Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1803.08534 · Physical review. D/Physical review. D.

Renormalization group flow in field theories with quenched disorder

Ofer Aharony A, Vladimir Narovlansky

Published Aug 15, 2018• 28 citations
arXivHTMLPDF

Abstract

In this paper, we analyze the renormalization group (RG) flow of field theories with quenched disorder, in which the couplings vary randomly in space.We analyze both classical (Euclidean) disorder and quantum disorder, emphasizing general properties rather than specific cases.The RG flow of the disorderaveraged theories takes place in the space of their coupling constants and also in the space of distributions for the disordered couplings, and the two mix together.We write down a generalization of the Callan-Symanzik equation for the flow of disorder-averaged correlation functions.We find that local operators can mix with the response of the theory to local changes in the disorder distribution and that the generalized Callan-Symanzik equation mixes the disorder averages of several different correlation functions.For classical disorder, we show that this can lead to new types of anomalous dimensions and to logarithmic behavior at fixed points.For quantum disorder, we find that the RG flow always generates a rescaling of time relative to space, which at a fixed point generically leads to Lifshitz scaling.The dynamical scaling exponent z behaves as an anomalous dimension (as in other nonrelativistic RG flows), and we compute it at leading order in perturbation theory in the disorder for a general theory.Our results agree with a previous perturbative computation by Boyanovsky and Cardy, and with a holographic disorder computation of Hartnoll and Santos.We also find in quantum disorder that local operators mix with nonlocal (in time) operators under the RG, and that there are critical exponents associated with the disorder distribution that have not previously been discussed.In large-N theories, the disorder averages may be computed exactly, and we verify that they are consistent with the generalized Callan-Symanzik equations.

DOI
10.1103/physrevd.98.045012

Related papers

Renormalization Group in Field Theories with Quantum Quenched Disorder

Vladimir Narovlansky, Ofer Aharony A

We study the renormalization group flow in general quantum field theories with quenched disorder, focusing on random quantum critical points. We show that in disorder-averaged correlation functions the flow mixes local and nonlocal operators. This leads to a new critical exponent related to the disorder (as in classical disorder). We show that the time coordinate is rescaled at each renormalization group step, leading to anisotropic spacetime scaling at critical points. We write a universal formula for the dynamical scaling exponent z for weak disorder.

arXiv:1803.08529•Physical Review Letters•Aug 15, 2018•18 citations

Strong disorder RG principles within a fixed-cell-size real space renormalization: application to the random transverse field Ising model on various fractal lattices

Cécile Monthus, Thomas Garel

Strong disorder renormalization is an energy-based renormalization that leads to a complicated renormalized topology for the surviving clusters as soon as d > 1. In this paper, we propose to include strong disorder renormalization ideas within the more traditional fixed-cell-size real space RG framework. We first consider the one-dimensional chain as a test for this fixed-cell-size procedure: we find that all exactly known critical exponents are reproduced correctly, except for the magnetic exponent β (because it is related to more subtle persistence properties of the full RG flow). We then apply numerically this fixed-cell-size procedure to two types of renormalizable fractal lattice: (i) the Sierpinski gasket of fractal dimension D = ln3/ln2, where there is no underlying classical ferromagnetic transition, so that the RG flow in the ordered phase is similar to what happens in d = 1; (ii) a hierarchical diamond lattice of fractal dimension D = 4/3, where there is an underlying classical ferromagnetic transition, so that the RG flow in the ordered phase is similar to what happens on hypercubic lattices of dimension d > 1. In both cases, we find that the transition is governed by an infinite disorder fixed point: besides the measure of the activated exponent ψ, we analyze the RG flow of various observables in the disordered and ordered phases, in order to extract the 'typical' correlation length exponents of these two phases which are different from the finite-size correlation length exponent.

arXiv:1201.6136•Journal of Statistical Mechanics Theory and Experiment•May 3, 2012•11 citations

