Phenomenological Renormalization Group Research Articles

In and Out of Criticality? State-Dependent Scaling in the Rat Visual Cortex

The presumed proximity to a critical point is believed to endow the brain with scale-invariant statistics, which are thought to confer various functional advantages in terms of its information processing, storage, and transmission capabilities. To assess the relationship between scaling and cortical states, we apply a phenomenological renormalization group analysis to 3-h spiking data recordings from the urethane-anesthetized rat's visual cortex. Under this type of anesthesia, cortical states dynamically shift across a spectrum of synchronization levels, defined by population spiking rate variability. By developing a scaling criterion based on the kurtosis of the momentum-space activity distribution, our study combines the coarse-graining method with state-dependent analysis. We find that scaling signatures only appear as spiking variability surpasses a specified threshold. Notably, within this regime, scaling exponents show relative stability. Conversely, subthreshold activity is primarily asynchronous and fails to meet the scaling criterion. Our results suggest that a wide range of cortical states corresponds to small deviations around a critical point, with the system fluctuating in and out of criticality, spending roughly three-quarters of the experiment duration within a scaling regime. Published by the American Physical Society 2024

Critical scaling of whole-brain resting-state dynamics

Scale invariance is a characteristic of neural activity. How this property emerges from neural interactions remains a fundamental question. Here, we studied the relation between scale-invariant brain dynamics and structural connectivity by analyzing human resting-state (rs-) fMRI signals, together with diffusion MRI (dMRI) connectivity and its approximation as an exponentially decaying function of the distance between brain regions. We analyzed the rs-fMRI dynamics using functional connectivity and a recently proposed phenomenological renormalization group (PRG) method that tracks the change of collective activity after successive coarse-graining at different scales. We found that brain dynamics display power-law correlations and power-law scaling as a function of PRG coarse-graining based on functional or structural connectivity. Moreover, we modeled the brain activity using a network of spins interacting through large-scale connectivity and presenting a phase transition between ordered and disordered phases. Within this simple model, we found that the observed scaling features were likely to emerge from critical dynamics and connections exponentially decaying with distance. In conclusion, our study tests the PRG method using large-scale brain activity and theoretical models and suggests that scaling of rs-fMRI activity relates to criticality.

Quasiuniversal scaling in mouse-brain neuronal activity stems from edge-of-instability critical dynamics

The brain is in a state of perpetual reverberant neural activity, even in the absence of specific tasks or stimuli. Shedding light on the origin and functional significance of such a dynamical state is essential to understanding how the brain transmits, processes, and stores information. An inspiring, albeit controversial, conjecture proposes that some statistical characteristics of empirically observed neuronal activity can be understood by assuming that brain networks operate in a dynamical regime with features, including the emergence of scale invariance, resembling those seen typically near phase transitions. Here, we present a data-driven analysis based on simultaneous high-throughput recordings of the activity of thousands of individual neurons in various regions of the mouse brain. To analyze these data, we construct a unified theoretical framework that synergistically combines a phenomenological renormalization group approach and techniques that infer the general dynamical state of a neural population, while designing complementary tools. This strategy allows us to uncover strong signatures of scale invariance that are "quasiuniversal" across brain regions and experiments, revealing that all the analyzed areas operate, to a greater or lesser extent, near the edge of instability.

Critical properties of the Anderson transition on random graphs: Two-parameter scaling theory, Kosterlitz-Thouless type flow, and many-body localization

The Anderson transition in random graphs has raised great interest, partly because of its analogy with the many-body localization (MBL) transition. Unlike the latter, many results for random graphs are now well established, in particular the existence and precise value of a critical disorder separating a localized from an ergodic delocalized phase. However, the renormalization group flow and the nature of the transition are not well understood. In turn, recent works on the MBL transition have made the remarkable prediction that the flow is of Kosterlitz-Thouless type. Here we show that the Anderson transition on graphs displays the same type of flow. Our work attests to the importance of rare branches along which wave functions have a much larger localization length $\xi_\parallel$ than the one in the transverse direction, $\xi_\perp$. Importantly, these two lengths have different critical behaviors: $\xi_\parallel$ diverges with a critical exponent $\nu_\parallel=1$, while $\xi_\perp$ reaches a finite universal value ${\xi_\perp^c}$ at the transition point $W_c$. Indeed, $\xi_\perp^{-1} \approx {\xi_\perp^c}^{-1} + \xi^{-1}$, with $\xi \sim (W-W_c)^{-\nu_\perp}$ associated with a new critical exponent $\nu_\perp = 1/2$, where $\exp( \xi)$ controls finite-size effects. The delocalized phase inherits the strongly non-ergodic properties of the critical regime at short scales, but is ergodic at large scales, with a unique critical exponent $\nu=1/2$. This shows a very strong analogy with the MBL transition: the behavior of $\xi_\perp$ is identical to that recently predicted for the typical localization length of MBL in a phenomenological renormalization group flow. We demonstrate these important properties for a smallworld complex network model and show the universality of our results by considering different network parameters and different key observables of Anderson localization.

