Understanding parton evolution in matter from renormalization group analysis

According to Landau criterion, a phase transition should be first order when cubic terms of order parameters are allowed in its effective Ginzburg-Landau free energy. Recently, it was shown by renormalization group (RG) analysis that continuous transition can happen at putatively first-order $Z_3$ transitions in 2D Dirac semimetals and such non-Landau phase transitions were dubbed "fermion-induced quantum critical points" (FIQCP) [Li et al., Nature Communications 8, 314 (2017)]. The RG analysis, controlled by the 1/$N$ expansion with $N$ the number of flavors of four-component Dirac fermions, shows that FIQCP occurs for $N\geq N_c$. Previous QMC simulations of a microscopic model of SU($N$) fermions on the honeycomb lattice showed that FIQCP occurs at the transition between Dirac semimetals and Kekule-VBS for $N\geq 2$. However, precise value of the lower bound $N_c$ has not been established. Especially, the case of $N=1$ has not been explored by studying microscopic models so far. Here, by introducing a generalized SU($N$) fermion model with $N=1$ (namely spinless fermions on the honeycomb lattice), we perform large-scale sign-problem-free Majorana quantum Monte Carlo simulations and find convincing evidence of FIQCP for $N=1$. Consequently, our results suggest that FIQCP can occur in 2D Dirac semimetals for all positive integers $N\geq 1$.

Before presenting a specific model, we would like to revisit in more depth and from a more general perspective the renormalization group (RG) formalism previously introduced in Chap. 2. The RG analysis, introduced in field theory and in critical phase transitions, is a very general mathematical (and conceptual) tool, which allows one to decompose the problem of finding the “macroscopic” behavior of a large number of interacting parts into a succession of simpler problems with a decreasing number of interacting parts, whose effective properties vary with the scale of observation. Technically, this is done as we will see by defining a mapping between the observational scale and the distance |T − T c| from the critical point. The term “observational scale” usually refers to the physical scale of an observation. In the spin context, the observational scale rather refers to the size of the block of spins that one analyzes within the system. The usefulness of the RG approach is based on the existence of scale invariance and self-similarity of the observables at the critical point. The purpose of the RG is to translate in mathematical language the concept that a critical point results from the aggregate response of an ensemble of elements. In addition, the RG formalism can be used as a tool of model construction [803]. In this presentation, we follow [629, 706].

Proliferation of defects is a mechanism that allows for topological phase transitions. Such a phase transition is found in two dimensions for the XY model, which lies in the Berezinskii-Kosterlitz-Thouless (BKT) universality class. The transition point can be found using renormalization group analysis. We apply renormalization group arguments to determine the nature of BKT transitions for the three-dimensional plaquette-dimer model, which is a model that exhibits fractonic mobility constraints. We show that an important part of this analysis demands a modified dimensional analysis that changes the interpretation of scaling dimensions upon coarse-graining. Using this modified dimensional analysis, we compute the beta functions of the model and predict a finite critical value above which the fractonic phase melts, proliferating dipoles. Importantly, the transition point is found through a renormalization group analysis that accounts for the phenomenon of UV/IR mixing, characteristic of fractonic models.

The heterogeneity of the subsurface permeability is considered as the most influential factor in determining groundwater flow and the transport of toxic contaminants. Numerical simulators cannot handle the large grids required to represent the small-scale variability of permeability, and thus explicit estimates of the large-scale behavior in terms of coarse-grained parameters are often required. Perturbation formulations of the effective permeability are based on simplifying assumptions that are valid only for certain probability distributions and weak heterogeneity. A generalized perturbation ansatz that involves higher orders has been proposed (Gelhar and Axness, 1983), but to our knowledge its validity has not been rigorously proved before in three dimensions. In this work we propose a general upscaling formulation valid for strong heterogeneity, general permeability distributions, and media with impermeable zones. We show that the effective permeability is determined by the self-energy series of the permeability fluctuations at zero frequency. Using the diagrammatic representation, we obtain a Dyson equation that involves only irreducible diagrams of the proper self-energy series. We develop a renormalization group (RG) analysis for isotropic lognormal media that proves the generalized perturbation ansatz to all orders. We show that the RG result accurately estimates laboratory permeability measurements in limestone (strong heterogeneity) and sandstone (weak heterogeneity). We also propose an explicit RG estimate for the preasymptotic effective permeability. We compare our results with an approach based on a leading order Green's function expansion (Paleologos et al., 1996), which, however, requires intensive numerical computations. Finally, we investigate the relation between the RG expression and the algebraic means used in numerical upscaling.

