Renormalization Group Formalism Research Articles

Gradient Flow Exact Renormalization Group for Scalar Quantum Electrodynamics

Gradient Flow Exact Renormalization Group (GF-ERG) is a framework to define the renormalization group flow of Wilsonian effective action utilizing coarse-graining along the diffusion equations. We apply it for Scalar Quantum Electrodynamics and derive flow equations for the Wilsonian effective action with the perturbative expansion in the gauge coupling. We focus on the quantum corrections to the correlation functions up to the second order of the gauge coupling and discuss the gauge invariance of the GF-ERG flow. We demonstrate that the anomalous dimension of the gauge field agrees with the standard perturbative computation and that the mass of the photon keeps vanishing in general spacetime dimensions. The latter is a noteworthy fact that contrasts with the conventional Exact Renormalization Group formalism in which an artificial photon mass proportional to a cutoff scale is induced. Our results imply that the GF-ERG can give a gauge-invariant renormalization group flow in a non-perturbative way.

Correction to: Chiral anomaly as a composite operator in the gradient flow exact renormalization group formalism

Correction to: Chiral anomaly as a composite operator in the gradient flow exact renormalization group formalism

Flexible DMRG-Based Framework for Anharmonic Vibrational Calculations.

We present a novel formulation of the vibrational density matrix renormalization group (vDMRG) algorithm tailored to strongly anharmonic molecules described by general, high-dimensional model representations of potential energy surfaces. For this purpose, we extend the vDMRG framework to support vibrational Hamiltonians expressed in the so-called n-mode second-quantization formalism. The resulting n-mode vDMRG method offers full flexibility with respect to both the functional form of the PES and the choice of the single-particle basis set. We leverage this framework to apply, for the first time, vDMRG based on an anharmonic modal basis set optimized with the vibrational self-consistent field algorithm on an on-the-fly constructed PES. We also extend the n-mode vDMRG framework to include excited-state-targeting algorithms in order to efficiently calculate anharmonic transition frequencies. We demonstrate the capabilities of our novel n-mode vDMRG framework for methyloxirane, a challenging molecule with 24 coupled vibrational modes.

Scale dependence of the Kondo interaction in the functional renormalization group formalism

Scale dependence of the Kondo interaction in the functional renormalization group formalism

Combining the in-medium similarity renormalization group with the density matrix renormalization group: Shell structure and information entropy

We propose a novel many-body framework combining the density matrix renormalization group (DMRG) with the valence-space (VS) formulation of the in-medium similarity renormalization group. This hybrid scheme admits for favorable computational scaling in large-space calculations compared to direct diagonalization. The capacity of the VS-DMRG approach is highlighted in ab initio calculations of neutron-rich nickel isotopes based on chiral two- and three-nucleon interactions, and allows us to perform converged ab initio computations of ground and excited state energies. We also study orbital entanglement in the VS-DMRG, and investigate nuclear correlation effects in oxygen, neon, and magnesium isotopes. The explored entanglement measures reveal nuclear shell closures as well as pairing correlations.

Chiral anomaly as a composite operator in the gradient flow exact renormalization group formalism

Abstract The gradient flow exact renormalization group (GFERG) is an idea that incorporates gauge-invariant gradient flows into the formalism of the exact renormalization group (ERG). GFERG introduces a Wilson action with a cutoff while keeping the vector gauge invariance manifestly. The details of the formalism are still to be worked out. In this paper, we apply GFERG to construct the Wilson action of massless Dirac fermions under the background chiral gauge fields. By formulating the chiral anomaly as a “composite operator”, we make the scale invariance of the anomaly manifest. We argue that the same result extends to QCD.

