Exact Renormalization Group Equation Research Articles

Holographic RG from an exact RG: Locality and general coordinate invariance in the bulk

In earlier papers it was shown that the correct kinetic term for scalar, vector gauge field and the spin two field in AdSD+1 space is obtained starting from the exact renormalization group (ERG) equation for a CFTD perturbed by scalar composite, conserved vector current, and conserved traceless energy momentum tensor respectively. In this paper interactions are studied, and it is shown that a flipped version of the Polchinski ERG equation that evolves toward the UV can be written down and is useful for making contact with the usual AdS/CFT prescriptions for correlation function calculations. The scalar-scalar-spin-2 interaction in the bulk is derived from the ERG equation in the large N semiclassical approximation. It is also shown that after mapping to AdS the interaction is local on a scale of the bare cutoff rather than the moving cutoff (which would have corresponded to the anti–de Sitter scale). The map to AdSD+1 plays a crucial role in this locality. The local nature of the coupling ensures that this interaction term in the bulk action is obtained by gauge fixing a general coordinate invariant scalar kinetic term in the bulk action. A wave function renormalization of the scalar field is found to be required for a mutually consistent map of the two fields to AdSD+1. Published by the American Physical Society 2024

Exact renormalization group for wave functionals

AbstractMotivated by the construction of continuum tensor networks for interacting field theories, which are relevant in understanding the emergence of space-time in the gauge/gravity correspondence, we derive a non-perturbative functional differential equation for wave functionals in scalar field theories from the exact renormalization group equation. We check the validity of the equation using the perturbation theory. We calculate the wave functional up to the first-order perturbation and verify that it satisfies the equation.

Exact renormalization group equation for lattice Ginzburg–Landau models adapted to the solution in the local potential approximation

The Wilson Green’s function approach and, alternatively, Feynman’s diffusion equation and the Hori representation have been used to derive an exact functional RG equation (EFRGE) that in the course of the RG flow interpolates between the interaction part of the lattice Ginzburg–Landau Hamiltonian and the logarithm of the generating functional of the S-matrix. Because the S-matrix vertices are the amputated correlation functions of the fluctuating field, it has been suggested that in the critical region the amputation of the long-range tails makes the S-matrix functional more localized and thus more amenable to the local potential approximation (LPA) than the renormalized free energy functional used in Wilson’s EFRGE. By means of a functional Legendre transform the S-matrix EFRGE has been converted into an EFRGE for the effective action (EA). It has been found that the field-dependent part of EA predicted by the equation is the same as calculated within the known EA EFRGE approaches but in addition it is accurately accounts for the field-independent terms. These are indispensable in calculation of such important quantities as the specific heat, the latent heat, etc. With the use of the derived EFRGE a closed expression for the renormalization counterterm has been obtained which when subtracted from the divergent solution of the Wetterich equation would lead to a finite exact expression for the EA thus making two approaches formally equivalent. The S-matrix equation has been found to be simply connected with a generalized functional Burgers’ equation which establishes a direct correspondence between the first order phase transitions and the shock wave solutions of the RG equation. The transparent semi-group structure of the S-matrix RG equation makes possible the use of different RG techniques at different stages of the RG flow in order to improve the LPA solution.

The exact renormalization group and dimensional regularization

The exact renormalization group (ERG) is formulated implementing the decimation of degrees of freedom by means of a particular momentum integration measure. The definition of this measure involves a distribution that links this decimation process with the dimensional regularization technique employed in field theory calculations. Taking the dimension [Formula: see text], the one-loop solutions to the ERG equations for the scalar field theory in this scheme are shown to coincide with the dimensionally regularized perturbative field theory calculation in the limit [Formula: see text], if a particular relation between the scale parameter [Formula: see text] and [Formula: see text] is employed. In general, it is shown that in this scheme the solutions to the ERG equations for the proper functions coincide when [Formula: see text] with the complete diagrammatic contributions appearing in field theory for these functions and this theory, provided that exact relations between [Formula: see text] and [Formula: see text] hold. In addition, a nonperturbative approximation is considered. This approximation consists in a truncation of the ERG equations, which by means of a low-momentum expansion leads to reasonable results.

