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

arXiv:1812.09318

Beyond the Imry-Ma Length: Scaling Behavior in the 3D Random Field $XY$ Model

Ronald Fisch

Published Dec 13, 2018• 0 citations
arXivHTMLPDF

Abstract

We have performed studies of the 3D random field $XY$ model on $L \times L \times L$ simple cubic lattices with periodic boundary conditions, with a random field strength of $h_r$ = 1.875, for $L =$ 64, 96 and 128, using a parallelized Monte Carlo algorithm. We present results for the angle-averaged magnetic structure factor, $S ( k )$ at $T$ = 1.00, which appears to be the temperature at which small jumps in the magnetization per spin and the energy per spin occur. The magnetization jump per spin scales with size roughly as $L^{- 3/4}$, while the energy jump per spin scales like $L^{- 3/2}$. The results also indicate the existence of an approximately logarithmic divergence of $S ( k )$ as $k \to 0$. The magnetic susceptibility, $χ(\vec{\bf k} = 0 )$, on the other hand, seems to have a value of about 14.2 under these conditions. This suggests the absence of a ferromagnetic phase, and that the lower critical dimension for long-range order in this model is three. Similar results are found for $L$ = 64 samples at $h_r$ = 2.0 and $T$ = 0.875. We expect that the behavior is qualitatively similar along the entire phase boundary, but the scaling exponents may not be universal. These results appear to be related to recent work on quantum disorder.

DOI
10.48550/arxiv.1812.09318

Related papers

Scaling Behavior in the 3D Random Field $XY$ Model

Ronald Fisch

We have performed studies of the 3D random field $XY$ model on $L \times L \times L$ simple cubic lattices with periodic boundary conditions, with a random field strength of $h_r$ = 1.875, for $L = 64$ and $L = 96$, using a parallelized Monte Carlo algorithm. We present results for the angle-averaged magnetic structure factor, $S ( k )$ at $T = 1.00$, which appears to be the temperature at which small jumps in the magnetization per spin and the energy per spin occur. The results indicate the existence of an approximately logarithmic divergence of $S ( k )$ as $k \to 0$. This suggests that the lower critical dimension for long range order in this model is three.

arXiv:1801.01817•Jan 5, 2018•0 citations

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 ${T}_{c}$.

arXiv:0902.4049•Physical Review B•Jun 24, 2009•8 citations

Three-dimensional random-field Ising magnet: Interfaces, scaling, and the nature of states

A. Alan Middleton, Daniel S. Fisher

The nature of the zero-temperature ordering transition in the three-dimensional Gaussian random-field Ising magnet is studied numerically, aided by scaling analyses. Various numerical calculations are used to consistently infer the location of the transition to a high precision. A variety of boundary conditions are imposed on large samples to study the order of the transition and the number of states in the large volume limit. In the ferromagnetic phase, where the domain walls have fractal dimension ${d}_{s}=2,$ the scaling of the roughness of the domain walls, $w\ensuremath{\sim}{L}^{\ensuremath{\zeta}},$ is consistent with the theoretical prediction $\ensuremath{\zeta}=2/3.$ As the randomness is increased through the transition, the probability distribution of the interfacial tension of domain walls scales in a manner that is clearly consistent with a single second-order transition. At the critical point, the fractal dimensions of domain walls and the fractal dimension of the outer surface of spin clusters are investigated: there are at least two distinct physically important fractal dimensions that describe domain walls. These dimensions are argued to be related by scaling to combinations of the energy scaling exponent $\ensuremath{\theta},$ which determines the violation of hyperscaling, the correlation length exponent $\ensuremath{\nu},$ and the magnetization exponent $\ensuremath{\beta}.$ The value $\ensuremath{\beta}=0.017\ifmmode\pm\else\textpm\fi{}0.005$ computed from finite-size scaling of the magnetization is very nearly zero: this estimate is supported by the study of the spin cluster size distribution at criticality. The variation of configurations in the interior of a sample with boundary conditions is consistent with the hypothesis that there is a single transition separating the disordered phase with one ground state from the ordered phase with two ground states. The array of results, including values for several exponents, are shown to be consistent with a scaling picture and a geometric description of the influence of boundary conditions on the spins. The details of the algorithm used and its implementation are also described.

arXiv:cond-mat/0107489•Physical review. B, Condensed matter•Mar 19, 2002•172 citations

