Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. E

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

Chi-Ning Chen, Chin‐Kun Hu, N. Sh. Izmailian, Ming‐Chya Wu

Published Jan 2, 2019• 4 citations
Publisher

Abstract

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).

DOI
10.1103/physreve.99.012102

Related papers

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

N. Sh. Izmailian, Ming-Chya Wu, Chin‐Kun Hu

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.

arXiv:1611.03200•Physical review. E•Nov 23, 2016•8 citations

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

N. Sh. Izmailian, Chin‐Kun Hu, Ralph Kenna B

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.

Physical Review E•Jun 29, 2015•3 citations

Dimer model on a triangular lattice

N. Sh. Izmailian, Ralph Kenna B

We analyze the partition function of the dimer model on an $\mathcal{M}\ifmmode\times\else\texttimes\fi{}\mathcal{N}$ triangular lattice wrapped on a torus obtained by Fendley, Moessner, and Sondhi [Phys. Rev. B 66, 214513 (2002)]. From a finite-size analysis we have found that the dimer model on such a lattice can be described by a conformal field theory having a central charge $c=\ensuremath{-}2$. The shift exponent for the specific heat is found to depend on the parity of the number of lattice sites $\mathcal{N}$ along a given lattice axis: e.g., for odd $\mathcal{N}$ we obtain the shift exponent $\ensuremath{\lambda}=1$, while for even $\mathcal{N}$ it is infinite ($\ensuremath{\lambda}=\ensuremath{\infty}$). In the former case, therefore, the finite-size specific-heat pseudocritical point is size dependent, while in the latter case it coincides with the critical point of the thermodynamic limit.

arXiv:1106.3376•Physical Review E•Aug 4, 2011•17 citations

Finite-size effect for the critical point of an anisotropic dimer model of domain walls

Somendra M. Bhattacharjee, John F. Nagle

The finite-size effect is studied in the Kasteleyn model of dimers on the brick lattice. This model is isomorphic to an anisotropic domain-wall model. Asymptotic analysis of the exact Pfaffian solution for the specific heat establishes that finite-size-scaling theory is valid near the critical point of this model. The finite-size-scaling function is a function of a scaled temperature variable \ensuremath{\tau} and a shape factor scrr=${N}^{2}$/M, where 2N is the number of lattice points in the direction perpendicular to the preferred axis for the domain walls and 2M is the number of lattice points parallel to the preferred axis. The scaled temperature variable \ensuremath{\tau} is given by ${\mathrm{MN}}^{2}$t/(M+${N}^{2}$), where t is the reduced temperature. As a function of \ensuremath{\tau} the scaling function scrP(\ensuremath{\tau},scrr) is a sequence of \ensuremath{\delta} functions in the limit scrr=0 and a smooth single-peaked function in the limit scrr=\ensuremath{\infty}. In the latter case the specific heat per lattice site can be written as ,\ensuremath{\infty}), where \ensuremath{\alpha} is the bulk specific-heat exponent with the known value of (1/2) and ${\ensuremath{\nu}}_{M}$ is found to have the value of 1. In the case scrr=0, the specific heat per lattice site can be written in an equivalent form by replacing M by N and ${\ensuremath{\nu}}_{m}$ by ${\ensuremath{\nu}}_{N}$ which takes the value ${\ensuremath{\nu}}_{N}$=(1/2). According to finite-size-scaling theory ${\ensuremath{\nu}}_{M}$ and ${\ensuremath{\nu}}_{N}$ may be interpreted to be the critical exponents ${\ensuremath{\nu}}_{\mathrm{y}}$ and ${\ensuremath{\nu}}_{\mathrm{x}}$, respectively, of the divergent length scales in the two principal directions. Our exact values of ${\ensuremath{\nu}}_{m}$ and ${\ensuremath{\nu}}_{N}$ are in agreement with the values of ${\ensuremath{\nu}}_{\mathrm{y}}$ and ${\ensuremath{\nu}}_{\mathrm{x}}$ predicted for general anisotropic domain-wall models.

Physical Review A•May 1, 1985•46 citations

Finite-size effects for the Ising model on helical tori

N. Sh. Izmailian, Chin‐Kun Hu

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.

Physical Review E•Oct 11, 2007•32 citations

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

N. Sh. Izmailian, Koryun B. Oganesyan, Ming‐Chya Wu +1 more

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.

Physical Review E•Jan 24, 2006•33 citations
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).
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.
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.
We analyze the partition function of the dimer model on an $\mathcal{M}\ifmmode\times\else\texttimes\fi{}\mathcal{N}$ triangular lattice wrapped on a torus obtained by Fendley, Moessner, and Sondhi [Phys. Rev. B 66, 214513 (2002)]. From a finite-size analysis we have found that the dimer model on such a lattice can be described by a conformal field theory having a central charge $c=\ensuremath{-}2$. The shift exponent for the specific heat is found to depend on the parity of the number of lattice sites N\mathcal{N}N along a given lattice axis: e.g., for odd N\mathcal{N}N we obtain the shift exponent $\ensuremath{\lambda}=1$, while for even N\mathcal{N}N it is infinite ($\ensuremath{\lambda}=\ensuremath{\infty}$). In the former case, therefore, the finite-size specific-heat pseudocritical point is size dependent, while in the latter case it coincides with the critical point of the thermodynamic limit.
The finite-size effect is studied in the Kasteleyn model of dimers on the brick lattice. This model is isomorphic to an anisotropic domain-wall model. Asymptotic analysis of the exact Pfaffian solution for the specific heat establishes that finite-size-scaling theory is valid near the critical point of this model. The finite-size-scaling function is a function of a scaled temperature variable \ensuremath{\tau} and a shape factor scrr=N2{N}^{2}N2/M, where 2N is the number of lattice points in the direction perpendicular to the preferred axis for the domain walls and 2M is the number of lattice points parallel to the preferred axis. The scaled temperature variable \ensuremath{\tau} is given by MN2{\mathrm{MN}}^{2}MN2t/(M+N2{N}^{2}N2), where t is the reduced temperature. As a function of \ensuremath{\tau} the scaling function scrP(\ensuremath{\tau},scrr) is a sequence of \ensuremath{\delta} functions in the limit scrr=0 and a smooth single-peaked function in the limit scrr=\ensuremath{\infty}. In the latter case the specific heat per lattice site can be written as ,\ensuremath{\infty}), where \ensuremath{\alpha} is the bulk specific-heat exponent with the known value of (1/2) and ${\ensuremath{\nu}}_{M}$ is found to have the value of 1. In the case scrr=0, the specific heat per lattice site can be written in an equivalent form by replacing M by N and ${\ensuremath{\nu}}_{m}$ by ${\ensuremath{\nu}}_{N}$ which takes the value ${\ensuremath{\nu}}_{N}$=(1/2). According to finite-size-scaling theory ${\ensuremath{\nu}}_{M}$ and ${\ensuremath{\nu}}_{N}$ may be interpreted to be the critical exponents ${\ensuremath{\nu}}_{\mathrm{y}}$ and ${\ensuremath{\nu}}_{\mathrm{x}}$, respectively, of the divergent length scales in the two principal directions. Our exact values of ${\ensuremath{\nu}}_{m}$ and ${\ensuremath{\nu}}_{N}$ are in agreement with the values of ${\ensuremath{\nu}}_{\mathrm{y}}$ and ${\ensuremath{\nu}}_{\mathrm{x}}$ predicted for general anisotropic domain-wall models.
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.
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.