Renormalization group in quantum critical theories with Harris-marginal disorder

Koushik Ganesan, Andrew Lucas, Leo Radzihovsky

We develop a renormalization group for weak Harris-marginal disorder in otherwise strongly interacting quantum critical theories, focusing on systems which have emergent conformal invariance. Using conformal perturbation theory, we argue that previously proposed random lines of fixed points with Lifshitz scaling in fact flow toward other universal fixed points, and this flow is captured by a ``one-loop'' analysis. Our approach appears best controlled in theories with only a few operators with low scaling dimension. In this regime, we compare our predictions for the flow of disorder to holographic models, and find complete agreement.

arXiv:2110.11978•Physical review. D/Physical review. D.•Mar 28, 2022•8 citations

Connecting finite-size scaling and renormalization-group flows in Anderson localization on random graphs

Ignacio García-Mata, Weitao Chen, John Martin +2 more

We investigate the relation between two complementary descriptions of the Anderson transition on small-world random graphs: finite-size scaling of eigenfunction moments and renormalization-group (RG) flows of multifractal dimensions $D_q$. The RG flows recover the qualitative structure previously reported for the information dimension $D_1$, while their extension to other $D_q$ with smaller moment orders $q<1/2$ confirms the strongly multifractal nature of the localized phase characterized by a clear separation between multifractal and localized behaviors at strong disorder. On the other hand, we show that we can construct another beta function characterizing the flow of eigenfunction moments, whose collapse onto a single function across different disorder strengths provides a clear confirmation of single-parameter scaling property. These results indicate that the family of RG trajectories observed for the information dimension $D_1$ need not imply two-parameter scaling and Kosterlitz-Thouless like critical behavior. To characterize these properties, one should consider different observables, such as large $q>1/2$ and small $q<1/2$ eigenfunction moments, which are associated with distinct critical behaviors, in particular different critical exponents.

cond-mat.dis-nnCross-listedarXiv:2609.31872•Sep 29, 2026

Renormalization Group for disordered Dirac Fermions

André LeClair

We use the recently proposed all orders beta functions to study localization problems in two dimensions.A classification of random Dirac hamiltonians is presented.Beta functions for the Chalker-Coddington network model are presented and it is described how the renormalization group flows are toward a true singularity.The main subject of this talk is localization-delocalization transitions in two dimensions.This is a subject with many challenging and unsolved problems, the most important being finding a theoretical description of the critical properties of the quantum Hall transition.The Chalker-Coddington network model is perhaps the simplest model believed to be in the universality class of the quantum Hall transition [1].It can be formulated as a theory of random Dirac fermions, and it is believed that the disorder averaged effective theory should possess an infra-red fixed point conformal field theory capable of predicting the critical exponents governing the transition, e.g. the delocalization length exponent[2, 3].I will describe an attempt at finding the expected fixed point based on a proposed all orders beta function [5,6].The all-orders beta functions for the network models are based on general formulas found in [4].These beta-functions generally describe current-current interactions in two dimensions.Though they remain conjectural, several strong checks of the formula were performed for the simplest su(2) case in [7].As I will describe, for reasons that are not understood at the time, but appear to be specific to the application of the general theory to disordered systems, the renormalizaton group flows toward a true singularity, and thus we did not find the expected fixed point.The plan of the talk is as follows.I will first describe a classification of Random Dirac Fermions in 2d.In the next section I explain how supersymmetric Effective Actions are obtained upon disorder averaging.These generally involve current-current interactions and in the next section I describe the renormalization group (RG) for general interactions of this kind.Using arguments independent of the detailed beta functions, I then propose a connection between the possible fixed points and the Virasoro Master equation.In the last sections these methods are applied to the network models for the quantum Hall transtions.Unfortuanately, as I will describe, the all orders beta functions that appear to be correct for su(2) predict flows toward true singularities in the disordered problems.

