arXiv:cond-mat/9609213 · Physical review. B, Condensed matter

Numerical study of competing spin-glass and ferromagnetic order

Mikhail V. Simkin

Published May 1, 1997• 3 citations

Abstract

Two- and three-dimensional random Ising models with a Gaussian distribution of couplings with variance J and nonvanishing mean value ${\mathrm{J}}_{0}$ are studied using the zero-temperature domain-wall renormalization group (DWRG). The DWRG trajectories in the (${\mathrm{J}}_{0}$,J) plane after rescaling can be collapsed on two curves: one for ${\mathrm{J}}_{0}$/J>${\mathrm{r}}_{\mathrm{c}}$ and the other for ${\mathrm{J}}_{0}$/J${\mathrm{r}}_{\mathrm{c}}$. In the first case the DWRG flows are toward the ferromagnetic fixed point both in two and three dimensions while in the second case flows are towards a paramagnetic fixed point and spin-glass fixed point in two and three dimensions, respectively.

DOI: 10.1103/physrevb.55.11405

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arXiv:cond-mat/9609213 · Physical review. B, Condensed matter

Numerical study of competing spin-glass and ferromagnetic order

Mikhail V. Simkin

Published May 1, 1997• 3 citations

arXiv HTML PDF

Abstract

Two- and three-dimensional random Ising models with a Gaussian distribution of couplings with variance J and nonvanishing mean value

{\mathrm{J}}_{0}

are studied using the zero-temperature domain-wall renormalization group (DWRG). The DWRG trajectories in the (

{\mathrm{J}}_{0}

,J) plane after rescaling can be collapsed on two curves: one for

{\mathrm{J}}_{0}

/J>

{\mathrm{r}}_{\mathrm{c}}

and the other for

{\mathrm{J}}_{0}

{\mathrm{r}}_{\mathrm{c}}

. In the first case the DWRG flows are toward the ferromagnetic fixed point both in two and three dimensions while in the second case flows are towards a paramagnetic fixed point and spin-glass fixed point in two and three dimensions, respectively.

DOI: 10.1103/physrevb.55.11405

Related papers

Zero-temperature spin-glass–ferromagnetic transition: Scaling analysis of the domain-wall energy

Cécile Monthus, Thomas Garel

For the Ising model with Gaussian random coupling of average ${J}_{0}$ and unit variance, the zero-temperature spin-glass--ferromagnetic transition as a function of the control parameter ${J}_{0}$ can be studied via the size- $L$ -dependent renormalized coupling defined as the domain-wall energy ${J}^{R}(L)\ensuremath{\equiv}{E}_{\text{GS}}^{(\text{AF})}(L)\ensuremath{-}{E}_{\text{GS}}^{(\text{F})}(L)$ (i.e., the difference between the ground-state energies corresponding to antiferromagnetic and ferromagnetic boundary conditions in one direction). We study numerically the critical exponents of this zero-temperature transition within the Migdal-Kadanoff approximation as a function of the dimension $d=2,3,4,5,6$ . We then compare with the mean-field spherical model. Our main conclusion is that in low dimensions, the critical stiffness exponent ${\ensuremath{\theta}}^{c}$ is clearly bigger than the spin-glass stiffness exponent ${\ensuremath{\theta}}^{\text{SG}}$, but that they turn out to coincide in high enough dimension and in the mean-field spherical model. We also discuss the finite-size scaling properties of the averaged value and of the width of the distribution of the renormalized couplings.For the Ising model with Gaussian random coupling of average ${J}_{0}$ and unit variance, the zero-temperature spin-glass--ferromagnetic transition as a function of the control parameter ${J}_{0}$ can be studied via the size-$L$-dependent renormalized coupling defined as the domain-wall energy ${J}^{R}(L)\ensuremath{\equiv}{E}_{\text{GS}}^{(\text{AF})}(L)\ensuremath{-}{E}_{\text{GS}}^{(\text{F})}(L)$ (i.e., the difference between the ground-state energies corresponding to antiferromagnetic and ferromagnetic boundary conditions in one direction). We study numerically the critical exponents of this zero-temperature transition within the Migdal-Kadanoff approximation as a function of the dimension $d=2,3,4,5,6$. We then compare with the mean-field spherical model. Our main conclusion is that in low dimensions, the critical stiffness exponent ${\ensuremath{\theta}}^{c}$ is clearly bigger than the spin-glass stiffness exponent ${\ensuremath{\theta}}^{\text{SG}}$, but that they turn out to coincide in high enough dimension and in the mean-field spherical model. We also discuss the finite-size scaling properties of the averaged value and of the width of the distribution of the renormalized couplings.

