Universal finite-size scaling functions with exact nonuniversal metric factors

Specific heat and partition function zeros for the dimer model on the checkerboard B lattice: Finite-size effects

There are three possible classifications of the dimer weights on the bonds of the checkerboard lattice and they are denoted as checkerboard A, B, and C lattices [Phys. Rev. E 91, 062139 (2015)PLEEE81539-375510.1103/PhysRevE.91.062139]. The dimer model on the checkerboard B and C lattices has much richer critical behavior compared to the dimer model on the checkerboard A lattice. In this paper we study in full detail the dimer model on the checkerboard B lattice. The dimer model on the checkerboard B lattice has two types of critical behavior. In one limit this model is the anisotropic dimer model on rectangular lattice with algebraic decay of correlators and in another limit it is the anisotropic generalized Kasteleyn model with radically different critical behavior. We analyze the partition function of the dimer model on a 2M×2N checkerboard B lattice wrapped on a torus. We find very unusual behavior of the partition function zeros and the specific heat of the dimer model. Remarkably, the partition function zeros of finite-size systems can have very interesting structures, made of rings, concentric circles, radial line segments, or even arabesque structures. We find out that the number of the specific heat peaks and the number of circles of the partition function zeros increases with the system size. The lattice anisotropy of the model has strong effects on the behavior of the specific heat, dominating the relation between the correlation length exponent ν and the shift exponent λ, and λ is generally unequal to 1/ν (λ≠1/ν).

Finite-size corrections and scaling for the dimer model on the checkerboard lattice

Lattice models are useful for understanding behaviors of interacting complex many-body systems. The lattice dimer model has been proposed to study the adsorption of diatomic molecules on a substrate. Here we analyze the partition function of the dimer model on a 2M×2N checkerboard lattice wrapped on a torus and derive the exact asymptotic expansion of the logarithm of the partition function. We find that the internal energy at the critical point is equal to zero. We also derive the exact finite-size corrections for the free energy, the internal energy, and the specific heat. Using the exact partition function and finite-size corrections for the dimer model on a finite checkerboard lattice, we obtain finite-size scaling functions for the free energy, the internal energy, and the specific heat of the dimer model. We investigate the properties of the specific heat near the critical point and find that the specific-heat pseudocritical point coincides with the critical point of the thermodynamic limit, which means that the specific-heat shift exponent λ is equal to ∞. We have also considered the limit N→∞ for which we obtain the expansion of the free energy for the dimer model on the infinitely long cylinder. From a finite-size analysis we have found that two conformal field theories with the central charges c=1 for the height function description and c=-2 for the construction using a mapping of spanning trees can be used to describe the dimer model on the checkerboard lattice.

Exact solution of the dimer model on the generalized finite checkerboard lattice

We present the exact closed-form expression for the partition function of a dimer model on a generalized finite checkerboard rectangular lattice under periodic boundary conditions. We investigate three different sets of dimer weights, each with different critical behaviors. We then consider different limits for the model on the three lattices. In one limit, the model for each of the three lattices is reduced to the dimer model on a rectangular lattice, which belongs to the c=-2 universality class. In another limit, two of the lattices reduce to the anisotropic Kasteleyn model on a honeycomb lattice, the universality class of which is given by c=1. The result that the dimer model on a generalized checkerboard rectangular lattice can manifest different critical behaviors is consistent with early studies in the thermodynamic limit and also provides insight into corrections to scaling arising from the finite-size versions of the model.

Exact finite-size corrections for the spanning-tree model under different boundary conditions

We express the partition functions of the spanning tree on finite square lattices under five different sets of boundary conditions in terms of a principal partition function with twisted-boundary conditions. Based on these expressions, we derive the exact asymptotic expansions of the logarithm of the partition function for each case. We have also established several groups of identities relating spanning-tree partition functions for the different boundary conditions. We also explain an apparent discrepancy between logarithmic correction terms in the free energy for a two-dimensional spanning-tree model with periodic and free-boundary conditions and conformal field theory predictions. We have obtained corner free energy for the spanning tree under free-boundary conditions in full agreement with conformal field theory predictions.

