Renormalization Group Procedure Research Articles

Favorable phase transitions induced by spinful electron–electron interactions in two-dimensional semimetal with a quadratic band crossing point

We study the effects of marginally spinful electron–electron interactions on the low-energy instabilities and favorable phase transitions in a two-dimensional (2D) spin-1/2 semimetal that owns a quadratic band crossing point (QBCP) parabolically touched by the upper and lower bands. In the framework of a renormalization group procedure, all sorts of interactions are treated on the equal footing to derive the coupled energy-dependent evolutions of all interaction couplings that govern the low-energy properties. Deciphering the essential physical information of such flows, we at first find that the tendencies of interaction parameters fall into three categories including Limit case, Special case, and General case based on the initial conditions. In addition, the 2D QBCP system is attracted to several distinct kinds of fixed points (FPs) in the interaction-parameter space, namely FP1+/FP2−, FP1±/ FP2±/FP3±, and FP1±/FP3±/FP41,42,43± with the subscripts characterizing the features of FPs for the Limit, Special, and General cases, respectively. Furthermore, as approaching these FPs, we demonstrate that the spinful fermion–fermion interactions can induce a number of favorable instabilities accompanied by certain phase transitions. Specifically, the quantum anomalous Hall (QAH), quantum spin Hall (QSH), and nematic (Nem.) site(bond) states are dominant for FP1±, FP2±, and FP3±, respectively. Rather, QSH becomes anisotropic nearby FP41,42,43± with one component leading and the others subleading. Besides, Nem.site(bond), chiral superconductivity, and nematic-spin-nematic (NSN.) site(bond) are subleading candidates around these FPs. Our findings provide valuable insights for further research into the 2D QBCP and similar systems.

Laplacian renormalization group: an introduction to heterogeneous coarse-graining

The renormalization group (RG) constitutes a fundamental framework in modern theoretical physics. It allows the study of many systems showing states with large-scale correlations and their classification into a relatively small set of universality classes. The RG is the most powerful tool for investigating organizational scales within dynamic systems. However, the application of RG techniques to complex networks has presented significant challenges, primarily due to the intricate interplay of correlations on multiple scales. Existing approaches have relied on hypotheses involving hidden geometries and based on embedding complex networks into hidden metric spaces. Here, we present a practical overview of the recently introduced Laplacian RG (LRG) for heterogeneous networks. First, we present a brief overview that justifies the use of the Laplacian as a natural extension of well-known field theories to analyze spatial disorder. We then draw an analogy to traditional real-space RG procedures, explaining how the LRG generalizes the concept of ‘Kadanoff supernodes’ as block nodes that span multiple scales. These supernodes help mitigate the effects of cross-scale correlations due to small-world properties. Additionally, we rigorously define the LRG procedure in momentum space in the spirit of the Wilson RG. Finally, we show different analyses for the evolution of network properties along the LRG flow following structural changes when the network is properly reduced.

Dynamics of heavy quarks in the Fock space

This paper concerns a method of describing hadrons that starts with the canonical front form Hamiltonian of QCD. The method is developed in the relatively simple context of QCD with only heavy quarks. We regulate its canonical Hamiltonian by introducing a vanishingly small gluon mass mg. For positive mg, the small-x gluon divergences become ultraviolet and hence they are renormalized in the same way the ultraviolet transverse divergences are. This is done using the renormalization group procedure for effective particles. Up to the second order of expansion of the renormalized Hamiltonian in powers of the quark-gluon coupling constant g, only the quark mass-squared and gluon-exchange divergences require counterterms. In these circumstances, we calculate an effective potential between quarks in heavy quarkonia in an elementary way, replacing all the quarkonium-state components with gluons of mass mg by only one component with just one gluon that is assigned a mass mG, comparable to or exceeding the scale of typical relative momenta of bound quarks. In the limit of mg→0 and large mG two results are obtained. (1) While the color-singlet quarkonium mass eigenvalue stays finite and physically reasonable in that limit, the eigenvalues for single quarks and octet quarkonia are infinite. (2) Besides the coulomb terms, the effective quark-antiquark potential is quadratic as a function of the distance and spherically symmetric for typical separations between quarks but becomes logarithmic and no longer spherically symmetric for large separations. Our conclusion indicates how to systematically improve upon the approximations made in this paper. Published by the American Physical Society 2024

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.

