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

Journal of Physics A Mathematical and General

Calculating dynamic structure factors with the real space renormalisation group

Anthony C. Maggs, Robin B. Stinchcombe

Published Sep 11, 1986• 4 citations
Publisher

Abstract

The authors develop exact methods, based on the real space renormalisation group, for calculating dynamic structure factors of Heisenberg ferromagnets. The method is applied to a pure chain, a Sierpinski gasket and a quasi-one-dimensional model of a dilute chain. The result for the Sierpinski gasket exhibits a rich structure because of the requirement of discrete scale invariance. By the use of the distribution scaling the method is generalised to treat a randomly diluted chain.

DOI
10.1088/0305-4470/19/13/028

Related papers

A renormalisation group treatment of the diluted spin-1/2Heisenberg ferromagnet

Robin B. Stinchcombe

The bond-diluted spin- 1 / 2 Heisenberg ferromagnet is treated by a position-space renormalisation group method. The method is briefly outlined and results are presented for one, two and three dimensions. The results include exponents and, for three-dimensional systems, the dependence of the transition temperature T c on concentration p.

Journal of Physics C Solid State Physics•May 28, 1979•5 citations

Dynamic Critical Behavior of Sierpinski Gasket Type Fractal

Wang Xiaomin, Ruibao Tao

We have applied the exact real space renormalisation group method to analyse the critical dynamics of Heisenberg spin system at the percolation threshold embedded in the two dimensional Sierpinski Gasket type fractal. Our analysis yields the dynamic critical exponent Z and the scaling behaviors of response function and characteristic frequency. We also show that the scaling exponent relation is accurately satisfied by the values of the Z and two other exponents.

Chinese Physics Letters•Aug 1, 1991•0 citations

Position space treatment of diluted classical Heisenberg ferromagnet

Robin B. Stinchcombe

The bond-diluted classical Heisenberg ferromagnet is treated by a position space renormalisation group method valid when the valid behaviour is at low temperatures. The method is outlined and used to obtain exact (one-dimensional) and approximate (two- and three-dimensional) results for the critical behaviour of the correlation length near the percolation limit. The procedure agrees in the pure limit with exact results of Polyakov and Migdal for the two-dimensional case. At general concentrations an exact correspondence with related treatments of random resistor networks is exhibited. This implies that the percolation-to-thermal crossover exponents of the Ising and Heisenberg systems are the same. The generalisations required for a proper comparison with neutron scattering experiments are briefly discussed.

Journal of Physics C Solid State Physics•Jul 14, 1979•30 citations

Dynamical structure factor of random antiferromagnetic Heisenberg spin chains

Zhaoxin Xu, He-Ping Ying, Xin Wan

Combining quantum Monte Carlo simulations with the maximum entropy method, we study the dynamical structure factor $S(k,ω)$ of spin-1/2 antiferromagnetic Heisenberg chains with various random bond distributions. We emphasize the crossover behavior in the dynamical properties from pure chain to disorder-dominated random singlet phase due to bond randomness. For the distribution corresponding to the infinite randomness fixed point, $S(k,ω)$ develops broad non-spinon excitations as well as the random-singlet peak near $(k, ω) = (π, 0)$, consistent with the known results obtained by the real-space renormalization group method. For weak disorder, however, we find clear signature of spinon excitations, reminiscent of pure spin chains, blurred by disorder. We discuss the implication for experiments on random-bond antiferromagnetic spin chains, realizable, e.g., in BaCu$_2$(Si$_x$Ge$_{1-x}$)$_2$O$_7$.

arXiv:cond-mat/0703772•Mar 29, 2007•0 citations

Static critical properties of the pure and diluted Heisenberg or Ising models

Mathew Raymond. Davies, Davies, Mathew Raymond

Real space renormalisation group scaling techniques are used to investigate the static critical behaviour of the pure and dilute, classical, anisotropic Heisenberg model. Transfer matrix methods are employed to obtain asymptotically exact expressions for the correlation lengths and susceptibilities of the one-dimensional system. The resulting scaling relationships are combined with an approximate bond moving scheme to treat pure and dilute models in higher dimensionalities. Detailed discussions are given for the dependence of correlation lengths and susceptibilities on temperature, anisotropy and concentration, and fcr the critical temperature on anisotropy and concentration. Particular emphasis is given to the weakly anisotropic system near percolation threshold and comparisons are made between the results of the present analysis and those of neutron-scattering experiments on dilute quasi-two- and three-dimensional systems.

