arXiv subjects

Ronald Fisch

Publications and source records attributed to Ronald Fisch.

At least 19 recordsLinked to original sources

arXiv2022

Model for Dipolar Glass and Relaxor Ferroelectric Behavior

Heat bath Monte Carlo simulations have been used to study a 12-state discretized Heisenberg model with a type of random field, for several values of the randomness coupling parameter $h_R$. The 12 states correspond to the [110] directions of a cube. Simple cubic lattices of size $128 \times 128 \times 128$ with periodic boundary conditions were used, and 32 samples were studied for each value of $h_R$. The model has the standard nonrandom two-spin exchange term with coupling energy $J$ and a field which adds an energy $h_R$ to two of the 12 spin states, chosen randomly and independently at each site. We provide results for the cases $h_R / J =$ -2.5, -2.0, -1.5, 3.0 and 4.0. For all these cases except $h_R / J =$ -2.5, we see an apparently sharp phase transition at a temperature $T_c$ where the specific heat and the longitudinal susceptibility are peaked. At $T_c$, the behavior of the peak in the structure factor, $S ({\bf k} )$, at small $|{\bf k}|$ is a straight line on a log-log plot. However, the value of the slope of this line is different for $h_R /J =$ -1.5 and 3.0 than it is for $h_R / J =$ -2.0 and 4.0. We believe that the first two cases are showing the behavior of a cubic fixed point in a weak random field, and the behavior of the second two cases are showing the behavior of an isotropic fixed point when the Imry-Ma length is smaller than the sample size. Below $T_c$, these $L = 128$ samples show ferroelectric order, and this order rapidly becomes oriented along one of the eight [111] directions as $T$ is reduced. This rotation of the ordering direction is caused by the cubic anisotropy. For $h_R / J =$ -2.5, we do not see clear evidence of a single well-defined $T_c$.

cond-mat.stat-mech↗

arXiv2021

Random Field Critical Scaling in a Model of Dipolar Glass and Relaxor Ferroelectric Behavior

Heat bath Monte Carlo simulations have been used to study a 12-state discretized Heisenberg model with a new type of random field on simple cubic lattices of size $128 \times 128 \times 128$. The 12 states correspond to the [110] directions of a cube. The model has the standard nonrandom two-spin exchange term with coupling energy $J$ and a random field which consists of adding an energy $h_R$ to two of the 12 spin states, chosen randomly and independently at each site. We report on the case $h_R / J = 3$, which has a sharp phase transition at about $T_c / J = 1.40625$. Below $T_c$, the model has long-range ferroelectric order oriented along one of the eight [111] directions. At $T_c$, the behavior of the peak in the structure factor, $S ({\bf k} )$, at small $|{\bf k}|$ is a straight line on a log-log plot, which gives the result $\bar{\eta} = 1.214 \pm 0.014$. The onset of orientational order below $T_c$ is very rapid for this value of $h_R$. There are peaks in the specific heat and longitudinal susceptibility at $T_c$. Below $T_c$ there is a strong correction to ordinary scaling, which is probably caused by the cubic anisotropy, which is a dangerous irrelevant variable.

cond-mat.dis-nn↗

arXiv2020

Behavior of the Random Field $XY$ Model on Simple Cubic Lattices at $h_r = 1.5$

We have performed studies of the 3D random field $XY$ model on 32 samples of $L \times L \times L$ simple cubic lattices with periodic boundary conditions, with a random field strength of $h_r$ = 1.5, for $L =$ 128, using a parallelized Monte Carlo algorithm. We present results for the sample-averaged magnetic structure factor, $S (\vec{\bf k})$ over a range of temperature, using both random hot start and ferromagnetic cold start initial states, and $\vec{\bf k}$ along the [1,0,0] and [1,1,1] directions. At $T =$ 1.875, $S (\vec{\bf k})$ shows a broad peak near $|\vec{\bf k}| = 0$, with a correlation length which is limited by thermal fluctuations, rather than the lattice size. As $T$ is lowered, this peak grows and sharpens. By $T =$ 1.5, it is clear that the correlation length is larger than $L =$ 128. The lowest temperature for which $S (\vec{\bf k})$ was calculated is $T =$ 1.421875, where the hot start and cold start initial conditions are usually not finding the same local minimum in the phase space. Our results are consistent with the idea that there is a finite value of $T$ below which $S (\vec{\bf k})$ diverges slowly as $|\vec{\bf k}|$ goes to zero. This divergence would imply that the relaxation time of the spins is also diverging. That is the signature of an ergodicity-breaking phase transition.