Long-range random transverse-field Ising model in three dimensions

István A. Kovács, Róbert Juhász, Ferenc Iglói

We consider the random transverse-field Ising model in $d=3$ dimensions with long-range ferromagnetic interactions which decay as a power $\ensuremath{\alpha}>d$ with the distance. Using a variant of the strong-disorder renormalization group method we study numerically the phase-transition point from the paramagnetic side. We find that the fixed point controlling the transition is of the strong-disorder type, and based on experience with other similar systems, we expect the results to be qualitatively correct, but probably not asymptotically exact. The distribution of the (sample dependent) pseudocritical points is found to scale with $1/lnL, L$ being the linear size of the sample. Similarly, the critical magnetization scales with ${(lnL)}^{\ensuremath{\chi}}/{L}^{d}$ and the excitation energy behaves as ${L}^{\ensuremath{-}\ensuremath{\alpha}}$. Using extreme-value statistics we argue that extrapolating from the ferromagnetic side the magnetization approaches a finite limiting value and thus the transition is of mixed order.

arXiv:1601.04206•Physical review. B./Physical review. B•May 31, 2016•29 citations

Critical behavior of a 3-D Ising model in a random field

David Paul Landau, H. H. Lee, W. P. Kao

A Monte Carlo method has been used to study a simple cubic Ising ferromagnet in a random quenched magnetic field. The Hamiltonian for this model is ℋ==JΣ(ij)σiσj−ΣiHiσi, where σi,σj=±1, J is the nearest-neighbor interaction constant, and the field Hi=tH is fixed at each site with ti=±1 at random and Σti=0. L×L×L lattices with periodic boundary conditions have been studied for a range of H and T. As expected we find a ferromagnetic ordered state which for small H undergoes a second order phase change to the paramagnetic state with increasing temperature. A finite size scaling analysis of the preliminary data suggests that the critical exponent β is substantially smaller (β∼0.2) than the usual 3-dim Ising value of 0.31. Results obtained for small lattices indicate that below kT/J-2 the transition becomes first order suggesting that a tricritical point appears on the critcal field curve.

Journal of Applied Physics•Mar 1, 1978•66 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. 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.