Functional truncations for the solution of the nonperturbative RG equations

We consider the Wetterich exact renormalization group (RG) equation. Approximate closed equations are obtained from it, applying certain truncation schemes for the effective average action. These equations are solved either purely numerically or by certain extra truncations for the potential and related quantities, called the functional truncations. Traditionally, the functional truncations consist of truncated expansions in powers of , where , φ is the averaged order parameter, and corresponds to the minimum of the dimensionless potential , depending on the infrared cut-off scale k. We propose a new approach of functional truncations, using the expansion , where , is an optimization parameter and µ is the exponent, describing the asymptotic. The newly developed method provides accurate estimates of the critical exponents η, ν and also ω. In the case of the local potential approximation (LPA), the discrepancy with the purely numerical solution is for ν and 0.0012 for ω at the s 13 order of truncation. We show that this method is advantageous for estimations beyond the LPA, especially, for the critical exponent ω. In particular, we have obtained , , and for the equation originally derived in 2020 J. Phys. A: Math. Theor. 53 415002 within a new truncation scheme for the effective action. The approach can also be used for the solution of other nonperturbative RG models that have a broad range of applications.

Derivative expansion for computing critical exponents of O(N) symmetric models at next-to-next-to-leading order.

We apply the derivative expansion of the effective action in the exact renormalization group equation up to fourth order to the Z_{2} and O(N) symmetric scalar models in d=3 Euclidean dimensions. We compute the critical exponents ν, η, and ω using polynomial expansion in the field. We obtain our predictions for the exponents employing two regulators widely used in exact renormalization group computations. We apply Wynn's epsilon algorithm to improve the predictions for the critical exponents, extrapolating beyond the next-to-next-to-leading order prediction of the derivative expansion.

Exact renormalization group, entanglement entropy, and black hole entropy

The study of black hole physics revealed a fundamental connection between thermodynamics, quantum mechanics, and gravity. Today, it is known that black holes are thermodynamical objects with well-defined temperature and entropy. Although black hole radiance gives us the mechanism from which we can associate a well-defined temperature to the black hole, the origin of its entropy remains a mystery. Here we investigate how the quantum fluctuations from the fields that render the black hole its temperature contribute to its entropy. By using the exact renormalization group equation for a self-interacting real scalar field in a spacetime possessing a bifurcate Killing horizon, we find the renormalization group flow of the total gravitational entropy. We show that throughout the flow one can split the quantum field contribution to the entropy into a part coming from the entanglement between field degrees of freedom inside and outside the horizon and a part due to the quantum corrections to the Wald entropy coming from the Noether charge. The renormalized black hole entropy is shown to be constant throughout the flow while the balance between the effective black hole entropy at low energies and the infra-red entanglement entropy changes. A similar conclusion is valid for the Wald entropy part of the total entropy. Additionally, our calculations show that there is no mismatch between the renormalization of the coupling constants coming from the effective action or the gravitational entropy, solving an apparent "puzzle" that appeared to exist for interacting fields.

The renormalization group and the diffusion equation

Abstract We study the relationship between the renormalization group and the diffusion equation. We consider the exact renormalization group equation for a scalar field that includes an arbitrary cutoff function and an arbitrary quadratic seed action. As a generalization of the result obtained by Sonoda and Suzuki, we find that the correlation functions of diffused fields with respect to the bare action agree with those of bare fields with respect to the effective action, where the diffused field obeys a generalized diffusion equation determined by the cutoff function and the seed action and agrees with the bare field at the initial time.

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.

Holographic RG and exact RG in O(N) model

In this paper an Exact Renormalization Group (ERG) equation is written for the critical O(N) model in D-dimensions (with D≈3) at the Wilson-Fisher fixed point perturbed by a scalar composite operator. The action is written in terms of an auxiliary scalar field and reproduces correlation functions of a scalar composite operator. The equation is derived starting from the Polchinski ERG equation for the fundamental scalar field. As described in Sathiapalan and Sonoda (2017) [1] an evolution operator for the Polchinski ERG equation can be written in the form of a functional integral, with a D+1 dimensional scalar field theory action. In the case of the fundamental scalar field this action only has a kinetic term and therefore looks quite different from Holographic RG where there are potential terms. But in the composite operator case discussed in this paper, the ERG equation and consequently the D+1 dimensional action contains higher order potential terms for the scalar field and is therefore very similar to the case of Holographic RG. Furthermore this action can be mapped to a scalar field action in AdSD+1 using the techniques of ([1]). The leading cubic term of the potential is computed in this paper for D≈3 and expectedly vanishes in D=3 in agreement with results in the AdS/CFT literature.

