\text{\ensuremath{-}}/\mathcal{A}$-theorem as well as how our work may help to construct truncations of RG flow equations in the future, including numerically stable schemes for solving the corresponding PDEs.

Perturbative renormalization of ϕ44 theory on the half space R+×R3 with flow equations

In this paper, we give a rigorous proof of the renormalizability of the massive ϕ44 theory on a half-space using renormalization group flow equations. We find that five counterterms are needed to make the theory finite, namely, ϕ2, ϕ∂zϕ, ϕ∂z2ϕ, ϕΔxϕ, and ϕ4 for (z,x)∈R+×R3. The amputated correlation functions are distributions in position space. We consider a suitable class of test functions and prove inductive bounds for the correlation functions folded with these test functions. The bounds are uniform in the cutoff and, thus, directly lead to renormalizability.

Structural aspects of FRG in quantum tunneling computations

We probe both the unidimensional quartic harmonic oscillator and the double well potential through a numerical analysis of the Functional Renormalization Group flow equations truncated at first order in the derivative expansion. The two partial differential equations for the potential Vk(φ) and the wave function renormalization Zk(φ), as obtained in different schemes and with distinct regulators, are studied down to k=0, and the energy gap between lowest and first excited state is computed, in order to test the reliability of the approach in a strongly non-perturbative regime. Our findings point out at least three ranges of the quartic coupling λ, one with higher λ where the lowest order approximation is already accurate, the intermediate one where the inclusion of the first correction produces a good agreement with the exact results and, finally, the one with smallest λ where presumably the higher order correction of the flow is needed. Some details of the specifics of the infrared regulator are also discussed.

Complex dynamical properties of coupled Van der Pol–Duffing oscillators with balanced loss and gain

We consider a Hamiltonian system of coupled Van der Pol–Duffing (VdPD) oscillators with balanced loss and gain. The system is analyzed perturbatively by using renormalization group (RG) techniques as well as multiple scale analysis (MSA). Both the methods produce identical results in the leading order of the perturbation. The RG flow equation is exactly solvable and the slow variation of amplitudes and phases in time can be computed analytically. The system is analyzed numerically and shown to admit periodic solutions in regions of parameter-space, confirming the results of the linear stability analysis and perturbation methods. The complex dynamical behavior of the system is studied in detail by using time-series, Poincaré-sections, power-spectra, auto-correlation function and bifurcation diagrams. The Lyapunov exponents are computed numerically. The numerical analysis reveals chaotic behaviour in the system beyond a critical value of the parameter that couples the two VdPD oscillators through linear coupling, thereby providing yet another example of Hamiltonian chaos in a system with balanced loss and gain. Further, we modify the nonlinear terms of the model to make it a non-Hamiltonian system of coupled VdPD oscillators with balanced loss and gain. The non-Hamiltonian system is analyzed perturbatively as well as numerically and shown to posses regular periodic as well as chaotic solutions. It is seen that the -symmetry is not an essential requirement for the existence of regular periodic solutions in both the Hamiltonian as well as non-Hamiltonian systems.

Multiloop flow equations for single-boson exchange fRG

The recently introduced single-boson exchange (SBE) decomposition of the four-point vertex of interacting fermionic many-body systems is a conceptually and computationally appealing parametrization of the vertex. It relies on the notion of reducibility of vertex diagrams with respect to the bare interaction U, instead of a classification based on two-particle reducibility within the widely used parquet decomposition. Here, we re-derive the SBE decomposition in a generalized framework (suitable for extensions to, e.g., inhomogeneous systems or real-frequency treatments) following from the parquet equations. We then derive multiloop functional renormalization group (mfRG) flow equations for the ingredients of this SBE decomposition, both in the parquet approximation, where the fully two-particle irreducible vertex is treated as an input, and in the more restrictive SBE approximation, where this role is taken by the fully U-irreducible vertex. Moreover, we give mfRG flow equations for the popular parametrization of the vertex in terms of asymptotic classes of the two-particle reducible vertices. Since the parquet and SBE decompositions are closely related, their mfRG flow equations are very similar in structure.Graphic abstract

Spin functional renormalization group for dimerized quantum spin systems

We investigate dimerized quantum spin systems using the spin functional renormalization group approach proposed by Krieg and Kopietz [Phys. Rev. B 99, 060403(R) (2019)] which directly focuses on the physical spin correlation functions and avoids the representation of the spins in terms of fermionic or bosonic auxiliary operators. Starting from decoupled dimers as initial condition for the renormalization group flow equations, we obtain the spectrum of the triplet excitations as well as the magnetization in the quantum paramagnetic, ferromagnetic, and thermally disordered phases at all temperatures. Moreover, we compute the full phase diagram of a weakly coupled dimerized spin system in three dimensions, including the correct mean field critical exponents at the two quantum critical points.

Stochastic dynamics of chemotactic colonies with logistic growth

The interplay between cellular growth and cell-cell signaling is essential for the aggregation and proliferation of bacterial colonies, as well as for the self-organization of cell tissues. To investigate this interplay, we focus here on the collective properties of dividing chemotactic cell colonies by studying their long-time and large-scale dynamics through a renormalization group (RG) approach. The RG analysis reveals that a relevant but unconventional chemotactic interaction —corresponding to a polarity-induced mechanism— is generated by fluctuations at macroscopic scales, even when an underlying mechanism is absent at the microscopic level. This emerges from the interplay of the well-known Keller-Segel (KS) chemotactic nonlinearity and cell birth and death processes. At one-loop order, we find no stable fixed point of the RG flow equations. We discuss a connection between the dynamics investigated here and the celebrated Kardar-Parisi-Zhang (KPZ) equation with long-range correlated noise, which points at the existence of a strong-coupling, nonperturbative fixed point.

Self-consistent O(4) model spectral functions from analytically continued functional renormalization group flows

In this paper we explore practicable ways for self-consistent calculations of spectral functions from analytically continued functional renormalization group (aFRG) flow equations. As a particularly straightforward one we propose to include parametrizations of self-energies based on explicit analytic one-loop expressions. To exemplify this scheme we calculate the spectral functions of pion and sigma meson of the $O(4)$ model at vanishing temperature in the broken phase. Comparing the results with those from previous aFRG calculations, we explicitly demonstrate how self-consistency at all momenta fixes the tight relation between particle masses and decay thresholds. In addition, the two-point functions from our new semianalytic FRG scheme have the desired domain of holomorphy built in and can readily be studied in the entire cut-complex frequency plane, on physical as well as other Riemann sheets. This is very illustrative and allows, for example, to trace the flow of the resonance pole of the sigma meson across an unphysical sheet. In order to assess the limitations due to the underlying one-loop structure, we also introduce a fully self-consistent numerical scheme based on spectral representations with scale-dependent spectral functions. The most notable improvement of this numerically involved calculation is that it describes the three-particle resonance decay of an off-shell pion, ${\ensuremath{\pi}}^{*}\ensuremath{\rightarrow}\ensuremath{\sigma}\ensuremath{\pi}\ensuremath{\rightarrow}3\ensuremath{\pi}$. Apart from this further conceptual improvement, overall agreement with the results from the considerably simpler semianalytic one-loop scheme is very encouraging, however. The latter can therefore provide a sound and practicable basis for self-consistent calculations of spectral functions in more realistic effective theories for warm and dense matter.