# 2. an infinite lattice structure that is a

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 real lattice space 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 usually represented in 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 the current distribution 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 between arbitrary lattice points is given by

(8)

where In following section we

use the aforementioned method to determine the two-point resistance on the

generalized decorated square lattice of identical resistors R. 3. Generalized decorated square network The 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 generalized decorated square 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 calcualted 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 by