The dimer model on the triangular lattice.
We analyze the partition function of the dimer model on an triangular lattice wrapped on 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 conformal field theory having central charge . The shift exponent for the specific heat is found to depend on the parity of the number of lattice sites along a given lattice axis: e.g., for odd we obtain the shift exponent , while for even it is infinite (). 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.
In experiments and in numerical studies of critical phenomena, it is essential to take into account finite-size effects in order to extract correct infinite-volume predictions from the data. Therefore, in recent decades there have been many investigations of finite-size scaling, finite-size corrections, and boundary effects for critical model systems. In the quest to improve our understanding of realistic systems of finite extent, two-dimensional models play a crucial role in statistical mechanics as they have long served as a testing ground to explore the general ideas of finite-size scaling under controlled conditions. Very few of them have been solved exactly, the Ising model ferdinand1969 (); JaKe02 (); izmailian2001 (); izmailian2002b (); izmailian2002 (); izmailian2002a (); Kenna2001 (); Kenna2002 (); Kenna2002a () and the dimer model ferdinand1967 (); izmailian2003 (); izmailian2005 (); kong2006 (); kong2006a (); izmailian2006 () being the most prominent examples.
The dimer model is a two-particle system. The main difference between it and one-particle systems such as Ising, Heisenberg, or Potts models etc., is that occupation of a given lattice site ensures that at least one of its nearest-neighbor sites is also occupied. It is well known that, due to this non-locality, the critical behavior exhibited by the dimer model can depend on the lattice structure and shape (square, triangle, honeycomb, etc). Previous studies have shown that finite-size corrections in the free energy can exhibit a strong dependence upon the parity of the lattice, and this has provoked controversial conclusions about the value of the central charge from to . One can expect that such unusual finite-size behavior should also hold for the dimer model on other lattices and here we investigate the triangular lattice in particular.
We are particularly interested in the finite-size scaling behavior of the specific-heat pseudocritical point. In finite systems the counterparts of the singularities which mark higher-order phase transitions in the thermodynamic limit are smooth peaks the shapes of which depend on the critical exponents. In particular, let be the specific heat at a reduced temperature given by for a system of linear extent characterized by . In the infinite-volume limit, diverges at the critical point . In finite volume, the analog to the divergence is a finite peak the shape of which is characterized by (i) its position (ii) its height and (iii) its value at the infinite-volume critical point . In particular, the position of the specific heat peak, , is a pseudocritical point which approaches as , where is called the shift exponent. In most models exhibiting higher-order phase transitions, the shift exponent coincides with the inverse of the correlation-length critical exponent , but this is not a direct conclusion of FSS theory is not always true.
For example, for the Ising model in two dimensions, Ferdinand and Fisher determined that behavior of the specific-heat pseudocritical point matches that of the correlation length with ferdinand1969 (). However, Ising models defined on two-dimensional lattices with other topologies have shift exponents which differ from the inverse correlation length critical exponent (see Ref.JaKe02 () and references therein). This is despite the fact that the critical properties on such lattices are the same as for the torus in the thermodynamic limit. A question we wish to address here is the corresponding status of the shift exponent in the dimer model.
In contrast to spin models, the critical behavior of dimer models are strongly influenced by the structure of the underlying lattice. For example the square lattice dimer model is critical with algebraic decay of correlators Fisher2 (); Fisher3 (), while the dimer model on the anisotropic honeycomb lattice, which is equivalent to five-vertex model on the square lattice Wu5 (), exhibits a KDP-type singularity and the dimer model on the Fisher-type lattice exhibits Ising-type transitions Fisher1 (). Thus, it appears that the dimer model itself has not a single critical behavior, but several critical behaviors associated with different classes of universality.
It has been shown explicitly Kasteleyn () that the free energy per site for the dimer model on the square lattice is insensitive to the precise form of the boundary conditions in the limit of a large lattice. This is in contrast to its finite-size counterpart, for which sensitivity to boundary conditions is notable feature, in particular to the parity of the number of lattice sites along a given lattice axis izmailian2003 (); Ferdinand (). Similar statements hold for the dimer model on the honeycomb and triangular lattices.
Very recently, it has been shown izmailian2005 () that the finite-size corrections of the dimer model on planar square lattices also depend crucially on the parity of and the boundary conditions and such unusual finite-size behavior can be fully explained in the framework of the logarithmic conformal field theory.
Our objective in this paper is to study the finite size properties of dimer model on the plane triangular lattice using the same techniques developed in papers izmailian2003 (); izmailian2002b (). The paper is organized as follows. In Sec. II we introduce the dimer model on the triangular lattice with periodic boundary conditions. In Sec. III we discuss the finite-size corrections for an infinitely long cylinder of circumference and find that the dimer model on the triangular lattice can be described by conformal field theory with central charge . In Sec. IV we investigate the properties of the specific heat near the critical point and find that the specific-heat shift exponent depends on the parity of the number of lattice sites along the lattice axis . For odd we obtain for the shift exponent , while for even we find that the shift exponent is infinity (). Our results are summarized and discussed in Sec. V.
