Mermin-Wagner theorem

In accord with the Hohenberg-Mermin-Wagner theorem [15] the considered two-dimensional system is in the paramagnetic state for T > 0. This result can also be obtained using Eq. (8). Also it can be shown [7] that in the infinite crystal at T = 0 the long-range antiferromagnetic order is destroyed for the hole concentrations x > xc 0.02. 

What is so special about low-dimensional solids The key element here is the Mermin-Wagner theorem [2]. It states [3] that systems with short-range forces that are at finite temperature cannot undergo any phase transition to a state that breaks (1) a discrete symmetry in one dimension (d = 1) or (2) a continuous symmetry in one or two dimensions (d = 2). Why is this so ... 

The impact of this is tremendous. No long-range order (LRO) can exist at finite temperature in one dimension no crystals, no magnets, no superconductors. Only special transitions are possible in two dimensions. The Ising model (n = 1 component) is an example [7]. The Kosterlitz-Thouless transition [8], without LRO, is another case for d = 2 and n = 2, discussed in Section V.C. The thermal fluctuations are very destructive in lower dimensions. Quantum fluctuations (i.e., those associated with the dynamics of a system) also tend to suppress LRO and can sometimes destroy it even at 0 K when the Mermin-Wagner theorem does not apply. Such is the case of the quantum spin- antiferromagnetic models [9] in one dimension. 

Mermin-Wagner theorem (5). This theorem forbids long-range order in one and two dimensions at finite temperature, permitting only superconducting fluctuations ("para superconductivity) to survive. With a smalt amount of interplanar hopping, the correlation length in the vertical direction increases from zero to a finite value. But in many other respects, the present mechanism is inherently... 

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