arXiv subjects

I. A. Campbell

Publications and source records attributed to I. A. Campbell.

At least 19 recordsLinked to original sources

arXiv2018

Hyperscaling violation in Ising Spin Glasses

In addition to the standard scaling rules relating critical exponents at second order transitions, hyperscaling rules involve the dimension of the model. It is well known that in canonical Ising models hyperscaling rules are modified above the upper critical dimension. It was shown by M. Schwartz in 1991 that hyperscaling can also break down in Ising systems with quenched random interactions; Random Field Ising models which are in this class have been intensively studied. Here numerical Ising Spin Glass data relating the scaling of the normalized Binder cumulant to that of the reduced correlation length are presented for dimensions 3, 4, 5 and 7. Hyperscaling is clearly violated in dimensions 3 and 4 as well as above the upper critical dimension $D=6$. Estimates are obtained for the "violation of hyperscaling exponent" values in the various models.

cond-mat.dis-nn↗

arXiv2018

The Ising universality class in dimension three: corrections to scaling

Simulation data are analyzed for four 3D spin-$1/2$ Ising models: on the FCC lattice, the BCC lattice, the SC lattice and the Diamond lattice. The observables studied are the susceptibility, the reduced second moment correlation length, and the normalized Binder cumulant. From measurements covering the entire paramagnetic temperature regime the corrections to scaling are estimated. We conclude that a correction term having an exponent which is consistent within the statistics with the bootstrap value of the universal subleading thermal confluent correction exponent, $θ_{2} \sim 2.454(3)$, is almost always present with a significant amplitude. In all four models, for the normalized Binder cumulant the leading confluent correction term has zero amplitude. This implies that the universal ratio of leading confluent correction amplitudes $a_{χ_{4}}/a_χ = 2$ in the 3D Ising universality class.

cond-mat.stat-mech↗

arXiv2018

The bimodal Ising spin glass in dimension two : the anomalous dimension $η$

Direct measurements of the spin glass correlation function $G(R)$ for Gaussian and bimodal Ising spin glasses in dimension two have been carried out in the temperature region $T \sim 1$. In the Gaussian case the data are consistent with the known anomalous dimension value $η\equiv 0$. For the bimodal spin glass in this temperature region $T > T^{*}(L)$, well above the crossover $T^{*}(L)$ to the ground state dominated regime, the effective exponent $η$ is clearly non-zero and the data are consistent with the estimate $η\sim 0.28(4)$ given by McMillan in 1983 from similar measurements. Measurements of the temperature dependence of the Binder cumulant $U_{4}(T,L)$ and the normalized correlation length $ξ(T,L)/L$ for the two models confirms the conclusion that the 2D bimodal model has a non-zero effective $η$ both below and above $T^{*}(L)$. The 2D bimodal and Gaussian interaction distribution Ising spin glasses are not in the same Universality class.

cond-mat.dis-nn↗

arXiv2017

Hyperscaling breakdown and Ising Spin Glasses: the Binder cumulant

Among the Renormalization Group Theory scaling rules relating critical exponents, there are hyperscaling rules involving the dimension of the system. It is well known that in Ising models hyperscaling breaks down above the upper critical dimension. It was shown by M. Schwartz [Europhys. Lett. {\bf 15}, 777 (1991)] that the standard Josephson hyperscaling rule can also break down in Ising systems with quenched random interactions. A related Renormalization Group Theory hyperscaling rule links the critical exponents for the normalized Binder cumulant and the correlation length in the thermodynamic limit. An appropriate scaling approach for analyzing measurements from criticality to infinite temperature is first outlined. Numerical data on the scaling of the normalized correlation length and the normalized Binder cumulant are shown for the canonical Ising ferromagnet model in dimension three where hyperscaling holds, for the Ising ferromagnet in dimension five (so above the upper critical dimension) where hyperscaling breaks down, and then for Ising spin glass models in dimension three where the quenched interactions are random. For the Ising spin glasses there is a breakdown of the normalized Binder cumulant hyperscaling relation in the thermodynamic limit regime, with a return to size independent Binder cumulant values in the finite-size scaling regime around the critical region.

