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

Physical review. B, Condensed matter

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

G. Grinstein, David Mukamel

Published Apr 1, 1983• 69 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.27.4503

Related papers

Exact solution of a one-dimensional XY model in a random field

Robert A. Pelcovits, David Mukamel

We present an exact solution of a one-dimensional $\mathrm{XY}$ model in a random magnetic field in the limit of strong-field pinning. The structure factor exhibits Lorentzian-squared behavior at nonzero temperatures. The scaling behavior of the correlation length as a function of randomness is different from that obtained in the weak-field pinning regime. We also study the dynamical behavior of the system using a simple relaxational model. The magnetization decays anomalously with time as ${t}^{\frac{7}{6}}{e}^{{\ensuremath{-}ct}^{\frac{1}{3}}}$.

Physical review. B, Condensed matter•Nov 1, 1983•10 citations

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

Gabor Forgács, W. F. Wolff, A Süt

The Ising model in a random magnetic field h is studied on a square lattice. h has the same value along a row, but varies randomly from row to row. h=\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty} with probability p/2 and is zero with probability 1-p. The quenched free energy F and specific heat C are calculated exactly. We find C\ensuremath{\sim}lnp for small p at the Onsager temperature ${T}_{c}$. It is shown that for p\ensuremath{\ne}0, F(p,T) is not analytic in T at T=${T}_{c}$, i.e., it has a Griffiths-like singularity.

Physical review. B, Condensed matter•May 1, 1985•4 citations

Correlation function of random-field Ising chains: is it Lorentzian or not?

J. M. Luck, Theo M. Nieuwenhuizen

The two-point connected correlation function, or wavevector-dependent susceptibility, of the ferromagnetic Ising chain in a random field is calculated exactly at any temperature, for a two-parameter family of diluted symmetric exponential distributions of the magnetic fields. Thermodynamic properties of this model have been derived in a previous work by the authors. Besides the correlation function itself, the solution provides exact results for the (usual) susceptibility, the correlation length and the Edwards-Anderson parameter. The low-temperature regime is examined in full detail: the authors obtain in closed form the limit value at zero temperature of various quantities, and the first correction to this limit, which behaves linearly with temperature. The correlation length is discontinuous at zero temperature. They also derive the scaling form of the correlation function for small p, where the dilution p is the probability for a spin to be subjected to a non-zero random field. Even in this limit, the correlation function is more complicated than the simple Lorentzian predicted, e.g., by mean-field theory.

Journal of Physics A Mathematical and General•Jun 21, 1989•18 citations

Two-dimensional Ising model in random magnetic fields

I. Morgenstern, Kurt Binder, Richard M. Hornreich

The statistical mechanics of finite $L\ifmmode\times\else\texttimes\fi{}L$ Ising square lattices in a field $\ifmmode\pm\else\textpm\fi{}h$ of random sign is investigated numerically by a modified recursive transfer matrix method for $6<~L<~16$. Our results are consistent with the absence of a spontaneous magnetization for $h\ensuremath{\ne}0$ even in the ground state. The singularities occurring at $T=0$ in the range $0<~h<~4J$, $J$ being the exchange constant, are discussed in terms of a cluster expansion. For nonzero $T$, less than the critical temperature of the pure two-dimensional Ising model, and $h$ sufficiently small, the system exhibits a nonzero spin-glass order parameter of the Mattis type although there is no magnetization. The ferromagnetic correlation function becomes long ranged for $h\ensuremath{\rightarrow}0$ and is calculated from the domain-wall density.

Physical review. B, Condensed matter•Jan 1, 1981•97 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 ${h}_{c}>0$.

arXiv:1307.3987•Physical Review B•Jul 31, 2014•10 citations

Correlation functions of the one-dimensional random-field Ising model at zero temperature

Edward Farhi, Sam Gutmann

We consider the one-dimensional random-field Ising model, where the spin-spin coupling J is ferromagnetic and the external field is chosen to be +h with probability p and -h with probability 1-p. At zero temperature, we calculate an exact expression for the correlation length of the quenched average of the correlation function 〈${\mathit{s}}_{0}$${\mathit{s}}_{\mathit{n}}$〉-〈${\mathit{s}}_{0}$〉〈${\mathit{s}}_{\mathit{n}}$〉 in the case that 2J/h is not an integer. The result is a discontinuous function of 2J/h. When p=1/2, we also place a bound on the correlation length of the quenched average of the correlation function 〈${\mathit{s}}_{0}$${\mathit{s}}_{\mathit{n}}$〉.