Baryon Masses Estimate in Heavy Flavor QCD

We apply the renormalization group procedure for effective particles (RGPEP) to the QCD eigenvalue problem for only heavy quarks. We derive the effective Hamiltonian that acts on the Fock space by solving the RGPEP equation up to second order in powers of the coupling constant. The eigenstates that contain three quarks and two or more gluons are eliminated by inserting a gluon-mass term in the component with one gluon and the eigenvalue problem for baryons is formulated. We estimate masses for bbb and ccc states and find that the results match the estimates obtained in lattice QCD and in quark models.

Renormalization group method for solving initial value problem of slow‐fast system

In this paper, we apply the singular renormalization group (RG) method to study the initial value problem of slow‐fast system. Based on some basic assumptions, we construct a uniformly valid approximate solution by an explicit renormalization group procedure, and then give a rigorous proof of the error estimate up to second order. Moreover, a classical example in optimal control is presented to illustrate the effectiveness. Our results extend the applications of RG method to some extent and are consistent with previous results as well.

Laplacian renormalization group for heterogeneous networks

The renormalization group is the cornerstone of the modern theory of universality and phase transitions and it is a powerful tool to scrutinize symmetries and organizational scales in dynamical systems. However, its application to complex networks has proven particularly challenging, owing to correlations between intertwined scales. To date, existing approaches have been based on hidden geometries hypotheses, which rely on the embedding of complex networks into underlying hidden metric spaces. Here we propose a Laplacian renormalization group diffusion-based picture for complex networks, which is able to identify proper spatiotemporal scales in heterogeneous networks. In analogy with real-space renormalization group procedures, we first introduce the concept of Kadanoff supernodes as block nodes across multiple scales, which helps to overcome detrimental small-world effects that are responsible for cross-scale correlations. We then rigorously define the momentum space procedure to progressively integrate out fast diffusion modes and generate coarse-grained graphs. We validate the method through application to several real-world networks, demonstrating its ability to perform network reduction keeping crucial properties of the systems intact.

Real-space numerical renormalization group computation of transport properties in side-coupled geometry

A real-space formulation of the numerical renormalization group (NRG) procedure is introduced. The real-space construction, dubbed eNRG, is more straightforward than the NRG discretization, and allows a faithful description of the coupling between quantum dots and conduction states even if the design of the couplings is intricate. General features of the two procedures are discussed comparatively. A more specific comparison is then developed, based on computations of the zero-bias transport properties for an Anderson-model description of a quantum wire side-coupled to a single quantum dot. An eNRG computation is shown to reproduce accurately the temperature-dependent electrical conductance for the uncorrelated model in the continuum limit, while significant deviations mark the corresponding NRG computation for thermal energies comparable to the conduction bandwidth. A combination of analytical and numerical results for the transport properties of the correlated model then provides a more exacting check on the accuracy of the eNRG procedure. A recent NRG analysis mapping the transport properties of a single-electron transistor onto universal functions of the temperature scaled by the Kondo temperature is extended to the side-coupled device, on the basis of eNRG reasoning. Numerical results for the electrical conductance, thermopower, and thermal conductance in side-coupled geometry are then shown to agree very well with the mappings. The numerical results are also checked against the thermal dependence of the thermopower measured by K\"ohler et al. [Phys. Rev. B 77, 104412 (2008)] in ${\mathrm{Lu}}_{0.9}^{\phantom{\ifmmode\dagger\else\textdagger\fi{}}}{\mathrm{Yb}}_{0.1}^{\phantom{\ifmmode\dagger\else\textdagger\fi{}}}{\mathrm{Rh}}_{2}^{\phantom{\ifmmode\dagger\else\textdagger\fi{}}}{\mathrm{Si}}_{2}^{\phantom{\ifmmode\dagger\else\textdagger\fi{}}}$, and the remarkably accurate, recent conductance measurements by Xu et al. [Chin. Phys. Lett. 38, 087101 (2021)].

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.

Asymptotic freedom using a gluon mass as a regulator

Front-Form Hamiltonian dynamics provides a framework in which QCD’s vacuum is simple and states are boost invariant. However, canonical expressions are divergent and must be regulated in order to establish well-defined eigenvalue problems. The Renormalization Group Procedure for Effective Particles (RGPEP) provides a systematic way of finding counterterms and obtaining regulated Hamiltonians. Among its achievements is the description of asymptotic freedom, with a running coupling constant defined as the coefficient in front of the three gluon-vertex operators in the regulated Hamiltonian. However, the obtained results need a deeper understanding, since the coupling exhibits a finite dependence on the regularization functions, at least at the third-order term in the perturbative expansion. Here we present a similar derivation using a different regularization scheme based on massive gluons. The procedure can be extended to incorporate contributions from virtual fermions.

