Renormalization Group Ideas Research Articles

Dynamics of packed swarms: Time-displaced correlators of two-dimensional incompressible flocks.

We analytically calculate the scaling exponents of a two-dimensional KPZ-like system: coherently moving incompressible polar active fluids. Using three different renormalization group approximation schemes, we obtain values for the roughness exponent χ and anisotropy exponent ζ that are extremely near the known exact results. This implies our prediction for the previously unknown dynamic exponent z is likely to be quantitatively accurate.

Emergent Spinon Dispersion and Symmetry Breaking in Two-Channel Kondo Lattices.

Two-channel Kondo lattice serves as a model for a growing family of heavy-fermion compounds. We employ a dynamical large-N technique and go beyond the independent bath approximation to study this model both numerically and analytically using renormalization group ideas. We show that the Kondo effect induces dynamic magnetic correlations that lead to an emergent spinon dispersion. Furthermore, we develop a quantitative framework that interpolates between infinite dimension where the channel-symmetry broken results of mean-field theory are confirmed, and one-dimension where the channel symmetry is restored and a critical fractionalized mode is found.

RG-Flow: a hierarchical and explainable flow model based on renormalization group and sparse prior

Flow-based generative models have become an important class of unsupervised learning approaches. In this work, we incorporate the key ideas of renormalization group (RG) and sparse prior distribution to design a hierarchical flow-based generative model, RG-Flow, which can separate information at different scales of images and extract disentangled representations at each scale. We demonstrate our method on synthetic multi-scale image datasets and the CelebA dataset, showing that the disentangled representations enable semantic manipulation and style mixing of the images at different scales. To visualize the latent representations, we introduce receptive fields for flow-based models and show that the receptive fields of RG-Flow are similar to those of convolutional neural networks. In addition, we replace the widely adopted isotropic Gaussian prior distribution by the sparse Laplacian distribution to further enhance the disentanglement of representations. From a theoretical perspective, our proposed method has complexity for inpainting of an image with edge length L, compared to previous generative models with complexity.

Wilsonian Renormalization in the 1970s

This paper examines the history of the renormalization group, a cornerstone of contemporary theoretical physics, focusing on the work of Kenneth Wilson (winner of the 1982 Nobel Prize in physics) and affiliated scholars in the 1970s. In particular, it reconstructs how studies of the renormalization group led to formative interactions between two distinct branches of physics, namely particle physics and condensed matter theory. Instead of explaining such intellectual coordination as the result of material and conceptual exchanges, as in Peter Galison’s widely influential discussion of the “trading zone,” my analysis emphasizes the pedagogical labor, social institutions, and political economic conditions that gave the renormalization group its mediating power. To that end, I show how early lectures and fast circulating pre-prints on the renormalization group created a population of physicists in the United States conversant in the rudiments of both condensed matter and particle theory. I then root the formation of a transatlantic network of renormalization group enthusiasts in the geopolitics of the Cold War, showing that the spread of Wilsonian ideas was made possible by a liberal internationalist program of academic exchanges and summer schools sponsored by the US state department and NATO. Finally, I argue that sharp cuts to basic science funding in the United States pushed young physicists seeking jobs in the 1970s to work across specializations, which visibly impacted how renormalization group ideas were interpreted and used—often against the objections of their original progenitors.

The mixed spin-3/2 and spin-2 Blume-Capel model by renormalization group approach

PurposeThe authors investigate the magnetic properties of a mixed spin-3/2 and spin-2 Blume-Capel model on square and cubic lattices with two different single-ion anisotropies.Design/methodology/approachTo study the critical behavior of a mixed spin-3/2 and spin-2 system, the authors have used a real space renormalization group approximation and specifically the Migdal-Kadanoff technique. The authors give the phase diagrams for two different cases: (1) on the (Δ/|J|, 1/|J|) plane with ΔA = ΔB = Δ, and (2) on the (ΔA/|J|, 1/|J|) and (ΔB/|J|, 1/|J|) planes for selected values of ΔB/|J| and ΔA/|J|, respectively.FindingsThe phase diagrams obtained show that the system exhibits both second- and first-order phase transitions as well as tricritical points for some values of the anisotropies. Moreover, using the variation of the free energy and its derivative at low temperatures, the authors have seen the appearance of first-order transitions at very low temperatures.Originality/valueFew investigations of mixed spin-3/2 and spin-2 systems with crystal field have been realized. For this reason, the authors use the renormalization group approach to complete the work done on these systems. In absence of an exact solution, this contributes to the synthesis of the approximation results on mixed spins models.

