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• 1. Basylko, S. A.
Condensed Matter Theory, Department of Theoretical Physics, AlbaNova University Center, KTH.
One-dimensional Kondo lattice model studied through numerical diagonalization2008In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 77, no 7, article id 073103Article in journal (Refereed)

The one-dimensional Kondo lattice model is studied by means of the numerical diagonalization method. By using massively parallel computations, we were able to study lattices large enough to obtain convergent results for electron densities n <= 2/3. For such densities, an additional ferromagnetic region is found inside the paramagnetic phase. Also, a region is found where the localized spins participate in the low-energy dynamics together with the conduction electrons, thus resulting in a large Fermi surface. These results are an independent confirmation of previous density matrix renormalization group results.

• 2. Belonoshko, A. B.
Properties of the fcc Lennard-Jones crystal model at the limit of superheating2007In: PHYSICAL REVIEW B, Vol. 76, no 6, article id 064121Article in journal (Refereed)

The face-centered-cubic (fcc) Lennard-Jones (LJ) model can be considered as a representative model of a simple solid. We investigate the mechanism of melting at the limit of superheating in the fcc LJ solid by means of the procedure recently developed by us [Phys. Rev. B 73, 012201 (2006)]. Insight into the mechanism of melting was gained by studying diffusion and defects in the fcc LJ solid by means of molecular dynamics simulations. We found that the limit of superheating achieved by us is likely to be the highest so far. We also found that the size of the cluster which ignites the melting is very small (down to five to six atoms, depending on the size of the supercell) and closely correlates with the linear size of a supercell when the number of atoms varies between 500 and 13 500.

• 3. Campbell, I. A.
Extended scaling analysis of the S=1/2 Ising ferromagnet on the simple cubic lattice2011In: PHYSICAL REVIEW B, Vol. 83, no 1, article id 014411Article in journal (Refereed)

It is often assumed that for treating numerical (or experimental) data on continuous transitions the formal analysis derived from the renormalization-group theory can only be applied over a narrow temperature range, the "critical region"; outside this region correction terms proliferate rendering attempts to apply the formalism hopeless. This pessimistic conclusion follows largely from a choice of scaling variables and scaling expressions, which is traditional but very inefficient for data covering wide temperature ranges. An alternative "extended scaling" approach can be made where the choice of scaling variables and scaling expressions is rationalized in the light of well established high-temperature series expansion developments. We present the extended scaling approach in detail, and outline the numerical technique used to study the three-dimensional (3D) Ising model. After a discussion of the exact expressions for the historic 1D Ising spin chain model as an illustration, an exhaustive analysis of high quality numerical data on the canonical simple cubic lattice 3D Ising model is given. It is shown that in both models, with appropriate scaling variables and scaling expressions (in which leading correction terms are taken into account where necessary), critical behavior extends from T-c up to infinite temperature.

• 4. Campbell, Ian A.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Hyperscaling Violation in Ising Spin Glasses2019In: Entropy, ISSN 1099-4300, E-ISSN 1099-4300, Vol. 21, no 10, article id 978Article in journal (Refereed)

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.

• 5.
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL, 60607-7045, USA; Berlin Mathematical School, Berlin, Germany.
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL, 60607-7045, USA . Department of Physics, AlbaNova University Center, KTH, 106 91, Stockholm, Sweden . Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On the Validations of the Asymptotic Matching Conjectures2008In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 133, no 3, p. 513-533Article in journal (Refereed)

In this paper we review the asymptotic matching conjectures for r-regular bipartite graphs, and their connections in estimating the monomer-dimer entropies in d-dimensional integer lattice and Bethe lattices. We prove new rigorous upper and lower bounds for the monomer-dimer entropies, which support these conjectures. We describe a general construction of infinite families of r-regular tori graphs and give algorithms for computing the monomer-dimer entropy of density p, for any p is an element of[0,1], for these graphs. Finally we use tori graphs to test the asymptotic matching conjectures for certain infinite r-regular bipartite graphs.

