I have to prove, considering the limits of high and low temperatures, duality equation: Z = q e 2 N K F ( e − K ) = q − N ( e K + q − 1 ) 2 N F ( e K − 1 e K + q − 1 ) for square lattice with Z = ∑ σ 1 , σ N ∏ &lt; i j &gt; e K δ σ i σ j

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I have to prove, considering the limits of high and low temperatures, duality equation:  Z  =  q      e          2      N      K        F      (          e              −        K              )    =      q          −      N                  (              e                  K                    +      q      −      1      )              2      N        F      (                            e                      K                          −        1                              e                      K                          +        q        −        1              )  for square lattice with  Z  =      ∑                  σ                  1                    ,              σ                  N                          ∏          &amp;lt;      i      j      &amp;gt;            e          K              δ                              σ                          i                                            σ                          j

ustalovatfog · Accepted Answer

The duality equation can be proved using the Fortuin-Kasteleyn representation of the Potts model. Introduce link variables       b          i      j        ∈  {  0  ,  1  } to rewrite the partition function as                    Z                    =                  ∑                      {            σ            }                                    ∏                      (            i            ,            j            )                                    e                      K                          δ                                                σ                  i                                ,                                  σ                  j                                                                                              =                  ∑                      {            σ            }                                    ∏                      (            i            ,            j            )                                                [                                    e          K                          δ                                    σ              i                        ,                          σ              j                                      +                  e          0                (        1        −                  δ                                    σ              i                        ,                          σ              j                                      )                              ]                                                    =                  ∑                      {            σ            }                                    ∏                      (            i            ,            j            )                                                [                                    ∑                                    b                              i                j                                      =            0                    1                                      (                          (                  e          K                −        1        )                  δ                                    σ              i                        ,                          σ              j                                                δ                                    b                              i                j                                      ,            1                          +                  δ                                    b                              i                j                                      ,            0                                                ]                              The variables       b          i      j       form a graph on the lattice. The sum over the spins   σ can now be performed leaving  Z  =      ∑          G            u          b      (      G      )            q          C      (      G      )      where   u  =      e    K    −  1 is the number of edges in the graph G and   C  (  G  ) is the number of clusters. The duality transformation consists in associating to each link       b          i      j       on the lattice a dual link       b          i      j        ∗    =  1  −      b          i      j       on the dual lattice. Each loops on the dual lattice encloses a cluster on the lattice. Using Euler relations, one finally gets the duality relation.

I have to prove, considering the limits of high and low temperatures, duality equation: Z=q e^(2NK) F(e^(-K))=q^(-N)(e^(K)+q-1)^(2 N)F((e^(K)-1)/(e^(K)+q-1))

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