cond-mat.dis-nn↗

arXiv2017

Ising spin glasses in dimension two; universality and non-universality

Following numerous earlier studies, extensive simulations and analyses were made on the continuous interaction distribution Gaussian model and the discrete bimodal interaction distribution Ising Spin Glass (ISG) models in dimension two (P.H. Lundow and I.A. Campbell, Phys. Rev. E {\bf 93}, 022119 (2016)). Here we further analyse the bimodal and Gaussian data together with data on two other continuous interaction distribution 2D ISG models, the uniform and the Laplacian models, and three other discrete interaction distribution models, a diluted bimodal model, an "anti-diluted" model, and a more exotic symmetric Poisson model. Comparisons between the three continuous distribution models show that not only do they share the same exponent $η\equiv 0$ but that to within the present numerical precision they share the same critical exponent $ν$ also, and so lie in a single universality class. On the other hand the critical exponents of the four discrete distribution models are not the same as those of the continuous distributions, and differ from one discrete distribution model to another. Discrete distribution ISG models in dimension two have non-zero values of the critical exponent $η$; they do not lie in a single universality class.

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arXiv2016

Ising Spin Glasses and Renormalization Group Theory: the Binder cumulant

Numerical data on scaling of the normalized Binder cumulant and the normalized correlation length are shown for the Thermodynamic limit regime, first for canonical Ising ferromagnet models and then for a range of Ising spin glass models. A fundamental Renormalization Group Theory rule linking the critical exponents for the two observables is well obeyed in the Ising models, but not for the Ising spin glasses in dimensions three and four. We conclude that there is a violation of a standard Josephson hyperscaling rule in Ising spin glasses.

cond-mat.dis-nn↗

arXiv2016

Ising Spin Glasses in dimension five

Ising spin glass models with bimodal, Gaussian, uniform and Laplacian interaction distributions in dimension five are studied through detailed numerical simulations. The data are analyzed in both the finite-size scaling regime and the thermodynamic limit regime. It is shown that the values of critical exponents and of dimensionless observables at criticality are model dependent. Models in a single universality class have identical values for each of these critical parameters, so Ising spin glass models in dimension five with different interaction distributions each lie in different universality classes. This result confirms conclusions drawn from measurements in dimension four and dimension two.

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arXiv2015

The bimodal and Gaussian Ising Spin Glasses in dimension two revisited

A new analysis is given of numerical simulation data on the archetype square lattice Ising Spin Glasses (ISG) with a bimodal ($\pm J$) and Gaussian interaction distributions. It is well established that the ordering temperature of both models is zero. The Gaussian has a non-degenerate ground state so exponent $η\equiv 0$ and it has a continuous distribution of energy levels. For the bimodal model, above a size dependent cross-over temperature $T^{*}(L)$ there is a regime of effectively continuous energy levels; below $T^{*}(L)$ there is a distinct regime dominated by the highly degenerate ground state plus an energy gap to the excited states. $T^{*}(L)$ tends to zero at very large $L$ leaving only the effectively continuous regime in the thermodynamic limit. We show that in this regime the critical exponent $η$ is not zero, so the effectively continuous regime $2$D bimodal ISG is not in the same universality class as the $2$D Gaussian ISG. The simulation data on both models are analyzed using a scaling variable $τ= T^2/(1+T^2)$ suitable for zero temperature transition ISGs, together with appropriate scaling expressions. Accurate simulation estimates can be obtained for the temperature dependence of the thermodynamic limit reduced susceptibility $χ(τ)$ and second moment correlation length $ξ(τ)$ over the entire range of temperature from zero to infinity. The Gaussian critical exponent from the simulations $ν= 3.5(1)$ is in full agreement with the well established value from the literature. The bimodal exponent from the thermodynamic limit regime analysis is $ν= 4.2(1)$, once again different from the Gaussian value.

