Physical review. B, Condensed matter

Ferromagnetism in theXYmodel with random threefold anisotropy

Ronald Fisch

Published Jul 1, 1992• 12 citations

Abstract

A Monte Carlo algorithm has been used to study the XY model with random threefold single-site anisotropy on simple cubic lattices, in the strong-anisotropy limit. The model has a pure XY-like ferromagnetic critical point at ${\mathit{T}}_{\mathit{c}}$/J=2.190\ifmmode\pm\else\textpm\fi{}0.005. The zero-field thermodynamic functions remain close to those of the simple XY model down to T\ensuremath{\approxeq}2.00J. The ground state has an energy per spin of ${\mathit{E}}_{0}$/J=-2.2305\ifmmode\pm\else\textpm\fi{}0.0010 and a magnetization per spin of ${\mathit{M}}_{0}$=0.720\ifmmode\pm\else\textpm\fi{}0.003.

DOI: 10.1103/physrevb.46.242

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Finite-size scaling critical behavior of randomly pinned spin-density waves

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A Monte Carlo Study of the Criticality of the 3D XY Model

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52Jan 1, 20150 citations

The three-dimensional XY model in P-fold random anisotropy

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The three-dimensional XY model in the presence of two- and threefold random anisotropy has been investigated by Monte Carlo simulation and finite-size scaling using system sizes 4 3 , 8 3 , 16 3 , 32 3 . The results are consistent with the existence of a conventional transition to a spin glass phase at T c =2.2 and with correlation exponent v=0.67. No evidence for quasiferromagnetism is found.

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Monte Carlo study of the phase transitions in the classical XY ferromagnets with random anisotropy

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Phase transitions in the anisotropic XY ferromagnet with quenched nonmagnetic impurity

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arXiv:2309.09659International Journal of Modern Physics CDec 17, 20232 citations

Physical review. B, Condensed matter

Ferromagnetism in theXYmodel with random threefold anisotropy

Ronald Fisch

Published Jul 1, 1992• 12 citations

Publisher

Abstract

{\mathit{T}}_{\mathit{c}}

/J=2.190\ifmmode\pm\else\textpm\fi{}0.005. The zero-field thermodynamic functions remain close to those of the simple XY model down to T\ensuremath{\approxeq}2.00J. The ground state has an energy per spin of

{\mathit{E}}_{0}

/J=-2.2305\ifmmode\pm\else\textpm\fi{}0.0010 and a magnetization per spin of

{\mathit{M}}_{0}

=0.720\ifmmode\pm\else\textpm\fi{}0.003.

DOI: 10.1103/physrevb.46.242

Related papers

Finite-size scaling critical behavior of randomly pinned spin-density waves

Ronald Fisch

We have performed Monte Carlo studies of the three-dimensional $XY$ model with random uniaxial anisotropy, which is a model for randomly pinned spin-density waves. We study $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ simple cubic lattices using $L$ values in the range 16--64, and with random anisotropy strengths of $D/2J=1$ , 2, 3, 6, and $\ensuremath{\infty}$. There is a well-defined finite-temperature critical point, ${T}_{c}$ , for each of these values of $D/2J$ . We present results for the angle-averaged magnetic structure factor $S(k)$ at ${T}_{c}$ for $L=64$ . We also use finite-size scaling analysis to study scaling functions for the critical behavior of the specific heat, the magnetization, and the longitudinal magnetic susceptibility. Good data collapse of the scaling functions over a wide range of $T$ is seen for $D/2J=6$ and $\ensuremath{\infty}$. For our finite values of $D/2J$ the scaled magnetization function increases with $L$ below ${T}_{c}$ and appears to approach an $L$ -independent limit for large $L$ . This suggests that the system is ferromagnetic below ${T}_{c}$ .We have performed Monte Carlo studies of the three-dimensional $XY$ model with random uniaxial anisotropy, which is a model for randomly pinned spin-density waves. We study $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ simple cubic lattices using $L$ values in the range 16--64, and with random anisotropy strengths of $D/2J=1$, 2, 3, 6, and $\ensuremath{\infty}$. There is a well-defined finite-temperature critical point, ${T}_{c}$, for each of these values of $D/2J$. We present results for the angle-averaged magnetic structure factor $S(k)$ at ${T}_{c}$ for $L=64$. We also use finite-size scaling analysis to study scaling functions for the critical behavior of the specific heat, the magnetization, and the longitudinal magnetic susceptibility. Good data collapse of the scaling functions over a wide range of $T$ is seen for $D/2J=6$ and $\ensuremath{\infty}$. For our finite values of $D/2J$ the scaled magnetization function increases with $L$ below ${T}_{c}$ and appears to approach an $L$-independent limit for large $L$. This suggests that the system is ferromagnetic below ${T}_{c}$.

