Physical Review Letters

Phase Transitions in Long-Range Ferromagnetic Chains

J. Michael Kosterlitz

Published Dec 6, 1976• 244 citations

Abstract

The critical behavior of one-dimensional ferromagnetic chains with isotropic long-range interactions between $n$-component spins is investigated near the critical range of ${r}^{\ensuremath{-}2}$. Different formulations of the renormalization group are used for $n>1$ and $n=1$. It is shown that the behavior is quite different in the two cases, with very close analogies to the short-range problem near two dimensions. Critical exponents are calculated to first order.

DOI: 10.1103/physrevlett.37.1577

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Physical Review Letters

Phase Transitions in Long-Range Ferromagnetic Chains

J. Michael Kosterlitz

Published Dec 6, 1976• 244 citations

Publisher

Abstract

The critical behavior of one-dimensional ferromagnetic chains with isotropic long-range interactions between

n

-component spins is investigated near the critical range of

{r}^{\ensuremath{-}2}

. Different formulations of the renormalization group are used for

n>1

and

n=1

. It is shown that the behavior is quite different in the two cases, with very close analogies to the short-range problem near two dimensions. Critical exponents are calculated to first order.

DOI: 10.1103/physrevlett.37.1577

Related papers

Critical Behavior of Isotropic Systems with Long-Range Interactions

Y. Yamazaki, Masuo Suzuki

Effects of the potential range on critical behavior in the second order phase transition are studied for isotropic N-component spin systems with long-range interactions by means of the Callan-Symanzik or renormalization-group equations. Critical exponents η, γ, φ, ν, α, δ and β and the scaling laws are derived. The equation of state, the correctional index ω and the logarithmic corrections in 2σ dimensions are also derived. The instability of the fixed point is discussed. The rigorous results for η and δ with an arbitrary N and for γ and φ in the spherical case N=∞ are obtained.Effects of the potential range on critical behavior in the second order phase transition are studied for isotropic N-component spin systems with long-range interactions by means of the Callan-Symanzik or renormalization-group equations. Critical exponents η, γ, φ, ν, α, δ and β and the scaling laws are derived. The equation of state, the correctional index ω and the logarithmic corrections in 2σ dimensions are also derived. The instability of the fixed point is discussed. The rigorous results for η and δ with an arbitrary N and for γ and φ in the spherical case N=∞ are obtained.

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Low-temperature renormalization group for ferromagnets with long-range interactions

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Ferromagnets with the number of spin components $n>2$ and with power-law interactions ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}$ are studied in dimensions $d=2+\ensuremath{\epsilon}$ for small $\ensuremath{\epsilon}$. The following theorem is proved: Let ${\ensuremath{\eta}}_{\mathrm{SR}}$ be the value of the critical exponent $\ensuremath{\eta}$ for short-range force (e.g., nearest-neighbor exchange). Then in the presence of the long-range power-law interaction the critical behavior is the same as for short-range interactions provided that the inequality $2\ensuremath{-}\ensuremath{\sigma}<{\ensuremath{\eta}}_{\mathrm{SR}}$ is satisfied. In this case the critical exponents depend on $d$ and $n$ but not on $\ensuremath{\sigma}$. In the opposite case, the value of $\ensuremath{\eta}$ is ${\ensuremath{\eta}}_{\mathrm{LR}}=2\ensuremath{-}\ensuremath{\sigma}$ and the critical exponent $\ensuremath{\nu}$ depends on $\ensuremath{\sigma}$ in addition to $d$ and $n$ .Ferromagnets with the number of spin components $n>2$ and with power-law interactions ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}$ are studied in dimensions $d=2+\ensuremath{\epsilon}$ for small $\ensuremath{\epsilon}$. The following theorem is proved: Let ${\ensuremath{\eta}}_{\mathrm{SR}}$ be the value of the critical exponent $\ensuremath{\eta}$ for short-range force (e.g., nearest-neighbor exchange). Then in the presence of the long-range power-law interaction the critical behavior is the same as for short-range interactions provided that the inequality $2\ensuremath{-}\ensuremath{\sigma}<{\ensuremath{\eta}}_{\mathrm{SR}}$ is satisfied. In this case the critical exponents depend on $d$ and $n$ but not on $\ensuremath{\sigma}$. In the opposite case, the value of $\ensuremath{\eta}$ is ${\ensuremath{\eta}}_{\mathrm{LR}}=2\ensuremath{-}\ensuremath{\sigma}$ and the critical exponent $\ensuremath{\nu}$ depends on $\ensuremath{\sigma}$ in addition to $d$ and $n$.