cond-mat.dis-nn↗

arXiv2015

Non-self-averaging in Ising spin glasses; hyperuniversality

Ising spin glasses with bimodal and Gaussian near-neighbor interaction distributions are studied through numerical simulations. The non-self-averaging (normalized inter-sample variance) parameter $U_{22}(T,L)$ for the spin glass susceptibility (and for higher moments $U_{nn}(T,L)$) is reported for dimensions 2, 3, 4, 5 and 7. In each dimension $d$ the non-self-averaging parameters in the paramagnetic regime vary with the sample size L and the correlation length $ξ(T,L)$ as $U_{nn}(β,L) = [K_{d}ξ(T,L)/L]^d$, and so follow a renormalization group law due to Aharony and Harris (1991). Empirically, it is found that the $K_{d}$ values are independent of d to within the statistics. The maximum values $[U_{nn}(T,L)]_{\max}$ are almost independent of L in each dimension, and remarkably the estimated thermodynamic limit critical $[U_{nn}(T,L)]_{\max}$ peak values are also dimension-independent to within the statistics and so are "hyperuniversal". These results show that the form of the spin-spin correlation function distribution at criticality in the large $L$ limit is independent of dimension within the ISG family. Inspection of published non-self-averaging data for 3D Heisenberg and XY spin glasses the light of the Ising spin glass non-self-averaging results show behavior incompatible with a spin-driven ordering scenario, but compatible with that expected on a chiral-driven ordering interpretation.

cond-mat.dis-nn↗

arXiv2015

The Ising Spin Glass in dimension four

The critical behaviors of the bimodal and Gaussian Ising spin glass (ISG) models in dimension four are studied through extensive numerical simulations, and from an analysis of high temperature series expansion (HTSE) data of Klein {\it et al.} (1991). The simulations include standard finite size scaling measurements, thermodynamic limit regime measurements, and analyses which provide estimates of critical exponents without any consideration of the critical temperature. The higher order HTSE series for the bimodal model provide accurate estimates of the critical temperature and critical exponents. These estimates are independent of and fully consistent with the simulation values. Comparisons between ISG models in dimension four show that the critical exponents and the critical constants for dimensionless observables depend on the form of the interaction distribution of the model.

cond-mat.dis-nn↗

arXiv2015

Evidence for non-universal scaling in dimension four Ising spin glasses

The critical behavior of the Binder cumulant for Ising spin glasses in dimension four are studied through simulation measurements. Data for the bimodal interaction model are compared with those for the Laplacian interaction model. Special attention is paid to scaling corrections. The limiting infinite size value at criticality for this dimensionless variable is a parameter characteristic of a universality class. This critical limit is estimated to be equal to $0.523(3)$ in the bimodal model and to $0.473(3)$ in the Laplacian model.

cond-mat.dis-nn↗

arXiv2014

The Ising Spin Glass in dimension four; non-universality

Extensive simulations are made on Ising Spin Glasses (ISG) with Gaussian, Laplacian and bimodal interaction distributions in dimension four. Standard finite size scaling analyses near and at criticality provide estimates of the critical inverse temperatures $β_c$, critical exponents, and critical values of a number of dimensionless parameters. Independent estimates are obtained for $β_c$ and the exponent $ν$ from thermodynamic derivative peak data. A detailed explanation is given of scaling in the thermodynamic limit with the ISG scaling variable $τ= 1-β^2/β_c^2$ and the appropriate scaling expressions. Data over the entire paramagnetic range of temperatures are analysed in order to obtain further estimates of the critical exponents together with correction to scaling terms. The Privman-Fisher ansatz then leads to compact scaling expressions for the whole paramagnetic regime and for all sample sizes L. Comparisons between the 4d ISG models show that the critical dimensionless parameters characteristic of a universality class, and the susceptibility and correlation length critical exponents $γ$ and $ν$, depend on the form of the interaction distribution. From these observations it can be deduced that critical exponents are not universal in ISGs, at least in dimension four.

cond-mat.dis-nn↗

arXiv2014

Effect of Disorder in the Frustrated Ising FCC Antiferromagnet: Phase Diagram and Stretched Exponential Relaxation