Workshop on Integrable Theories, Solitons and Duality•Dec 1, 2002•0 citations

Stability of renormalization group fixed points and decay of correlations

Jean Zinn‐Justin

Abstract Chapter 7 is devoted to a discussion of the renormalization group (RG) flow when the effective field theory that describes universal properties of critical phenomena depends on several coupling constants. The universal properties of a large class of macroscopic phase transitions with short range interactions can be described by statistical field theories involving scalar fields with quartic interactions. The simplest critical systems have an O(N) orthogonal symmetry and, therefore, the corresponding field theory has only one quartic interaction. However, in more general physical systems, the flow of quartic interactions is more complicated. This chapter examines these systems from the RG viewpoint. RG beta functions are shown to generate a gradient flow. Some examples illustrate the notion of emergent symmetry. The local stability of fixed points is related to the value of the scaling field dimension.

•Jun 19, 2019•0 citations
In this paper, we analyze the renormalization group (RG) flow of field theories with quenched disorder, in which the couplings vary randomly in space.We analyze both classical (Euclidean) disorder and quantum disorder, emphasizing general properties rather than specific cases.The RG flow of the disorderaveraged theories takes place in the space of their coupling constants and also in the space of distributions for the disordered couplings, and the two mix together.We write down a generalization of the Callan-Symanzik equation for the flow of disorder-averaged correlation functions.We find that local operators can mix with the response of the theory to local changes in the disorder distribution and that the generalized Callan-Symanzik equation mixes the disorder averages of several different correlation functions.For classical disorder, we show that this can lead to new types of anomalous dimensions and to logarithmic behavior at fixed points.For quantum disorder, we find that the RG flow always generates a rescaling of time relative to space, which at a fixed point generically leads to Lifshitz scaling.The dynamical scaling exponent z behaves as an anomalous dimension (as in other nonrelativistic RG flows), and we compute it at leading order in perturbation theory in the disorder for a general theory.Our results agree with a previous perturbative computation by Boyanovsky and Cardy, and with a holographic disorder computation of Hartnoll and Santos.We also find in quantum disorder that local operators mix with nonlocal (in time) operators under the RG, and that there are critical exponents associated with the disorder distribution that have not previously been discussed.In large-N theories, the disorder averages may be computed exactly, and we verify that they are consistent with the generalized Callan-Symanzik equations.
We study the renormalization group flow in general quantum field theories with quenched disorder, focusing on random quantum critical points. We show that in disorder-averaged correlation functions the flow mixes local and nonlocal operators. This leads to a new critical exponent related to the disorder (as in classical disorder). We show that the time coordinate is rescaled at each renormalization group step, leading to anisotropic spacetime scaling at critical points. We write a universal formula for the dynamical scaling exponent z for weak disorder.
Strong disorder renormalization is an energy-based renormalization that leads to a complicated renormalized topology for the surviving clusters as soon as d > 1. In this paper, we propose to include strong disorder renormalization ideas within the more traditional fixed-cell-size real space RG framework. We first consider the one-dimensional chain as a test for this fixed-cell-size procedure: we find that all exactly known critical exponents are reproduced correctly, except for the magnetic exponent β (because it is related to more subtle persistence properties of the full RG flow). We then apply numerically this fixed-cell-size procedure to two types of renormalizable fractal lattice: (i) the Sierpinski gasket of fractal dimension D = ln3/ln2, where there is no underlying classical ferromagnetic transition, so that the RG flow in the ordered phase is similar to what happens in d = 1; (ii) a hierarchical diamond lattice of fractal dimension D = 4/3, where there is an underlying classical ferromagnetic transition, so that the RG flow in the ordered phase is similar to what happens on hypercubic lattices of dimension d > 1. In both cases, we find that the transition is governed by an infinite disorder fixed point: besides the measure of the activated exponent ψ, we analyze the RG flow of various observables in the disordered and ordered phases, in order to extract the 'typical' correlation length exponents of these two phases which are different from the finite-size correlation length exponent.