•Jan 1, 1982•0 citations

Real-space dynamic renormalization group: Systematics for T>~Tc

Gene F. Mazenko, Oriol T. Valls

In this paper we systematize the real-space dynamic-renormalization-group method we developed elsewhere. We treat the two-dimensional kinetic Ising model with relaxational dynamics defined on a square lattice. We show how one can set up a systematic perturbation-theory expansion which treats the coupling between cells of spins as a small parameter. The novel feature of the expansion is that it provides a method for determining the appropriate effective interaction between spins in a set of uncoupled cells. By obtaining a good zeroth-order approximation for a set of uncoupled cells, one finds that the effective interaction between cells is reduced. Similar ideas are also applied to the dynamics of the problem and we indicate how one can obtain a good zeroth-order approximation in terms of the dynamics of uncoupled cells. We couple these perturbation theory ideas to the real-space dynamic-renormalization-group method and carry out explicit calculations of a variety of observable quantities including the magnetic susceptibility, specific heat, static structure factor, time-dependent spin-autocorrelation function, and the dynamic structure factor. In every case we obtain excellent results in keeping with the scaling behavior near the phase transition and the high-temperature limit.

Physical review. B, Condensed matter•Aug 1, 1981•19 citations
The authors develop exact methods, based on the real space renormalisation group, for calculating dynamic structure factors of Heisenberg ferromagnets. The method is applied to a pure chain, a Sierpinski gasket and a quasi-one-dimensional model of a dilute chain. The result for the Sierpinski gasket exhibits a rich structure because of the requirement of discrete scale invariance. By the use of the distribution scaling the method is generalised to treat a randomly diluted chain.
The bond-diluted classical Heisenberg ferromagnet is treated by a position space renormalisation group method valid when the valid behaviour is at low temperatures. The method is outlined and used to obtain exact (one-dimensional) and approximate (two- and three-dimensional) results for the critical behaviour of the correlation length near the percolation limit. The procedure agrees in the pure limit with exact results of Polyakov and Migdal for the two-dimensional case. At general concentrations an exact correspondence with related treatments of random resistor networks is exhibited. This implies that the percolation-to-thermal crossover exponents of the Ising and Heisenberg systems are the same. The generalisations required for a proper comparison with neutron scattering experiments are briefly discussed.
Combining quantum Monte Carlo simulations with the maximum entropy method, we study the dynamical structure factor S(k,ω)S(k,ω)S(k,ω) of spin-1/2 antiferromagnetic Heisenberg chains with various random bond distributions. We emphasize the crossover behavior in the dynamical properties from pure chain to disorder-dominated random singlet phase due to bond randomness. For the distribution corresponding to the infinite randomness fixed point, S(k,ω)S(k,ω)S(k,ω) develops broad non-spinon excitations as well as the random-singlet peak near (k,ω)=(π,0)(k, ω) = (π, 0)(k,ω)=(π,0), consistent with the known results obtained by the real-space renormalization group method. For weak disorder, however, we find clear signature of spinon excitations, reminiscent of pure spin chains, blurred by disorder. We discuss the implication for experiments on random-bond antiferromagnetic spin chains, realizable, e.g., in BaCu2_22​(Six_xx​Ge1−x_{1-x}1−x​)2_22​O7_77​.
Real space renormalisation group scaling techniques are used to investigate the static critical behaviour of the pure and dilute, classical, anisotropic Heisenberg model. Transfer matrix methods are employed to obtain asymptotically exact expressions for the correlation lengths and susceptibilities of the one-dimensional system. The resulting scaling relationships are combined with an approximate bond moving scheme to treat pure and dilute models in higher dimensionalities. Detailed discussions are given for the dependence of correlation lengths and susceptibilities on temperature, anisotropy and concentration, and fcr the critical temperature on anisotropy and concentration. Particular emphasis is given to the weakly anisotropic system near percolation threshold and comparisons are made between the results of the present analysis and those of neutron-scattering experiments on dilute quasi-two- and three-dimensional systems.
In this paper we systematize the real-space dynamic-renormalization-group method we developed elsewhere. We treat the two-dimensional kinetic Ising model with relaxational dynamics defined on a square lattice. We show how one can set up a systematic perturbation-theory expansion which treats the coupling between cells of spins as a small parameter. The novel feature of the expansion is that it provides a method for determining the appropriate effective interaction between spins in a set of uncoupled cells. By obtaining a good zeroth-order approximation for a set of uncoupled cells, one finds that the effective interaction between cells is reduced. Similar ideas are also applied to the dynamics of the problem and we indicate how one can obtain a good zeroth-order approximation in terms of the dynamics of uncoupled cells. We couple these perturbation theory ideas to the real-space dynamic-renormalization-group method and carry out explicit calculations of a variety of observable quantities including the magnetic susceptibility, specific heat, static structure factor, time-dependent spin-autocorrelation function, and the dynamic structure factor. In every case we obtain excellent results in keeping with the scaling behavior near the phase transition and the high-temperature limit.