2. The two-site resistance : a theorem

 2.The two-site resistance : a theorem   Consider an infinite latticestructure 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 positionvector of a unit cell inthe  is given by , where  are the unit cell vectors and  areintegers. then, each lattice site can be  characterized by the position ofits cell, , and its position inside the cell, as . Thus, one can write anylattice site as .    Let and denote theelectric 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 firstBrillouin zone ,the unit cell in the reciprocal lattice, with the boundaries According to Kirchhoff’s current rule and Ohm’s law, the total currententering 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 connectsthese points to the two terminals of an external source and measure the currentgoing through the source while no other lattice points are connected to externalsources.

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Then, the two-point resistance is given by Ohm’slaw:                                                                (5)      The computation of the two-point resistance isnow 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 structureof 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 resistanceon 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 byintroducing a resistor between the decorating sites ( see Fig.

1).  In , the antiferromagnetic Potts model has been studied on the generalizeddecorated 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 canbe 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. Thegeneralized decorated square lattice of the resistor network.   By a combination of  Kirchhoff’s current rule and Ohm’slaw, 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 fromEq.(7), we have              (16) where is thedeterminant of the matrix .     The equivalent resistancebetween the origin and lattice site in the generalized decorated square latticecan be calculated from Eq.(8) for d =2: (17) Applying this equation, weanalytically and numerically calculate some resistances: Example 1.

Theresistance between the lattice sites and is given by Example 2. The resistancebetween the lattice sites and is given by Example 3. The resistance between thelattice sites and is given by Example 4. From the symmetry of the lattice oneobtains Example 5. The resistance between the lattice sites and is given 

x

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