2. The two-site resistance : a theorem

Consider

an infinite lattice structure that is a uniform tiling of d-dimensional

space with 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

lattice

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 the 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