Critical Thinking

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

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