Chain breaking and Kosterlitz-Thouless scaling at the many-body localization transition in the random-field Heisenberg spin chain

Despite tremendous theoretical efforts to understand subtleties of the many-body localization (MBL) transition, many questions remain open, in particular concerning its critical properties. Here we make the key observation that MBL in one dimension is accompanied by a spin freezing mechanism which causes chain breakings in the thermodynamic limit. Using analytical and numerical approaches, we show that such chain breakings directly probe the typical localization length, and that their scaling properties at the MBL transition agree with the Kosterlitz-Thouless scenario predicted by phenomenological renormalization group approaches.

Scaling and criticality in a phenomenological renormalization group

We present a systematic study to test a recently introduced phenomenological renormalization group, proposed to coarse-grain data of neural activity from their correlation matrix. The approach allows, at least in principle, to establish whether the collective behavior of the network of spiking neurons is described by a non-Gaussian critical fixed point. We test this renormalization procedure in a variety of models focusing in particular on the contact process, which displays an absorbing phase transition at $\lambda = \lambda_c$ between a silent and an active state. We find that the results of the coarse-graining do not depend on the presence of long-range interactions, but some scaling features persist in the super-critical system up to a distance of $10\%$ from $\lambda_c$. Our results provide insights on the possible subtleties that one needs to consider when applying such phenomenological approaches directly to data to infer signatures of criticality.

Critical Dynamics of Transverse-field Quantum Ising Model using Finite-Size Scaling and Matrix Product States

The study of phase transition is usually done by numerical simulation of finite system. Conventional methods such as Monte Carlo simulations and phenomenological renormalization group methods obtain the critical exponents without obtaining the quantum wavefunction of the system. The Matrix Product States formalism allows one to obtain accurate numerical wavefunctions of short ranged interacting quantum many-body systems. In this study we combine the Finite Size Scaling theory and Matrix Product States formalism to study the critical dynamics of one-dimensional quantum Ising model. Finite size simulations of 20, 40, 60, 80, 100 and 120 spins are done using the Density Matrix Renormalization Group to obtain the ground state wavefunction of the system. The thermodynamic quantities such as the magnetization, susceptibility and correlation function are calculated. The critical exponents independently calculated are respectively β/ν = 0.1235(1), γ/ν = 1.7351(2), and η = 0.249(1). They conform with the theoretical values from analytical solution and fulfil the hyperscaling relation. We showed that both methods combined can reliably study the critical dynamics of one-dimensional Ising-like quantum lattice systems. Application of the study on water-ice phase transition of single-file water in nanopores is proposed.

Multiscale entanglement clusters at the many-body localization phase transition

We study numerically the formation of entanglement clusters across the many-body localization phase transition. We observe a crossover from strong many-body entanglement in the ergodic phase to weak local correlations in the localized phase, with contiguous clusters throughout the phase diagram. Critical states close to the transition have a structure compatible with fractal or multiscale-entangled states, characterized by entanglement at multiple levels: small strongly entangled clusters are weakly entangled together to form larger clusters. The critical point therefore features subthermal entanglement and a power-law distributed cluster size, while the localized phase presents an exponentially decreasing cluster distribution. These results are consistent with some of the recently proposed phenomenological renormalization-group schemes characterizing the many-body localized critical point, and may serve to constrain other such schemes.