We present a renormalization group (RG) analysis of the problem of Anderson localization on a random regular graph (RRG) which generalizes the RG of Abrahams, Anderson, Licciardello, and Ramakrishnan to infinite-dimensional graphs. The RG equations necessarily involve two parameters (one being the changing connectivity of subtrees), but we show that the one-parameter scaling hypothesis is recovered for sufficiently large system sizes for both eigenstates and spectrum observables. We also explain the nonmonotonic behavior of dynamical and spectral quantities as a function of the system size for values of disorder close to the transition, by identifying two terms in the beta function of the running fractal dimension of different signs and functional dependence. Our theory provides a simple and coherent explanation for the unusual scaling behavior observed in numerical data of the Anderson model on RRG and of many-body localization.

Bethe-Salpeter approach to the top mode standard model

Recent experiments have observed correlated insulating and possible superconducting phases in twisted homobilayer transition metal dichalcogenides (TMDs). Besides the spin-valley locked moir\'e bands due to the intrinsic Ising spin-orbit coupling, homobilayer moir\'e TMDs also possess either logarithmic or power-law divergent Van Hove singularities (VHSs) near the Fermi surface, controllable by an external displacement field. The former and the latter are dubbed conventional and higher-order VHSs, respectively. Here, we perform a perturbative renormalization group (RG) analysis to unbiasedly study the dominant instabilities in homobilayer TMDs for both the conventional and higher-order VHS cases. We find that the spin-valley locking largely alters the RG flows and leads to instabilities unexpected in the corresponding extensively studied graphene-based moir\'e systems, such as spin- and valley-polarized ferromagnetism and topological superconductivity with mixed parity. In particular, for the case with two higher-order VHSs, we find a spin-valley-locking-driven metallic state with no symmetry breaking in the TMDs despite the diverging bare susceptibility. Our results show how the spin-valley locking significantly affects the RG analysis and demonstrate that moir\'e TMDs are suitable platforms to realize various interaction-induced spin-valley locked phases, highlighting physics fundamentally different from the well-studied graphene-based moir\'e systems.

The renormalization group (RNG) analysis of the Kosterlitz--Thouless (KT) phase transition is a basic paradigm in statistical physics and a well-known success of the RNG approach. It was recently shown that the derivation of the RNG parameter flow for the KT problem rests on the assumption of one-sided polarization, which is implausible and cannot be derived from first principles. We extend this analysis by exhibiting simple self-consistent alternate assumptions that lead to different parameter flows. We then study the KT transition numerically and show that the properties of the transition are well described by the standard RNG analysis (up to some minor paradoxes). The reason for the success of the assumption of one-sided polarization remains unknown; the problem exemplifies the difficulties in the mathematical analysis of the RNG. Related issues in turbulence theory are pointed out.

A new classification scheme for relevant couplings in renormalization group transformations

Renormalization-group (RG) methods developed previously for the study of the growth of order in unstable systems are extended to treat the spinodal decomposition of the two-dimensional spin-exchange kinetic Ising model. The conservation of the order parameter and fixed-length sum rule are properly preserved in the theory. Various correlation functions in both coordinate and momentum space are calculated as functions of time. The scaling function for the structure factor is extracted. We compare our results with direct Monte Carlo (MC) simulations and find them in good agreement. The time rescaling parameter entering the RG analysis is temperature dependent, as was determined in previous work through a RG analysis of MC simulations. The results exhibit a long-time logarithmic growth law for the typical domain size, both analytically and numerically. In the time region where MC simulations have previously been performed, the logarithmic growth law can be fitted to a power law with an effective exponent. This exponent is found to be in excellent agreement with the result of MC simulations. The logarithmic growth law agrees with a physical model of interfacial motion which involves an interplay between the local curvature and an activated jump across the interface.

We extend our Wilsonian renormalization group (RG) analysis on the pionless nuclear effective field theory in the two-nucleon sector in two ways; on the one hand, (1) we enlarge the space of operators up to including those of [Formula: see text] in the S waves, and, on the other hand, (2) we consider the RG flows in higher partial waves (P and D waves). In the larger space calculations, we find, in addition to nontrivial fixed points, two "fixed lines" and a "fixed surface" which are related to marginal operators. In the higher partial wave calculations, we find similar phase structures to that of the S waves, but there are two relevant directions in the P waves at the nontrivial fixed points and three in the D waves. We explain the physical meaning of the P-wave phase structure by explicitly calculating the low-energy scattering amplitude. We also discuss the relation between the Legendre flow equation which we employ and the RG equation by Birse, McGovern and Richardson, and possible implementation of power divergence subtraction in higher partial waves.

