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

Cut-off scalings for one-dimensional order

Aleksander L. Owczarek, Philippe M. Binder, D. Y. K. Ko

Published Jan 7, 1992• 1 citation
Publisher

Abstract

The authors consider one-dimensional Ising systems of length L with an interaction cut-off R(L) in the limit R to infinity . They present free energy arguments for order at low temperatures provided R(L))(A In L) 1 kappa /, where kappa is a new exponent; this opens the possibility of one-dimensional pseudo-order in macroscopic systems.
DOI
10.1088/0305-4470/25/1/004

Related papers

One-dimensional Ising model with long-range interactions: A renormalization-group treatment

Sergio Alejandro Cannas

The one-dimensional Ising model with ferromagnetic interactions which decay as 1/${\mathit{r}}^{\mathrm{\ensuremath{\alpha}}}$ is considered. Using a real-space renormalization group scheme (RG) we calculate the critical temperature and the correlation-length critical exponent as a function of \ensuremath{\alpha}. General asymptotic properties are obtained for arbitrary values of the rescaling length b of the RG transformation. Several rigorous results are recovered exactly in the limit b\ensuremath{\rightarrow}\ensuremath{\infty}. We obtain a b=\ensuremath{\infty} extrapolation of the critical temperature for arbitrary values of \ensuremath{\alpha}>1, which we conjecture aproximates with high precision the exact one. In particular, we obtain the value ${\mathit{T}}_{\mathit{c}}$/J=${\mathrm{\ensuremath{\pi}}}^{2}$/12 for the 1/${\mathit{r}}^{2}$ model.

Physical review. B, Condensed matter•Aug 1, 1995•75 citations

Critical exponents of small one-dimensional ising magnetic

D.V. Spirin, V. N. Udodov

Within the framework of a generalized Ising model, a one-dimensional magnetic of a finite length with free ends is considered. The correlation length exponent, dynamic critical exponent z of the magnet is calculated taking into account the next nearest neighbor interactions and the external field.

arXiv:cond-mat/0703639•Mar 24, 2007•0 citations

Universality Class for Spin Model with Generalized Ising Spin on One-Dimensional Lattice

T. Horiguchi

We generalize the Ising spin σi to a function σ(xi), which is a mapping from [-a, a] into [-1, 1]. We obtain low-temperature expansions of the free energy, the correlation length and so on for a system of σ(xi) with nearest-neighbor interaction on a one-dimensional lattice. The critical behavior of the system is determined by the nature of σ(xi) around its maximum absolute value.

Progress of Theoretical Physics•Sep 1, 1991•4 citations

One-dimensional models

K. H. Fischer, John Hertz

As in many problems, one-dimensional models can give some insight into the physics of spin glasses because they are easier to solve than those in higher dimensionality, but this advantage can be limited because the physics of the one-dimensional problem may be essentially different. Typically, for example, the higher-dimensional problem we really want to study has a broken symmetry phase at low temperature, while one-dimensional systems (with short-range interactions) do not exhibit broken symmetry at finite temperature. Nevertheless, short-range one-dimensional systems can have broken symmetry at zero temperature, so one can study this relatively simple kind of order and the approach to it as the temperature is lowered to zero. This zero-temperature phase transition can be compared with finite-temperature transitions in higher dimensionality. This is the motivation for Section 12.1, where we will see that the critical point at T = h = 0 in a one-dimensional Ising spin glass with nearest-neighbour interactions has interesting features in common with finite-temperature spin glass transitions, and that they can be calculated analytically, sometimes rather simply. Further insight can be gained by studying one-dimensional systems with long-range (typically power law) interactions. Then if the exponent describing the falloff of the interaction strength with distance is small enough, finite-temperature phase transitions are possible. Though such problems are harder to solve than nearest-neighbour ones, they can still be easier than the full higher-dimensional ones.

Cambridge University Press eBooks•May 2, 1991•1 citation

Long-Range Order in One-Dimensional Ising Systems

D. J. Thouless

The argument of Landau and Lifshitz for the absence of long-range order in one-dimensional systems is used to show that order is absent if the interaction energy falls off faster than ${n}^{\ensuremath{-}2}$. When the interaction falls off as ${n}^{\ensuremath{-}2}$, the order cannot go continuously to zero.

