arXiv subjects

P. H. Lundow

Publications and source records attributed to P. H. Lundow.

At least 19 recordsLinked to original sources

arXiv2023

Revising the universality class of the four-dimensional Ising model

The aim of this paper is to determine the behavior of the specific heat of the 4-dimensional Ising model at the critical temperature, and via that determine if the Ising model and the $ϕ^4$-model belong to the same universality class in dimension 4. In order to do this we have carried out what is currently the largest scale simulations of the 4-dimensional Ising model, extending the lattices size up to $L=256$ and the number of samples per size by several orders of magnitude compared to earlier works, keeping track of data for both the canonical and microcanonical ensembles. Our conclusion is that the Ising model has a bounded specific heat, while the $ϕ^4$-model is known to have a logarithmic divergence at the critical point. Hence the two models belong to distinct universality classes in dimension 4.

cond-mat.stat-mech↗

arXiv2022

Damage spreading in the random cluster model

We investigate the damage spreading effect in the Fortuin-Kasteleyn random cluster model for 2- and 3-dimensional grids with periodic boundary. For 2D the damage function has a global maximum at $p=\sqrt{q}/(1+\sqrt{q})$ for all $q>0$ and also local maxima at $p=1/2$ and $p=q/(1+q)$ for $q\lesssim 0.75$. For 3D we observe a local maximum at $p=q/(1+q)$ for $q\lesssim 0.46$ and a global maximum at $p=1/2$ for $q\lesssim 4.5$. The chaotic phase of the model's $(p,q)$-parameter space is where the coupling time is of exponential order and we locate points on its boundary. For 3-dimensional grids the lower bound of this phase may be equal to the corresponding critical point of the $q$-state Potts model for $q\ge 3$.

cond-mat.stat-mech↗

arXiv2021

Boundary effects on finite-size scaling for the 5-dimensional Ising model

High-dimensional ($d\ge 5$) Ising systems have mean-field critical exponents. However, at the critical temperature the finite-size scaling of the susceptibility $χ$ depends on the boundary conditions. A system with periodic boundary conditions then has $χ\propto L^{5/2}$. Deleting the $5L^4$ boundary edges we receive a system with free boundary conditions and now $χ\propto L^2$. In the present work we find that deleting the $L^4$ boundary edges along just one direction is enough to have the scaling $χ\propto L^2$. It also appears that deleting $L^3$ boundary edges results in an intermediate scaling, here estimated to $χ\propto L^{2.275}$. We also study how the energy and magnetisation distributions change when deleting boundary edges.

cond-mat.stat-mech↗

arXiv2020

Efficient computation of permanents, with applications to boson sampling and random matrices

In order to find the outcome probabilities of quantum mechanical systems like the optical networks underlying Boson sampling, it is necessary to be able to compute the permanents of unitary matrices, a computationally hard task. Here we first discuss how to compute the permanent efficiently on a parallel computer, followed by algorithms which provide an exponential speed-up for sparse matrices and linear run times for matrices of limited bandwidth. The parallel algorithm has been implemented in a freely available software package, also available in an efficient serial version. As part of the timing runs for this package we set a new world record for the matrix order on which a permanent has been computed. Next we perform a simulation study of several conjectures regarding the distribution of the permanent for random matrices. Here we focus on permanent anti-concentration conjecture, which has been used to find the classical computational complexity of Boson sampling. We find a good agreement with the basic versions of these conjectures and based on our data we propose refined versions of some of them. For small systems we also find noticable deviations from a propose strengthening of a bound for the number of photons in a Boson sampling system.

quant-ph↗

arXiv2018

Revisiting the cavity-method threshold for random 3-SAT

A detailed Monte Carlo-study of the satisfiability threshold for random 3-SAT has been undertaken. In combination with a monotonicity assumption we find that the threshold for random 3-SAT satisfies $α_3 \leq 4.262$. If the assumption is correct, this means that the actual threshold value for $k=3$ is lower than that given by the cavity method. In contrast the latter has recently been shown to give the correct value for large $k$. Our result thus indicate that there are distinct behaviors for $k$ above and below some critical $k_c$, and the cavity method may provide a correct mean-field picture for the range above $k_c$.

cond-mat.stat-mech↗

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.

cond-mat.dis-nn↗

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.

cond-mat.dis-nn↗

arXiv2016

The scaling window of the 5D Ising model with free boundary conditions

The five-dimensional Ising model with free boundary conditions has recently received a renewed interest in a debate concerning the finite-size scaling of the susceptibility near the critical temperature. We provide evidence in favour of the conventional scaling picture, where the susceptibility scales as $O(L^2)$ inside a critical scaling window of width $O(1/L^2)$. Our results are based on Monte Carlo data gathered on system sizes up to $L=79$ (ca. three billion spins) for a wide range of temperatures near the critical point. We analyse the magnetisation distribution, the susceptibility and also the scaling and distribution of the size of the Fortuin-Kasteleyn cluster containing the origin. The probability of this cluster reaching the boundary determines the correlation length, and its behaviour agrees with the mean field critical exponent $δ=3$, that the scaling window has width $O(1/L^2)$.

cond-mat.stat-mech↗

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↗

arXiv2015

The discontinuity of the specific heat for the 5D Ising model

In this paper we investigate the behaviour of the specific heat around the critical point of the Ising model in dimension 5 to 7. We find a specific heat discontinuity, like that for the mean field Ising model, and provide estimates for the left and right hand limits of the specific heat at the critical point. We also estimate the singular exponents, describing how the specific heat approaches those limits. Additionally, we make a smaller scale investigation of the same properties in dimension 6 and 7, and provide strongly improved estimates for the critical termperature $K_c$ in $d=5,6,7$ which bring the best MC-estimate closer to those obtained by long high temperature series expanions.

cond-mat.stat-mech↗