Kosterlitz-Thouless scaling at many-body localization phase transitions

We propose a scaling theory for the many-body localization (MBL) phase transition in one dimension, building on the idea that it proceeds via a 'quantum avalanche'. We argue that the critical properties can be captured at a coarse-grained level by a Kosterlitz-Thouless (KT) renormalization group (RG) flow. On phenomenological grounds, we identify the scaling variables as the density of thermal regions and the lengthscale that controls the decay of typical matrix elements. Within this KT picture, the MBL phase is a line of fixed points that terminates at the delocalization transition. We discuss two possible scenarios distinguished by the distribution of rare, fractal thermal inclusions within the MBL phase. In the first scenario, these regions have a stretched exponential distribution in the MBL phase. In the second scenario, the near-critical MBL phase hosts rare thermal regions that are power-law distributed in size. This points to the existence of a second transition within the MBL phase, at which these power-laws change to the stretched exponential form expected at strong disorder. We numerically simulate two different phenomenological RGs previously proposed to describe the MBL transition. Both RGs display a universal power-law length distribution of thermal regions at the transition with a critical exponent $\alpha_c=2$, and continuously varying exponents in the MBL phase consistent with the KT picture.

Semi-directed percolation in two dimensions

We studied a model of semi-directed percolation on finite strips of the square and triangular lattices. Using the transfer-matrix method, combined with phenomenological renormalization group approach, we obtain good numerical estimates for critical probabilities and correlation lengths critical exponents. Our results confirm the conjecture that semi-directed percolation belongs to the universality class of the usual fully-directed percolation model.

Finding critical points using improved scaling Ansätze

Analyzing in detail the first corrections to the scaling hypothesis, we develop accelerated methods for the determination of critical points from finite size data. The output of these procedures is sequences of pseudo-critical points which rapidly converge towards the true critical points. The convergence is faster than that obtained with the fastest method available to date, which consists of estimating the location of the gap's closure (the so called phenomenological renormalization group). Having fast converging sequences at our disposal allows us to draw conclusions on the basis of shorter system sizes. This can be extremely important in particularly hard cases such as two-dimensional quantum systems with frustrations, or in Monte Carlo simulations when the sign problem occurs. After reviewing the most efficient techniques available to date, we test the effectiveness of the proposed methods both analytically on the basis of the one-dimensional XY model and numerically at phase transitions occurring in non-integrable spin models. In particular, we show how a new Homogeneity Condition Method is able to produce fast converging sequences in correspondence to the Berezinskii–Kosterlitz–Thouless (BKT) transition simply by making use of ground-state quantities on relatively small systems. Remarkably, our method tested on the frustrated spin-1/2 Heisenberg model gives a BKT critical point which is incompatible with the ones present in the past literature based on different methods. This discrepancy raises the fundamental question of determining the correct renormalization group approach and scaling assumptions that yield to the sequences converging to the true critical point. Finally, we formulate a general prescription that allows us to analyze and efficiently locate critical points in a variety of cases, without knowing in advance the universality class of the tested transition. Even if our methods are tested here in one dimension, we expect them to be valid in any spatial dimensionality and both for quantum and classical statistical systems.

Phenomenological Renormalization Group and Cluster Approximation

We propose a modification of the Nightingale renormalization group for lattice spin and gauge models by combining it with the cluster decimation approximation. Essential ingredients of our approach are: 1) exact calculation of the partition and correlation functions on a finite lattice strip; 2) preservation of the mass gap or the second moment correlation length, computed in the infinite strip length limit, on each decimation step. The method is applied to studying the general two- and three-dimensional Z(N) models. A perfect agreement with exact results (whenever available) is found. An extension of the method to models with a continuous symmetry is briefly discussed.

Real-space renormalization group method for quantum 1/2 spins on the pyrochlore lattice

A simple phenomenological real-space renormalization group method for quantum Heisenberg spins with nearest and next nearest neighbour interactions on a pyrochlore lattice is presented. Assuming a scaling law for the order parameter of two clusters of different sizes, a set of coupled equations that gives the fixed points of the renormalization group transformation and, thus, the critical temperatures and ordered phases of the system is found. The particular case of spins 1/2 is studied in detail. Furthermore, to simplify the mathematical details, from all the possible phases arising from the renormalization group transformation, only those phases in which the magnetic lattice is commensurate with a subdivision of the crystal lattice into four interlocked face-centred cubic sublattices are considered. These correspond to a quantum spin liquid, ferromagnetic order, or non-collinear order in which the total magnetic moment of a tetrahedral unit is zero. The corresponding phase diagram is constructed and the differences with respect to the classical model are analysed. It is found that this method reproduces fairly well the phase diagram of the pyrochlore lattice under the aforementioned constraints.