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Physical review. B, Condensed matterJul 1, 1984185 citations

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Journal of Applied PhysicsNov 15, 199129 citations

Domain-wall renormalization-group study of the three-dimensional random Ising model at finite temperature

W. L. McMillan

The three-dimensional random Ising model with a Gaussian distribution of nearest-neighbor interactions is studied for the pure spin-glass case where the average interaction vanishes. The distribution of domain-wall free energies at finite temperature is calculated with the use of the Metropolis Monte Carlo algorithm for finite lattices. A renormalization-group transformation is set up which preserves the domain-wall free-energy distribution when the lattice parameter is changed. The spin-glass transition temperature is found to be ${T}_{g}$ =(1.0\ifmmode\pm\else\textpm\fi{}0.2)J\ifmmode \tilde{}\else \~{}\fi{} where J\ifmmode \tilde{}\else \~{}\fi{} is the variance of the Gaussian interaction distribution. The thermal exponent is \ensuremath{\nu}=1.8\ifmmode\pm\else\textpm\fi{}0.5 and the heat-capacity exponent is \ensuremath{\alpha}=-3.4\ifmmode\pm\else\textpm\fi{}1.5. The heat capacity exhibits a rounded peak at higher temperatures.The three-dimensional random Ising model with a Gaussian distribution of nearest-neighbor interactions is studied for the pure spin-glass case where the average interaction vanishes. The distribution of domain-wall free energies at finite temperature is calculated with the use of the Metropolis Monte Carlo algorithm for finite lattices. A renormalization-group transformation is set up which preserves the domain-wall free-energy distribution when the lattice parameter is changed. The spin-glass transition temperature is found to be ${T}_{g}$=(1.0\ifmmode\pm\else\textpm\fi{}0.2)J\ifmmode \tilde{}\else \~{}\fi{} where J\ifmmode \tilde{}\else \~{}\fi{} is the variance of the Gaussian interaction distribution. The thermal exponent is \ensuremath{\nu}=1.8\ifmmode\pm\else\textpm\fi{}0.5 and the heat-capacity exponent is \ensuremath{\alpha}=-3.4\ifmmode\pm\else\textpm\fi{}1.5. The heat capacity exhibits a rounded peak at higher temperatures.

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Physical review. B, Condensed matterJan 1, 198599 citations

Domain-wall energies and magnetization of the two-dimensional random-bond Ising model

