2. the current going through the source while no

 

2. The two-site resistance : a theorem

 

Consider
an infinite lattice structure that is a uniform tiling of d-dimensional
space with resistors. Let

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 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