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/0406334

Ma-Dasgupta renormalization studies of various disordered systems

Cécile Monthus

Published Jun 15, 2004• 0 citations
arXivHTMLPDF

Abstract

The Ma-Dasgupta real-space renormalization methods allow to study disordered systems which are governed by strong disorder fixed points. After a general introduction to the qualitative ideas and to the quantitative renormalization rules, we describe the explicit exact results that can be obtained in various one-dimensional models, either classical or quantum, either for dynamics or statics. The main part of this dissertation is devoted to statistical physics models, with special attention to (i) the off-equilibrium dynamics of a particle diffusing in a Brownian potential or in a trap landscape, (ii) the coarsening dynamics and the equilibrium of classical disordered spin chains, (iii) the delocalization transition of a random polymer at an interface. The last part of the dissertation deals with two disordered quantum spin chains which exhibit a zero-temperature phase transition as the disorder varies, namely (a) the random antiferromagnetic S=1 spin chain, (b) the random transverse field Ising chain.

DOI
10.48550/arxiv.cond-mat/0406334

Related papers

Griffiths-McCoy Singularities in Random Quantum Spin Chains: Exact Results through Renormalization

Ferenc Iglói, Róbert Juhász, Péter Lajkó

The Ma-Dasgupta-Hu renormalization group (RG) scheme is used to study singular quantities in the Griffiths phase of random quantum spin chains. For the random transverse-field Ising spin chain we have extended Fisher's analytical solution to the off-critical region and calculated the dynamical exponent exactly. Concerning other random chains we argue by scaling considerations that the RG method generally becomes asymptotically exact for large times, both at the critical point and in the whole Griffiths phase. This statement is checked via numerical calculations on the random Heisenberg and quantum Potts models by the density matrix renormalization group method.

arXiv:cond-mat/0005375•Physical Review Letters•Feb 12, 2001•49 citations

Renormalization Group Study on Coupled Random Antiferromagnetic Spin-1/2 Chains

This dissertation is a systematic theoretical effort to investigate the low-energy properties of random antiferromagnetic quantum spin-1/2 systems beyond one dimension. We use the celebrated real space renormalization group (RSRG) technique developed by Dasgupta, Ma, and Hu to understand the static and dynamic properties of random antiferromagnetic spin-1/2 ladders and weakly coupled random antiferromagnetic spin-1/2 chains. Pure antiferromagnetic spin-1/2 ladders are known to be in the spin liquid phase with a ¯nite energy gap in the excitation spectrum and a short range spin-spin correlation. We are interested in investigating the nature of these systems when disorder, controlled by bond randomness and the presence of impurities, is introduced. Using real-space renormalization group method we are able to find that when disorder comes from bond randomness only, the system flows into a Griffith phase which is characterized by non-universal diverging spin susceptibility and short-range spin-spin correlation. When impurities with spin 0 or 1 are introduced into the system, the system flows into a different fixed point which is controlled by large effective spins whose susceptibility is characterized by a universal Curie-like 1=T behavior. This conclusion holds for any impurities with integer spin. We also study the low-energy collective excitations and dynamical response functions of weakly coupled random antiferromagnetic spin-1/2 chains at low temperature whose low energy properties are governed by the strong-randomness fixed point. By combining RSRG technique to tackle the intrachain couplings and Random Phase Approximation (RPA) formalism for the interchain couplings, we show that the system supports collective spin wave excitations with linear dispersions and calculate the spin wave velocity in terms of microscopic parameters of the chain. Our result agrees with the measured dispersion for Zn-doped CuGeO3 where it shows a linear dispersion along the chain. We predict the spectra weight within RPA which can be qualitatively compared to the scattering intensity in the Inelastic Neutron Scattering experiment.

•Jan 1, 2005•0 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

Real-Space Renormalization Group Study of Effects of Anisotropy on S=1 Random Antiferromagnetic Chain

Shigeru Koikegami, Synge Todo, Hajime Takayama

We investigate S=1 antiferromagnetic quantum spin chain, whose exchange couplings are strongly disordered. By the real-space renomalization group method, introduced by Ma, Dasgupta, and Hu, the renormalization group flows are analyzed numerically in a plain of the anisotoropy of the exchange coupling vs. the staggered magnetic field. As the result, the Heisenberg point, which has a zero average of the exchange coupling anisotropy, is specified as the unstable fixed point against the anisotropy.

arXiv:cond-mat/0106289•Jun 15, 2001•0 citations

RENORMALIZATION-GROUP INVESTIGATION OF THE S = 1 RANDOM ANTIFERROMAGNETIC HEISENBERG CHAIN

Péter Lajkó

We introduce variants of the Ma-Dasgupta renormalization-group (RG) approach for random quantum spin chains, in which the energy-scale is reduced by decimation built on either perturbative or non-perturbative principles. In one non-perturbative version of the method, we require the exact invariance of the lowest gaps, while in a second class of perturbative Ma-Dasgupta techniques, different decimation rules are utilized. For the S = 1 random antiferromagnetic Heisenberg chain, both type of methods provide the same type of disorder dependent phase diagram, which is in agreement with density-matrix renormalization-group calculations and previous studies.