Renormalization group method for the turbulent mass transport two-equation model

Turbulent mass transport is an essential phenomenon in chemical reactors. A two-equation model of turbulent mass transport has been proposed for less numerical simulation error in previous work. However, there are still some model constants that have undetermined values introduced by empirical modeling rules. In the present paper, the renormalization group (RNG) method is applied to the turbulent mass transport process. Through the RNG procedure, the small scale turbulent components in Navier-Stokes and convection diffusion equations can be eliminated and combined into large scale sources. The corresponding model constants are derived and applied to a grid turbulent diffusion simulation case. In this paper, the grid turbulence diffusion experiment with laser induced fluorescence (LIF) is carried out to verify the simulation results. A satisfactory agreement of concentration distribution between simulation and experiment indicates that the model constants obtained by the RNG method are reliable.

Theory of multi-orbital topological superconductivity in transition metal dichalcogenides

We study possible superconducting states in transition metal dichalcogenide (TMD) monolayers, assuming an on-site pairing potential that includes both intra- and inter-orbital terms. We find that if the mirror symmetry with respect to the system’s plane is broken (e.g., by a substrate), this type of pairing can give rise to unconventional superconductivity, including time-reversal-invariant nodal and fully gapped topological phases. Using a multi-orbital renormalization group procedure, we show how these phases may result from the interplay between the local Coulomb repulsion, Hund’s rule coupling, and phonon-mediated attraction. In particular, for a range of interaction parameters, the system transitions from a trivial phase to a nodal phase and finally to a gapped topological phase upon increasing the strength of the mirror symmetry breaking term.

Neural Monte Carlo renormalization group

The key idea behind the renormalization group (RG) transformation is that properties of physical systems with very different microscopic makeups can be characterized by a few universal parameters. However, finding the optimal RG transformation remains difficult due to the many possible choices of the weight factors in the RG procedure. Here we show, by identifying the conditional distribution in the restricted Boltzmann machine (RBM) and the weight factor distribution in the RG procedure, an optimal real-space RG transformation can be learned without prior knowledge of the physical system. This neural Monte Carlo RG algorithm allows for direct computation of the RG flow and critical exponents. This scheme naturally generates a transformation that maximizes the real-space mutual information between the coarse-grained region and the environment. Our results establish a solid connection between the RG transformation in physics and the deep architecture in machine learning, paving the way to further interdisciplinary research.

Elementary example of exact effective-Hamiltonian computation

We present an exact computation of effective Hamiltonians for an elementary model obtained from the Yukawa theory by going to the limit of bare fermions being infinitely heavy and bare bosons being at rest with respect to the fermions that emit or absorb them. The coupling constant can be arbitrarily large. The Hamiltonians are computed by solving the differential equation of the renormalization group procedure for effective particles (RGPEP). Physical fermions, defined in the model as eigenstates of the effective Hamiltonians, are obtained in the form of an effective fermion dressed with a coherent state of effective bosons. The model computation illustrates the method that can be used in perturbative computations of effective Hamiltonians for realistic theories. It shows the mechanism by which the perturbative expansion and Tamm-Dancoff approximation increase in accuracy along the RGPEP evolution.

Curvature function renormalization, topological phase transitions, and multicriticality

A recently proposed curvature renormalization group scheme for topological phase transitions defines a generic `curvature function' as a function of the parameters of the theory and shows that topological phase transitions are signalled by the divergence of this function at certain parameters values, called critical points, in analogy with usual phase transitions. A renormalization group procedure was also introduced as a way of flowing away from the critical point towards a fixed point, where an appropriately defined correlation function goes to zero and topological quantum numbers characterising the phase are easy to compute. In this paper, using two independent models - a model in the AIII symmetry class and a model in the BDI symmetry class - in one dimension as examples, we show that there are cases where the fixed point curve and the critical point curve appear to intersect, which turn out to be multi-critical points, and focus on understanding its implications.