Critical behavior of 3D [formula omitted] lattice gauge theories at zero temperature

Three-dimensional Z(N) lattice gauge theories at zero temperature are studied for various values of N. Using a modified phenomenological renormalization group, we explore the critical behavior of the generalized Z(N) model for N=2,3,4,5,6,8. Numerical computations are used to simulate vector models for N=2,3,4,5,6,8,13,20 for lattices with linear extension up to L=96. We locate the critical points of phase transitions and establish their scaling with N. The values of the critical indices indicate that the models with N>4 belong to the universality class of the three-dimensional XY model. However, the exponent α derived from the heat capacity is consistent with the Ising universality class. We discuss a possible resolution of this puzzle.

Large scale behavior of a two-dimensional model of anisotropic branched polymers

We study critical properties of anisotropic branched polymers modeled by semi-directed lattice animals on a triangular lattice. Using the exact transfer-matrix approach on strips of quite large widths and phenomenological renormalization group analysis, we obtained pretty good estimates of various critical exponents in the whole high-temperature region, including the point of collapse transition. Our numerical results suggest that this collapse transition belongs to the universality class of directed percolation.

Ground state phase diagram of anS=12two-leg Heisenberg spin ladder system with negative four-spin interaction

We investigate the ground-state phase diagram and critical properties of the $S=1/2$ two-leg Heisenberg spin ladder system with a negative four-spin interaction using a numerical exact diagonalization method. Using the perturbation theory in the strong negative rung-coupling limit, we derive an $S=1$ bilinear-biquadratic chain as an effective model. We discuss the ground-state phase diagram in this limit. Next we numerically determine a phase boundary between the rung singlet phase and the columnar dimer (CD) phase by the phenomenological renormalization group method, and one between the CD phase and the Haldane phase by the twisted boundary condition method. We confirm that the phase transition between the CD phase and the Haldane phase is of second order and this universality class is described by the $k=2$ $SU(2)$ Wess-Zumino-Novikov-Witten nonlinear $\ensuremath{\sigma}$ model, calculating the central charge and scaling dimensions.

Upper transition point for percolation on the enhanced binary tree: A sharpened lower bound

Hyperbolic structures are obtained by tiling a hyperbolic surface with negative Gaussian curvature. These structures generally exhibit two percolation transitions: a system-wide connection can be established at a certain occupation probability p = pc1, and there emerges a unique giant cluster at pc2 > pc1. There have been debates about locating the upper transition point of a prototypical hyperbolic structure called the enhanced binary tree (EBT), which is constructed by adding loops to a binary tree. This work presents its lower bound as pc2 ≳ 0.55 by using phenomenological renormalization-group methods and discusses some solvable models related to the EBT.

Self-similar analytical solution of the critical fluctuations problem for the Bose–Einstein condensation in an ideal gas

An exact analytical solution for the universal probability distribution of the order parameter fluctuations as well as for the universal statistical and thermodynamic functions of an ideal gas in the whole critical region of Bose–Einstein condensation (BEC) is obtained. A universal constraint nonlinearity is found that is responsible for all nontrivial critical phenomena of the BEC phase transition. Simple analytical approximations, which describe the universal structure of the critical region in terms of confluent hypergeometric or parabolic cylinder functions, as well as asymptotics of the exact solution are derived. The results for the order parameter, all higher order moments of BEC fluctuations and thermodynamic quantities, including specific heat, perfectly match the known asymptotics outside critical region as well as the phenomenological renormalization-group ansatz with known critical exponents in the close vicinity of the critical point. Thus, a full analytical solution to a long-standing problem of finding a universal structure of the λ point for BEC in an ideal gas is found.

The critical behaviour of the 1-dimensional XY model with power law decay long range interactions

We discuss the critical behaviour of the 1-dimensional classical XY model with power law decay long range interactions by using Monte Carlo simulation. It has been predicted by renormalization group calculation that the critical behaviour changes with the power of interactions. From the simulation result of temperature dependence of the specific heat, we confirm that the system shows the critical behaviour of the mean field type and the non-trivial long range order. We also estimate the critical exponent η for the system with the inverse square long range interactions, and by using the phenomenological renormalization group plot, we confirm the 'Berezinskii-Kosterlitz-Thouless (BKT)-like' transition which has been predicted by the spin wave calculations.

Finite-Size scaling at Yang-Lee singularities of 2D discrete spin models

We numerically study the Hamiltonian limits of the two dimensional (2D) Ising and 3-state Potts models in complex magneticfields. From the Phenomenological Renormalization Group, we find the criticalfield values associated with the Yang-Lee singularity (YLS) in these models. We also determine the low-lying part of the excitation spectrum at the YLS. Finally, we compare the resulting patterns of energy levels to predictions for conformal non-unitary minimal models.