Physical review. B, Solid state

Random-bond Ising chain in a magnetic field

Julio F. Fernández

Published Dec 1, 1977• 17 citations

Abstract

The magnetization $m$ as a function of magnetic field $h$ for various temperatures is numerically obtained for the one-dimensional random-bond Ising model with each bond energy either $+J$ or $\ensuremath{-}J$ with equal probability. For high temperatures ($\frac{\mathrm{kT}}{J}>~0.6$) the results obtained agree with those of Landau and Blume. We have obtained $m$ vs $h$ for low temperatures as a numerical approach to the zero-temperature limit.

DOI: 10.1103/physrevb.16.5125

Related papers

Random-bond Ising chain in a magnetic field at low temperatures

Alexander Vilenkin

Physical review. B, Condensed matterAug 1, 197826 citations

Zero temperature magnetization of the random axis chain

Bernard Derrida, Annette Zippelius

We have calculated the zero temperature magnetization of the random axis chain in the limit of strong anisotropy and small magnetic field h. For the uniform distribution of random axes we obtain m = Ch1/ δ with the same critical exponent δ = 3 as in the one dimensional Ising spinglass with a Gaussian distribution of bonds.

Journal de physiqueJan 1, 19862 citations

Exact solution of a one-dimensional Ising model in a random magnetic field

G. Grinstein, David Mukamel

Physical review. B, Condensed matterApr 1, 198369 citations

Exact ground states of one-dimensional long-range random-field Ising magnets

Timo Dewenter, Alexander K. Hartmann

arXiv:1307.3987Physical Review BJul 31, 201410 citations

Ground-state numerical study of the three-dimensional random-field Ising model

Ilija Dukovski, Jonathan L. Machta

arXiv:cond-mat/0207438Physical review. B, Condensed matterJan 21, 200340 citations

Random Ising models: Convergence of successive approximations towards exact results

A. Benyoussef, N. Boccara

Journal of Applied PhysicsMar 15, 19840 citations

Physical review. B, Solid state

Random-bond Ising chain in a magnetic field

Julio F. Fernández

Published Dec 1, 1977• 17 citations

Publisher

Abstract

The magnetization

m

as a function of magnetic field

h

for various temperatures is numerically obtained for the one-dimensional random-bond Ising model with each bond energy either

+J

\ensuremath{-}J

with equal probability. For high temperatures (

\frac{\mathrm{kT}}{J}>~0.6

) the results obtained agree with those of Landau and Blume. We have obtained

m

h

for low temperatures as a numerical approach to the zero-temperature limit.

DOI: 10.1103/physrevb.16.5125

Related papers

Random-bond Ising chain in a magnetic field at low temperatures

Alexander Vilenkin

The low-temperature behavior ($T\ensuremath{\ll}h, J$) of a random-bond Ising chain in a magnetic field is considered [$H=\ensuremath{-}(\frac{1}{2})J\ensuremath{\Sigma}{T}_{i}{\ensuremath{\sigma}}_{i}{\ensuremath{\sigma}}_{i+1}\ensuremath{-}h\ensuremath{\Sigma}{\ensuremath{\sigma}}_{i}, {\ensuremath{\sigma}}_{i}=\ifmmode\pm\else\textpm\fi{}1$, ${{T}_{i}}$ is a fixed random sequence of numbers +1 and -1 with concentrations ${c}_{1}$ and ${c}_{2}=1\ensuremath{-}{c}_{1}$, respectively]. The ground-state energy ${E}_{0}$ , magnetization ${\ensuremath{\mu}}_{0}$ and zero-point entropy ${S}_{0}$ are calculated exactly. It is shown that ${\ensuremath{\mu}}_{0}$ and ${S}_{0}$ are discontinuous functions of magnetic field having jumps at $h=\frac{J}{n}$ , $n=1, 2, \dots{}$ .The low-temperature behavior ($T\ensuremath{\ll}h, J$) of a random-bond Ising chain in a magnetic field is considered [$H=\ensuremath{-}(\frac{1}{2})J\ensuremath{\Sigma}{T}_{i}{\ensuremath{\sigma}}_{i}{\ensuremath{\sigma}}_{i+1}\ensuremath{-}h\ensuremath{\Sigma}{\ensuremath{\sigma}}_{i}, {\ensuremath{\sigma}}_{i}=\ifmmode\pm\else\textpm\fi{}1$, ${{T}_{i}}$ is a fixed random sequence of numbers +1 and -1 with concentrations ${c}_{1}$ and ${c}_{2}=1\ensuremath{-}{c}_{1}$, respectively]. The ground-state energy ${E}_{0}$, magnetization ${\ensuremath{\mu}}_{0}$ and zero-point entropy ${S}_{0}$ are calculated exactly. It is shown that ${\ensuremath{\mu}}_{0}$ and ${S}_{0}$ are discontinuous functions of magnetic field having jumps at $h=\frac{J}{n}$, $n=1, 2, \dots{}$.