cond-mat.stat-mech↗

arXiv2019

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

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.

cond-mat.stat-mech↗

arXiv2018

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

cond-mat.dis-nn↗

arXiv2013

Resonant Interactions Along the Critical Line of the Riemann Zeta Function

We have studied some properties of the special Gram points of the Riemann zeta function which lie on contour lines ${\bf Im}(ζ( s )) = 0$ which do not contain zeroes of $ζ( s )$. We find that certain functions of these points, which all lie on the critical line ${\bf Re}( s ) = 1/2$, are correlated in remarkable and unexpected ways. We have data up to a height of $t = 10^4$, where $s = σ+ it$.

math.NT↗

arXiv2012

Evidence of Long Range Order in the Riemann Zeta Function

We have done a statistical analysis of some properties of the contour lines Im$(ζ(s))$ = 0 of the Riemann zeta function. We find that this function is broken up into strips whose average width on the critical line does not appear to vary with height. We also compute the position of the primary zero for the lowest 200 strips, and find that this probability distribution also appears to be scale invariant.

math.NT↗

arXiv2010

Glassy Freezing and Long-Range Order in the 3D Random Field $XY$ Model

Monte Carlo studies of the 3D random field $XY$ model on simple cubic lattices of size $64^3$, using two different isotropic random-field probability distributions of moderate strength, show a glassy freezing behavior above $T_c$, and long-range order below $T_c$, consistent with our earlier results for weaker and stronger random fields. This model should describe random pinning in vortex lattices in type-II superconducting alloys, charge-density wave materials, and decagonal quasicrystals.

cond-mat.dis-nn↗

arXiv2009

Finite-Size Scaling Critical Behavior of Randomly Pinned Spin-Density Waves

We have performed Monte Carlo studies of the 3D $XY$ model with random uniaxial anisotropy, which is a model for randomly pinned spin-density waves. We study $L \times L \times L$ simple cubic lattices, using $L$ values in the range 16 to 64, and with random anisotropy strengths of $D / 2 J$ = 1, 2, 3, 6 and $\infty$. There is a well-defined finite temperature critical point, $T_c$, for each these values of $D / 2 J$. 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 $D / 2 J$ = 6 and $\infty$. For our finite values of $D / 2 J$ 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 $T_c$.

cond-mat.dis-nn↗

arXiv2007

Structure Factor of the 3D Random Field XY Model

We have performed Monte Carlo studies of the 3D random field XY model on $L \times L \times L$ simple cubic lattices, with random field strengths of $h_r$ = 1 and 2. We present results for the angle-averaged magnetic structure factor, $S (k)$ at $L = 64$. Our results appear to indicate a phase transition into a ferromagnetic state. This is made possible by the existence of a Griffiths singularity. It appears that at the phase transition $M^2$ jumps to zero discontinuously, with a latent heat which is probably subextensive.

cond-mat.dis-nn↗

arXiv2007

Aspect-Ratio Scaling of Domain Wall Entropy for the 2D $\pm J$ Ising Spin Glass

The ground state entropy of the 2D Ising spin glass with +1 and -1 bonds is studied for $L \times M$ square lattices with $L \le M$ and $p$ = 0.5, where $p$ is the fraction of negative bonds, using periodic and/or antiperiodic boundary conditions. From this we obtain the domain wall entropy as a function of $L$ and $M$. It is found that for domain walls which run in the short, $L$ direction, there are finite-size scaling functions which depend on the ratio $M / L^{d_S}$, where $d_S = 1.22 \pm 0.01$. When $M$ is larger than $L$, very different scaling forms are found for odd $L$ and even $L$. For the zero-energy domain walls, which occur when $L$ is even, the probability distribution of domain wall entropy becomes highly singular, and apparently multifractal, as $M / L^{d_S}$ becomes large.