Functional renormalization group for multilinear disordered Langevin dynamics II:Revisiting the p = 2 spin dynamics for Wigner and Wishart ensembles

In this paper, we investigate the large-time behavior for a slightly modified version of the standard p = 2 soft spins dynamics model, including a quartic or higher potential. The equilibrium states of such a model correspond to an effective field theory, which has been recently considered as a novel paradigm for signal detection in data science based on the renormalization group argument. We consider a Langevin-like equation, including a disorder term that leaves in the Wigner or Wishart ensemble. Then we construct a nonperturbative renormalization group formalism valid in the large N limit, where eigenvalues distributions for the disorder can be replaced by their analytic limits, namely the Wigner and Marchenko-Pastur laws. One of the main advantages of this approach is that the interactions remain local in time, avoiding the non-locality arising from the approaches that integrate out the disorder at the partition function level.

Field-theoretic functional renormalization group formalism for non-Fermi liquids and its application to the antiferromagnetic quantum critical metal in two dimensions

To capture the universal low-energy physics of metals within effective field theories, one has to generalize the usual notion of scale invariance and renormalizable field theory due to the presence of intrinsic scales (Fermi momenta). In this paper, we develop a field-theoretic functional renormalization group formalism for full low-energy effective field theories of non-Fermi liquids that include all gapless modes around the Fermi surface. The formalism is applied to the non-Fermi liquid that arises at the antiferromagnetic quantum critical point in two space dimensions. In the space of coupling functions, an interacting fixed point arises at a point with momentum-independent couplings and vanishing nesting angle. In theories deformed with non-zero nesting angles, coupling functions acquire universal momentum profiles controlled by the bare nesting angles at low energies before flowing to superconducting states in the low-energy limit. The superconducting instability is unavoidable because lukewarm electrons that are coherent enough to be susceptible to pairing end up being subject to a renormalized attractive interaction with its minimum strength set by the nesting angle irrespective of the bare four-fermion coupling. Despite the inevitable superconducting instability, theories with small bare nesting angles and bare four-fermion couplings that are repulsive or weakly attractive must pass through the region with slow RG flow due to the proximity to the non-Fermi liquid fixed point. The bottleneck region controls the scaling behaviours of the normal state and the quasi-universal pathway from the non-Fermi liquid to superconductivity. In the limit that the nesting angle is small, the non-Fermi liquid scaling dictates the physics over a large window of energy scale above the superconducting transition temperature.

Survival in two-species reaction-diffusion system with Lévy flights: renormalization group treatment and numerical simulations

We analyze the two-species reaction-diffusion system including trapping reaction as well as coagulation/annihilation reactions where particles of both species are performing Lévy flights with control parameter , known to lead to superdiffusive behavior. The density as well as the correlation function for target particles B in such systems are known to scale with nontrivial universal exponents at space dimension . Applying the renormalization group formalism we calculate these exponents in a case of superdiffusion below the critical dimension . The numerical simulations in one-dimensional case are performed as well. The quantitative estimates for the decay exponent of the density of survived particles B are in a good agreement with our analytical results. In particular, it is found that the surviving probability of the target particles in superdiffusive regime is higher than that in a system with ordinary diffusion.

Computational Method for Evaluating the Thermoelectric Power Factor for Organic Materials Modeled by the Holstein Model: A Time-Dependent Density Matrix Renormalization Group Formalism.

Organic/polymeric materials are of emerging importance for thermoelectric conversion. The soft nature of these materials implies strong electron-phonon coupling, often leading to carrier localization. This poses great challenges for the conventional Boltzmann transport description based on relaxation time approximation and band structure calculations. In this work, combining the Kubo formula with the finite-temperature time-dependent density matrix renormalization group (FT-TD-DMRG) in the grand canonical ensemble, we developed a nearly exact algorithm to calculate the thermoelectric power factor PF = α2 σ, where α is the Seebeck coefficient and σ is the electrical conductivity, and apply the algorithm to Holstein Hamiltonian with electron-phonon coupling to model organic materials. Our algorithm can provide a unified description covering the weak coupling limit described by the bandlike Boltzmann transport to the strong coupling hopping limit.

