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:cond-mat/9702146 · Physical review. B, Condensed matter

Real-space renormalization of the Chalker-Coddington model

Daniel P. Arovas, Martin Janßen, Boris Shapiro

Published Aug 15, 1997• 35 citations
arXivHTMLPDF

Abstract

We study a number of hierarchical network models related to the Chalker-Coddington model of quantum percolation. Our aim is to describe the physics of the quantum Hall transition. The hierarchical network models are constructed by combining series and parallel composition of quantum resistors. The localization-delocalization transition occurring in these models is treated by real-space renormalization techniques. Essentially, the localization-delocalization transition is due to a competition between two one-dimensional localization mechanisms.

DOI
10.1103/physrevb.56.4751

Related papers

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

Chalker-Coddington Network Model is Quantum Critical

John Bradley Marston, Shan-Wen Tsai

We show that the localization transition in the integer quantum Hall effect as described by the Chalker-Coddington network model is quantum critical. We first map the anisotropic network model to the problem of diagonalizing a one-dimensional non-Hermitian noncompact supersymmetric lattice Hamiltonian of interacting bosons and fermions. Its behavior is investigated numerically using the density matrix renormalization group method, and critical behavior is found at the plateau transition. This result is confirmed by a generalization of the Lieb-Schultz-Mattis theorem.

arXiv:cond-mat/9812261•Physical Review Letters•Jun 14, 1999•34 citations

Action formulation of the network model of plateau-plateau transitions in the quantum Hall effect

A. Sedrakyan

The Lagrangian (action) formulation of the Chalker-Coddington network model for plateau-plateau transitions in the quantum Hall effect is presented based on a model of fermions hopping on Manhattan lattice (ML). The dimensionless Landauer resistance is considered and its average is calculated over the random U(1) phases with constant distribution on the circle. The Lagrangian of the resultant model on ML is found and the corresponding R matrix is written down. It appeared, that this model is integrable, raising hope to investigate physics of plateau- plateau transitions by the exact method of powerful algebraic Bethe ansatz.

arXiv:cond-mat/0303349•Physical review. B, Condensed matter•Dec 29, 2003•14 citations

Localization-delocalization transitions in a two-dimensional quantum percolation model: von Neumann entropy studies

Longyan Gong, Peiqing Tong

In two-dimensional quantum site-percolation square lattice models, the von Neumann entropy is extensively studied numerically. At a certain eigenenergy, the localization-delocalization transition is reflected by the derivative of von Neumann entropy which is maximal at the quantum percolation threshold ${p}_{q}$. The phase diagram of localization-delocalization transitions is deduced in the extrapolation to infinite system sizes. The nonmonotonic eigenenergies dependence of ${p}_{q}$ and the lowest value ${p}_{q}\ensuremath{\simeq}0.665$ are found. At localized-delocalized transition points, the finite scaling analysis for the von Neumann entropy is performed and it is found the critical exponents $\ensuremath{\nu}$ not to be universal. These studies provide an evidence that the existence of a quantum percolation threshold ${p}_{q}<1$ in the two-dimensional quantum percolation problem.

arXiv:0911.5531•Physical Review B•Nov 20, 2009•19 citations

Three-dimensional network model for strong topological insulator transitions

Jun Ho Son, S. Raghu

We construct a three-dimensional (3D), time-reversal symmetric generalization of the Chalker-Coddington network model for the integer quantum Hall transition. In addition to a weak topological insulator phase already accommodated by the network model framework in the preexisting literature, our network model hosts strong topological insulator phases. We unambiguously demonstrate that strong topological insulator phases emerge as intermediate phases between a trivial insulator phase and a weak topological phase. Additionally, we found a duality that relates a trivial insulator phase and a weak topological phase in our network model. Remarkably, strong topological phases are mapped to themselves under this duality. We show that upon adding sufficiently strong disorder, the strong topological insulator phases undergo phase transitions into a metallic phase. We numerically determine the critical exponent of the insulator-metal transition. Our network model explicitly shows how a semiclassical percolation picture of topological phase transitions in two dimensions can be generalized to 3D and opens up a venue for studying 3D topological phase transitions.