We analyze the antiferromagnetic spin-1/2 XXZ model on the kagome lattice at finite external magnetic field with the help of a non-perturbative zero-temperature renormalization group (RG) technique. The exact nature of the ground and excited state properties (e.g. gapped or gapless spectrum etc) of this system are still debated. Approximate methods have typically been adopted towards understanding the low-energy spectrum. Following the work of Kumar et al (2014 Phys. Rev. B 90 174409), we use a Jordan–Wigner transformation to map the spin problem into one of spinless fermions (spinons) in the presence of a statistical gauge field, and with nearest-neighbor interactions. While the work of Kumar et al was confined mostly to the plateau at 1/3-filling (magnetization per site) in the XY regime, we analyze the role of inter-spinon interactions in shaping the phases around this plateau in the entire XXZ model. The RG phase diagram obtained contains three spin liquid phases whose position is determined as a function of the exchange anisotropy and the energy scale for fluctuations arising from spinon scattering. Two of these spins liquids are topologically ordered states of matter with gapped, degenerate states on the torus. The gap for one of these phases corresponds to the one-spinon band gap of the Azbel–Hofstadter spectrum for the XY part of the Hamiltonian, while the other arises from two-spinon interactions. The Heisenberg point of this problem is found to lie within the interaction gapped spin liquid phase, in broad agreement with a recent experimental finding. The third phase is an algebraic spin liquid with a gapless Dirac spectrum for spinon excitations, and possess properties that show departures from the Fermi liquid paradigm. The three phase boundaries correspond to critical theories, and meet at a SU(2)-symmetric multicritical point. This special critical point agrees well with the gap-closing transition point predicted by Kumar et al. We discuss the relevance of our findings to various recent experiments, as well as results obtained from other theoretical analyses.

We describe and discuss the recent progress in the study of propagation and localization of acoustic and elastic waves in heterogeneous media. The heterogeneity is represented by a spatial distribution of the local elastic moduli. Both randomly distributed elastic moduli as well as those with long-range correlations with a nondecaying power-law correlation function, are considered. The motivation for the study is twofold. One is that recent analysis of experimental data for the spatial distribution of the elastic moduli of rock indicated that the distribution is characterized by the type of long-range correlations that we consider in this study. The second motivation for the problem is to understand whether localization of electrons (which, in quantum mechanics, are described by wave functions) has any analogy in the propagation of classical waves in disordered media. The problem is studied by two approaches. One of them is based on developing a dynamic renormalization group (RG) approach to analytical analysis of the governing equations for wave propagation. The RG analysis indicates that, depending on the type of the disorder (correlated vs. uncorrelated), one may have a transition between localized and extended regimes in any spatial dimension. The second approach utilizes numerical simulations of the governing equations in two- and three-dimensional media. The results obtained by the two approaches are in agreement with each other. Using numerical simulations, we also describe how the characteristics of a propagating wave may be used for probing the differences between heterogeneous media with short- and long-range correlations. To do so, we study the evolution of several distinct characteristics of the waves, such as the amplitude of the coherent wave front, its width, the spectral densities, the scalogram (wavelet transformation of the waves’ amplitudes at different scales and times), and the dispersion relation. It is demonstrated that such properties have completely different characteristics in uncorrelated and correlated media. Finally, it is shown how wave propagation may be used for establishing a link between the static and dynamical properties of heterogeneous media.

We investigate approximate emergent nonabelian symmetry in a class of weakly first order `deconfined' phase transitions using Monte Carlo simulations and a renormalization group analysis. We study a transition in a 3D classical loop model that is analogous to a deconfined 2+1D quantum phase transition in a magnet with reduced lattice symmetry. The transition is between the N\'eel phase and a twofold degenerate valence bond solid (lattice-symmetry-breaking) phase. The combined order parameter at the transition is effectively a four-component superspin. It has been argued that in some weakly first order `pseudocritical' deconfined phase transitions, the renormalization group flow can take the system very close to the ordered fixed point of the symmetric $O(N)$ sigma model, where $N$ is the total number of `soft' order parameter components, despite the fact that $O(N)$ is not a microscopic symmetry. This yields a first order transition with unconventional phenomenology. We argue that this occurs in the present model, with $N=4$. This means that there is a regime of lengthscales in which the transition resembles a `spin-flop' transition in the ordered $O(4)$ sigma model. We give numerical evidence for (i) the first order nature of the transition, (ii) the emergence of $O(4)$ symmetry to an accurate approximation, and (iii) the existence of a regime in which the emergent $O(4)$ is `spontaneously broken', with distinctive features in the order parameter probability distribution. These results may be relevant for other models studied in the literature, including 2+1D QED with two flavours, the `easy-plane' deconfined critical point, and the N\'eel--VBS transition on the rectangular lattice.

The totally asymmetric simple exclusion process along with particle adsorption and evaporation kinetics is a model of boundary-induced nonequilibrium phase transition. In the continuum limit, the average particle density across the system is described by a singular differential equation involving multiple scales which lead to the formation of boundary layers (BL) or shocks. A renormalization group analysis is developed here by using the location and the width of the BL as the renormalization parameters. It not only allows us to cure the large distance divergences in the perturbative solution for the BL but also generates, from the BL solution, an analytical form for the global density profile. The predicted scaling form is checked against numerical solutions for finite systems.