We study the phase transition in a face-centered-cubic antiferromagnet with Ising spins as a function of the concentration $p$ of ferromagnetic bonds randomly introduced into the system. Such a model describes the spin-glass phase at strong bond disorder. Using the standard Monte Carlo simulation and the powerful Wang-Landau flat-histogram method, we carry out in this work intensive simulations over the whole range of $p$. We show that the first-order transition disappears with a tiny amount of ferromagnetic bonds, namely $p\sim 0.01$, in agreement with theories and simulations on other 3D models. The antiferromagnetic long-range order is also destroyed with a very small $p$ ($\simeq 5%$). With increasing $p$, the system changes into a spin glass and then to a ferromagnetic phase when $p>0.65$. The phase diagram in the space ($T_c,p$) shows an asymmetry, unlike the case of the $\pm J$ Ising spin glass on the simple cubic lattice. We calculate the relaxation time around the spin-glass transition temperature and we show that the spin autocorrelation follows a stretched exponential relaxation law where the factor $b$ is equal to $\simeq 1/3$ at the transition as suggested by the percolation-based theory. This value is in agreement with experiments performed on various spin glasses and with Monte Carlo simulations on different SG models.

cond-mat.stat-mech↗

arXiv2013

Critical exponents of the binomial Ising Spin Glass in dimension four; non-universality

Extensive simulations are made on the bimodal Ising Spin Glass (ISG) in dimension four. The transition temperature is established using a combination of standard finite size scaling and of thermodynamic derivative peak data. Measurements in the thermodynamic limit regime are analysed so as to estimate critical exponents and confluent correction terms. Comparisons with results on other 4d ISGs show that the susceptibility and correlation length critical exponents $γ$ and $ν$ depend on the form of the interaction distribution. From this observation it can be deduced that critical exponents are not universal in ISGs.

cond-mat.dis-nn↗

arXiv2013

Critical exponents in Ising Spin Glasses

Extensive simulations are made of the spin glass susceptibility and correlation length in five dimension Ising Spin Glasses (ISGs) with Gaussian and bimodal interaction distributions. Once the transition temperature is accurately established using a standard criterion, critical exponents and correction terms can be readily estimated by extrapolating measurements made in the thermodynamic limit regime. The data show that the critical exponents of the susceptibility $γ$ and of the correlation length $ν$ depend on the form of the interaction distribution. This observation implies that quite generally critical exponents are not universal in ISGs.

cond-mat.dis-nn↗

arXiv2013

Link overlaps at Criticality and Universality in Ising Spin Glasses

Extensive simulations are made of link and spin overlaps in four and five dimensional Ising Spin Glasses (ISGs). Moments and moment ratios of the mean link overlap distributions (the variance, the kurtosis and the skewness) show clear critical behavior around the ISG ordering temperature. The link overlap measurements can be used to identify the ISG transition accurately; the link overlap is often a more efficient tool in this context than the spin overlap because the link overlap inter-sample variability is much weaker. Once the transition temperature is accurately established, critical exponents can be readily estimated by extrapolating measurements made in the thermodynamic limit regime. The data show that the bimodal and Gaussian spin glass susceptibility exponents $γ$ are different from each other, both in dimension 5 and in dimension 4. Hence ISG critical exponents are not universal in a given dimension, but depend on the form of the interaction distribution.

cond-mat.dis-nn↗

arXiv2013

The Ising ferromagnet in dimension five: link and spin overlaps

In the simple [hyper]cubic five dimension near neighbor interaction Ising ferromagnet, extensive simulation measurements are made of the link overlap and the spin overlap distributions. These "two replica" measurements are standard in the Spin Glass context but are not usually recorded in ferromagnet simulations. The moments and moment ratios of these distributions (the variance, the kurtosis and the skewness) show clear critical behaviors at the known ordering temperature of the ferromagnet. Analogous overlap effects can be expected quite generally in Ising ferromagnets in any dimension. The link overlap results in particular, with peaks at criticality in the kurtosis and the skewness, also have implications for Spin Glasses.

cond-mat.stat-mech↗

arXiv2012

The Ising Spin Glass in dimension five: link overlaps

Extensive simulations are made of the link overlap in five dimensional Ising Spin Glasses (ISGs) through and below the ordering transition. Moments of the mean link overlap distributions (the kurtosis and the skewness) show clear critical maxima at the ISG ordering temperature. These criteria can be used as efficient tools to identify a freezing transition quite generally and in any dimension. In the ISG ordered phase the mean link overlap distribution develops a strong two peak structure, with the link overlap spectra of individual samples becoming very heterogeneous. There is no tendency towards a "trivial" universal single peak distribution in the range of size and temperature covered by the data.

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