Physical Review•Nov 10, 1969•285 citations

Aperiodic extended surface perturbations in the Ising model

Loïc Turban

We study the influence of an aperiodic extended surface perturbation on the surface critical behaviour of the two-dimensional Ising model in the extreme anisotropic limit. The perturbation decays as a power κ of the distance l from the free surface with an oscillating amplitude $A_l = ( - 1)^{f_l } A$ where f l = 0, 1 follows some aperiodic sequence with an asymptotic density equal to 1/2 so that the mean amplitude vanishes. The relevance of the perturbation is discussed by combining scaling arguments of Cordery and Burkhardt for the Hilhorst-van Leeuwen model and Luck for aperiodic perturbations. The relevance-irrelevance criterion involves the decay exponent κ, the wandering exponent ω which governs the fluctuation of the sequence and the bulk correlation length exponent ν. Analytical results are obtained for the surface magnetization which displays a rich variety of critical behaviours in the (κ, ω)-plane. The results are checked through a numerical finite-size-scaling study. They show that second-order effects must be taken into account in the discussion of the relevance-irrelevance criterion. The scaling behaviours of the first gap and the surface energy are also discussed.

arXiv:cond-mat/9811386•The European Physical Journal B•Jan 1, 1999•2 citations
The one-dimensional Ising model with ferromagnetic interactions which decay as 1/${\mathit{r}}^{\mathrm{\ensuremath{\alpha}}}$ is considered. Using a real-space renormalization group scheme (RG) we calculate the critical temperature and the correlation-length critical exponent as a function of \ensuremath{\alpha}. General asymptotic properties are obtained for arbitrary values of the rescaling length b of the RG transformation. Several rigorous results are recovered exactly in the limit b\ensuremath{\rightarrow}\ensuremath{\infty}. We obtain a b=\ensuremath{\infty} extrapolation of the critical temperature for arbitrary values of \ensuremath{\alpha}>1, which we conjecture aproximates with high precision the exact one. In particular, we obtain the value Tc{\mathit{T}}_{\mathit{c}}Tc​/J=${\mathrm{\ensuremath{\pi}}}^{2}$/12 for the 1/r2{\mathit{r}}^{2}r2 model.
As in many problems, one-dimensional models can give some insight into the physics of spin glasses because they are easier to solve than those in higher dimensionality, but this advantage can be limited because the physics of the one-dimensional problem may be essentially different. Typically, for example, the higher-dimensional problem we really want to study has a broken symmetry phase at low temperature, while one-dimensional systems (with short-range interactions) do not exhibit broken symmetry at finite temperature. Nevertheless, short-range one-dimensional systems can have broken symmetry at zero temperature, so one can study this relatively simple kind of order and the approach to it as the temperature is lowered to zero. This zero-temperature phase transition can be compared with finite-temperature transitions in higher dimensionality. This is the motivation for Section 12.1, where we will see that the critical point at T = h = 0 in a one-dimensional Ising spin glass with nearest-neighbour interactions has interesting features in common with finite-temperature spin glass transitions, and that they can be calculated analytically, sometimes rather simply. Further insight can be gained by studying one-dimensional systems with long-range (typically power law) interactions. Then if the exponent describing the falloff of the interaction strength with distance is small enough, finite-temperature phase transitions are possible. Though such problems are harder to solve than nearest-neighbour ones, they can still be easier than the full higher-dimensional ones.
We study the influence of an aperiodic extended surface perturbation on the surface critical behaviour of the two-dimensional Ising model in the extreme anisotropic limit. The perturbation decays as a power κ of the distance l from the free surface with an oscillating amplitude Al=(−1)flAA_l = ( - 1)^{f_l } AAl​=(−1)fl​A where f l = 0, 1 follows some aperiodic sequence with an asymptotic density equal to 1/2 so that the mean amplitude vanishes. The relevance of the perturbation is discussed by combining scaling arguments of Cordery and Burkhardt for the Hilhorst-van Leeuwen model and Luck for aperiodic perturbations. The relevance-irrelevance criterion involves the decay exponent κ, the wandering exponent ω which governs the fluctuation of the sequence and the bulk correlation length exponent ν. Analytical results are obtained for the surface magnetization which displays a rich variety of critical behaviours in the (κ, ω)-plane. The results are checked through a numerical finite-size-scaling study. They show that second-order effects must be taken into account in the discussion of the relevance-irrelevance criterion. The scaling behaviours of the first gap and the surface energy are also discussed.