• 6. Friedland, S.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
The 1-vertex transfer matrix and accurate estimation of channel capacity2010In: IEEE Transactions on Information Theory, ISSN 0018-9448, E-ISSN 1557-9654, Vol. 57, no 8, p. 3692-3699Article in journal (Refereed)

The notion of a 1-vertex transfer matrix for multidimensional codes is introduced. It is shown that the capacity of such codes, or the topological entropy, can be expressed as the limit of the logarithm of spectral radii of 1-vertex transfer matrices. Storage and computations using the 1-vertex transfer matrix are much smaller than storage and computations needed for the standard transfer matrix. The method is applied to estimate the first 15 digits of the entropy of the 2-D (0, 1) run length limited channel. A large-scale computation of eigenvalues for the (0, 1) run length limited channel in 2-D and 3-D have been carried out. This was done in order to be able to compare the computational cost of the new method with the standard transfer matrix and have rigorous bounds to compare the estimates with. This in turn leads to improvements on the best previous lower and upper bounds for these channels.

• 7.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Condensed Matter Theory, Department of Physics, Royal Institute of Technology, SE-106 91, Stockholm, Sweden.
The Ising partition function for 2D grids with periodic boundary: computation and analysis2002In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 108, no 3-4, p. 429-457Article in journal (Refereed)

The Ising partition function for a graph counts the number of bipartitions of the vertices with given sizes, with a given size of the induced edge cut. Expressed as a 2-variable generating function it is easily translatable into the corresponding partition function studied in statistical physics. In the current paper a comparatively efficient transfer matrix method is described for computing the generating function for the n×n grid with periodic boundary. We have applied the method to up to the 15×15 grid, in total 225 vertices. We examine the phase transition that takes place when the edge cut reaches a certain critical size. From the physical partition function we extract quantities such as magnetisation and susceptibility and study their asymptotic behaviour at the critical temperature.

• 8.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Department of Physics, AlbaNova University Center, KTH, SE-106 91 Stockholm, Sweden. Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Department of Physics, AlbaNova University Center, KTH, SE-106 91 Stockholm, Sweden. Department of Physics, AlbaNova University Center, KTH, SE-106 91 Stockholm, Sweden. Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Computation of the Ising partition function for two-dimensional square grids2004In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, ISSN 1063-651X, E-ISSN 1095-3787, Vol. 69, no 4, p. 19-, article id 046104Article in journal (Refereed)

An improved method for obtaining the Ising partition function for $n \times n$ square grids with periodic boundary is presented. Our method applies results from Galois theory in order to split the computation into smaller parts and at the same time avoid the use of numerics. Using this method we have computed the exact partition function for the $320 \times 320$-grid, the $256 \times 256$-grid, and the $160 \times 160$-grid, as well as for a number of smaller grids. We obtain scaling parameters and compare with what theory prescribes.

• 9.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Department of Physics, AlbaNova University Center, KTH, SE-106 91, Stockholm, Sweden . Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Department of Physics, AlbaNova University Center, KTH, SE-106 91, Stockholm, Sweden; Department of Biosciences at Novum, Karolinska institutet, SE-141 57, Huddinge, Sweden. Department of Physics, AlbaNova University Center, KTH, SE-106 91, Stockholm, Sweden . Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
A Monte Carlo sampling scheme for the Ising model2004In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 114, no 1-2, p. 455-480Article in journal (Refereed)

In this paper we describe a Monte Carlo sampling scheme for the Ising model and similar discrete-state models. The scheme does not involve any particular method of state generation but rather focuses on a new way of measuring and using the Monte Carlo data. We show how to reconstruct the entropy S of the model, from which, e.g., the free energy can be obtained. Furthermore we discuss how this scheme allows us to more or less completely remove the effects of critical fluctuations near the critical temperature and likewise how it reduces critical slowing down. This makes it possible to use simple state generation methods like the Metropolis algorithm also for large lattices.