Carlo Amoruso, Alexander K. Hartmann

We study ground-state properties of the two-dimensional random-bond Ising model with couplings having a concentration $p∊[0,1]$ of antiferromagnetic and $(1\ensuremath{-}p)$ of ferromagnetic bonds. We apply an exact matching algorithm which enables us the study of systems with linear dimension $L$ up to $700$ . We study the behavior of the domain-wall energies and of the magnetization. We find that the paramagnet-ferromagnet transition occurs at ${p}_{c}\ensuremath{\sim}0.103$ compared to the concentration ${p}_{n}\ensuremath{\sim}0.109$ at the Nishimori point, which means that the phase diagram of the model exhibits a re-entrance. Furthermore, we find no indications for an (intermediate) spin-glass ordering at finite temperature.We study ground-state properties of the two-dimensional random-bond Ising model with couplings having a concentration $p∊[0,1]$ of antiferromagnetic and $(1\ensuremath{-}p)$ of ferromagnetic bonds. We apply an exact matching algorithm which enables us the study of systems with linear dimension $L$ up to $700$. We study the behavior of the domain-wall energies and of the magnetization. We find that the paramagnet-ferromagnet transition occurs at ${p}_{c}\ensuremath{\sim}0.103$ compared to the concentration ${p}_{n}\ensuremath{\sim}0.109$ at the Nishimori point, which means that the phase diagram of the model exhibits a re-entrance. Furthermore, we find no indications for an (intermediate) spin-glass ordering at finite temperature.

arXiv:cond-mat/0401464Physical Review BOct 29, 200448 citations

Two- and three-dimensional random Ising models with a Gaussian distribution of couplings with variance J and nonvanishing mean value

{\mathrm{J}}_{0}

are studied using the zero-temperature domain-wall renormalization group (DWRG). The DWRG trajectories in the (

{\mathrm{J}}_{0}

,J) plane after rescaling can be collapsed on two curves: one for

{\mathrm{J}}_{0}

/J>

{\mathrm{r}}_{\mathrm{c}}

and the other for

{\mathrm{J}}_{0}

{\mathrm{r}}_{\mathrm{c}}

For the Ising model with Gaussian random coupling of average

{J}_{0}

and unit variance, the zero-temperature spin-glass--ferromagnetic transition as a function of the control parameter

{J}_{0}

can be studied via the size-

L

-dependent renormalized coupling defined as the domain-wall energy ${J}^{R}(L)\ensuremath{\equiv}{E}_{\text{GS}}^{(\text{AF})}(L)\ensuremath{-}{E}_{\text{GS}}^{(\text{F})}(L)$ (i.e., the difference between the ground-state energies corresponding to antiferromagnetic and ferromagnetic boundary conditions in one direction). We study numerically the critical exponents of this zero-temperature transition within the Migdal-Kadanoff approximation as a function of the dimension

d=2,3,4,5,6

. We then compare with the mean-field spherical model. Our main conclusion is that in low dimensions, the critical stiffness exponent ${\ensuremath{\theta}}^{c}$ is clearly bigger than the spin-glass stiffness exponent ${\ensuremath{\theta}}^{\text{SG}}$, but that they turn out to coincide in high enough dimension and in the mean-field spherical model. We also discuss the finite-size scaling properties of the averaged value and of the width of the distribution of the renormalized couplings.

{T}_{g}

=(1.0\ifmmode\pm\else\textpm\fi{}0.2)J\ifmmode \tilde{}\else \~{}\fi{} where J\ifmmode \tilde{}\else \~{}\fi{} is the variance of the Gaussian interaction distribution. The thermal exponent is \ensuremath{\nu}=1.8\ifmmode\pm\else\textpm\fi{}0.5 and the heat-capacity exponent is \ensuremath{\alpha}=-3.4\ifmmode\pm\else\textpm\fi{}1.5. The heat capacity exhibits a rounded peak at higher temperatures.

We study ground-state properties of the two-dimensional random-bond Ising model with couplings having a concentration

p∊[0,1]

of antiferromagnetic and $(1\ensuremath{-}p)$ of ferromagnetic bonds. We apply an exact matching algorithm which enables us the study of systems with linear dimension

L

up to

700

. We study the behavior of the domain-wall energies and of the magnetization. We find that the paramagnet-ferromagnet transition occurs at ${p}_{c}\ensuremath{\sim}0.103$ compared to the concentration ${p}_{n}\ensuremath{\sim}0.109$ at the Nishimori point, which means that the phase diagram of the model exhibits a re-entrance. Furthermore, we find no indications for an (intermediate) spin-glass ordering at finite temperature.