arXiv:0902.4049Physical Review BJun 24, 20098 citations

A Monte Carlo Study of the Criticality of the 3D XY Model

Sindre Vaskinn

In this master thesis the criticality of the 3D XY model with cubic anisotropy has been examined by the means of Monte Carlo simulation. The 3D XY model is a special case of the $O(n)$ spin-vector model which describe numerous phenomena's. Here, $n$ is the number of vector dimensions and for the XY model $n=2$ . The $O(n)$ -model is invariant for global rotations. This means that it is arbitrary which direction in the symmetry breaking occurs. In real system one should always expect perturbations from a perfect system, here it is done by adding a cubic anisotropy (order $p=4$ ) then the symmetry of the Hamiltonian is lowered and it will be advantageous for the spins to be ordered in directions $90^o$ apart. This symmetry is called $C_{4}$ , which is a part of the cyclic group, and it restricts the symmetry breaking to one of these four directions. \n\nThe criticality of this model is when the spins suddenly align as the temperature is lowered to a critical temperature. Sufficiently close to the critical temperature quantities like the magnetization, which is the sum of spins, will show a power law growth characterised by critical exponents specified by the class of universality of the transition. The critical exponents can be obtained by finite-size scaling. By applying renormalization theory on the critical region one can show that some of parameters in the Hamiltonian are: relevant and effect the scaling, irrelevant and does not effect the scaling and dangerously irrelevant meaning that they only affect some of the critical exponents. Previous Monte Carlo and renormalization calculus show that anisotropy which is relevant for $p<4$ becomes dangerously irrelevant for $p\\geq4$ . The question of interest is what kind of effect the dangerously irrelevant parameter in the Hamiltonian has on the scaling. This is interesting because the effect of dangerously irrelevant parameters is hardly studied and anisotropy arise in many crystal systems because of the lattice geometry. \n\nThe results are obtained by Monte Carlo simulations and the critical slowing down close to the phase transition is solved by the method of parallel tempering. This method was chosen because it is simple to implement in a parallel code.\n\nThe Binder cumulant for different quantities and the peak of the magnetic susceptibility was used support a critical temperature $T_c = 2.2019 \\ \\textnormal k$ for no field applied and no anisotropy. The known critical exponents for the no field and no anisotropy was used. The scaling made a good collapse, meaning that the right $T_c$ and the correct critical exponents where used. The scaling did not take into account a magnetic field and therefore a good collapse of curves where not obtained in this case. 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The 3D XY model is a special case of the $O(n)$ spin-vector model which describe numerous phenomena's. Here, $n$ is the number of vector dimensions and for the XY model $n=2$. The $O(n)$-model is invariant for global rotations. This means that it is arbitrary which direction in the symmetry breaking occurs. In real system one should always expect perturbations from a perfect system, here it is done by adding a cubic anisotropy (order $p=4$) then the symmetry of the Hamiltonian is lowered and it will be advantageous for the spins to be ordered in directions $90^o$ apart. This symmetry is called $C_{4}$, which is a part of the cyclic group, and it restricts the symmetry breaking to one of these four directions. \n\nThe criticality of this model is when the spins suddenly align as the temperature is lowered to a critical temperature. Sufficiently close to the critical temperature quantities like the magnetization, which is the sum of spins, will show a power law growth characterised by critical exponents specified by the class of universality of the transition. The critical exponents can be obtained by finite-size scaling. By applying renormalization theory on the critical region one can show that some of parameters in the Hamiltonian are: relevant and effect the scaling, irrelevant and does not effect the scaling and dangerously irrelevant meaning that they only affect some of the critical exponents. Previous Monte Carlo and renormalization calculus show that anisotropy which is relevant for $p<4$ becomes dangerously irrelevant for $p\\geq4$. The question of interest is what kind of effect the dangerously irrelevant parameter in the Hamiltonian has on the scaling. This is interesting because the effect of dangerously irrelevant parameters is hardly studied and anisotropy arise in many crystal systems because of the lattice geometry. \n\nThe results are obtained by Monte Carlo simulations and the critical slowing down close to the phase transition is solved by the method of parallel tempering. This method was chosen because it is simple to implement in a parallel code.\n\nThe Binder cumulant for different quantities and the peak of the magnetic susceptibility was used support a critical temperature $T_c = 2.2019 \\ \\textnormal k$ for no field applied and no anisotropy. The known critical exponents for the no field and no anisotropy was used. The scaling made a good collapse, meaning that the right $T_c$ and the correct critical exponents where used. The scaling did not take into account a magnetic field and therefore a good collapse of curves where not obtained in this case. 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52Jan 1, 20150 citations