Fixed-point quantum circuits for quantum field theories

Renormalization group ideas and effective operators are used to efficiently determine localized unitaries for preparing the ground states of non-interacting scalar field theories on digital quantum devices. With these methods, classically computed ground states in a small spatial volume can be used to determine operators for preparing the ground state in a beyond-classical quantum register, even for interacting scalar field theories. Due to the exponential decay of correlation functions and the double exponential suppression of digitization artifacts, the derived quantum circuits are expected to be relevant already for near-term quantum devices.

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.

Tensor networks for complex quantum systems

Tensor network states and methods have erupted in recent years. Originally developed in the context of condensed matter physics and based on renormalization group ideas, tensor networks lived a revival thanks to quantum information theory and the understanding of entanglement in quantum many-body systems. Moreover, it has been not-so-long realized that tensor network states play a key role in other scientific disciplines, such as quantum gravity and artificial intelligence. In this context, here we provide an overview of basic concepts and key developments in the field. In particular, we briefly discuss the most important tensor network structures and algorithms, together with a sketch on advances related to global and gauge symmetries, fermions, topological order, classification of phases, entanglement Hamiltonians, AdS/CFT, artificial intelligence, the 2d Hubbard model, 2d quantum antiferromagnets, conformal field theory, quantum chemistry, disordered systems, and many-body localization.

Examples of renormalization group transformations for image sets

Using the example of configurations generated with the worm algorithm for the two-dimensional Ising model, we propose renormalization group (RG) transformations, inspired by the tensor RG, that can be applied to sets of images. We relate criticality to the logarithmic divergence of the largest principal component. We discuss the changes in link occupation under the RG transformation, suggest ways to obtain data collapse, and compare with the two state tensor RG approximation near the fixed point.

Finite Temperature Phase Diagrams of the Mixed Spin-1 and Spin-2 Blume–Capel Model by Renormalization Group Approach

The Blume–Capel model with mixed spins S = 1 and S = 2, on d-dimensional hypercubic lattice, is studied using the real space renormalization group approximation and specifically the Migdal–Kadanoff technique. We give the phase diagrams on the (Δ1/|J|, 1/|J|) and (Δ2/|J|, 1/|J|) planes which are studied for selected values of Δ2/|J| and Δ1/|J| respectively, with first and second order phase transitions and tricritical points. Also, the associated fixed points are drawn up in a table, and by linearizing the transformation at the vicinity of these points, we determine the critical exponents for d = 2 and d = 3. In particular, we find that the system under study may exhibit reentrant phenomenon.

Mixed Spin-1/2 and Spin-5/2 Model by Renormalization Group Theory: Recursion Equations and Thermodynamic Study

Using the renormalization group approximation, specifically the Migdal-Kadanoff technique, we investigate the Blume-Capel model with mixed spins S = 1/2 and S = 5/2 on d-dimensional hypercubic lattice. The flow in the parameter space of the Hamiltonian and the thermodynamic functions are determined. The phase diagram of this model is plotted in the (anisotropy, temperature) plane for both cases d = 2 and d = 3 in which the system exhibits the first and second order phase transitions and critical end-points. The associated fixed points are drawn up in a table, and by linearizing the transformation at the vicinity of these points, we determine the critical exponents for d = 2 and d = 3. We have also presented a variation of the free energy derivative at the vicinity of the first and second order transitions. Finally, this work is completed by a discussion and comparison with other approximation.

