Ising model and auxiliary fields

Lattice systems IAt each site x n = an we haveH(φ) = tφ n−1 + µBφ n + tφ n+1Introducing normal coordinates (periodic: x N+1 = x 1 )φ n = √ 1 ∑e iqan φ(q)NThis diagonalizes the HamiltonianHφ(q) =q≡ǫ(q){ }} {(2t cos(qa) + µB)φ(q)⎛⎜⎝On matrix form:⎞µB t 0 0 ...⎛t µB t 0 ...0 t µB t ... ⎟⎠ → ⎜⎝.... ..ǫ(q 1 ) 00 ǫ(q 2 ) 00 0 ǫ(q 3 ) 00 0 0 ǫ(q 3 )⎞⎟⎠