By Johannes Berg, Gerold Busch

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

**Sample text**

M is finite for t → 0. 36) You can proof the following scaling relations: α + 2β + γ = 2 α + β(1 + δ) = 2 that are valid for all system with a single relevant thermal eigenvalue and a single relevant symmetry breaking field. 46 6 Disordered Systems Much of physics concerns homogeneous systems: • homogeneous and isotropic space in classical mechanics and electrodynamics • regular lattices (crystals) in solid state physics • ferromagnets (all interactions between neighboring spins are of the same kind) in statistical mechanics Common tools and concepts to treat homogeneous systems are: • inverse space methods: Fourier transform • statistical mechanics: transfer matrix • mean field: each degree of freedom is in the same ”environment“ as all the others Frequently, ”homogeneity“ and ”order“ are valid idealizations.

All systems that flow onto the same fixed points are called to be in the same universality class. We will now define the reduced free energy (per spin): f = − N1 ln Z. Then Z = e−N f ({k}) The partition function Z of a system is invariant under RG Z = Tr {s} e −H[s] = Tr {s } e −H [s ] =Z Thus, Z = const × e−N f ({k}) with N = b−d N with f ({k}) = g({k}) + b−d f ({k }) g({k}) represent the ”analytic first modes“ and f ({k}) = b−d f ({k }) defines the singular behavior around fixed point. For the universality class of the Ising model ut , uh are relevant (yt , yh > 0) and f (ut , uh ) = b−d f ( ut , uh ) = ...

F = −kB T ln L + 2J 27 −kB T ln L → ∞ as L → ∞ if T > 0. The free energy of a system with an additional domain wall is thus lower than a system without it. The so called Peierls-argument shows that the ordered phase is unstable against the introduction of domains provided T > 0 (no finite T phase transition). Consider now a domain of down-spins with domain length l in d = 2. The energy difference is ≈ 2Jl. As at each point you can go in three directions, the entropy difference is ≈ ln[number of random walks of length l], therefore ≈ ln[µ]l = l ln µ with µ ≈ 3.

### Advanced Statistical Physics: Lecture Notes (Wintersemester 2011/12) by Johannes Berg, Gerold Busch

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