The three-dimensional XY model in P-fold random anisotropy

P Reed

Journal of Physics A Mathematical and GeneralFeb 7, 19916 citations

Monte Carlo study of the phase transitions in the classical XY ferromagnets with random anisotropy

Olivia Mallick, Muktish Acharyya

The three-dimensional anisotropic classical XY ferromagnet has been investigated by extensive Monte Carlo simulation using the Metropolis single spin flip algorithm. The magnetisation (M) and the susceptibility (χ) are measured and studied as functions of the temperature of the system. For constant anisotropy, the ferro-para phase transition has been found to take place at a higher temperature than that observed in the isotropic case. The system gets ordered at higher temperatures for higher values of the strength of anisotropy. The opposite scenario is observed in the case of random anisotropy. For all three different kinds of statistical distributions (uniform, Gaussian, and bimodal) of random anisotropy, the system gets ordered at lower temperatures for higher values of the width of the distribution of anisotropy. We have provided the phase boundaries in the case of random anisotropy. The critical exponents for the scaling laws M∼L−βν and χ∼Lγν are estimated through the finite size analysis.The three-dimensional anisotropic classical XY ferromagnet has been investigated by extensive Monte Carlo simulation using the Metropolis single spin flip algorithm. The magnetisation (M) and the susceptibility (χ) are measured and studied as functions of the temperature of the system. For constant anisotropy, the ferro-para phase transition has been found to take place at a higher temperature than that observed in the isotropic case. The system gets ordered at higher temperatures for higher values of the strength of anisotropy. The opposite scenario is observed in the case of random anisotropy. For all three different kinds of statistical distributions (uniform, Gaussian, and bimodal) of random anisotropy, the system gets ordered at lower temperatures for higher values of the width of the distribution of anisotropy. We have provided the phase boundaries in the case of random anisotropy. The critical exponents for the scaling laws M∼L−βν and χ∼Lγν are estimated through the finite size analysis.

arXiv:2208.10109Phase TransitionsAug 15, 20237 citations

Power-law correlated phase in random-field XY models and randomly pinned charge-density waves

