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By Johannes Berg, Gerold Busch

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Since T is translation-invariant, Bloch-states are eigenstates of T. )t / N √ √ We get x|q = eiqx / N , q|x = e−iqx / N , x|x = x|q q|x = q 1 N eiq(x−x ) = δx,x q Proof that |q is an eigenstate of T: x|T |q = x|T |x x = x |q √ δx ,x+1 + δx ,x−1 eiqx / N x √ √ = eiq(x+1) + eiq(x−1) / N = eiq + e−iq eiqx / N = (2 cosh q) x|q ≡ T (q) x|q Now we can compute 0|W (l)|0 = 0|T l |0 = 0|T l |q q|0 q T l (q) 0|q q|0 = = q 1 N T l (q) q This step uses that the Bloch states are also eigenstates of the product Tl .

The energy contribution of a domain wall is ∆H /N ∝ (∆θ)2 ≈ (1/ldomain )2 where ∆θ is the angle between neighboring spins. Taking the domain wall size to be comparable to the size of the domain itself (minimum domain size), we find −2 d−2 d ∆H = ldomain ldomain = ldomain d−1 in discrete systems. This means in continuous systems, This is different from the scaling ldomain domains are easier to introduce because they cost less energy. Already in d = 2, the ordered state is destroyed by fluctuations due to the introduction of domains.

Pushing the series to higher orders allows to extract aspects of critical behaviour. 2 High-temperature approximation (general) For βH[s] 1∀s (high temperatures), the Taylor expansion exp(−βH(s)) ≈ 1−βH[s]+β 2 /2H2 [s]+ ... is dominated by the first few terms. We exploit this expansion to approximate Z: ln Z = ln Tr e−βH = ln Tr 1 − βH[s] + β 2 /2H2 [s] + ... Tr H β 2 Tr H2 + + ... Tr 1 2 Tr 1 Tr H β 2 Tr H2 = ln (Tr (1)) + ln 1 − β + + ... Tr 1 2 Tr 1 β2 H2 0 + ... = ln 2N + ln 1 − β H 0 + 2 = ln (Tr (1)) 1 − β For small β we can make another approximation ln(1 − x) = −x − x2 /2 + ...

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