Random-Field Ising Model Criticality: A Ground-State Determination

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

Critical aspects of the random-field Ising model

Nikolaos G. Fytas, Panagiotis E. Theodorakis, Ioannis Georgiou +1 more

The European Physical Journal BJun 1, 201311 citations

Ground-state optimization and hysteretic demagnetization: The random-field Ising model

Mikko J. Alava, Vittorio Basso, Francesca Colaiori +4 more

We compare the ground state of the random-field Ising model with Gaussian distributed random fields, with its nonequilibrium hysteretic counterpart, the demagnetized state. This is a low-energy state obtained by a sequence of slow magnetic-field oscillations with decreasing amplitude. The main concern is how optimized the demagnetized state is with respect to the best-possible ground state. Exact results for the energy in $d=1$ show that in a paramagnet, with finite spin-spin correlations, there is a significant difference in the energies if the disorder is not so strong that the states are trivially almost alike. We use numerical simulations to better characterize the difference between the ground state and the demagnetized state. For $d\ensuremath{\geqslant}3$, the random-field Ising model displays a disorder induced phase transition between a paramagnetic and a ferromagnetic state. The locations of the critical points ${R}_{c}^{(\mathrm{DS})}$ and ${R}_{c}^{(\mathrm{GS})}$ differ for the demagnetized state and ground state. We argue based on the numerics that in $d=3$ the scaling at the transition is the same in both states. This claim is corroborated by the exact solution of the model on the Bethe lattice, where the critical points are also different.We compare the ground state of the random-field Ising model with Gaussian distributed random fields, with its nonequilibrium hysteretic counterpart, the demagnetized state. This is a low-energy state obtained by a sequence of slow magnetic-field oscillations with decreasing amplitude. The main concern is how optimized the demagnetized state is with respect to the best-possible ground state. Exact results for the energy in $d=1$ show that in a paramagnet, with finite spin-spin correlations, there is a significant difference in the energies if the disorder is not so strong that the states are trivially almost alike. We use numerical simulations to better characterize the difference between the ground state and the demagnetized state. For $d\ensuremath{\geqslant}3$, the random-field Ising model displays a disorder induced phase transition between a paramagnetic and a ferromagnetic state. The locations of the critical points ${R}_{c}^{(\mathrm{DS})}$ and ${R}_{c}^{(\mathrm{GS})}$ differ for the demagnetized state and ground state. We argue based on the numerics that in $d=3$ the scaling at the transition is the same in both states. This claim is corroborated by the exact solution of the model on the Bethe lattice, where the critical points are also different.

arXiv:cond-mat/0407297Physical Review BFeb 28, 200519 citations

Static critical behavior of the random field Ising model

Z. Slanič

Jan 1, 19980 citations

Numerical study of the three-dimensional random-field Ising model at zero and positive temperature

Yong Wu, Jonathan L. Machta

In this paper the three-dimensional random-field Ising model is studied at both zero temperature and positive temperature. Critical exponents are extracted at zero temperature by finite size scaling analysis of large discontinuities in the bond energy. The heat capacity exponent $\ensuremath{\alpha}$ is found to be near zero. The ground states are determined for a range of external field and disorder strength near the zero temperature critical point and the scaling of ground state tilings of the field-disorder plane is discussed. At positive temperature the specific heat and the susceptibility are obtained using the Wang-Landau algorithm. It is found that sharp peaks are present in these physical quantities for some realizations of systems sized ${16}^{3}$ and larger. These sharp peaks result from flipping large domains and correspond to large discontinuities in ground state bond energies. Finally, zero temperature and positive temperature spin configurations near the critical line are found to be highly correlated suggesting a strong version of the zero temperature fixed point hypothesis.In this paper the three-dimensional random-field Ising model is studied at both zero temperature and positive temperature. Critical exponents are extracted at zero temperature by finite size scaling analysis of large discontinuities in the bond energy. The heat capacity exponent $\ensuremath{\alpha}$ is found to be near zero. The ground states are determined for a range of external field and disorder strength near the zero temperature critical point and the scaling of ground state tilings of the field-disorder plane is discussed. At positive temperature the specific heat and the susceptibility are obtained using the Wang-Landau algorithm. It is found that sharp peaks are present in these physical quantities for some realizations of systems sized ${16}^{3}$ and larger. These sharp peaks result from flipping large domains and correspond to large discontinuities in ground state bond energies. Finally, zero temperature and positive temperature spin configurations near the critical line are found to be highly correlated suggesting a strong version of the zero temperature fixed point hypothesis.

arXiv:cond-mat/0602314Physical Review BAug 21, 200640 citations

Critical Properties of the Random Field Ising Model

Jacques Villain, Thomas Nattermann, Moshe Schwartz

Springer proceedings in physicsJan 1, 19870 citations

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.

d=1

show that in a paramagnet, with finite spin-spin correlations, there is a significant difference in the energies if the disorder is not so strong that the states are trivially almost alike. We use numerical simulations to better characterize the difference between the ground state and the demagnetized state. For $d\ensuremath{\geqslant}3$, the random-field Ising model displays a disorder induced phase transition between a paramagnetic and a ferromagnetic state. The locations of the critical points

{R}_{c}^{(\mathrm{DS})}

and

{R}_{c}^{(\mathrm{GS})}

differ for the demagnetized state and ground state. We argue based on the numerics that in

d=3

the scaling at the transition is the same in both states. This claim is corroborated by the exact solution of the model on the Bethe lattice, where the critical points are also different.

{16}^{3}

and larger. These sharp peaks result from flipping large domains and correspond to large discontinuities in ground state bond energies. Finally, zero temperature and positive temperature spin configurations near the critical line are found to be highly correlated suggesting a strong version of the zero temperature fixed point hypothesis.