Ronald Fisch

Monte Carlo simulations have been used to study the ${\mathrm{Z}}_{\mathrm{N}}$ ferromagnet in a random field on simple cubic lattices for N=6 and N=12. The random field is chosen to have infinite strength and random direction on a fraction x of the sites of the lattice, and to be zero on the remaining sites. For N=6 and x=1/16 there are two phase transitions. At low temperature there is a ferromagnetic phase, which is stabilized by the sixfold nonrandom anisotropy. The intermediate temperature phase is characterized by a |k${\mathrm{|}}^{\mathrm{\ensuremath{-}}3}$ decay of two-spin correlations, but no true ferromagnetic order. At the transition between the power-law correlated phase and the paramagnetic phase the magnetic susceptibility diverges, and the two-spin correlations decay approximately as |k${\mathrm{|}}^{\mathrm{\ensuremath{-}}2.87}$. There is no evidence for a latent heat at either transition, but the magnetization seems to disappear discontinuously. For N=6 and x=1/8 the correlation length never exceeds 12, and the paramagnetic phase goes directly into the ferromagnetic phase; the two-spin correlation function is peaked at small |k|, but the only divergence is the ferromagnetic \ensuremath{\delta} function at |k|=0. Results in the paramagnetic and power-law correlated phases for N=12 are essentially identical to those for N=6, so the power-law correlated phase should exist in the limit N\ensuremath{\rightarrow}\ensuremath{\infty}. The ferromagnetic phase terminates near x=1/6 for N=6, but the limit of ferromagnetic stability for N=12 is less than x=1/16.Monte Carlo simulations have been used to study the ${\mathrm{Z}}_{\mathrm{N}}$ ferromagnet in a random field on simple cubic lattices for N=6 and N=12. The random field is chosen to have infinite strength and random direction on a fraction x of the sites of the lattice, and to be zero on the remaining sites. For N=6 and x=1/16 there are two phase transitions. At low temperature there is a ferromagnetic phase, which is stabilized by the sixfold nonrandom anisotropy. The intermediate temperature phase is characterized by a |k${\mathrm{|}}^{\mathrm{\ensuremath{-}}3}$ decay of two-spin correlations, but no true ferromagnetic order. At the transition between the power-law correlated phase and the paramagnetic phase the magnetic susceptibility diverges, and the two-spin correlations decay approximately as |k${\mathrm{|}}^{\mathrm{\ensuremath{-}}2.87}$. There is no evidence for a latent heat at either transition, but the magnetization seems to disappear discontinuously. For N=6 and x=1/8 the correlation length never exceeds 12, and the paramagnetic phase goes directly into the ferromagnetic phase; the two-spin correlation function is peaked at small |k|, but the only divergence is the ferromagnetic \ensuremath{\delta} function at |k|=0. Results in the paramagnetic and power-law correlated phases for N=12 are essentially identical to those for N=6, so the power-law correlated phase should exist in the limit N\ensuremath{\rightarrow}\ensuremath{\infty}. The ferromagnetic phase terminates near x=1/6 for N=6, but the limit of ferromagnetic stability for N=12 is less than x=1/16.

arXiv:cond-mat/9608036Physical review. B, Condensed matterApr 1, 199712 citations

Phase transitions in the anisotropic XY ferromagnet with quenched nonmagnetic impurity