Ii Partition Function
In the present work, we consider the dimer model on triangular lattice under periodic boundary conditions. The partition function is given by
where summation is taken over all dimer covering configurations, where , and are, respectively, dimer weights in the horizontal, vertical and diagonal directions, and where , and are, respectively, the number of horizontal, vertical and diagonal dimers (Fig.1). The dimer model on the triangular lattice undergoes phase transition at the point (and likewise for and ), where the partition function is singular. Thus the general triangular lattice model is critical in the square lattice limit. The dimer weight plays a role similar to the reduced temperature in the Ising model. In what follows, we will set .
An explicit expression for the partition function of the dimer model on an triangular lattice wrapped on torus has been obtained by Fendley, Moessner and Sondhi Moesner () and can be written as izmailian2006 ()
for even . The notation corresponds to periodic boundary conditions for the underlying free fermion in the -direction while represents anti-periodic boundary conditions. The boundary conditions in the -direction are similarly controlled by the parameter .
Since the total number of sites must be even if the lattice is to be completely covered by dimers, we will consider two cases, namely even-even (ee) case when and , and even-odd (eo) case when and . Note that due to the symmetry of the lattice the odd-even case (oe) (, ) can be obtained from even-odd case by simple transformation , where is a aspect ratio.
a). Dimers on lattices
In the even-even case where , the second product in (3) may be compactly written as where the function is given by
Splitting this product into two parts,
shifting the index in the second part from to , and noting the translation symmetry , this may be expressed as
Defining the partition function with twisted boundary conditions
Note also that for all , the partition function is even with respect to its argument t. Hence, near the critical point () we have
The only exception is the point where both and are equal to zero. This case has to be treated separately since at this point () the partition function vanishes. As a result, we have
At the critical point we have
Taking the derivative of Eq.(7) with respect to variable and then considering limit we obtain
b). Dimers on lattices
Thus we can see that partition function for the dimer model on triangular lattice under periodic boundary conditions can be expressed in terms of the only one subject, namely, with and .
Iii Dimer on the infinitely long strip
Conformal invariance of the model in the continuum scaling limit would dictate that the asymptotic finite-size scaling behavior of the critical free energy of an infinitely long two-dimensional strip of finite width has the form
where is the bulk free energy, is a surface free energy and is constant. Unlike the free energy densities and , the constant A is universal. The value of is related to the central charge and the highest conformal weight of the underlying conformal theory, and depends on the boundary conditions in the transversal direction. These two dependencies combine into a function of the effective central charge Blote (); Affleck (); Cardy (),
Let us now consider the dimer model on the infinitely long strip of width under periodic boundary conditions.
Considering the logarithm of the partition function given by Eq.(13), we note that it can be transformed as
The second sum here vanishes in the formal limit . The asymptotic expansion of the first sum can be found with the help of the Euler-Maclaurin summation formula
where , is Catalan’s constant and are so-called Bernoulli polynomials and . We have also used the symmetry property, , of the lattice dispersion relation and its Taylor expansion
where , , , etc.
Thus one can easily write down all the terms of the exact asymptotic expansion for the
From , we can obtain the asymptotic expansion of the free energy per bond of an infinitely long cylinder of circumference . Since the expression for the partition function is different for even and odd , we will consider these two cases separately. For even (), we have
and for odd
The bulk free energy is the same for even and odd cases. Thus we find that the finite-size corrections in a crucial way depend on the parity of . In particular it means that due to the certain non-local features present in the dimer model, a change of parity of induces a change of boundary condition. The similar situation also happen in the dimer model on the square lattice, see izmailian2005 (), where a detailed analysis of boundary conditions and parity dependence effects has been carried out in this context.
Since the effective central charge merely determines some combination of and , one cannot obtain the values of both without some assumption about one of them. This assumption can be a posteriori justified if the conformal description obtained from it is fully consistent. Surprisingly, there are two consistent values of that can be used to describe the dimer model, namely and . For example for the dimer model on an infinitely long cylinder of even circumference one can obtained from Eqs.(18), (20) and (27) that the central charge and the highest conformal weight can take values and or and . For the dimer model on an infinitely long cylinder of odd circumference one can obtained from Eqs.(18), (20) and (28) that the central charge and the highest conformal weight can take values and . It turns out in this case that another consistent conformal description exists, with and izmailian2005 (). In particular, it has been shown (for more details see izmailian2005 ()) that although the dimer model is originally defined on a cylinder with odd circumference , it shows the finite-size corrections expected on a strip and must really be viewed as a model on a strip.