Renormalization group approach for an enzyme‐substrate‐inhibitor kinetics

The renormalization group (RG) method of Chen, Goldenfeld, Oono et al. offers a comprehensive approach to formally computing asymptotic expansions of the solutions to singular perturbation problems. A renormalization group formalism developed recently for the Michaelis–Menten (MM) model is extended to competitive systems. We will first revisit the singular perturbation analysis for the standard quasi‐steady‐state approximation (sQSSA) of enzyme‐substrate‐inhibitor kinetics. Our result shows that the RG method leads to a new prediction for the concentrations of substrate, inhibitor, and complexes, which can reproduce or surpass the results of matched asymptotics.

Unconventional superconductivity from weak coupling

We develop a joint formalism and numerical framework for analyzing the superconducting instability of metals from a weak coupling perspective. This encompasses the Kohn–Luttinger formulation of weak coupling renormalization group for superconductivity as well as the random phase approximation imposed on the diagrammatic expansion of the two-particle Green’s function. The central quantity to resolve is the effective interaction in the Cooper channel, for which we develop an optimized numerical framework. Our code is capable of treating generic multi-orbital models in two as well as three spatial dimensions and, in particular, arbitrary avenues of spin-orbit coupling.Graphic abstract

Functional renormalization group for multilinear disordered Langevin dynamics I Formalism and first numerical investigations at equilibrium

This paper aims at using the functional renormalization group formalism to study the equilibrium states of a stochastic process described by a quench–disordered multilinear Langevin equation. Such an equation characterizes the evolution of a time-dependent N-vector q(t) = {q 1(t), ⋯ q N (t)} and is traditionally encountered in the dynamical description of glassy systems at and out of equilibrium through the so-called Glauber model. From the connection between Langevin dynamics and quantum mechanics in imaginary time, we are able to coarse-grain the path integral of the problem in the Fourier modes, and to construct a renormalization group flow for effective Euclidean action. In the large N-limit we are able to solve the flow equations for both matrix and tensor disorder. The numerical solutions of the resulting exact flow equations are then investigated using standard local potential approximation, taking into account the quench disorder. In the case where the interaction is taken to be matricial, for finite N the flow equations are also solved. However, the case of finite N and taking into account the non-equilibrum process will be considered in a companion investigation.

One-particle irreducible Wilson action in the gradient flow exact renormalization group formalism

Abstract We define a one-particle irreducible (1PI) Wilson action in the gradient flow exact renormalization group (GFERG) formalism as the Legendre transform of a Wilson action. We consider quantum electrodynamics in particular, and show that the GFERG flow equation preserves the invariance of the 1PI Wilson action (excluding the gauge-fixing term) under the conventionalU(1) gauge transformation. This is in contrast to the invariance of the original Wilson action under a modified U(1) gauge transformation. The global chiral transformation also takes the conventional form for the 1PI Wilson action. Despite the complexity of the GFERG flow equation, the conventional form of the gauge and global chiral transformations may allow us to introduce a non-perturbative Ansatz for gauge and chiral invariant 1PI Wilson actions.

Gaussian information bottleneck and the non-perturbative renormalization group

The renormalization group (RG) is a class of theoretical techniques used to explain the collective physics of interacting, many-body systems. It has been suggested that the RG formalism may be useful in finding and interpreting emergent low-dimensional structure in complex systems outside of the traditional physics context, such as in biology or computer science. In such contexts, one common dimensionality-reduction framework already in use is information bottleneck (IB), in which the goal is to compress an ‘input’ signal X while maximizing its mutual information with some stochastic ‘relevance’ variable Y. IB has been applied in the vertebrate and invertebrate processing systems to characterize optimal encoding of the future motion of the external world. Other recent work has shown that the RG scheme for the dimer model could be ‘discovered’ by a neural network attempting to solve an IB-like problem. This manuscript explores whether IB and any existing formulation of RG are formally equivalent. A class of soft-cutoff non-perturbative RG techniques are defined by families of non-deterministic coarsening maps, and hence can be formally mapped onto IB, and vice versa. For concreteness, this discussion is limited entirely to Gaussian statistics (GIB), for which IB has exact, closed-form solutions. Under this constraint, GIB has a semigroup structure, in which successive transformations remain IB-optimal. Further, the RG cutoff scheme associated with GIB can be identified. Our results suggest that IB can be used to impose a notion of ‘large scale’ structure, such as biological function, on an RG procedure.