Olivia Mallick, Muktish Acharyya

The equilibrium behaviors of the anisotropic XY ferromagnet, with nonmagnetic impurity, have been investigated in three dimensions by Monte Carlo simulation using Metropolis algorithm. Two different types of anisotropy, namely, the bilinear exchange type and single-site anisotropy are considered here. The thermodynamic behaviors of the components of the magnetizations (M), susceptibility ([Formula: see text]) and the specific heat (C) have been studied systematically through extensive Monte Carlo simulations. The ferro–para phase transition has been observed to take place at a lower temperature for impure anisotropic XY ferromagnet. The pseudocritical temperature ([Formula: see text]) has been found to decrease as the system gets more and more impure (impurity concentration p increases). In the case of bilinear exchange type of anisotropy ([Formula: see text]), the pseudocritical temperature ([Formula: see text]) increases linearly with [Formula: see text] for any given concentration of nonmagnetic impurity (p). The slope of this linear function has been found to depend on the impurity concentration (p). The slope decreases linearly with the impurity concentration (p). In the case of the single-site anisotropy (D), the pseudocritical temperature ([Formula: see text]) has been found to decrease linearly with p for fixed D. The critical temperature (for a fixed set of parameter values) has been estimated from the temperature variation of fourth-order Binder cumulants ([Formula: see text]) for different system sizes (L). The critical magnetization ([Formula: see text]) and the maximum value of the susceptibility ([Formula: see text]) are calculated for different system sizes (L). The critical exponents for the assumed scaling laws, [Formula: see text] and [Formula: see text], are estimated through the finite size analysis. We have estimated, [Formula: see text], equals [Formula: see text] and [Formula: see text] for bilinear exchange and single-site anisotropy, respectively. We have also estimated, [Formula: see text] equals [Formula: see text] and [Formula: see text] for bilinear exchange and single-site anisotropy, respectively.The equilibrium behaviors of the anisotropic XY ferromagnet, with nonmagnetic impurity, have been investigated in three dimensions by Monte Carlo simulation using Metropolis algorithm. Two different types of anisotropy, namely, the bilinear exchange type and single-site anisotropy are considered here. The thermodynamic behaviors of the components of the magnetizations (M), susceptibility ([Formula: see text]) and the specific heat (C) have been studied systematically through extensive Monte Carlo simulations. The ferro–para phase transition has been observed to take place at a lower temperature for impure anisotropic XY ferromagnet. The pseudocritical temperature ([Formula: see text]) has been found to decrease as the system gets more and more impure (impurity concentration p increases). In the case of bilinear exchange type of anisotropy ([Formula: see text]), the pseudocritical temperature ([Formula: see text]) increases linearly with [Formula: see text] for any given concentration of nonmagnetic impurity (p). The slope of this linear function has been found to depend on the impurity concentration (p). The slope decreases linearly with the impurity concentration (p). In the case of the single-site anisotropy (D), the pseudocritical temperature ([Formula: see text]) has been found to decrease linearly with p for fixed D. The critical temperature (for a fixed set of parameter values) has been estimated from the temperature variation of fourth-order Binder cumulants ([Formula: see text]) for different system sizes (L). The critical magnetization ([Formula: see text]) and the maximum value of the susceptibility ([Formula: see text]) are calculated for different system sizes (L). The critical exponents for the assumed scaling laws, [Formula: see text] and [Formula: see text], are estimated through the finite size analysis. We have estimated, [Formula: see text], equals [Formula: see text] and [Formula: see text] for bilinear exchange and single-site anisotropy, respectively. We have also estimated, [Formula: see text] equals [Formula: see text] and [Formula: see text] for bilinear exchange and single-site anisotropy, respectively.

arXiv:2309.09659International Journal of Modern Physics CDec 17, 20232 citations

{\mathit{T}}_{\mathit{c}}

{\mathit{E}}_{0}

/J=-2.2305\ifmmode\pm\else\textpm\fi{}0.0010 and a magnetization per spin of

{\mathit{M}}_{0}

=0.720\ifmmode\pm\else\textpm\fi{}0.003.

We have performed Monte Carlo studies of the three-dimensional

XY

model with random uniaxial anisotropy, which is a model for randomly pinned spin-density waves. We study $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ simple cubic lattices using

L

values in the range 16--64, and with random anisotropy strengths of

D/2J=1

, 2, 3, 6, and $\ensuremath{\infty}$. There is a well-defined finite-temperature critical point,

{T}_{c}

, for each of these values of

D/2J

. We present results for the angle-averaged magnetic structure factor

S(k)

{T}_{c}

for

L=64

. We also use finite-size scaling analysis to study scaling functions for the critical behavior of the specific heat, the magnetization, and the longitudinal magnetic susceptibility. Good data collapse of the scaling functions over a wide range of

T

is seen for

D/2J=6

and $\ensuremath{\infty}$. For our finite values of

D/2J

the scaled magnetization function increases with

L

below

{T}_{c}

and appears to approach an

L

-independent limit for large

L

. This suggests that the system is ferromagnetic below

{T}_{c}

In this master thesis the criticality of the 3D XY model with cubic anisotropy has been examined by the means of Monte Carlo simulation. The 3D XY model is a special case of the

O(n)

spin-vector model which describe numerous phenomena's. Here,

n

is the number of vector dimensions and for the XY model

n=2

. The

O(n)