Physical review. B, Condensed matterAug 1, 197826 citations

Zero temperature magnetization of the random axis chain

Bernard Derrida, Annette Zippelius

Journal de physiqueJan 1, 19862 citations

Exact solution of a one-dimensional Ising model in a random magnetic field

G. Grinstein, David Mukamel

Analytic results for the free energy, magnetic structure factor, and Edwards-Anderson order parameter of a one-dimensional ferromagnetic Ising model in a random magnetic field are obtained. The structure factor consists of both Lorentzian and Lorentzian-squared terms at all temperatures ( $T$ ) greater than zero; the Lorentzian-squared terms vanish at $T=0$ . The calculated correlation length agrees with predictions of the Imry-Ma domain argument.Analytic results for the free energy, magnetic structure factor, and Edwards-Anderson order parameter of a one-dimensional ferromagnetic Ising model in a random magnetic field are obtained. The structure factor consists of both Lorentzian and Lorentzian-squared terms at all temperatures ($T$) greater than zero; the Lorentzian-squared terms vanish at $T=0$. The calculated correlation length agrees with predictions of the Imry-Ma domain argument.

Physical review. B, Condensed matterApr 1, 198369 citations

Exact ground states of one-dimensional long-range random-field Ising magnets

Timo Dewenter, Alexander K. Hartmann

We investigate the one-dimensional long-range random-field Ising magnet with Gaussian distribution of the random fields. In this model, a ferromagnetic bond between two spins is placed with a probability $p\ensuremath{\sim}{r}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\sigma}}$, where $r$ is the distance between these spins and $\ensuremath{\sigma}$ is a parameter to control the effective dimension of the model. Exact ground states at zero temperature are calculated for system sizes up to $L={2}^{19}$ via graph theoretical algorithms for four different values of $\ensuremath{\sigma}\ensuremath{\in}{0.25,0.4,0.5,1.0}$ while varying the strength $h$ of the random fields. For each of these values several independent physical observables are calculated, i.e., magnetization, Binder parameter, susceptibility, and a specific-heat-like quantity. The ferromagnet-paramagnet transitions at critical values ${h}_{c}(\ensuremath{\sigma})$ as well as the corresponding critical exponents are obtained. The results agree well with theory, and interestingly we find for $\ensuremath{\sigma}=1/2$ the data is compatible with a critical random-field strength ${h}_{c}>0$ .We investigate the one-dimensional long-range random-field Ising magnet with Gaussian distribution of the random fields. In this model, a ferromagnetic bond between two spins is placed with a probability $p\ensuremath{\sim}{r}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\sigma}}$, where $r$ is the distance between these spins and $\ensuremath{\sigma}$ is a parameter to control the effective dimension of the model. Exact ground states at zero temperature are calculated for system sizes up to $L={2}^{19}$ via graph theoretical algorithms for four different values of $\ensuremath{\sigma}\ensuremath{\in}{0.25,0.4,0.5,1.0}$ while varying the strength $h$ of the random fields. For each of these values several independent physical observables are calculated, i.e., magnetization, Binder parameter, susceptibility, and a specific-heat-like quantity. The ferromagnet-paramagnet transitions at critical values ${h}_{c}(\ensuremath{\sigma})$ as well as the corresponding critical exponents are obtained. The results agree well with theory, and interestingly we find for $\ensuremath{\sigma}=1/2$ the data is compatible with a critical random-field strength ${h}_{c}>0$.