arXiv:hep-th/9304158•Physical review. B, Condensed matter•Oct 1, 1993•9 citations
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 (TTT) greater than zero; the Lorentzian-squared terms vanish at T=0T=0T=0. The calculated correlation length agrees with predictions of the Imry-Ma domain argument.
We present an exact solution of a one-dimensional XY\mathrm{XY}XY model in a random magnetic field in the limit of strong-field pinning. The structure factor exhibits Lorentzian-squared behavior at nonzero temperatures. The scaling behavior of the correlation length as a function of randomness is different from that obtained in the weak-field pinning regime. We also study the dynamical behavior of the system using a simple relaxational model. The magnetization decays anomalously with time as ${t}^{\frac{7}{6}}{e}^{{\ensuremath{-}ct}^{\frac{1}{3}}}$.
The Ising model in a random magnetic field h is studied on a square lattice. h has the same value along a row, but varies randomly from row to row. h=\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty} with probability p/2 and is zero with probability 1-p. The quenched free energy F and specific heat C are calculated exactly. We find C\ensuremath{\sim}lnp for small p at the Onsager temperature Tc{T}_{c}Tc​. It is shown that for p\ensuremath{\ne}0, F(p,T) is not analytic in T at T=Tc{T}_{c}Tc​, i.e., it has a Griffiths-like singularity.
The two-point connected correlation function, or wavevector-dependent susceptibility, of the ferromagnetic Ising chain in a random field is calculated exactly at any temperature, for a two-parameter family of diluted symmetric exponential distributions of the magnetic fields. Thermodynamic properties of this model have been derived in a previous work by the authors. Besides the correlation function itself, the solution provides exact results for the (usual) susceptibility, the correlation length and the Edwards-Anderson parameter. The low-temperature regime is examined in full detail: the authors obtain in closed form the limit value at zero temperature of various quantities, and the first correction to this limit, which behaves linearly with temperature. The correlation length is discontinuous at zero temperature. They also derive the scaling form of the correlation function for small p, where the dilution p is the probability for a spin to be subjected to a non-zero random field. Even in this limit, the correlation function is more complicated than the simple Lorentzian predicted, e.g., by mean-field theory.
The statistical mechanics of finite $L\ifmmode\times\else\texttimes\fi{}L$ Ising square lattices in a field $\ifmmode\pm\else\textpm\fi{}h$ of random sign is investigated numerically by a modified recursive transfer matrix method for 6< L< 166<~L<~166< L< 16. Our results are consistent with the absence of a spontaneous magnetization for $h\ensuremath{\ne}0$ even in the ground state. The singularities occurring at T=0T=0T=0 in the range 0< h< 4J0<~h<~4J0< h< 4J, JJJ being the exchange constant, are discussed in terms of a cluster expansion. For nonzero TTT, less than the critical temperature of the pure two-dimensional Ising model, and hhh sufficiently small, the system exhibits a nonzero spin-glass order parameter of the Mattis type although there is no magnetization. The ferromagnetic correlation function becomes long ranged for $h\ensuremath{\rightarrow}0$ and is calculated from the domain-wall density.
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 rrr 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=219L={2}^{19}L=219 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 hhh 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 hc>0{h}_{c}>0hc​>0.
We consider the one-dimensional random-field Ising model, where the spin-spin coupling J is ferromagnetic and the external field is chosen to be +h with probability p and -h with probability 1-p. At zero temperature, we calculate an exact expression for the correlation length of the quenched average of the correlation function 〈s0{\mathit{s}}_{0}s0​sn{\mathit{s}}_{\mathit{n}}sn​〉-〈s0{\mathit{s}}_{0}s0​〉〈sn{\mathit{s}}_{\mathit{n}}sn​〉 in the case that 2J/h is not an integer. The result is a discontinuous function of 2J/h. When p=1/2, we also place a bound on the correlation length of the quenched average of the correlation function 〈s0{\mathit{s}}_{0}s0​sn{\mathit{s}}_{\mathit{n}}sn​〉.