Revisited functional renormalization group approach for random matrices in the large- N limit

The nonperturbative renormalization group has been considered as a solid framework to investigate fixed point and critical exponents for matrix and tensor models, expected to correspond with the so-called double scaling limit. In this paper, we focus on matrix models and address the question of the compatibility between the approximations used to solve the exact renormalization group equation and the modified Ward identities coming from the regulator. We show in particular that standard local potential approximation strongly violates the Ward identities, especially in the vicinity of the interacting fixed point. Extending the theory space including derivative couplings, we recover an interacting fixed point with a critical exponent not so far from the exact result, but with a nonzero value for derivative couplings, evoking a strong dependence concerning the regulator. Finally, we consider a modified regulator, allowing to keep the flow not so far from the ultralocal region and recover the results of the literature up to a slight improvement.

Wilson action for the O(N) model

In this paper the fixed-point Wilson action for the critical O(N) model in D=4−ϵ dimensions is written down in the ϵ expansion to order ϵ2. It is obtained by solving the fixed-point Polchinski Exact Renormalization Group equation (with anomalous dimension) in powers of ϵ. This is an example of a theory that has scale and conformal invariance despite having a finite UV cutoff. The energy-momentum tensor for this theory is also constructed (at zero momentum) to order ϵ2. This is done by solving the Ward-Takahashi identity for the fixed point action. It is verified that the trace of the energy-momentum tensor is proportional to the violation of scale invariance as given by the exact RG, i.e., the β function. The vanishing of the trace at the fixed point ensures conformal invariance. Some examples of calculations of correlation functions are also given.

Wilsonian renormalization group in the functional non-perturbative approach

We consider a functional relation between a given Wilsonian renormalization group (RG) flow, which has to be related to a specific coarse-graining procedure, and an infinite family of (UV cutoff) scale-dependent field redefinitions. Within this framework, one can define a family of Wilsonian proper-time (PT) exact RG equations associated to an arbitrary regulator function. New applications of these RG flow schemes to the Ising Universality class in three dimensions in the derivative expansion are shortly illustrated.

On the low density regime of homogeneous electron gas

We investigate the low density limit of the Homogeneous Electron system, often called the Strictly Correlated regime. We begin with a systematic presentation of the expansion around infinite rS, based on the first quantized treatments suggested in the existing literature. We show that the expansion is asymptotic in the parameter rS1∕4 and that the leading order result contains exponential corrections that are significant even for rS∼100. Thus, the systematic expansion is of limited utility. As a byproduct of this analysis, we find that there is no Wigner Crystal (WC) in one spatial dimension. This is an example of the Mermin–Wagner theorem, but was not appreciated in some earlier literature. More modern work (Schulz, 1993 [1]) has come to conclusions identical to ours. Note that the long range Coulomb potential modifies the dispersion relation of phonons in one dimension, but still leads to the instability of the crystal, due to a very weak infrared divergence. We then propose a new approximation scheme based on renormalization group ideas. We show that the Wegner–Houghton–Wilson–Polchinski exact renormalization group equation reduces, in the low density limit, to a classical equation for scale dependent electron and plasmon fields. In principle, this should allow us to lower the wave number cutoff of the model to a point where Wigner’s intuitive argument for dominance of the classical Coulomb forces becomes rigorously correct.

Commutative–non-commutative dualities

We show that it is in principle possible to construct dualities between commutative and non-commutative theories in a systematic way. This construction exploits a generalization of the exact renormalization group equation (ERG). We apply this to the simple case of the Landau problem and then generalize it to the free and interacting non-canonical scalar field theory. This constructive approach offers the advantage of tracking the implementation of the Lorentz symmetry in the non-commutative dual theory. In principle, it allows for the construction of completely consistent non-commutative and non-local theories where the Lorentz symmetry and unitarity are still respected, but may be implemented in a highly non-trivial and non-local manner.