Thus from the finite size analyzes we can see that two conformal field theories with the central charges and can be used to described the dimer model on the triangular lattice. But since the general triangular lattice model is critical in the square lattice limit and the dimer model on the square lattice belongs to universality class izmailian2005 (), we come to the conclusion that the dimer model on the triangular lattice can also be described by conformal field theory having central charge .
Iv Specific heat near the critical point
Let us now consider the behavior of the specific heat near the critical point. The specific heat of the dimer model on triangular lattice is defined as
where is free energy of the system
and where is the lattice area.
The pseudocritical point is the value of the temperature at which the specific heat has its maximum for finite lattice. One can determine this quantity as the point where the derivative of vanishes. The pseudocritical point approaches the critical point as in a manner dictated by the shift exponent ,
where is the characteristic size of the system. The coincidence of with , where is the correlation lengths exponent, is common to most models, but it is not a direct consequence of finite-size scaling and is not always true.
Since the expression for the partition function is different for even and odd , we consider these two cases separately.
where is the critical specific heat, and . We have
From Eq.(32), the first derivative of the specific heat on a finite lattice near the infinite volume critical point can be found, and it seen to vanish when
To find the exponent one need the finite-size corrections to and . The exact asymptotic expansions of and can be found along the same lines as in Ref.izmailian2007 () and leading finite size behavior is
where is the characteristic size of the system. Thus from Eqs. (31) and (39) we find that the shift exponent is for odd . In Fig. 2(a) and 2(b) we plot the dependence of the specific heat for lattice. We can see from Fig. 2(b) that the position of the specific heat peak is shifted from zero.
The partition function given by Eq. (1) is a polynomial function of its argument . In the case of even and the partition function is an even function with respect of its argument and hence the number of diagonal bonds also should be even. There is also a simple geometrical explanation why, in the case of even-even lattices, the number of diagonal dimers should be even. Let us consider an even-even () lattice with periodic boundary conditions in the vertical and horizontal directions. Such a lattice can be divided into two sublattice A and B, as shown in Fig. 3. Each sublattice consists of sites in a such way that every horizontal or vertical edge connects a site in sublattice A to one in sublattice B, while diagonal edges connect two sites within sublattice A or two sites within sublattice B. Note that such a division into A and B sublattices is impossible for even-odd () lattices with periodic boundary conditions, since in that case, one can always find a horizontal or vertical edge which connects two sites in the same sublattice A or B. Thus, in the even-even case, each horizontal or vertical dimer occupies one site from sublattice A and another site from sublattice B, while diagonal dimer occupies two sites from sublattice A or B. If one diagonal dimer occupies two sites from sublattice A one should have another diagonal dimer which occupies two sites of sublattice B in order to insure that remaining sites can be occupied by horizontal and vertical dimers. Thus, in the case of even M and N, only even number of diagonal bonds are allowed. Similar geometrical considerations for the dimer model on lattice with free boundary conditions lead to the conclusion that for both even-even and even-odd lattices the number of diagonal bonds should be even.
Thus the first derivative of vanishes exactly at
In Fig. 4(a) and 4(b) we plot the dependence of the specific heat for a lattice. We can see from Fig. 3(b) that the position of the specific heat peak is equal exactly to zero.
Therefore the maximum of the specific heat (the pseudocritical point ) always occurs at vanishing reduced temperature for any finite lattice and coincides with the critical point at the thermodynamic limit. From Eqs. (31) and (41) we find that the shift exponent is for even .
Thus we have found that the shift exponent for the specific heat depend in a crucial way on the parity of the number of lattice sites along the lattice axis . For odd we obtain for the shift exponent , while for even we have found that the shift exponent is infinity ( ).
We analyze the partition function of the dimer model on triangular lattice wrapped on torus obtained by Fendley, Moessner and Sondhi Moesner (). From a finite-size analysis we have found that the dimer model on the triangular lattice can be described by conformal field theory having central charge . Thus we have shown that the dimer model on the triangular lattice belongs to the same universality class as the dimer model on the square lattice, while the dimer model on the honeycomb lattice belongs to another universality class. In addition, we have found that the shift exponent depends in a crucial way on the parity of the number of lattice sites along the lattice axis : for odd we obtain , while for even we have found that . 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. This adds to the catalog of anomalous circumstances where the shift exponent is not coincident with the correlation-length critical exponent. The present circumstance manifests the additional feature that the shift-exponent is boundary-condition dependent.
One of us (N.Sh.I) thanks the Statistical Physics Group at the Departamento de Ciencias Exatas, Universidade Federal de Lavras, Brazil, for hospitality during completion of this work. This work was supported by the EU Programme FP7-People-2010-IRSES (Project No 269139) and partially supported by FAPEMIG (BPV-00061-10).
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