Functional renormalization group and Kohn–Sham scheme in density functional theory

Deriving accurate energy density functional is one of the central problems in condensed matter physics, nuclear physics, and quantum chemistry. We propose a novel method to deduce the energy density functional by combining the idea of the functional renormalization group and the Kohn–Sham scheme in density functional theory. The key idea is to solve the renormalization group flow for the effective action decomposed into the mean-field part and the correlation part. Also, we propose a simple practical method to quantify the uncertainty associated with the truncation of the correlation part. By taking the φ4 theory in zero dimension as a benchmark, we demonstrate that our method shows extremely fast convergence to the exact result even for the highly strong coupling regime.

RG-inspired machine learning for lattice field theory

Machine learning has been a fast growing field of research in several areas dealing with large datasets. We report recent attempts to use renormalization group (RG) ideas in the context of machine learning. We examine coarse graining procedures for perceptron models designed to identify the digits of the MNIST data. We discuss the correspondence between principal components analysis (PCA) and RG flows across the transition for worm configurations of the 2D Ising model. Preliminary results regarding the logarithmic divergence of the leading PCA eigenvalue were presented at the conference. More generally, we discuss the relationship between PCA and observables in Monte Carlo simulations and the possibility of reducing the number of learning parameters in supervised learning based on RG inspired hierarchical ansatzes.

The spin-2 Blume-Capel model by position space renormalization group

Using the renormalization group approximation and specifically the Migdal-Kadanoff technique, we study the spin-2 Blume-Capel model on d-dimensional hypercubic lattice. The phase diagram at the ground state of this model is determined. For T≠0K, we give also the phase diagram in the (temperature, anisotropy) plane, with first and second-order phase transitions and tricritical points. The associated fixed points are drawn up in a table, and by linearizing the transformation at the vicinity of these points, we determine the critical exponents for d=2 and d=3. Finally, this work is completed by a discussion and comparison with other approximation.

Adiabatically deformed ensemble: Engineering nonthermal states of matter

We propose a route towards engineering non-thermal states of matter, which show largely unexplored physics. The main idea relies on the adiabatic passage of a thermal ensemble under slow variations of the system Hamiltonian. If the temperature of the initial thermal ensemble is either zero or infinite the ensemble after the passage is a simple thermal one with the same vanishing or infinite temperature. However, for any finite non-zero temperature intriguing non-thermal ensembles can be achieved. We exemplify this in: (a) a single oscillator (b) a dimerized interacting one dimensional chain of spinless fermions, (c) a BCS-type superconductor and (d) the topological Kitaev chain. We solve these models with a combination of methods; either exactly, numerically using the density matrix renormalization group (DMRG) or within an approximate functional renormalization group (FRG) scheme. The designed states show strongly non-thermal behavior in each of the considered models. For example, for the chain of spinless fermions we exemplify how long ranged non-thermal power-law correlations can be stabilized and for the Kitaev chain we elucidate how the non-thermal ensemble can largely alter the transition temperature separating topological and trivial phases.

On background-independent renormalization of spin foam models

In this article we discuss an implementation of renormalization group ideas to spin foam models, where there is no a priori length scale with which to define the flow. In the context of the continuum limit of these models, we show how the notion of cylindrical consistency of path integral measures gives a natural analogue of Wilson’s RG flow equations for background-independent systems. We discuss the conditions for the continuum measures to be diffeomorphism-invariant, and consider both exact and approximate examples.