We develop a renormalization group for weak Harris-marginal disorder in otherwise strongly interacting quantum critical theories, focusing on systems which have emergent conformal invariance. Using conformal perturbation theory, we argue that previously proposed random lines of fixed points with Lifshitz scaling in fact flow toward other universal fixed points, and this flow is captured by a ``one-loop'' analysis. Our approach appears best controlled in theories with only a few operators with low scaling dimension. In this regime, we compare our predictions for the flow of disorder to holographic models, and find complete agreement.
We investigate the relation between two complementary descriptions of the Anderson transition on small-world random graphs: finite-size scaling of eigenfunction moments and renormalization-group (RG) flows of multifractal dimensions DqD_qDq​. The RG flows recover the qualitative structure previously reported for the information dimension D1D_1D1​, while their extension to other DqD_qDq​ with smaller moment orders q<1/2q<1/2q<1/2 confirms the strongly multifractal nature of the localized phase characterized by a clear separation between multifractal and localized behaviors at strong disorder. On the other hand, we show that we can construct another beta function characterizing the flow of eigenfunction moments, whose collapse onto a single function across different disorder strengths provides a clear confirmation of single-parameter scaling property. These results indicate that the family of RG trajectories observed for the information dimension D1D_1D1​ need not imply two-parameter scaling and Kosterlitz-Thouless like critical behavior. To characterize these properties, one should consider different observables, such as large q>1/2q>1/2q>1/2 and small q<1/2q<1/2q<1/2 eigenfunction moments, which are associated with distinct critical behaviors, in particular different critical exponents.
We use the recently proposed all orders beta functions to study localization problems in two dimensions.A classification of random Dirac hamiltonians is presented.Beta functions for the Chalker-Coddington network model are presented and it is described how the renormalization group flows are toward a true singularity.The main subject of this talk is localization-delocalization transitions in two dimensions.This is a subject with many challenging and unsolved problems, the most important being finding a theoretical description of the critical properties of the quantum Hall transition.The Chalker-Coddington network model is perhaps the simplest model believed to be in the universality class of the quantum Hall transition [1].It can be formulated as a theory of random Dirac fermions, and it is believed that the disorder averaged effective theory should possess an infra-red fixed point conformal field theory capable of predicting the critical exponents governing the transition, e.g. the delocalization length exponent[2, 3].I will describe an attempt at finding the expected fixed point based on a proposed all orders beta function [5,6].The all-orders beta functions for the network models are based on general formulas found in [4].These beta-functions generally describe current-current interactions in two dimensions.Though they remain conjectural, several strong checks of the formula were performed for the simplest su(2) case in [7].As I will describe, for reasons that are not understood at the time, but appear to be specific to the application of the general theory to disordered systems, the renormalizaton group flows toward a true singularity, and thus we did not find the expected fixed point.The plan of the talk is as follows.I will first describe a classification of Random Dirac Fermions in 2d.In the next section I explain how supersymmetric Effective Actions are obtained upon disorder averaging.These generally involve current-current interactions and in the next section I describe the renormalization group (RG) for general interactions of this kind.Using arguments independent of the detailed beta functions, I then propose a connection between the possible fixed points and the Virasoro Master equation.In the last sections these methods are applied to the network models for the quantum Hall transtions.Unfortuanately, as I will describe, the all orders beta functions that appear to be correct for su(2) predict flows toward true singularities in the disordered problems.
Abstract Chapter 7 is devoted to a discussion of the renormalization group (RG) flow when the effective field theory that describes universal properties of critical phenomena depends on several coupling constants. The universal properties of a large class of macroscopic phase transitions with short range interactions can be described by statistical field theories involving scalar fields with quartic interactions. The simplest critical systems have an O(N) orthogonal symmetry and, therefore, the corresponding field theory has only one quartic interaction. However, in more general physical systems, the flow of quartic interactions is more complicated. This chapter examines these systems from the RG viewpoint. RG beta functions are shown to generate a gradient flow. Some examples illustrate the notion of emergent symmetry. The local stability of fixed points is related to the value of the scaling field dimension.