Simplified lattice model for polypeptide fibrillar transitions

Polypeptide fibrillar transitions are studied using a simplified lattice model, modified from the three-state Potts model, where uniform residues as spins, placed on a cubic lattice, can interact with neighbors to form coil, helical, sheet, or fibrillar structure. Using the transfer matrix method and numerical calculations, we analyzed the partition function and construct phase diagrams. The model manifests phase transitions among coil, helix, sheet, and fibril through parameterizing bond coupling energy ɛh,ɛs,ɛf, structural entropies sh,ss,sf of helical, sheet, and fibrillar states, and number density ρ. The phase diagrams show the transition sequence is basically governed by ɛh, ɛs, and ɛf, while the transition temperature is determined by the competition among ɛh, ɛs, and ɛf, as well as sh, ss, sf, and ρ. Furthermore, the fibrillation is accompanied with an abrupt phase transition from coil, helix, or sheet to fibril even for short polypeptide length, resembling the feature of nucleation-growth process. The finite-size effect in specific heat at transitions for the nonfibrillation case can be described by the scaling form of lattice model. With rich phase-transition properties, our model provides a useful reference for protein aggregation experiments and modeling.

Criticality in alternating layered Ising models. I. Effects of connectivity and proximity

The specific heats of exactly solvable alternating layered planar Ising models with strips of width m_{1} lattice spacings and "strong" couplings J_{1} sandwiched between strips of width m_{2} and "weak" coupling J_{2}, have been studied numerically to investigate the effects of connectivity and proximity. We find that the enhancements of the specific heats of the strong layers and of the overall or "bulk" critical temperature T_{c}(J_{1},J_{2};m_{1},m_{2}) arising from the collective effects reflect the observations of Gasparini and co-workers in experiments on confined superfluid helium. Explicitly, we demonstrate that finite-size scaling holds in the vicinity of the upper limiting critical point T_{1c} (∝J_{1}/k_{B}) and close to the corresponding lower critical limit T_{2c} (∝J_{2}/k_{B}) when m_{1} and m_{2} increase. However, the residual enhancement, defined via appropriate subtractions of leading contributions from the total specific heat, is dominated (away from T_{1c} and T_{2c}) by a decay factor 1/(m_{1}+m_{2}) arising from the seams (or boundaries) separating the strips; close to T_{1c} and T_{2c} the decay is slower by a factor lnm_{1} and lnm_{2}, respectively.

Criticality in alternating layered Ising models. II. Exact scaling theory

Part I of this article studied the specific heats of planar alternating layered Ising models with strips of strong coupling J_{1} sandwiched between strips of weak coupling J_{2}, to illustrate qualitatively the effects of connectivity, proximity, and enhancement in analogy to those seen in extensive experiments on superfluid helium by Gasparini and co-workers. It was demonstrated graphically that finite-size scaling descriptions hold in a variety of temperature regions including in the vicinity of the two specific heat maxima. Here we provide exact theoretical analyses and asymptotics of the specific heat that support and confirm the graphical findings. Specifically, at the overall or bulk critical point, the anticipated (and always present) logarithmic singularity is shown to vanish exponentially fast as the width of the stronger strips increases.

Amplitude ratios for critical systems in the c=-2 universality class

We study the finite-size corrections of the critical dense polymer (CDP) and the dimer models on ∞×N rectangular lattice. We find that the finite-size corrections in the CDP and dimer models depend in a crucial way on the parity of N, and a change of the parity of N is equivalent to the change of boundary conditions. We present a set of universal amplitude ratios for amplitudes in finite-size correction terms of critical systems in the universality class with central charge c=-2. The results are in perfect agreement with a perturbated conformal field theory under the assumption that all analytical corrections coming from the operators which belongs to the tower of the identity. Our results inspire many interesting problems for further studies.