arXiv:1307.3987Physical Review BJul 31, 201410 citations

Ground-state numerical study of the three-dimensional random-field Ising model

Ilija Dukovski, Jonathan L. Machta

The random field Ising model in three dimensions with Gaussian random fields is studied at zero temperature for system sizes up to ${60}^{3}.$ For each realization of the normalized random fields, the strength of the random field, $\ensuremath{\Delta}$ and a uniform external, H is adjusted to find the finite-size critical point. The finite-size critical point is identified as the point in the $H\ensuremath{-}\ensuremath{\Delta}$ plane where three degenerate ground states have the largest discontinuities in the magnetization. The discontinuities in the magnetization and bond energy between these ground states are used to calculate the magnetization and specific heat critical exponents and both exponents are found to be near zero.The random field Ising model in three dimensions with Gaussian random fields is studied at zero temperature for system sizes up to ${60}^{3}.$ For each realization of the normalized random fields, the strength of the random field, $\ensuremath{\Delta}$ and a uniform external, H is adjusted to find the finite-size critical point. The finite-size critical point is identified as the point in the $H\ensuremath{-}\ensuremath{\Delta}$ plane where three degenerate ground states have the largest discontinuities in the magnetization. The discontinuities in the magnetization and bond energy between these ground states are used to calculate the magnetization and specific heat critical exponents and both exponents are found to be near zero.

arXiv:cond-mat/0207438Physical review. B, Condensed matterJan 21, 200340 citations

Random Ising models: Convergence of successive approximations towards exact results

A. Benyoussef, N. Boccara

We studied a new and simple method which gives successive approximate expressions for the magnetization (or any order parameter) of a random Ising model as a function of temperature, site, or bond dilution, frustration, magnetic field,... To obtain the various approximations a trace is performed on finite clusters increasing in size. In particular, we investigated a honeycomb Ising model with random nearest-neighbor interactions and calculated sequences of critical temperatures, percolation thresholds,... which converge towards exact known results.We studied a new and simple method which gives successive approximate expressions for the magnetization (or any order parameter) of a random Ising model as a function of temperature, site, or bond dilution, frustration, magnetic field,... To obtain the various approximations a trace is performed on finite clusters increasing in size. In particular, we investigated a honeycomb Ising model with random nearest-neighbor interactions and calculated sequences of critical temperatures, percolation thresholds,... which converge towards exact known results.

Journal of Applied PhysicsMar 15, 19840 citations

The magnetization

m

as a function of magnetic field

h

for various temperatures is numerically obtained for the one-dimensional random-bond Ising model with each bond energy either

+J

\ensuremath{-}J

with equal probability. For high temperatures (

\frac{\mathrm{kT}}{J}>~0.6

) the results obtained agree with those of Landau and Blume. We have obtained

m

h

for low temperatures as a numerical approach to the zero-temperature limit.

{{T}_{i}}

is a fixed random sequence of numbers +1 and -1 with concentrations

{c}_{1}

and ${c}_{2}=1\ensuremath{-}{c}_{1}$, respectively]. The ground-state energy

{E}_{0}

, magnetization ${\ensuremath{\mu}}_{0}$ and zero-point entropy

{S}_{0}

are calculated exactly. It is shown that ${\ensuremath{\mu}}_{0}$ and

{S}_{0}

are discontinuous functions of magnetic field having jumps at

h=\frac{J}{n}

n=1, 2, \dots{}

T

) greater than zero; the Lorentzian-squared terms vanish at

T=0

. The calculated correlation length agrees with predictions of the Imry-Ma domain argument.

r

is the distance between these spins and $\ensuremath{\sigma}$ is a parameter to control the effective dimension of the model. Exact ground states at zero temperature are calculated for system sizes up to

L={2}^{19}

via graph theoretical algorithms for four different values of $\ensuremath{\sigma}\ensuremath{\in}{0.25,0.4,0.5,1.0}$ while varying the strength

h

of the random fields. For each of these values several independent physical observables are calculated, i.e., magnetization, Binder parameter, susceptibility, and a specific-heat-like quantity. The ferromagnet-paramagnet transitions at critical values ${h}_{c}(\ensuremath{\sigma})$ as well as the corresponding critical exponents are obtained. The results agree well with theory, and interestingly we find for $\ensuremath{\sigma}=1/2$ the data is compatible with a critical random-field strength

{h}_{c}>0

The random field Ising model in three dimensions with Gaussian random fields is studied at zero temperature for system sizes up to

{60}^{3}.

For each realization of the normalized random fields, the strength of the random field, $\ensuremath{\Delta}$ and a uniform external, H is adjusted to find the finite-size critical point. The finite-size critical point is identified as the point in the $H\ensuremath{-}\ensuremath{\Delta}$ plane where three degenerate ground states have the largest discontinuities in the magnetization. The discontinuities in the magnetization and bond energy between these ground states are used to calculate the magnetization and specific heat critical exponents and both exponents are found to be near zero.