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 Applied Physics

Breakdown of scaling in the 1D spin-1/2 Heisenberg ferromagnet

Jill C. Bonner, Gerhard H. Muller

Published Apr 15, 1987• 4 citations
Publisher

Abstract

The problem of the critical behavior of the spin-1/2 Heisenberg ferromagnetic chain has challenged many research workers since 1964. We discuss the critical exponents α and γ recently determined by Schlottmann and by Takahashi and Yamada in the context of previously known Tc=0 critical properties of the 1D s=1/2 XXZ model and the classical Heisenberg model, and we comment on the manifest breakdown of scaling in the s=1/2 Heisenberg ferromagnet.

DOI
10.1063/1.338399

Related papers

Critical Behavior of the Isotropic Ferromagnetic Quantum Heisenberg Chain

P. Schlottmann

The thermodynamic Bethe-Ansatz equations for the isotropic ferromagnetic $S=\frac{1}{2}$ Heisenberg chain have been solved numerically. At low temperatures, I find a power-law dependence on $T$ for the specific heat and the susceptibility with critical exponents $\ensuremath{\alpha}=\ensuremath{-}0.49\ifmmode\pm\else\textpm\fi{}0.02$, $\ensuremath{\gamma}=2.00\ifmmode\pm\else\textpm\fi{}0.02$, and $\ensuremath{\Delta}\ensuremath{\simeq}\ensuremath{\gamma}$. The exponent $\ensuremath{\alpha}$ is compatible with effective spin waves and $\ensuremath{\gamma}$ and $\ensuremath{\Delta}$ are the exponents of the classical chain. Amplitudes and corrections to scaling are obtained and differences with previous results are discussed.

Physical Review Letters•May 13, 1985•70 citations

Critical behavior of anisotropic spin-SHeisenberg chains

Francisco Castilho Alcaraz, Adriana Moreo

Using a Lanczos method, we study the spectra of Heisenberg chains with spin S=1, 3/2, 2, 5/2, and 3, as a function of the anisotropy parameter \ensuremath{\lambda} and the number of sites of the chain. We use periodic, twisted, and open boundary conditions. We found that for all values of the spin when -1\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}0, these models are massless with critical behavior described by a conformal theory with central charge c=1. We discuss the whole operator content. In particular, we found that the critical exponent ${\mathrm{\ensuremath{\eta}}}_{\mathit{x}}$ follows the behavior ${\mathrm{\ensuremath{\eta}}}_{\mathit{x}}$=(\ensuremath{\pi}-\ensuremath{\gamma})/2\ensuremath{\pi}S and ${\mathrm{\ensuremath{\eta}}}_{\mathit{z}}$=2 where \ensuremath{\gamma}=${\mathrm{cos}}^{\mathrm{\ensuremath{-}}1}$(\ensuremath{\lambda}). For \ensuremath{\lambda}>0, our results indicate that the S=1 model is massive, while for S>1, the massless behavior characterized by the critical exponents described above continues up to \ensuremath{\lambda}=${\ensuremath{\lambda}}^{\mathrm{*}}$(S). For ${\ensuremath{\lambda}}^{\mathrm{*}}$\ensuremath{\le}1, our results are consistent with a massive phase for the integer-spin models and a massless phase with a rapid variation of the critical exponents for the half-integer-spin models.

Physical review. B, Condensed matter•Aug 1, 1992•50 citations

Study on the Critical Properties of the One-Dimensional Quantum XXZ Model with General Spin S Using the Quantum Renormalization Group Method

M. Inoue, Masuo Suzuki

For the isotropic Heisenberg model with an arbitrary spin 5, the correlation exponent l/ defined by ~ ~ T-v near the critical point Tc=O is evaluated as l/=1 with the use of the real space renormalization group method.This yields y=2 for the susceptibility exponent y defined by Xo~ T-r , using the scaling relation y=l + (2-1]) l/= 1 + l/.Our general result y=2 for all values of spin 5 includes Fisher's result in the classical limit 5-->00.For the anisotropic Heisenberg model (XXZ model) with a spin 5=1/2, an anisotropy scaling law is derived.

