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

Измаилян, Н. Ш.; Kenna, Ralph

doi:10.1103/physreve.91.022129

Cited by 3 publications

(6 citation statements)

References 49 publications

(78 reference statements)

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“…The subleading correction terms f p in the asymptotic expansion of the free energy have been obtained for many models, including the Ising [3,[25][26][27][28] and dimer models [3,4,14,[29][30][31] on different lattices under various boundary conditions, the Gaussian model [3], the spanning-tree model [32], and resistor networks [33,34]. In all these papers the subleading correction terms f p (for all p ) are expressed in terms of Kronecker's double series, which in turn are directly related to elliptic theta functions.…”

Section: Finite-size Scaling Theorymentioning

confidence: 99%

Exact finite-size corrections in the dimer model on a planar square lattice

Измаилян¹,

Papoyan²,

Ziff³

2019

J. Phys. A: Math. Theor.

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We consider the dimer model on the rectangular 2M × 2N lattice with free boundary conditions. We derive exact expressions for the coefficients in the asymptotic expansion of the free energy in terms of the elliptic theta functions (θ 2 , θ 3 , θ 4 ) and the elliptic integral of second kind (E), up to 22nd order. Surprisingly we find that ratio of the coefficients f p in the free energy expansion for strip ( f strip p ) and square ( f sq p ) geometries r p = f strip p /f sq p in the limit of large p tends to 1/2. Furthermore, we predict that the ratio of the coefficients f p in the free energy expansion for rectangular ( f p (ρ)) for aspect ratio ρ > 1 to the coefficients of the free energy for square geometries, multiplied by ρ −p−1 , that is r p = ρ −p−1 f p (ρ)/f sq p , is also equal to 1/2 in the limit of p → ∞. With these results, one can find the asymptotic behavior of the f p (ρ) from that of the f strip p , whose asymptotic behavior is derived explicitly here. We find that the corner contribution to the free energy for the dimer model on rectangular 2M × 2N lattices with free boundary conditions is equal to zero and explain that result in the framework of conformal field theory, with two consistent values of the central charge, namely, c = −2 for the construction of a conformal field theory using a mapping of spanning trees and c = 1 for the height function description. We also derive a simple exact expression for the free energy of open strips of arbitrary width.

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Section: Finite-size Scaling Theorymentioning

confidence: 99%

Exact finite-size corrections in the dimer model on a planar square lattice

Измаилян¹,

Papoyan²,

Ziff³

2019

J. Phys. A: Math. Theor.

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“…They verify the predictions of conformal field theory presented in by Equation (3) with central charge c = 1/2. Later, the conformal field theory prediction Equation (3) was confirmed [28] for the dimer model on odd-odd square lattices with one monomer on the boundary, for which the central charge is c = −2 and for another model in the c = −2 universality class, i.e., the spanning-tree model with free boundary conditions [29]. Moreover, the non-universal additive constant f nonuniv has been determined in rectangular geometry in the Ising universality class [35] and in c = −2 universality class for the dimer model [28] and spanning tree model [29].…”

Section: Introductionmentioning

confidence: 89%

“…It has been shown [26,27,29,45] that the exact partition functions of the Ising model, dimer model and spanning tree model on different planar lattices under free, cylindrical and periodic boundary conditions can be written in terms of the single expression…”

Section: Spanning Tree On the Cobweb Networkmentioning

confidence: 99%

“…In 2015, Izmailian and Kenna considered five different sets of boundary conditions for the spanning tree on finite square lattices, and expressed the partition functions in terms of a principal partition function with twisted-boundary conditions. In each case, they also derived the exact asymptotic expansions of the logarithm of the partition function [29]. Izmailian, Kenna and Wu also derived the spanning-tree-generating function for cobweb and fan networks [14,15].…”

Section: Introductionmentioning

confidence: 99%

“…There is little evidence to support these predictions from either exact solutions or numerical determinations [28,29,35]. An efficient bond propagation algorithm was recently used to compute the partition function of the Ising model with free edges and corners in two dimensions on a rectangular lattice [35].…”

Section: Introductionmentioning

confidence: 99%

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Universality and Exact Finite-Size Corrections for Spanning Trees on Cobweb and Fan Networks

Измаилян

Kenna

2019

Entropy

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The concept of universality is a cornerstone of theories of critical phenomena. It is very well understood in most systems, especially in the thermodynamic limit. Finite-size systems present additional challenges. Even in low dimensions, universality of the edge and corner contributions to free energies and response functions is less investigated and less well understood. In particular, the question arises how universality is maintained in correction-to-scaling in systems of the same universality class but with very different corner geometries. Two-dimensional geometries deliver the simplest such examples that can be constructed with and without corners.To investigate how the presence and absence of corners manifest universality, we analyze the spanning tree generating function on two different finite systems, namely the cobweb and fan networks. The corner free energies of these configurations have stimulated significant interest precisely because of expectations regarding their universal properties and we address how this can be delivered given that the finite-size cobweb has no corners while the fan has four.To answer, we appeal to the Ivashkevich-Izmailian-Hu approach which unifies the generating functions of distinct networks in terms of a single partition function with twisted boundary conditions. This unified approach shows that the contributions to the individual corner free energies of the fan network sum to zero so that it precisely matches that of the web. It therefore also matches conformal theory (in which the central charge is found to be c = -2) and finite-size scaling predictions.Correspondence in each case with results established by alternative means for both networks verifies the soundness of the Ivashkevich-Izmailian-Hu algorithm. Its broad range of usefulness is demonstrated by its application to hitherto unsolved problems namely the exact asymptotic expansions of the logarithms of the generating functions and the conformal partition functions for fan and cobweb geometries. We also investigate strip geometries, again confirming the predictions of conformal field theory.Thus the resolution of a universality puzzle demonstrates the power of the algorithm and opens up new applications in the future.

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Finite size and boundary effects in critical two-dimensional free-fermion models

Измаилян

2017

Eur. Phys. J. B

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

Cited by 3 publications

References 49 publications

Exact finite-size corrections in the dimer model on a planar square lattice

Exact finite-size corrections in the dimer model on a planar square lattice

Universality and Exact Finite-Size Corrections for Spanning Trees on Cobweb and Fan Networks

Finite size and boundary effects in critical two-dimensional free-fermion models

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