• 10.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On the Ising model for the simple cubic lattice2007In: Advances in Physics, ISSN 0001-8732, E-ISSN 1460-6976, Vol. 56, no 5, p. 653-755Article, review/survey (Refereed)

The Ising model was introduced in 1920 to describe a uniaxial system of magnetic moments, localized on a lattice, interacting via nearest-neighbour exchange interaction. It is the generic model for a continuous phase transition and arguably the most studied model in theoretical physics. Since it was solved for a two-dimensional lattice by Onsager in 1944, thereby representing one of the very few exactly solvable models in dimensions higher than one, it has served as a testing ground for new developments in analytic treatment and numerical algorithms. Only series expansions and numerical approaches, such as Monte Carlo simulations, are available in three dimensions. This review focuses on Monte Carlo simulation. We build upon a data set of unprecedented size. A great number of quantities of the model are estimated near the critical coupling. We present both a conventional analysis and an analysis in terms of a Puiseux series for the critical exponents. The former gives distinct values of the high- and low-temperature exponents; by means of the latter we can get these exponents to be equal at the cost of having true asymptotic behaviour being found only extremely close to the critical point. The consequences of this for simulations of lattice systems are discussed at length.

• 11. Lundow, P. H.
Alternative phenomenological coupling parameter for finite-size analysis of numerical data at criticality2010In: PHYSICAL REVIEW B, Vol. 82, no 2, article id 024414Article in journal (Refereed)

We introduce a parameter W(beta,L) =(pi <vertical bar m vertical bar > (2)/< m(2)>- 2/(pi-2) which like the kurtosis (Binder cumulant) is a phenomenological coupling characteristic of the shape of the distribution p(m) of the order parameter m. To demonstrate the use of the parameter we analyze extensive numerical data obtained from density-of-states measurements on the canonical simple-cubic spin-1/2 Ising ferromagnet, for sizes L=4 to L=256. Using the W parameter accurate estimates are obtained for the critical inverse temperature beta(c)=0.2216541(2), and for the thermal exponent nu=0.6308(4). In this system at least, corrections to finite-size scaling are significantly weaker for the W parameter than for the Binder cumulant.

• 12. Lundow, P. H.
Critical scaling to infinite temperature2011In: PHYSICAL REVIEW B, Vol. 83, no 18, article id 184408Article in journal (Refereed)

Three-dimensional Ising model ferromagnets on different lattices with nearest-neighbor interactions, and on simple-cubic lattices with equivalent interactions out to further neighbors, are studied numerically. The susceptibility data for all these systems are analyzed using the critical renormalization-group theory formalism over the entire temperature range above T-c with an appropriate choice of scaling variable and scaling expressions. Representative experimental data on a metallic ferromagnet (Ni) and an elementary fluid (Xe) are interpreted in the same manner so as to estimate effective coordination numbers.

• 13. Lundow, P. H.
Fortuin-Kasteleyn and damage-spreading transitions in random-bond Ising lattices2012In: PHYSICAL REVIEW E, Vol. 86, no 4, article id 041121Article in journal (Refereed)

The Fortuin-Kasteleyn and heat-bath damage-spreading temperatures T-FK(p) and T-DS(p) are studied on random-bond Ising models of dimensions 2-5 and as functions of the ferromagnetic interaction probability p; the conjecture that T-DS(p) similar to T-FK(p) is tested. It follows from a statement by Nishimori that in any such system, exact coordinates can be given for the intersection point between the Fortuin-Kasteleyn T-FK(p) transition line and the Nishimori line [p(NL,FK), T-NL,T-FK]. There are no finite-size corrections for this intersection point. In dimension 3, at the intersection concentration [p(NL,FK)], the damage spreading T-DS(p) is found to be equal to T-FK(p) to within 0.1%. For the other dimensions, however, T-DS(p) is observed to be systematically a few percent lower than T-FK(p).

• 14. Lundow, P. H.
Ising ferromagnet in dimension five: Link and spin overlaps2013In: PHYSICAL REVIEW E, Vol. 87, no 2, article id 022102Article in journal (Refereed)

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. DOI: 10.1103/PhysRevE.87.022102

• 15. Lundow, P. H.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Broken-cycle-free subgraphs and the log-concavity conjecture for chromatic polynomials2006In: Experimental Mathematics, ISSN 1058-6458, E-ISSN 1944-950X, Vol. 15, no 3, p. 343-353Article in journal (Refereed)

This paper concerns the coefficients of the chromatic polynomial of a graph. We first report on a computational verification of the strict log-concavity conjecture for chromatic polynomials for all graphs on at most 11 vertices, as well as for certain cubic graphs. In the second part of the paper we give a number of conjectures and theorems regarding the behavior of the coefficients of the chromatic polynomial, in part motivated by our computations. Here our focus is on epsilon(G), the average size of a broken-cycle-free subgraph of the graph G, whose behavior under edge deletion and contraction is studied.