arXiv:2008.03315•Physical review. B./Physical review. B•Sep 27, 2021•4 citations

Unifying Model for Several Classes of Two-Dimensional Phase Transition

Yonatan Dubi, Yigal Meir, Y. Avishai

A relatively simple and physically transparent model based on quantum percolation and dephasing is employed to construct a global phase diagram which encodes and unifies the critical physics of the quantum Hall, "two-dimensional metal-insulator," classical percolation and, to some extent, superconductor-insulator transitions. Using real-space renormalization group techniques, crossover functions between critical points are calculated. The critical behavior around each fixed point is analyzed and some experimentally relevant puzzles are addressed.

arXiv:cond-mat/0409759•Physical Review Letters•Apr 22, 2005•28 citations
We study a number of hierarchical network models related to the Chalker-Coddington model of quantum percolation. Our aim is to describe the physics of the quantum Hall transition. The hierarchical network models are constructed by combining series and parallel composition of quantum resistors. The localization-delocalization transition occurring in these models is treated by real-space renormalization techniques. Essentially, the localization-delocalization transition is due to a competition between two one-dimensional localization mechanisms.
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.
We show that the localization transition in the integer quantum Hall effect as described by the Chalker-Coddington network model is quantum critical. We first map the anisotropic network model to the problem of diagonalizing a one-dimensional non-Hermitian noncompact supersymmetric lattice Hamiltonian of interacting bosons and fermions. Its behavior is investigated numerically using the density matrix renormalization group method, and critical behavior is found at the plateau transition. This result is confirmed by a generalization of the Lieb-Schultz-Mattis theorem.
The Lagrangian (action) formulation of the Chalker-Coddington network model for plateau-plateau transitions in the quantum Hall effect is presented based on a model of fermions hopping on Manhattan lattice (ML). The dimensionless Landauer resistance is considered and its average is calculated over the random U(1) phases with constant distribution on the circle. The Lagrangian of the resultant model on ML is found and the corresponding R matrix is written down. It appeared, that this model is integrable, raising hope to investigate physics of plateau- plateau transitions by the exact method of powerful algebraic Bethe ansatz.
In two-dimensional quantum site-percolation square lattice models, the von Neumann entropy is extensively studied numerically. At a certain eigenenergy, the localization-delocalization transition is reflected by the derivative of von Neumann entropy which is maximal at the quantum percolation threshold pq{p}_{q}pq​. The phase diagram of localization-delocalization transitions is deduced in the extrapolation to infinite system sizes. The nonmonotonic eigenenergies dependence of pq{p}_{q}pq​ and the lowest value ${p}_{q}\ensuremath{\simeq}0.665$ are found. At localized-delocalized transition points, the finite scaling analysis for the von Neumann entropy is performed and it is found the critical exponents $\ensuremath{\nu}$ not to be universal. These studies provide an evidence that the existence of a quantum percolation threshold pq<1{p}_{q}<1pq​<1 in the two-dimensional quantum percolation problem.
We construct a three-dimensional (3D), time-reversal symmetric generalization of the Chalker-Coddington network model for the integer quantum Hall transition. In addition to a weak topological insulator phase already accommodated by the network model framework in the preexisting literature, our network model hosts strong topological insulator phases. We unambiguously demonstrate that strong topological insulator phases emerge as intermediate phases between a trivial insulator phase and a weak topological phase. Additionally, we found a duality that relates a trivial insulator phase and a weak topological phase in our network model. Remarkably, strong topological phases are mapped to themselves under this duality. We show that upon adding sufficiently strong disorder, the strong topological insulator phases undergo phase transitions into a metallic phase. We numerically determine the critical exponent of the insulator-metal transition. Our network model explicitly shows how a semiclassical percolation picture of topological phase transitions in two dimensions can be generalized to 3D and opens up a venue for studying 3D topological phase transitions.
A relatively simple and physically transparent model based on quantum percolation and dephasing is employed to construct a global phase diagram which encodes and unifies the critical physics of the quantum Hall, "two-dimensional metal-insulator," classical percolation and, to some extent, superconductor-insulator transitions. Using real-space renormalization group techniques, crossover functions between critical points are calculated. The critical behavior around each fixed point is analyzed and some experimentally relevant puzzles are addressed.