Progress of Theoretical Physics•May 1, 1986•1 citation

Universal Finite-Size Scaling Function of the Ferromagnetic Heisenberg Chain in a Magnetic Field

Hiroaki Nakamura, Naomichi Hatano, Minoru Takahashi

The finite-size scaling function of the magnetization of the ferromagnetic Heisenberg chain is argued to be universal with respect to the magnitude of the spin. The finite-size scaling function is given explicitly by an analytical calculation in the classical limit $S=\infty.$ The universality is checked for $S=1/2$ and $1$ by means of numerical calculations. Critical exponents are obtained as well. It is concluded that this universal scaling function originates in the universal behavior of the correlation function.

arXiv:cond-mat/9501013•Journal of the Physical Society of Japan•Jun 15, 1995•7 citations

Finite-size scaling and integer-spin Heisenberg chains

Jill C. Bonner, Gerhard H. Muller

Finite-size scaling (phenomenological renormalization) techniques are trusted and widely applied in low-dimensional magnetism and, particularly, in lattice gauge field theory. Recently, investigations have begun which subject the theoretical basis to systematic and intensive scrutiny to determine the validity of finite-size scaling in a variety of situations. The 2D ANNNI model is an example of a situation where finite-size scaling methods encounter difficulty, related to the occurrence of a disorder line (one-dimensional line). A second example concerns the behavior of the spin-1/2 antiferromagnetic XXZ model where the T=0 critical behavior is exactly known and features an essential singularity at the isotropic Heisenberg point. Standard finite-size scaling techniques do not convincingly reproduce the exact phase behavior and this is attributable to the essential singularity. The point is relevant in connection with a finite-size scaling analysis of a spin-one antiferromagnetic XXZ model, which claims to support a conjecture by Haldane that the T=0 phase behavior of integer-spin Heisenberg chains is significantly different from that of half-integer-spin Heisenberg chains.

Journal of Applied Physics•Mar 15, 1984•5 citations

Ferromagnetic Heisenberg chains: A description of the magnetic susceptibility from a noncritical scaling approach

Jean Souletie, Pierre Rabu, Marc Drillon

A noncritical scaling model is reported to describe the behavior of low-dimensional magnetic systems. Unlike the classical phase transition approach, nonsingular solutions are deduced that are worthwhile when correlations exist, but which are not large enough to trigger a long range order at $T\ensuremath{\ne}0$. The notion of ``universality class'' is extended to these systems that stand at or below a lower critical dimensionality, and illustrated for the quantum Heisenberg ferromagnetic chain. In this system, the $T$ dependence of $\ensuremath{\chi}T$ exhibits a power law with a negative critical exponent, $\ensuremath{-}1.25S$, and a negative critical temperature.