cond-mat.dis-nn↗

arXiv2007

Subextensive singularity in the 2D $\pm J$ Ising spin glass

The statistics of low energy states of the 2D Ising spin glass with +1 and -1 bonds are studied for $L \times L$ square lattices with $L \le 48$, and $p$ = 0.5, where $p$ is the fraction of negative bonds, using periodic and/or antiperiodic boundary conditions. The behavior of the density of states near the ground state energy is analyzed as a function of $L$, in order to obtain the low temperature behavior of the model. For large finite $L$ there is a range of $T$ in which the heat capacity is proportional to $T^{5.33 \pm 0.12}$. The range of $T$ in which this behavior occurs scales slowly to $T = 0$ as $L$ increases. Similar results are found for $p$ = 0.25. Our results indicate that this model probably obeys the ordinary hyperscaling relation $d ν= 2 - α$, even though $T_c = 0$. The existence of the subextensive behavior is attributed to long-range correlations between zero-energy domain walls, and evidence of such correlations is presented.

cond-mat.dis-nn↗

arXiv2007

Comment on "Conformal invariance and stochastic Loewner evolution processes in two-dimensional Ising spin glasses"

By combining the scaling relation of Amoruso {\it et al.}, PRL {\bf 97}, 267202 (2006) with standard droplet model assumptions, a value $θ= (\sqrt{6} - 3) / 2$ is obtained. This conjecture is reasonably consistent with the best existing numerical calculations, and may be exact.

cond-mat.dis-nn↗

arXiv2007

Ground-State and Domain-Wall Energies in the Spin-Glass Region of the 2D $\pm J$ Random-Bond Ising Model

The statistics of the ground-state and domain-wall energies for the two-dimensional random-bond Ising model on square lattices with independent, identically distributed bonds of probability $p$ of $J_{ij}= -1$ and $(1-p)$ of $J_{ij}= +1$ are studied. We are able to consider large samples of up to $320^2$ spins by using sophisticated matching algorithms. We study $L \times L$ systems, but we also consider $L \times M$ samples, for different aspect ratios $R = L / M$. We find that the scaling behavior of the ground-state energy and its sample-to-sample fluctuations inside the spin-glass region ($p_c \le p \le 1 - p_c$) are characterized by simple scaling functions. In particular, the fluctuations exhibit a cusp-like singularity at $p_c$. Inside the spin-glass region the average domain-wall energy converges to a finite nonzero value as the sample size becomes infinite, holding $R$ fixed. Here, large finite-size effects are visible, which can be explained for all $p$ by a single exponent $ω\approx 2/3$, provided higher-order corrections to scaling are included. Finally, we confirm the validity of aspect-ratio scaling for $R \to 0$: the distribution of the domain-wall energies converges to a Gaussian for $R \to 0$, although the domain walls of neighboring subsystems of size $L \times L$ are not independent.

cond-mat.dis-nn↗

arXiv2006

Finite-Size Scaling of the Domain Wall Entropy Distributions for the 2D $\pm J$ Ising Spin Glass

The statistics of domain walls for ground states of the 2D Ising spin glass with +1 and -1 bonds are studied for $L \times L$ square lattices with $L \le 48$, and $p$ = 0.5, where $p$ is the fraction of negative bonds, using periodic and/or antiperiodic boundary conditions. When $L$ is even, almost all domain walls have energy $E_{dw}$ = 0 or 4. When $L$ is odd, most domain walls have $E_{dw}$ = 2. The probability distribution of the entropy, $S_{dw}$, is found to depend strongly on $E_{dw}$. When $E_{dw} = 0$, the probability distribution of $|S_{dw}|$ is approximately exponential. The variance of this distribution is proportional to $L$, in agreement with the results of Saul and Kardar. For $E_{dw} = k > 0$ the distribution of $S_{dw}$ is not symmetric about zero. In these cases the variance still appears to be linear in $L$, but the average of $S_{dw}$ grows faster than $\sqrt{L}$. This suggests a one-parameter scaling form for the $L$-dependence of the distributions of $S_{dw}$ for $k > 0$.