Universality of the Ising and the S=1 model on Archimedean lattices: a Monte Carlo determination

The Ising models S=1/2 and S=1 are studied by efficient Monte Carlo schemes on the (3,4,6,4) and the (3,3,3,3,6) Archimedean lattices. The algorithms used, a hybrid Metropolis-Wolff algorithm and a parallel tempering protocol, are briefly described and compared with the simple Metropolis algorithm. Accurate Monte Carlo data are produced at the exact critical temperatures of the Ising model for these lattices. Their finite-size analysis provide, with high accuracy, all critical exponents which, as expected, are the same with the well-known 2D Ising model exact values. A detailed finite-size scaling analysis of our Monte Carlo data for the S=1 model on the same lattices provides very clear evidence that this model obeys, also very well, the 2D Ising model critical exponents. As a result, we find that recent Monte Carlo simulations and attempts to define effective dimensionality for the S=1 model on these lattices are misleading. Accurate estimates are obtained for the critical amplitudes of the logarithmic expansions of the specific heat for both models on the two Archimedean lattices.

Asymptotic expansion for the resistance between two maximally separated nodes on an M by N resistor network

We analyze the exact formulas for the resistance between two arbitrary notes in a rectangular network of resistors under free, periodic and cylindrical boundary conditions obtained by Wu [J. Phys. A 37, 6653 (2004)]. Based on such expression, we then apply the algorithm of Ivashkevich, Izmailian, and Hu [J. Phys. A 35, 5543 (2002)] to derive the exact asymptotic expansions of the resistance between two maximally separated nodes on an M×N rectangular network of resistors with resistors r and s in the two spatial directions. Our results is 1/s (R(M×N))(r,s) = c(ρ)ln S + c(0)(ρ,ξ) + ∑(p=1)(∞) (c(2p)(ρ,ξ))/S(p) with S = MN, ρ = r/s and ξ = M/N. The all coefficients in this expansion are expressed through analytical functions. We have introduced the effective aspect ratio ξeff = square root(ρ)ξ for free and periodic boundary conditions and ξeff = square root(ρ)ξ/2 for cylindrical boundary condition and show that all finite-size correction terms are invariant under transformation ξeff→1/ξeff.

Critical behavior of the pure and random-bond two-dimensional triangular Ising ferromagnet

We investigate the effects of quenched bond randomness on the critical properties of the two-dimensional ferromagnetic Ising model embedded in a triangular lattice. The system is studied in both the pure and disordered versions by the same efficient two-stage Wang-Landau method. In the first part of our study, we present the finite-size scaling behavior of the pure model, for which we calculate the critical amplitude of the specific heat's logarithmic expansion. For the disordered system, the numerical data and the relevant detailed finite-size scaling analysis along the lines of the two well-known scenarios-logarithmic corrections versus weak universality--strongly support the field-theoretically predicted scenario of logarithmic corrections. A particular interest is paid to the sample-to-sample fluctuations of the random model and their scaling behavior that are used as a successful alternative approach to criticality.

Mapping functions and critical behavior of percolation on rectangular domains

The existence probability E_{p} and the percolation probability P of bond percolation on rectangular domains with different aspect ratios R are studied via the mapping functions between systems with different aspect ratios. The superscaling behavior of E_{p} and P for such systems with exponents a and b , respectively, found by Watanabe [Phys. Rev. Lett. 93, 190601 (2004)] can be understood from the lower-order approximation of the mapping functions f_{R} and g_{R} for E_{p} and P , respectively; the exponents a and b can be obtained from numerically determined mapping functions f_{R} and g_{R} , respectively.

Self-similarity in the classification of finite-size scaling functions for toroidal boundary conditions

The conventional periodic boundary conditions in two dimensions are extended to general boundary conditions, prescribed by primitive vector pairs that may not coincide with the coordinate axes. This extension is shown to be unambiguously specified by the twisting scheme. Equivalent relations between different twist settings are constructed explicitly. The classification of finite-size scaling functions is discussed based on the equivalent relations. A self-similar pattern for distinct classes of finite-size scaling functions is shown to appear on the plane that parametrizes the toroidal geometry.