• 16. Lundow, P. H.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Critical behavior of the Ising model on the four-dimensional cubic lattice2009In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, E-ISSN 1550-2376, Vol. 80, no 3, p. 031104-4 pagesArticle in journal (Refereed)

In this paper we investigate the nature of the singularity of the Ising model of the four-dimensional cubic lattice. It is rigorously known that the specific heat has critical exponent alpha = 0 but a nonrigorous field-theory argument predicts an unbounded specific heat with a logarithmic singularity at T(c). We find that within the given accuracy the canonical ensemble data are consistent both with a logarithmic singularity and a bounded specific heat but that the microcanonical ensemble lends stronger support to a bounded specific heat. Our conclusion is that either much larger system sizes are needed for Monte Carlo studies of this model in four dimensions or the field-theory prediction of a logarithmic singularity is wrong.

• 17. Lundow, P. H.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Non-vanishing boundary effects and quasi-first-order phase transitions in high dimensional Ising models2011In: Nuclear Physics B, ISSN 0550-3213, E-ISSN 1873-1562, Vol. 845, no 1, p. 120-139Article in journal (Refereed)

In order to gain a better understanding of the Ising model in higher dimensions we have made a comparative study of how the boundary, open versus cyclic, of a d-dimensional simple lattice, for d = 1,...,5, affects the behaviour of the specific heat C and its microcanonical relative, the entropy derivative -partial derivative(2)S/partial derivative U(2). In dimensions 4 and 5 the boundary has a strong effect on the critical region of the model and for cyclic boundaries in dimension 5 we find that the model displays a quasi-first-order phase transition with a bimodal energy distribution. The latent heat decreases with increasing systems size but for all system sizes used in earlier papers the effect is clearly visible once a wide enough range of values for K is considered. Relations to recent rigorous results for high dimensional percolation and previous debates on simulation of Ising models and gauge fields are discussed. (C) 2010 Elsevier B.V. All rights reserved.

• 18. Lundow, P. H.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
The Ising model for the bcc, fcc and diamond lattices: A comparison2009In: Philosophical Magazine, ISSN 1478-6435, E-ISSN 1478-6443, Vol. 89, no 22-24, p. 2009-2042Article in journal (Refereed)

A large-scale Monte Carlo simulation study of the Ising model for the simple cubic lattice was recently performed by us. In this paper, we complement that study with the bcc, fcc and diamond lattices. Both the canonical and microcanonical ensembles are employed. We give estimates of the critical temperature and also other quantities in the critical region. An analysis of the critical behaviour points to distinct high-and low-temperature exponents, especially for the specific heat, as was also obtained for the simple cubic lattice, although the agreement is good between the different lattices. The source of this discrepancy is briefly discussed.

• 19. Lundow, P. H.
On the p, q-binomial distribution and the Ising model2010In: PHILOSOPHICAL MAGAZINE, Vol. 90, no 24, p. 3313-3353, article id PII 923766914Article in journal (Refereed)

We employ p, q-binomial coefficients, a generalisation of the binomial coefficients, to describe the magnetisation distributions of the Ising model. For the complete graph this distribution corresponds exactly to the limit case p = q. We apply our investigation to the simple d-dimensional lattices for d = 1, 2, 3, 4, 5 and fit p, q-binomial distributions to our data, some of which are exact but most are sampled. For d = 1 and d = 5, the magnetisation distributions are remarkably well-fitted by p,q-binomial distributions. For d = 4 we are only slightly less successful, while for d = 2, 3 we see some deviations (with exceptions!) between the p, q-binomial and the Ising distribution. However, at certain temperatures near Tc the statistical moments of the fitted distribution agree with the moments of the sampled data within the precision of sampling. We begin the paper by giving results of the behaviour of the p, q-distribution and its moment growth exponents given a certain parameterisation of p, q. Since the moment exponents are known for the Ising model (or at least approximately for d = 3) we can predict how p, q should behave and compare this to our measured p, q. The results speak in favour of the p, q-binomial distribution's correctness regarding its general behaviour in comparison to the Ising model. The full extent to which they correctly model the Ising distribution, however, is not settled.

