2. The two-site resistance : a theorem Consider an infinite lattice structure that is a uniform tiling of resistors. Let is the number of latticesites in the unit cell of the lattice and labeled by . If the position vector of a unit cell in the is given by , where are the unit cell vectors and are integers. then, each lattice site can be characterized by the position of its cell, , and its position inside the cell, as . Thus, one can write any lattice site as .
Let and denote the electric potential and current at any site. The electric potential and current at site are the form of their inverse Fourier transforms as (1) (2) where is the volume of the unit cell and is the vector of the reciprocal lattice in d-dimensions and is limited to the first Brillouin zone ,the unit cell in the reciprocal lattice, with the boundaries According to Kirchhoff’s current rule and Ohm’s law, the total current entering the lattice point in the unit cell can be written as (3) where is a s by s usually called lattice Laplacian matrix. Then, the two-point resistance is given by Ohm’s law: (5) The computation of the two-point resistance is now reduced to solving Eq. (5) for and by using the lattice Green’s function with given by (6) In physics the lattice Green function of the Laplacian matrix L is formally defined as (7) The general resistance formula can be stated as a theorem. Theorem. Consider an infinite lattice structure of resistor network that is a uniform tiling of space in d- dimensions. Then the resistance lattice points is given by (8) where In we use the aforementioned method to determine the two-point resistance on the generalized decorated square lattice of identical resistors R.
3. decorated well- studied decorated square network is formed by introducing extra sites in the middle of each side of a square lattice. Here we compute the two-site resistance on the generalized decorated square lattice obtained by introducing a resistor between the decorating sites ( see Fig. 1). In , the antiferromagnetic Potts model has been studied on the generalized decorated square lattice. In each unit cell there are three lattice sites labeled by ? = A,B, and C as shown in Fig.1.
In two dimensions the lattice site can be characterized by ,where . To find resistances on the lattice, we make use of the formulation given in Ref. 15. The electric potential and current at any site are (9) (10) Fig. 1.
The generalized decorated square lattice of the resistor network. By a combination of Kirchhoff’s current rule and Ohm’s law, the currents entering the lattice sites , from outside the lattice ,are (11) (12) (13) Substituting Eqs. (9) and (10) into (11)- (13), we have (14) where and is the Fourier transform of the Laplacian matrix given by (15) The Fourier transform of the Green’s function can be obtained from Eq.(7), we have (16) where is the determinant of the matrix . The equivalent resistance between the origin and lattice site in the generalized decorated square lattice can be calculated from Eq.
(8) for d =2: (17) Applying this equation, we analytically and numerically calculate someresistances: Example 1. The resistance between the lattice sites and is given by Example 2. The resistance between the lattice sites and isgiven by Example 3. The resistance between the lattice sites and is given by Example 4.
From the symmetry of the lattice one obtains Example 5. The resistance between the lattice sites and is given by