cond-mat.stat-mech↗

arXiv2006

Finite-size scaling of the Domain Wall Entropy for the 2D \pm J Ising Spin Glass

The statistics of domain walls for ground states of the 2D Ising spin glass with +1 and -1 bonds are studied for $L \times L$ square lattices with $L \le 20$, and $x$ = 0.25 and 0.5, where $x$ is the fraction of negative bonds, using periodic and/or antiperiodic boundary conditions. Under these conditions, almost all domain walls have an energy $E_{dw}$ equal to 0 or 4. The probability distribution of the entropy, $S_{dw}$, is found to depend strongly on $E_{dw}$. The results for $S_{dw}$ when $E_{dw} = 4$ agree with the prediction of the droplet model. Our results for $S_{dw}$ when $E_{dw} = 0$ agree with those of Saul and Kardar. In addition, we find that the distributions do not appear to be Gaussian in that case. The special role of $E_{dw} = 0$ domain walls is discussed, and the discrepancy between the prediction of Amoruso, Hartmann, Hastings and Moore and the result of Saul and Kardar is explained.

cond-mat.dis-nn↗

arXiv2003

Random Field Models for Relaxor Ferroelectric Behavior

Heat bath Monte Carlo simulations have been used to study a four-state clock model with a type of random field on simple cubic lattices. The model has the standard nonrandom two-spin exchange term with coupling energy $J$ and a random field which consists of adding an energy $D$ to one of the four spin states, chosen randomly at each site. This Ashkin-Teller-like model does not separate; the two random-field Ising model components are coupled. When $D / J = 3$, the ground states of the model remain fully aligned. When $D / J \ge 4$, a different type of ground state is found, in which the occupation of two of the four spin states is close to 50%, and the other two are nearly absent. This means that one of the Ising components is almost completely ordered, while the other one has only short-range correlations. A large peak in the structure factor $S (k)$ appears at small $k$ for temperatures well above the transition to long-range order, and the appearance of this peak is associated with slow, "glassy" dynamics. The phase transition into the state where one Ising component is long-range ordered appears to be first order, but the latent heat is very small.

cond-mat.dis-nn↗

arXiv1998

Quasi-Long-Range Order in Random-Anisotropy Heisenberg Models

Monte Carlo simulations have been used to study a discretized Heisenberg ferromagnet (FM) with random uniaxial single-site anisotropy on $L \times L \times L$ simple cubic lattices, for $L$ up to 64. The spin variable on each site is chosen from the twelve [110] directions. The random anisotropy has infinite strength and a random direction on a fraction $x$ of the sites of the lattice, and is zero on the remaining sites. In many respects the behavior of this model is qualitatively similar to that of the corresponding random-field model. Due to the discretization, for small $x$ at low temperature there is a [110] FM phase. For $x>0$ there is an intermediate quasi-long-range ordered (QLRO) phase between the paramagnet and the ferromagnet, which is characterized by a $|k|^{-3}$ divergence of the magnetic structure factor S$(k)$ for small $k$, but no true FM order. At the transition between the paramagnetic and QLRO phases S$(k)$ diverges like $|k|^{-2}$. The limit of stability of the QLRO phase is somewhat greater than $x=0.5$. For $x$ close to 1 the low temperature form of S$(k)$ can be fit by a Lorentzian, with a correlation length estimated to be $11 \pm 1$ at $x=1.0$ and $25 \pm 5$ at $x=0.75$.

cond-mat.dis-nn↗