• 20. Lundow, Per Håkan
The p,q-binomial distribution applied to the 5d Ising model2013In: PHILOSOPHICAL MAGAZINE, Vol. 93, no 14, p. 1755-1770Article in journal (Refereed)

The leading order form of the magnetization distribution is well-known for the 5d Ising model close to . Its corrections-to-scaling are not known though. Since we have earlier established that this distribution is extremely well-fitted by a -binomial distribution, we report considerably longer series expansions for its moments in terms of three parameters, providing new details on the scaling behaviour of the Ising distribution and its moments near . As applications, we give for example the scaling formulas for the ratios , and the full distribution at .

• 21.
Umeå University, Faculty of Science and Technology, Department of mathematics.
Compression of transfer matrices2001In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 231, no 1-3, p. 321-329Article in journal (Refereed)

We present a method for reducing the size of transfer matrices by exploiting symmetry. For example, the transfer matrix for enumeration of matchings in the graph C-4 x C-4 x P-n can be reduced from order 65536 to 402 simply due to the 384 automorphisms of C-4 x C-4. The matrix for enumeration of perfect matchings can be still further reduced to order 93, all in a straightforward and mechanical way. As an application we report an improved upper bound for the three-dimensional dimer problem. (C) 2001 Elsevier Science B.V. All rights reserved.

• 22.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Bimodal and Gaussian Ising spin glasses in dimension two2016In: Physical Review E, ISSN 2470-0045, Vol. 93, no 2, article id 022119Article in journal (Refereed)

An analysis is given of numerical simulation data to size L = 128 on the archetype square lattice Ising spin glasses (ISGs) with bimodal (+/- J) and Gaussian interaction distributions. It is well established that the ordering temperature of both models is zero. The Gaussian model has a nondegenerate ground state and thus a critical exponent. = 0, and a continuous distribution of energy levels. For the bimodal model, above a size-dependent crossover 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. The simulation data on both models are analyzed with the conventional scaling variable t = T and with a scaling variable tau(b) = T-2 /(1 + T 2) suitable for zero-temperature transition ISGs, together with appropriate scaling expressions. The data for the temperature dependence of the reduced susceptibility x(tau(b), L) and second moment correlation length xi(tau(b), L) in the thermodynamic limit regime are extrapolated to the tau(b) = 0 critical limit. The Gaussian critical exponent estimates from the simulations, eta= 0 and nu= 3.55(5), are in full agreement with the well-established values in the literature. The bimodal critical exponents, estimated from the thermodynamic limit regime analyses using the same extrapolation protocols as for the Gaussian model, are eta= 0.20(2) and nu= 4.8(3), distinctly different from the Gaussian critical exponents.

• 23.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Bimodal Ising spin glass in two dimensions: the anomalous dimension η2018In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 97, no 2, article id 024203Article in journal (Refereed)

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 ∼ 1. In the Gaussian case the data are consistent with the known anomalous dimension value η ≡ 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 nonzero and the data are consistent with the estimate η ∼ 0.28(4) given by McMillan in 1983 from similar measurements. Measurements of the temperature dependence of the Binder cumulant U4(T, L) and the normalized correlation length ξ(T, L)/L for the two models confirms the conclusion that the two-dimensional (2D) bimodal model has a nonzero 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.

• 24.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS–Université de Montpellier, Montpellier, France.
Evidence for nonuniversal scaling in dimension-four Ising spin glasses2015In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, E-ISSN 1550-2376, Vol. 91, no 4, article id 042121Article in journal (Refereed)

The critical behavior of the Binder cumulant for Ising spin glasses in dimension four is 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.

• 25.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Hyperscaling breakdown and Ising spin glasses: The Binder cumulant2018In: Physica A: Statistical Mechanics and its Applications, ISSN 0378-4371, E-ISSN 1873-2119, Vol. 492, p. 1838-1852Article in journal (Refereed)

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 Schwartz (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.

• 26.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Ising spin glasses in dimension five2017In: Physical Review E, ISSN 2470-0045, Vol. 95, no 1, article id 012112Article in journal (Refereed)

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.