A Renormalization-Group Study of Interacting Bose–Einstein Condensates: Absence of the Bogoliubov Mode below Four (T > 0) and Three (T = 0) Dimensions

We derive exact renormalization-group equations for the $n$-point vertices ($n=0,1,2,\cdots$) of interacting single-component Bose-Einstein condensates based on the vertex expansion of the effective action. They have a notable feature of automatically satisfying Goldstone's theorem (I), which yields the Hugenholtz-Pines relation $\Sigma(0)-\mu=\Delta(0)$ as the lowest-order identity. Using them, it is found that the anomalous self-energy $\Delta(0)$ vanishes below $d_{\rm c}=4$ ($d_{\rm c}=3$) dimensions at finite temperatures (zero temperature), contrary to the Bogoliubov theory predicting a finite "sound-wave" velocity $v_{\rm s}\propto [\Delta(0)]^{1/2}>0$. It is also argued that the one-particle density matrix $\rho({\bf r})\equiv\langle\hat\psi^\dagger({\bf r}_1)\hat\psi({\bf r}_1+{\bf r})\rangle$ for $d<d_{\rm c}$ dimensions approaches the off-diagonal-long-range-order value $N_{\bf 0}/V$ asymptotically as $r^{-d+2-\eta}$ with an exponent $\eta>0$. The anomalous dimension $\eta$ at finite temperatures is predicted to behave for $d=4-\epsilon$ dimensions ($0<\epsilon\ll 1$) as $\eta\propto\epsilon^2$. Thus, the interacting Bose-Einstein condensates are subject to long-range fluctuations similar to those at the second-order transition point, and their excitations in the one-particle channel are distinct from the Nambu-Goldstone mode with a sound-wave dispersion in the two-particle channel.

A functional RG approach for the BFKL Pomeron

In this paper we encode the perturbative BFKL leading logarithmic resummation, relevant for the Regge limit behavior of QCD scattering amplitudes, in the IR-regulated effective action which satisfies exact functional renormalization group equations. This is obtained using a truncation with a specific infinite set of non local vertices describing the multi-Regge kinematics (MRK). The goal is to use this framework to study, in the high energy limit and at larger transverse distances the transition to a much simpler effective local reggeon field theory, whose critical properties were recently investigated in the same framework. We perform a numerical analysis of the spectrum of the BFKL Pomeron deformed by the introduction of a Wilsonian infrared regulator to understand the properties of the leading poles (states) contributing to the high energy scattering.

Strong Interactions in the Regge Limit and Infrared Region

In this talk, we encode the perturbative BFKL leading logarithmic resummation, relevant for the Regge limit behavior of QCD scattering amplitudes, in the IR regulated effective action, which satisfies exact functional renormalization group equations. The goal is to use this framework to study, in the high-energy limit and at larger transverse distances the transition to a much simpler effective local Reggeon field theory, whose critical properties were recently investigated in the same framework. We perform a numerical analysis of the spectrum of the BFKL Pomeron by the introduction of a Wilsonian infrared regulator to understand the properties of the leading poles (Pomeron states) contributing to the high-energy scattering

Derivation of exact flow equations from the self-consistent parquet relations

We exploit the parquet formalism to derive exact flow equations for the two-particle-reducible four-point vertices, the self-energy, and typical response functions, circumventing the reliance on higher-point vertices. This includes a concise, algebraic derivation of the multiloop flow equations, which have previously been obtained by diagrammatic considerations. Integrating the multiloop flow for a given input of the totally irreducible vertex is equivalent to solving the parquet equations with that input. Hence, one can tune systems from solvable limits to complicated situations by variation of one-particle parameters, staying at the fully self-consistent solution of the parquet equations throughout the flow. Furthermore, we use the resulting differential form of the Schwinger–Dyson equation for the self-energy to demonstrate one-particle conservation of the parquet approximation and to construct a conserving two-particle vertex via functional differentiation of the parquet self-energy. Our analysis gives a unified picture of the various many-body relations and exact renormalization group equations.

A Wilsonian energy-momentum tensor

For local conformal field theories, it is shown how to construct an expression for the energy-momentum tensor in terms of a Wilsonian effective Lagrangian. Tracelessness implies a single, unintegrated equation which enforces both the Exact Renormalization Group equation and its partner encoding invariance under special conformal transformations.