-model is invariant for global rotations. This means that it is arbitrary which direction in the symmetry breaking occurs. In real system one should always expect perturbations from a perfect system, here it is done by adding a cubic anisotropy (order

p=4

) then the symmetry of the Hamiltonian is lowered and it will be advantageous for the spins to be ordered in directions

90^o

apart. This symmetry is called

C_{4}

, which is a part of the cyclic group, and it restricts the symmetry breaking to one of these four directions. \n\nThe criticality of this model is when the spins suddenly align as the temperature is lowered to a critical temperature. Sufficiently close to the critical temperature quantities like the magnetization, which is the sum of spins, will show a power law growth characterised by critical exponents specified by the class of universality of the transition. The critical exponents can be obtained by finite-size scaling. By applying renormalization theory on the critical region one can show that some of parameters in the Hamiltonian are: relevant and effect the scaling, irrelevant and does not effect the scaling and dangerously irrelevant meaning that they only affect some of the critical exponents. Previous Monte Carlo and renormalization calculus show that anisotropy which is relevant for

p<4

becomes dangerously irrelevant for

p\\geq4

. The question of interest is what kind of effect the dangerously irrelevant parameter in the Hamiltonian has on the scaling. This is interesting because the effect of dangerously irrelevant parameters is hardly studied and anisotropy arise in many crystal systems because of the lattice geometry. \n\nThe results are obtained by Monte Carlo simulations and the critical slowing down close to the phase transition is solved by the method of parallel tempering. This method was chosen because it is simple to implement in a parallel code.\n\nThe Binder cumulant for different quantities and the peak of the magnetic susceptibility was used support a critical temperature

T_c = 2.2019 \\ \\textnormal k

for no field applied and no anisotropy. The known critical exponents for the no field and no anisotropy was used. The scaling made a good collapse, meaning that the right

T_c

and the correct critical exponents where used. The scaling did not take into account a magnetic field and therefore a good collapse of curves where not obtained in this case. When the cubic anisotropy was added the critical exponent for the magnetic susceptibility and the cubic order parameter changed, while the other exponents where unchanged. This supports the view that cubic anisotropy is dangerously irrelevant for the 3D XY model. The scaling of the magnetic susceptibility showed a possible new kind of scaling, a scaling which is different on the two sides of the transition. Although a more accurate determination of the critical exponent in the ordered phase require more extensive simulations. It appears that this exponent is larger than the exponent in the disordered phase, where one recovered the isotropic exponent. The difference in the critical exponents has been predicted to increase with

p

and an investigation of the hexagonal anisotropy

p=6

has been started.

Monte Carlo simulations have been used to study the

{\mathrm{Z}}_{\mathrm{N}}

ferromagnet in a random field on simple cubic lattices for N=6 and N=12. The random field is chosen to have infinite strength and random direction on a fraction x of the sites of the lattice, and to be zero on the remaining sites. For N=6 and x=1/16 there are two phase transitions. At low temperature there is a ferromagnetic phase, which is stabilized by the sixfold nonrandom anisotropy. The intermediate temperature phase is characterized by a |k${\mathrm{|}}^{\mathrm{\ensuremath{-}}3}$ decay of two-spin correlations, but no true ferromagnetic order. At the transition between the power-law correlated phase and the paramagnetic phase the magnetic susceptibility diverges, and the two-spin correlations decay approximately as |k${\mathrm{|}}^{\mathrm{\ensuremath{-}}2.87}$. There is no evidence for a latent heat at either transition, but the magnetization seems to disappear discontinuously. For N=6 and x=1/8 the correlation length never exceeds 12, and the paramagnetic phase goes directly into the ferromagnetic phase; the two-spin correlation function is peaked at small |k|, but the only divergence is the ferromagnetic \ensuremath{\delta} function at |k|=0. Results in the paramagnetic and power-law correlated phases for N=12 are essentially identical to those for N=6, so the power-law correlated phase should exist in the limit N\ensuremath{\rightarrow}\ensuremath{\infty}. The ferromagnetic phase terminates near x=1/6 for N=6, but the limit of ferromagnetic stability for N=12 is less than x=1/16.