Finite-size effects for the Ising model on helical tori

We analyze the exact partition function of the Ising model on a square lattice under helical boundary conditions obtained by Liaw [Phys. Rev. E 73, 055101(R) (2006)]. Based on such an expression, we then extend the algorithm of Ivashkevich, Izmailian, and Hu [J. Phys. A 35, 5543 (2002)] to derive an exact asymptotic expansion of the logarithm of the partition function and its first to fourth derivatives at the critical point. From such results, we find that the shift exponent for the specific heat is lambda=1 for all values of the helicity factor d . We also find that finite-size corrections for the free energy, the internal energy, and the specific heat of the model in a crucial way depend on the helicity factor of the lattice.

Exact finite-size scaling functions for the interfacial tensions of the Ising model on planar lattices

Exact finite-size scaling functions of the interfacial tensions are obtained for the Ising model with isotropic coupling on a set of M x N planar lattices, including square (sq), plane triangular (pt), and honeycomb (hc) lattices. The analyses of transitive behaviors at criticality revise the knowledge of the interfacial tensions as a function of the aspect ratio defined by R = M/N for R approaching to zero gradually. The amplitudes of the interfacial tensions for the sq, pt, and hc lattices are further shown to have relative proportions 1:square root of 3 / 2:square root of 3 which are related to the aspect ratios for the three lattices to have similar domains.

Finite-size corrections and scaling for the triangular lattice dimer model with periodic boundary conditions

We analyze the partition function of the dimer model on M x N triangular lattice wrapped on the torus obtained by Fendley, Moessner, and Sondhi [Phys. Rev. B 66, 214513, (2002)]. Based on such an expression, we then extend the algorithm of Ivashkevich, Izmailian, and Hu [J. Phys. A 35, 5543 (2002)] to derive the exact asymptotic expansion of the first and second derivatives of the logarithm of the partition function at the critical point and find that the aspect-ratio dependence of finite-size corrections and the finite-size scaling functions are sensitive to the parity of the number of lattice sites along the lattice axis.

Logarithmic conformal field theory and boundary effects in the dimer model

We study the finite-size corrections of the dimer model on a square lattice with two different boundary conditions: free and periodic. We find that the finite-size corrections depend in a crucial way on the parity of ; we also show that such unusual finite-size behavior can be fully explained in the framework of the logarithmic conformal field theory.

Exact multileg correlation functions for the dense phase of branching polymers in two dimensions

We consider branching polymers on the planar square lattice with open boundary conditions and exactly calculate correlation functions of k polymer chains that connect two lattice sites with a large distance r apart for odd number of polymer chains k. We find that besides the standard power-law factor the leading term also has a logarithmic multiplier.

Exact finite-size corrections of the free energy for the square lattice dimer model under different boundary conditions

We express the partition functions of the dimer model on finite square lattices under five different boundary conditions (free, cylindrical, toroidal, Möbius strip, and Klein bottle) obtained by others (Kasteleyn, Temperley and Fisher, McCoy and Wu, Brankov and Priezzhev, and Lu and Wu) in terms of the partition functions with twisted boundary conditions Z(alpha, beta) with (alpha, beta)=(1/2,0), (0,1/2) and (1/2,1/2). Based on such expressions, we then extend the algorithm of Ivashkevich, Izmailian, and Hu [J. Phys. A 35, 5543 (2002)] to derive the exact asymptotic expansion of the logarithm of the partition function for all boundary conditions mentioned above. We find that the aspect-ratio dependence of finite-size corrections is sensitive to boundary conditions and the parity of the number of lattice sites along the lattice axis. We have also established several groups of identities relating dimer partition functions for the different boundary conditions.