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**Additional resources for Advanced Statistical Physics: Lecture Notes (Wintersemester 2011/12)**

**Sample text**

Thus 1 1 ln Z = ln(2 cosh2 k) + N 2N log det 1 − tT ∗ (qx , qy ) q using tr(log M ) = α ln λα = log α λα = log det M . e. small qx , qy . Let us therefore consider qx , qy → 0. The integrand becomes log (1 + t2 )2 − 2t(1 − t2 )(cos qx + cos qy ) → log (1 + t)2 − 2t(1 − t2 ) = ln 1 + 2t2 + t4 − 4t + 4t3 = ln(1 − t2 − 2t)2 ≥ 0 √ The minimum of (1−t2 −2t)2 is reached at the critical temperature (upper root of) t = tc = −1± 2 which leads us to a critical β of √ log(1 + 2) βc = 2J Expansion t = tc + ∆t and cos qx + cos qy = 1 − 12 (qx2 + qy2 ) + ...

This leads to a sum of 1 walk. For a single bond we have the walk between these to points. Additionally, there is an infinite number of walks that contain the bond, but for every of these walk, another one can be generated by cutting the bond and exchanging the lines which gives us a sign of −1. 2 Multiple Loops all (weighted) walks = 1 + 1 loop walks + 2 loop walks + 3 loop walks + ... 1 1 = 1 + Ξ + (Ξ)2 + (Ξ)3 + ... 2 3! e. (−1)nc . 3 Walks on the lattice and transfer matrices We show how walks on the lattice can be enumerated using transfer matrices.

Each lattice point x, y and direction µ specifies a particular state vector. Now also the weighted walks allow a transfer-matrix approach x, y, µ|W ∗ (l)|x , y , µ = x, y, µ|W ∗ (l − 1)|x , y , µ x , y , µ |W ∗ (l = 1)|x , y , µ 35 Let us define with |µ ∈ {|ket→, |↑ , |← , |↓ } and calculate x, y, µ|W ∗ (l = 1)|x , y , µ x, y, µ|T ∗ |x , y , µ x, y|x + 1, y x, y|x + 1, y eiπ/4 = T∗ = 0 x, y|x + 1, y e−iπ/4 ≡ where x, y|x , y = δxx δyy . x, y, µ|T ∗ |x , y , µ Fourier basis (diagonal up to the µµ -blocks).