• 27.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Ising spin glasses in two dimensions: Universality and nonuniversality2017In: Physical review. E, ISSN 2470-0045, E-ISSN 2470-0053, Vol. 95, no 4, article id 042107Article in journal (Refereed)

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 two dimensions [Lundow and Campbell, Phys. Rev. E 93, 022119 (2016)]. Here we further analyze the bimodal and Gaussian data together with data on two other continuous interaction distribution two-dimensional ISG models, the uniform and the Laplacian models, and three other discrete interaction distribution models, a diluted bimodal model, an "antidiluted" 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 eta equivalent to 0 but that to within the present numerical precision they share the same critical exponent. 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 the present data strongly indicate that they differ from one discrete distribution model to another. This is evidence that discrete distribution ISG models in two dimensions have nonzero values of the critical exponent. and do not lie in a single universality class.

• 28.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Non-self-averaging in Ising spin glasses and hyperuniversality2016In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, E-ISSN 1550-2376, Vol. 93, no 1, article id 012118Article in journal (Refereed)

Ising spin glasses with bimodal and Gaussian near-neighbor interaction distributions are studied through numerical simulations. The non-self-averaging (normalized intersample variance) parameter U-22(T, L) for the spin glass susceptibility [and for higher moments Unn (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 xi(T, L) as U-nn(beta, L) = [K-d xi (T, L)/L](d) and so follow a renormalization group law due to Aharony and Harris [Phys. Rev. Lett. 77, 3700 (1996)]. Empirically, it is found that the Kd 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 practically 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 three-dimensional Heisenberg and XY spin glasses the light of the Ising spin glass non-self-averaging results show behavior which appears to be compatible with that expected on a chiral-driven ordering interpretation but incompatible with a spin-driven ordering scenario.

• 29.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
The Ising Spin Glass in dimension four2015In: Physica A: Statistical Mechanics and its Applications, ISSN 0378-4371, E-ISSN 1873-2119, Vol. 434, p. 181-193Article in journal (Refereed)

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 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.

• 30.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
The Ising universality class in dimension three: Corrections to scaling2018In: Physica A: Statistical Mechanics and its Applications, ISSN 0378-4371, E-ISSN 1873-2119, Vol. 511, p. 40-53Article in journal (Refereed)

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, theta(2) similar to 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(x4)/a(x) = 2 in the 3D Ising universality class.

• 31.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Complete graph asymptotics for the Ising and random-cluster models on five-dimensional grids with a cyclic boundary2015In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, E-ISSN 1550-2376, Vol. 91, article id 022112Article in journal (Refereed)

The finite-size scaling behavior for the Ising model in five dimensions, with either free or cyclic boundary, has been the subject of a long-running debate. The older papers have been based on ideas from, e.g., field theory or renormalization. In this paper we propose a detailed and exact scaling picture for critical region of the model with cyclic boundary. Unlike the previous papers our approach is based on a comparison with the existing exact and rigorous results for the FK-random-cluster model on a complete graph. Based on those results we predict several distinct scaling regions in an L  -dependent window around the critical point. We test these predictions by comparing with data from Monte Carlo simulations and find a good agreement. The main feature which differs between the complete graph and the five-dimensional model with free boundary is the existence of a bimodal energy distribution near the critical point in the latter. This feature was found by the same authors in an earlier paper in the form of a quasi-first-order phase transition for the same Ising model.

• 32.
KTH Physics, AlbaNova University Center, SE-106 91 Stockholm, Sweden.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Exact and Approximate Compression of Transfer Matrices for Graph Homomorphisms2008In: LMS Journal of Computation and Mathematics, ISSN 1461-1570, E-ISSN 1461-1570, Vol. 11, p. 1-14Article in journal (Refereed)

The aim of this paper is to extend the previous work on transfer matrix compression in the case of graph homomorphisms. For H-homomorphisms of lattice-like graphs we demonstrate how the automorphisms of H, as well as those of the underlying lattice, can be used to reduce the size of the relevant transfer matrices. As applications of this method we give currently best known bounds for the number of 4- and 5-colourings of the square grid, and the number of 3- and 4-colourings of the three-dimensional cubic lattice. Finally, we also discuss approximate compression of transfer matrices.