Wannier band transitions in disordered π -flux ladders

Boundary obstructed topological insulators are an unusual class of higher-order topological insulators with topological characteristics determined by the so-called Wannier bands. Boundary obstructed phases can harbor hinge/corner modes, but these modes can often be destabilized by a phase transition on the boundary instead of the bulk. While there has been much work on the stability of topological insulators in the presence disorder, the topology of a disordered Wannier band, and disorder-induced Wannier transitions have not been extensively studied. In this work, we focus on the simplest example of a Wannier topological insulator: a mirror-symmetric $\pi$-flux ladder in 1D. We find that the Wannier topology is robust to disorder, and derive a real-space renormalization group procedure to understand a new type of strong disorder-induced transition between non-trivial and trivial Wannier topological phases. We also establish a connection between the Wannier topology of the ladder and the energy band topology of a related system with a physical boundary cut, something which has generally been conjectured for clean models, but has not been studied in the presence of disorder.

Modeling sequences with quantum states: a look under the hood

Classical probability distributions on sets of sequences can be modeled using quantum states. Here, we do so with a quantum state that is pure and entangled. Because it is entangled, the reduced densities that describe subsystems also carry information about the complementary subsystem. This is in contrast to the classical marginal distributions on a subsystem in which information about the complementary system has been integrated out and lost. A training algorithm based on the density matrix renormalization group (DMRG) procedure uses the extra information contained in the reduced densities and organizes it into a tensor network model. An understanding of the extra information contained in the reduced densities allow us to examine the mechanics of this DMRG algorithm and study the generalization error of the resulting model. As an illustration, we work with the even-parity dataset and produce an estimate for the generalization error as a function of the fraction of the dataset used in training.

Scaling theory for Mott–Hubbard transitions: I. T = 0 phase diagram of the 1/2-filled Hubbard model

We present a T = 0 K renormalization group (RG) phase diagram for the electronic Hubbard model in two dimensions on the square lattice at half filling. The RG procedure treats quantum fluctuations in the single particle occupation number nonperturbatively via the unitarily decoupling of one electronic state at every RG step. The resulting phase diagram thus possesses the quantum fluctuation energy scale (ω) as one of its axes. A relation is derived between ω and the effective temperature scale upto which gapless, as well as emergent gapped phases can be obtained. We find that the normal and insulating phases of the half-filled Hubbard model correspond, for any on-site repulsion, to a marginal Fermi liquid normal phase and a topologically-ordered gapped Mott insulating liquid respectively. The marginal Fermi liquid is found to arise from singular forward scattering in directions normal to the nested Fermi surface, while singular backscattering events lead to Mott liquid state. The transition between these two phases involves passage through a pseudogapped phase bookended by Fermi surface topology-changing Lifshitz transitions. The pseudogap phase is observed to arise from the electronic differentiation encoded within the nested Fermi surface, and involves the gradual gapping of the Fermi surface from antinodes to nodes via charge and spin excitations that are mutually entangled. We obtain effective Hamiltonians for various phases, as well as wavefunctions for the low-energy many-body eigenstates of the Mott liquid. Benchmarking of the ground-state energy per particle and the double-occupancy fraction for the Mott liquid against existing numerical results yields excellent agreement. Presence of a Néel ordering symmetry-breaking perturbation in the RG leads to an antiferromagnetic spin-ordered charge insulating Mott state. Our results thus offer novel insights on a variety of aspects of the Mott–Hubbard problem, and can be extended to the doped system.

Computation of effective front form Hamiltonians for massive Abelian gauge theory

Renormalization group procedure for effective particles (RGPEP) is applied in terms of a second-order perturbative computation to an Abelian gauge theory, as an example of application worth studying on the way toward derivation of a dynamical connection between the spectroscopy of bound states and their parton-model picture in the front form of Hamiltonian dynamics. In addition to the ultraviolet transverse divergences that are handled using the RGPEP in previously known ways, the small-x divergences are handled by introducing a mass parameter and a third polarization state for gauge bosons using a mechanism analogous to spontaneous breaking of global gauge symmetry, in a special limit that simplifies the theory to Soper's front form of massive QED. The resulting orders of magnitude of scales involved in the dynamics of effective constituents or partons in the simplified theory are identified for the fermion and boson mass counter terms, effective masses and self-interactions, as well as for the Coulomb-like effective interactions in bound states of fermions. Computations in orders higher than second are mentioned but not described in this article.

Probabilistic Modeling with Matrix Product States

Inspired by the possibility that generative models based on quantum circuits can provide a useful inductive bias for sequence modeling tasks, we propose an efficient training algorithm for a subset of classically simulable quantum circuit models. The gradient-free algorithm, presented as a sequence of exactly solvable effective models, is a modification of the density matrix renormalization group procedure adapted for learning a probability distribution. The conclusion that circuit-based models offer a useful inductive bias for classical datasets is supported by experimental results on the parity learning problem.