Physical Review B•Dec 15, 2005•26 citations
The problem of the critical behavior of the spin-1/2 Heisenberg ferromagnetic chain has challenged many research workers since 1964. We discuss the critical exponents α and γ recently determined by Schlottmann and by Takahashi and Yamada in the context of previously known Tc=0 critical properties of the 1D s=1/2 XXZ model and the classical Heisenberg model, and we comment on the manifest breakdown of scaling in the s=1/2 Heisenberg ferromagnet.
The thermodynamic Bethe-Ansatz equations for the isotropic ferromagnetic S=12S=\frac{1}{2}S=21​ Heisenberg chain have been solved numerically. At low temperatures, I find a power-law dependence on TTT for the specific heat and the susceptibility with critical exponents $\ensuremath{\alpha}=\ensuremath{-}0.49\ifmmode\pm\else\textpm\fi{}0.02$, $\ensuremath{\gamma}=2.00\ifmmode\pm\else\textpm\fi{}0.02$, and $\ensuremath{\Delta}\ensuremath{\simeq}\ensuremath{\gamma}$. The exponent $\ensuremath{\alpha}$ is compatible with effective spin waves and $\ensuremath{\gamma}$ and $\ensuremath{\Delta}$ are the exponents of the classical chain. Amplitudes and corrections to scaling are obtained and differences with previous results are discussed.
Using a Lanczos method, we study the spectra of Heisenberg chains with spin S=1, 3/2, 2, 5/2, and 3, as a function of the anisotropy parameter \ensuremath{\lambda} and the number of sites of the chain. We use periodic, twisted, and open boundary conditions. We found that for all values of the spin when -1\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}0, these models are massless with critical behavior described by a conformal theory with central charge c=1. We discuss the whole operator content. In particular, we found that the critical exponent ${\mathrm{\ensuremath{\eta}}}_{\mathit{x}}$ follows the behavior ${\mathrm{\ensuremath{\eta}}}_{\mathit{x}}$=(\ensuremath{\pi}-\ensuremath{\gamma})/2\ensuremath{\pi}S and ${\mathrm{\ensuremath{\eta}}}_{\mathit{z}}$=2 where \ensuremath{\gamma}=${\mathrm{cos}}^{\mathrm{\ensuremath{-}}1}$(\ensuremath{\lambda}). For \ensuremath{\lambda}>0, our results indicate that the S=1 model is massive, while for S>1, the massless behavior characterized by the critical exponents described above continues up to \ensuremath{\lambda}=${\ensuremath{\lambda}}^{\mathrm{*}}$(S). For ${\ensuremath{\lambda}}^{\mathrm{*}}$\ensuremath{\le}1, our results are consistent with a massive phase for the integer-spin models and a massless phase with a rapid variation of the critical exponents for the half-integer-spin models.
For the isotropic Heisenberg model with an arbitrary spin 5, the correlation exponent l/ defined by ~ ~ T-v near the critical point Tc=O is evaluated as l/=1 with the use of the real space renormalization group method.This yields y=2 for the susceptibility exponent y defined by Xo~ T-r , using the scaling relation y=l + (2-1]) l/= 1 + l/.Our general result y=2 for all values of spin 5 includes Fisher's result in the classical limit 5-->00.For the anisotropic Heisenberg model (XXZ model) with a spin 5=1/2, an anisotropy scaling law is derived.
The finite-size scaling function of the magnetization of the ferromagnetic Heisenberg chain is argued to be universal with respect to the magnitude of the spin. The finite-size scaling function is given explicitly by an analytical calculation in the classical limit S=∞.S=\infty.S=∞. The universality is checked for S=1/2S=1/2S=1/2 and 111 by means of numerical calculations. Critical exponents are obtained as well. It is concluded that this universal scaling function originates in the universal behavior of the correlation function.
Finite-size scaling (phenomenological renormalization) techniques are trusted and widely applied in low-dimensional magnetism and, particularly, in lattice gauge field theory. Recently, investigations have begun which subject the theoretical basis to systematic and intensive scrutiny to determine the validity of finite-size scaling in a variety of situations. The 2D ANNNI model is an example of a situation where finite-size scaling methods encounter difficulty, related to the occurrence of a disorder line (one-dimensional line). A second example concerns the behavior of the spin-1/2 antiferromagnetic XXZ model where the T=0 critical behavior is exactly known and features an essential singularity at the isotropic Heisenberg point. Standard finite-size scaling techniques do not convincingly reproduce the exact phase behavior and this is attributable to the essential singularity. The point is relevant in connection with a finite-size scaling analysis of a spin-one antiferromagnetic XXZ model, which claims to support a conjecture by Haldane that the T=0 phase behavior of integer-spin Heisenberg chains is significantly different from that of half-integer-spin Heisenberg chains.
A noncritical scaling model is reported to describe the behavior of low-dimensional magnetic systems. Unlike the classical phase transition approach, nonsingular solutions are deduced that are worthwhile when correlations exist, but which are not large enough to trigger a long range order at $T\ensuremath{\ne}0$. The notion of ``universality class'' is extended to these systems that stand at or below a lower critical dimensionality, and illustrated for the quantum Heisenberg ferromagnetic chain. In this system, the TTT dependence of $\ensuremath{\chi}T$ exhibits a power law with a negative critical exponent, $\ensuremath{-}1.25S$, and a negative critical temperature.