Signal Detection in Nearly Continuous Spectra and ℤ2-Symmetry Breaking

The large scale behavior of systems having a large number of interacting degrees of freedom is suitably described using the renormalization group from non-Gaussian distributions. Renormalization group techniques used in physics are then expected to provide a complementary point of view on standard methods used in data science, especially for open issues. Signal detection and recognition for covariance matrices having nearly continuous spectra is currently an open issue in data science and machine learning. Using the field theoretical embedding introduced in Entropy, 23(9), 1132 to reproduce experimental correlations, we show in this paper that the presence of a signal may be characterized by a phase transition with Z2-symmetry breaking. For our investigations, we use the nonperturbative renormalization group formalism, using a local potential approximation to construct an approximate solution of the flow. Moreover, we focus on the nearly continuous signal build as a perturbation of the Marchenko-Pastur law with many discrete spikes.

Field-Theoretic Functional Renormalization Group Formalism For Non-Fermi Liquids and its Application To The Antiferromagnetic Quantum Critical Metal in Two Dimensions

Field-Theoretic Functional Renormalization Group Formalism For Non-Fermi Liquids and its Application To The Antiferromagnetic Quantum Critical Metal in Two Dimensions

Field Theoretical Approach for Signal Detection in Nearly Continuous Positive Spectra II: Tensorial Data.

The tensorial principal component analysis is a generalization of ordinary principal component analysis focusing on data which are suitably described by tensors rather than matrices. This paper aims at giving the nonperturbative renormalization group formalism, based on a slight generalization of the covariance matrix, to investigate signal detection for the difficult issue of nearly continuous spectra. Renormalization group allows constructing an effective description keeping only relevant features in the low “energy” (i.e., large eigenvalues) limit and thus providing universal descriptions allowing to associate the presence of the signal with objectives and computable quantities. Among them, in this paper, we focus on the vacuum expectation value. We exhibit experimental evidence in favor of a connection between symmetry breaking and the existence of an intrinsic detection threshold, in agreement with our conclusions for matrices, providing a new step in the direction of a universal statement.

Functional renormalization group and 2PI effective action formalism

We combine two non-perturbative approaches, one based on the two-particle-irreducible (2PI) action, the other on the functional renormalization group (fRG), in an effort to develop new non-perturbative approximations for the field theoretical description of strongly coupled systems. In particular, we exploit the exact 2PI relations between the two-point and four-point functions in order to truncate the infinite hierarchy of equations of the functional renormalization group. The truncation is ”exact” in two ways. First, the solution of the resulting flow equation is independent of the choice of the regulator. Second, this solution coincides with that of the 2PI equations for the two-point and the four-point functions, for any selection of two-skeleton diagrams characterizing a so-called Φ-derivable approximation. The transformation of the equations of the 2PI formalism into flow equations offers new ways to solve these equations in practice, and provides new insight on certain aspects of their renormalization. It also opens the possibility to develop approximation schemes going beyond the strict Φ-derivable ones, as well as new truncation schemes for the fRG hierarchy.

The operator algebra at the Gaussian fixed-point

We consider the multiple products of relevant and marginal scalar composite operators at the Gaussian fixed-point in [Formula: see text] dimensions. This amounts to perturbative construction of the [Formula: see text] theory where the parameters of the theory are momentum-dependent sources. Using the exact renormalization group (ERG) formalism, we show how the scaling properties of the sources are given by the short-distance singularities of the multiple products.