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Physical review. B, Solid stateMar 1, 197417 citations

Effect of long-range interactions on the critical behavior of the continuous Ising model

E. Bayong, H. T. Diep

Critical behavior of the one- and two-dimensional general continuous Ising models with long-range ferromagnetic interactions decaying as ${1/r}^{d+\ensuremath{\sigma}}$ is studied using a histogram Monte Carlo technique. A continuous Ising model means that a spin can take any value between $\ensuremath{-}1$ and 1. It is found that the system exhibits a second-order phase transition with nonstandard critical exponents which depend on $\ensuremath{\sigma}.$ Results for various values of $\ensuremath{\sigma}$ will be shown and compared to predictions from renormalization-group theory. Though there is an agreement with the overall tendency predicted, there are several fundamental differences. Discussion is given.Critical behavior of the one- and two-dimensional general continuous Ising models with long-range ferromagnetic interactions decaying as ${1/r}^{d+\ensuremath{\sigma}}$ is studied using a histogram Monte Carlo technique. A continuous Ising model means that a spin can take any value between $\ensuremath{-}1$ and 1. It is found that the system exhibits a second-order phase transition with nonstandard critical exponents which depend on $\ensuremath{\sigma}.$ Results for various values of $\ensuremath{\sigma}$ will be shown and compared to predictions from renormalization-group theory. Though there is an agreement with the overall tendency predicted, there are several fundamental differences. Discussion is given.

Physical review. B, Condensed matterMay 1, 199928 citations

Relations between short-range and long-range Ising models

Maria Chiara Angelini, Giorgio Parisi A, Federico Ricci‐Tersenghi

We perform a numerical study of the long-range (LR) ferromagnetic Ising model with power law decaying interactions (J∝r{-d-σ}) on both a one-dimensional chain (d=1) and a square lattice (d=2). We use advanced cluster algorithms to avoid the critical slowing down. We first check the validity of the relation connecting the critical behavior of the LR model with parameters (d,σ) to that of a short-range (SR) model in an equivalent dimension D. We then study the critical behavior of the d=2 LR model close to the lower critical σ, uncovering that the spatial correlation function decays with two different power laws: The effect of the subdominant power law is much stronger than finite-size effects and actually makes the estimate of critical exponents very subtle. By including this subdominant power law, the numerical data are consistent with the standard renormalization group (RG) prediction by Sak [Phys. Rev. B 8, 281 (1973)], thus making not necessary (and unlikely, according to Occam's razor) the recent proposal by Picco [arXiv:1207.1018] of having a new set of RG fixed points in addition to the mean-field one and the SR one.We perform a numerical study of the long-range (LR) ferromagnetic Ising model with power law decaying interactions (J∝r{-d-σ}) on both a one-dimensional chain (d=1) and a square lattice (d=2). We use advanced cluster algorithms to avoid the critical slowing down. We first check the validity of the relation connecting the critical behavior of the LR model with parameters (d,σ) to that of a short-range (SR) model in an equivalent dimension D. We then study the critical behavior of the d=2 LR model close to the lower critical σ, uncovering that the spatial correlation function decays with two different power laws: The effect of the subdominant power law is much stronger than finite-size effects and actually makes the estimate of critical exponents very subtle. By including this subdominant power law, the numerical data are consistent with the standard renormalization group (RG) prediction by Sak [Phys. Rev. B 8, 281 (1973)], thus making not necessary (and unlikely, according to Occam's razor) the recent proposal by Picco [arXiv:1207.1018] of having a new set of RG fixed points in addition to the mean-field one and the SR one.

arXiv:1401.6805Physical Review EJun 12, 2014109 citations

Influence of long-range interactions on the critical behavior of the Ising model

Thibault Blanchard, Marco Picco, M. A. Rajabpour

arXiv:1211.6758Europhysics Letters (EPL)Mar 1, 201365 citations

The critical behavior of one-dimensional ferromagnetic chains with isotropic long-range interactions between

n

-component spins is investigated near the critical range of

{r}^{\ensuremath{-}2}

. Different formulations of the renormalization group are used for

n>1

and

n=1

. It is shown that the behavior is quite different in the two cases, with very close analogies to the short-range problem near two dimensions. Critical exponents are calculated to first order.

Ferromagnets with the number of spin components

n>2

and with power-law interactions ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}$ are studied in dimensions $d=2+\ensuremath{\epsilon}$ for small $\ensuremath{\epsilon}$. The following theorem is proved: Let ${\ensuremath{\eta}}_{\mathrm{SR}}$ be the value of the critical exponent $\ensuremath{\eta}$ for short-range force (e.g., nearest-neighbor exchange). Then in the presence of the long-range power-law interaction the critical behavior is the same as for short-range interactions provided that the inequality $2\ensuremath{-}\ensuremath{\sigma}<{\ensuremath{\eta}}_{\mathrm{SR}}$ is satisfied. In this case the critical exponents depend on

d

and

n

but not on $\ensuremath{\sigma}$. In the opposite case, the value of $\ensuremath{\eta}$ is ${\ensuremath{\eta}}_{\mathrm{LR}}=2\ensuremath{-}\ensuremath{\sigma}$ and the critical exponent $\ensuremath{\nu}$ depends on $\ensuremath{\sigma}$ in addition to

d

and

n

The critical behavior of

d

-dimensional ferromagnets with quadrupole-quadrupole short- and long-range interactions is discussed on the basis of renormalization-group calculations to orber ${\ensuremath{\epsilon}}^{2}$, with $\ensuremath{\epsilon}=4\ensuremath{-}d$. Such interactions may lead to a first-order transition, to characteristic "cubic" critical behavior, or to the usual isotropic critical behavior.