arXiv:0903.0462•International Journal of Modern Physics C•Dec 1, 2006•1 citation

Disorder Induced Cross-Over Effects at Quantum Critical Points

Enrico Carlon, Péter Lajkó, Ferenc Iglói

Critical properties of quantum spin chains with varying degrees of disorder are studied at zero temperature by analytical and extensive density matrix renormalization methods. Generally the phase diagram is found to contain three phases. The weak disorder regime, where the critical behavior is controlled by the fixed points of the pure system, and the strong disorder regime, which is attracted by an infinite randomness fixed point, are separated by an intermediate disorder regime, where dynamical scaling is anisotropic and the static and dynamical exponents are disorder dependent.

arXiv:cond-mat/0106325•Physical Review Letters•Dec 12, 2001•51 citations
The Ma-Dasgupta real-space renormalization methods allow to study disordered systems which are governed by strong disorder fixed points. After a general introduction to the qualitative ideas and to the quantitative renormalization rules, we describe the explicit exact results that can be obtained in various one-dimensional models, either classical or quantum, either for dynamics or statics. The main part of this dissertation is devoted to statistical physics models, with special attention to (i) the off-equilibrium dynamics of a particle diffusing in a Brownian potential or in a trap landscape, (ii) the coarsening dynamics and the equilibrium of classical disordered spin chains, (iii) the delocalization transition of a random polymer at an interface. The last part of the dissertation deals with two disordered quantum spin chains which exhibit a zero-temperature phase transition as the disorder varies, namely (a) the random antiferromagnetic S=1 spin chain, (b) the random transverse field Ising chain.
The Ma-Dasgupta-Hu renormalization group (RG) scheme is used to study singular quantities in the Griffiths phase of random quantum spin chains. For the random transverse-field Ising spin chain we have extended Fisher's analytical solution to the off-critical region and calculated the dynamical exponent exactly. Concerning other random chains we argue by scaling considerations that the RG method generally becomes asymptotically exact for large times, both at the critical point and in the whole Griffiths phase. This statement is checked via numerical calculations on the random Heisenberg and quantum Potts models by the density matrix renormalization group method.
This dissertation is a systematic theoretical effort to investigate the low-energy properties of random antiferromagnetic quantum spin-1/2 systems beyond one dimension. We use the celebrated real space renormalization group (RSRG) technique developed by Dasgupta, Ma, and Hu to understand the static and dynamic properties of random antiferromagnetic spin-1/2 ladders and weakly coupled random antiferromagnetic spin-1/2 chains. Pure antiferromagnetic spin-1/2 ladders are known to be in the spin liquid phase with a ¯nite energy gap in the excitation spectrum and a short range spin-spin correlation. We are interested in investigating the nature of these systems when disorder, controlled by bond randomness and the presence of impurities, is introduced. Using real-space renormalization group method we are able to find that when disorder comes from bond randomness only, the system flows into a Griffith phase which is characterized by non-universal diverging spin susceptibility and short-range spin-spin correlation. When impurities with spin 0 or 1 are introduced into the system, the system flows into a different fixed point which is controlled by large effective spins whose susceptibility is characterized by a universal Curie-like 1=T behavior. This conclusion holds for any impurities with integer spin. We also study the low-energy collective excitations and dynamical response functions of weakly coupled random antiferromagnetic spin-1/2 chains at low temperature whose low energy properties are governed by the strong-randomness fixed point. By combining RSRG technique to tackle the intrachain couplings and Random Phase Approximation (RPA) formalism for the interchain couplings, we show that the system supports collective spin wave excitations with linear dispersions and calculate the spin wave velocity in terms of microscopic parameters of the chain. Our result agrees with the measured dispersion for Zn-doped CuGeO3 where it shows a linear dispersion along the chain. We predict the spectra weight within RPA which can be qualitatively compared to the scattering intensity in the Inelastic Neutron Scattering experiment.
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 introduce variants of the Ma-Dasgupta renormalization-group (RG) approach for random quantum spin chains, in which the energy-scale is reduced by decimation built on either perturbative or non-perturbative principles. In one non-perturbative version of the method, we require the exact invariance of the lowest gaps, while in a second class of perturbative Ma-Dasgupta techniques, different decimation rules are utilized. For the S = 1 random antiferromagnetic Heisenberg chain, both type of methods provide the same type of disorder dependent phase diagram, which is in agreement with density-matrix renormalization-group calculations and previous studies.
Critical properties of quantum spin chains with varying degrees of disorder are studied at zero temperature by analytical and extensive density matrix renormalization methods. Generally the phase diagram is found to contain three phases. The weak disorder regime, where the critical behavior is controlled by the fixed points of the pure system, and the strong disorder regime, which is attracted by an infinite randomness fixed point, are separated by an intermediate disorder regime, where dynamical scaling is anisotropic and the static and dynamical exponents are disorder dependent.