Duality, Gauge Symmetries, Renormalization Groups and the BKT Transition

In this chapter, I will briefly review, from my own perspective, the situation within theoretical physics at the beginning of the 1970s, and the advances that played an important role in providing a solid theoretical and experimental foundation for the Berezinskii–Kosterlitz–Thouless theory (BKT). Over this period, it became clear that the Abelian gauge symmetry of the 2D-XY model had to be preserved to get the right phase structure of the model. In previous analyses, this symmetry was broken when using low order calculational approximations. Duality transformations at that time for two-dimensional models with compact gauge symmetries were introduced by José, Kadanoff, Nelson and Kirkpatrick (JKKN). Their goal was to analyze the phase structure and excitations of XY and related models, including symmetry breaking fields which are experimentally important. In a separate context, Migdal had earlier developed an approximate Renormalization Group (RG) algorithm to implement Wilson’s RG for lattice gauge theories. Although Migdal’s RG approach, later extended by Kadanoff, did not produce a true phase transition for the XY model, it almost did asymptotically in terms of a non-perturbative expansion in the coupling constant with an essential singularity. Using these advances, including work done on instantons (vortices), JKKN analyzed the behavior of the spin–spin correlation functions of the 2D XY-model in terms of an expansion in temperature and vortex-pair fugacity. Their analysis led to a perturbative derivation of RG equations for the XY model which are the same as those first derived by Kosterlitz for the two-dimensional Coulomb gas. JKKN’s results gave a theoretical formulation foundation and justification for BKT’s sound physical assumptions and for the validity of their calculational approximations that were, in principle, strictly valid only at very low temperatures, away from the critical [Formula: see text] temperature. The theoretical predictions were soon tested successfully against experimental results on superfluid helium films. The success of the BKT theory also gave one of the first quantitative proofs of the validity of the RG theory.

Towards rigorous analysis of the Levitov–Mirlin–Evers recursion

This paper aims to develop a rigorous asymptotic analysis of an approximate renormalization group recursion for inverse participation ratios Pq of critical powerlaw random band matrices. The recursion goes back to the work by Mirlin and Evers (2000 Phys. Rev. B 62 7920) and earlier works by Levitov (1990 Phys. Rev. Lett. 64 547, 1999 Ann. Phys. 8 697–706) and is aimed to describe the ensuing multifractality of the eigenvectors of such matrices. We point out both similarities and dissimilarities between the LME recursion and those appearing in the theory of multiplicative cascades and branching random walks and show that the methods developed in those fields can be adapted to the present case. In particular the LME recursion is shown to exhibit a phase transition, which we expect is a freezing transition, where the role of temperature is played by the exponent q. However, the LME recursion has features that make its rigorous analysis considerably harder and we point out several open problems for further study.

Starobinsky-Like Inflation and Running Vacuum in the Context of Supergravity

We describe the primeval inflationary phase of the early Universe within a quantum field theoretical (QFT) framework that can be viewed as the effective action of vacuum decay in the early times. Interestingly enough, the model accounts for the “graceful exit” of the inflationary phase into the standard radiation regime. The underlying QFT framework considered here is supergravity (SUGRA), more specifically an existing formulation in which the Starobinsky-type inflation (de Sitter background) emerges from the quantum corrections to the effective action after integrating out the gravitino fields in their (dynamically induced) massive phase. We also demonstrate that the structure of the effective action in this model is consistent with the generic idea of re-normalization group (RG) running of the cosmological parameters; specifically, it follows from the corresponding RG equation for the vacuum energy density as a function of the Hubble rate, ρ Λ ( H ) . Overall, our combined approach amounts to a concrete-model realization of inflation triggered by vacuum decay in a fundamental physics context, which, as it turns out, can also be extended for the remaining epochs of the cosmological evolution until the current dark energy era.

Berezinskii-Kosterlitz-Thouless phase transitions in two-dimensional non-Abelian spin models.

It is argued that two-dimensional U(N) spin models for any N undergo a Berezinskii-Kosterlitz-Thouless (BKT)-like phase transition, similarly to the famous XY model. This conclusion follows from the Berezinskii-like calculation of the two-point correlation function in U(N) models, approximate renormalization group analysis, and numerical investigations of the U(2) model. It is shown, via Monte Carlo simulations, that the universality class of the U(2) model coincides with that of the XY model. Moreover, preliminary numerical results point out that two-dimensional SU(N) spin models with the fundamental and adjoint terms and N>4 exhibit two phase transitions of BKT type, similarly to Z(N) vector models.