52•Jan 1, 2015•0 citations
Beyond the Imry-Ma Length: Scaling Behavior in the 3D Random Field XYXYXY Model
We have performed studies of the 3D random field XYXYXY model on L×L×LL \times L \times LL×L×L simple cubic lattices with periodic boundary conditions, with a random field strength of hrh_rhr​ = 1.875, for L=L =L= 64, 96 and 128, using a parallelized Monte Carlo algorithm. We present results for the angle-averaged magnetic structure factor, S(k)S ( k )S(k) at TTT = 1.00, which appears to be the temperature at which small jumps in the magnetization per spin and the energy per spin occur. The magnetization jump per spin scales with size roughly as L−3/4L^{- 3/4}L−3/4, while the energy jump per spin scales like L−3/2L^{- 3/2}L−3/2. The results also indicate the existence of an approximately logarithmic divergence of S(k)S ( k )S(k) as k→0k \to 0k→0. The magnetic susceptibility, χ(k⃗=0)χ(\vec{\bf k} = 0 )χ(k=0), on the other hand, seems to have a value of about 14.2 under these conditions. This suggests the absence of a ferromagnetic phase, and that the lower critical dimension for long-range order in this model is three. Similar results are found for LLL = 64 samples at hrh_rhr​ = 2.0 and TTT = 0.875. We expect that the behavior is qualitatively similar along the entire phase boundary, but the scaling exponents may not be universal. These results appear to be related to recent work on quantum disorder.
Scaling Behavior in the 3D Random Field XYXYXY Model
We have performed studies of the 3D random field XYXYXY model on L×L×LL \times L \times LL×L×L simple cubic lattices with periodic boundary conditions, with a random field strength of hrh_rhr​ = 1.875, for L=64L = 64L=64 and L=96L = 96L=96, using a parallelized Monte Carlo algorithm. We present results for the angle-averaged magnetic structure factor, S(k)S ( k )S(k) at T=1.00T = 1.00T=1.00, which appears to be the temperature at which small jumps in the magnetization per spin and the energy per spin occur. The results indicate the existence of an approximately logarithmic divergence of S(k)S ( k )S(k) as k→0k \to 0k→0. This suggests that the lower critical dimension for long range order in this model is three.
We have performed Monte Carlo studies of the three-dimensional XYXYXY 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 LLL values in the range 16--64, and with random anisotropy strengths of D/2J=1D/2J=1D/2J=1, 2, 3, 6, and $\ensuremath{\infty}$. There is a well-defined finite-temperature critical point, Tc{T}_{c}Tc​, for each of these values of D/2JD/2JD/2J. We present results for the angle-averaged magnetic structure factor S(k)S(k)S(k) at Tc{T}_{c}Tc​ for L=64L=64L=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 TTT is seen for D/2J=6D/2J=6D/2J=6 and $\ensuremath{\infty}$. For our finite values of D/2JD/2JD/2J the scaled magnetization function increases with LLL below Tc{T}_{c}Tc​ and appears to approach an LLL-independent limit for large LLL. This suggests that the system is ferromagnetic below Tc{T}_{c}Tc​.
The nature of the zero-temperature ordering transition in the three-dimensional Gaussian random-field Ising magnet is studied numerically, aided by scaling analyses. Various numerical calculations are used to consistently infer the location of the transition to a high precision. A variety of boundary conditions are imposed on large samples to study the order of the transition and the number of states in the large volume limit. In the ferromagnetic phase, where the domain walls have fractal dimension ds=2,{d}_{s}=2,ds​=2, the scaling of the roughness of the domain walls, $w\ensuremath{\sim}{L}^{\ensuremath{\zeta}},$ is consistent with the theoretical prediction $\ensuremath{\zeta}=2/3.$ As the randomness is increased through the transition, the probability distribution of the interfacial tension of domain walls scales in a manner that is clearly consistent with a single second-order transition. At the critical point, the fractal dimensions of domain walls and the fractal dimension of the outer surface of spin clusters are investigated: there are at least two distinct physically important fractal dimensions that describe domain walls. These dimensions are argued to be related by scaling to combinations of the energy scaling exponent $\ensuremath{\theta},$ which determines the violation of hyperscaling, the correlation length exponent $\ensuremath{\nu},$ and the magnetization exponent $\ensuremath{\beta}.$ The value $\ensuremath{\beta}=0.017\ifmmode\pm\else\textpm\fi{}0.005$ computed from finite-size scaling of the magnetization is very nearly zero: this estimate is supported by the study of the spin cluster size distribution at criticality. The variation of configurations in the interior of a sample with boundary conditions is consistent with the hypothesis that there is a single transition separating the disordered phase with one ground state from the ordered phase with two ground states. The array of results, including values for several exponents, are shown to be consistent with a scaling picture and a geometric description of the influence of boundary conditions on the spins. The details of the algorithm used and its implementation are also described.
We consider the random transverse-field Ising model in d=3d=3d=3 dimensions with long-range ferromagnetic interactions which decay as a power $\ensuremath{\alpha}>d$ with the distance. Using a variant of the strong-disorder renormalization group method we study numerically the phase-transition point from the paramagnetic side. We find that the fixed point controlling the transition is of the strong-disorder type, and based on experience with other similar systems, we expect the results to be qualitatively correct, but probably not asymptotically exact. The distribution of the (sample dependent) pseudocritical points is found to scale with 1/lnL,L1/lnL, L1/lnL,L being the linear size of the sample. Similarly, the critical magnetization scales with ${(lnL)}^{\ensuremath{\chi}}/{L}^{d}$ and the excitation energy behaves as ${L}^{\ensuremath{-}\ensuremath{\alpha}}$. Using extreme-value statistics we argue that extrapolating from the ferromagnetic side the magnetization approaches a finite limiting value and thus the transition is of mixed order.
A Monte Carlo method has been used to study a simple cubic Ising ferromagnet in a random quenched magnetic field. The Hamiltonian for this model is ℋ==JΣ(ij)σiσj−ΣiHiσi, where σi,σj=±1, J is the nearest-neighbor interaction constant, and the field Hi=tH is fixed at each site with ti=±1 at random and Σti=0. L×L×L lattices with periodic boundary conditions have been studied for a range of H and T. As expected we find a ferromagnetic ordered state which for small H undergoes a second order phase change to the paramagnetic state with increasing temperature. A finite size scaling analysis of the preliminary data suggests that the critical exponent β is substantially smaller (β∼0.2) than the usual 3-dim Ising value of 0.31. Results obtained for small lattices indicate that below kT/J-2 the transition becomes first order suggesting that a tricritical point appears on the critcal field curve.
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)O(n)O(n) spin-vector model which describe numerous phenomena's. Here, nnn is the number of vector dimensions and for the XY model n=2n=2n=2. The O(n)O(n)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=4p=4p=4) then the symmetry of the Hamiltonian is lowered and it will be advantageous for the spins to be ordered in directions 90o90^o90o apart. This symmetry is called C4C_{4}C4​, 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<4p<4p<4 becomes dangerously irrelevant for pgeq4p\\geq4pgeq4. 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 Tc=2.2019textnormalkT_c = 2.2019 \\ \\textnormal kTc​=2.2019textnormalk 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 TcT_cTc​ 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 ppp and an investigation of the hexagonal anisotropy p=6p=6p=6 has been started.