Spins and fermions on arbitrary lattices (Q1070012)

Consider a symmetric and connected graph L with N as a set of all its vertices and a mapping of N into the set of finite-dimensional fermionic operators fulfilling the requirement: \[ \{\Phi^+_ n,\Phi_ m)=\delta^ n_ m,\quad \{\Phi_ n,\Phi_ m\}=0,\quad n,m\in N. \] Enter the Clifford variables: \[ X_ n=\Phi^+_ n+\Phi_ n,\quad Y_ n=i(\Phi^+_ n-\Phi_ n),\quad n\in N, \] and the double-folding \(\tilde L\) of the graph L (vertex set \(\tilde N=Nx\{X,Y\}\), ((m,W),(n,Z))\(\in \tilde L\) iff \(m=n\) and \(W\neq Z\) or (m,n)\(\in L)\). Now assigning to every link l of \(\tilde L\) an operator \(S(l)=iW_ mZ_ n\) for \(l=(W_ m,Z_ n)\) one gets the following algebra: 1) \(S(l)^+=S(l)\), 2) \(S(l)^ 2=1\), 3) \([S(l),S(l')]=0\) unless l and l' overlap when we have: \(\{S(l).S(l')\}=0\), 4) \(S(l_ 1)S(l_ 2)\ldots S(l_ k)=i\) where links \(l_ 1,l_ 2,\ldots,l_ k\) form a closed path on the graph. It is proved that this algebra is uniquely determined provided that the additional requirement holds: Tr(\(\prod_{n\in N}S((n,X),(n,Y))=0\). An example of constructing the discussed algebra is put forward using the tensor product of generalized Dirac matrices assigned to every vertex of the graph respectively and then restricting the whole space to a certain subspace. Also bilinear Hamiltonian: \(H=\sum_{n\neq m}Im b_{(n,m)}\Phi^+_ n\Phi_ m+c_{(n,m)}Tr \Phi^+_ n\Phi_ m+\sum_{n}d_ n\Phi^+_ n\Phi_ n\) is expressed in terms of generalized Dirac matrices.