• 33.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Finite size scaling of the 5D Ising model with free boundary conditions2014In: Nuclear Physics B, ISSN 0550-3213, E-ISSN 1873-1562, Vol. 889, p. 249-260Article in journal (Refereed)

There has been a long running debate on the finite size scaling for the Ising model with free boundary conditions above the upper critical dimension, where the standard picture gives an L2 scaling for the susceptibility and an alternative theory has promoted an L5/2 scaling, as would be the case for cyclic boundary. In this paper we present results from simulation of the far largest systems used so far, up to sideL=160 and find that this data clearly supports the standard scaling. Further we present a discussion of why rigorous results for the random-cluster model provide both supports for the standard scaling picture and a clear explanation of why the scalings for free and cyclic boundary should be different.

• 34.
Department of theoretical physics, AlbaNova University Center, KTH, SE-106 91 Stockholm, Sweden.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Reconstruction of the finite size canonical ensemble from incomplete micro-canonical data2009In: Central European Journal of Physics, ISSN 1895-1082, E-ISSN 1644-3608, Vol. 7, no 3, p. 490-502Article in journal (Refereed)

In this paper we discuss how partial knowledge of the density of states for a model can be used to give good approximations of the energy distributions in a given temperature range. From these distributions one can then obtain the statistical moments corresponding to e. g. the internal energy and the specific heat. These questions have gained interest apropos of several recent methods for estimating the density of states of spin models. As a worked example we finally apply these methods to the 3-state Potts model for cubic lattices of linear order up to 128. We give estimates of e. g. latent heat and critical temperature, as well as the micro-canonical properties of interest.

• 35.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Revisiting the cavity-method threshold for random 3-SAT2019In: Physical review. E, ISSN 2470-0045, E-ISSN 2470-0053, Vol. 99, no 2, article id 022106Article in journal (Refereed)

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 α3≤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 kc, and the cavity method may provide a correct mean-field picture for the range above kc.

• 36.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
The discontinuity of the specific heat for the 5D Ising model2015In: Nuclear Physics B, ISSN 0550-3213, E-ISSN 1873-1562, Vol. 895, p. 305-318Article in journal (Refereed)

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 temperature K-c in d = 5, 6, 7 which bring the best MC-estimate closer to those obtained by long high temperature series expansions.

• 37.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
The scaling window of the 5D Ising model with free boundary conditions2016In: Nuclear Physics B, ISSN 0550-3213, E-ISSN 1873-1562, Vol. 911, p. 163-172Article in journal (Refereed)

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 L-2 inside a critical scaling window of width 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 delta = 3, that the scaling window has width 1/L-2.

• 38. Rosengren, A.
Isotope effect on superconductivity in Josephson coupled stripes in underdoped cuprates2008In: PHYSICAL REVIEW B, Vol. 77, no 13, article id 134508Article in journal (Refereed)

Inelastic neutron scattering data for YBaCuO as well as for LaSrCuO indicate incommensurate neutron scattering peaks with an incommensuration delta(x) away from the (pi,pi) point. T(c)(x) can be replotted as a linear function of the incommensuration for these materials. This linear relation implies that the constant that relates these two quantities, where one is the incommensuration (momentum) and the other is T(c)(x) (energy), has the dimension of velocity, which we denote by v(*): k(B)T(c)(x)=hv(*)delta(x). We argue that this experimentally determined relation can be obtained in a simple model of Josephson coupled stripes. Within this framework, we address the role of the O(16)-> O(18) isotope effect on T(c)(x). We assume that the incommensuration is set by the doping of the sample and is not sensitive to the oxygen isotope given the fixed doping. We find therefore that the only parameter that can change with the O isotope substitution in the relation T(c)(x)similar to delta(x) is the velocity v(*). We predict an oxygen isotope effect on v(*) and expect it to be similar or equal to 5%.

• 39.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
A counterexample to a conjecture by Blocki-Zwonek2018In: Experimental Mathematics, ISSN 1058-6458, E-ISSN 1944-950X, Vol. 27, no 1, p. 119-124Article in journal (Refereed)
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