2.

The two-site resistance : a theorem Consider an infinite lattice

structure that is a uniform tiling of resistors. Let is the number of lattice sites 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 site ,respectively. 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. In matrix notation Eq.(3) can be written in form:

(4)

To calculate the resistance between two lattice points and ,one connects

these points to the two terminals of an external source and measure the current

going through the source while no other lattice points are connected to external

sources. 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 expression 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 lattice 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 compute 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 some resistances: Example 1. The

resistance between the lattice sites and is given by Example 2. The resistance

between the lattice sites and is given 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

x

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