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Destruction of long range order in one-dimensional and two-dimensional systems having a continuous symmetry group. I. Classical systems

Sep, 1970
8 pages
Published in:
  • Sov.Phys.JETP 32 (1971) 493-500,
  • Zh.Eksp.Teor.Fiz. 59 (1971) 907-920
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Abstract:
The low-temperature state of two-dimensional classical systems, which in the three-dimensional case have an ordered phase with a spontaneous violation of a continuous symmetry (magnetic substances, crystals), is considered. It is shown that for arbitrary dimension the long-range correlations are determined by an expression for the energy of the long wavelength fluctuations, which is quadratic with respect to the gradients. The distinctive feature of the one- and two-dimensional cases is that the fluctuation deflections grow with distance and at sufficiently large distances may reach a finite value, which leads to the necessity to take account of the effects associated with these. Thus, for a lattice of plane classical spins (Sec. 1) the contribution from configurations, where the spin vector on a path between sufficiently distant points is turned through an angle containing several complete revolutions, becomes essential. The following new results are contained in this article: A complete description of the low-temperature state is obtained for a lattice of plane classical spins (Sec. 1) and two-dimensional crystals (Sec. 3 ), i.e., all of the n-point distribution functions are found, and the method is generalized to an arbitrary lattice system with a commutative continuous group; the two-point distribution function and the transformation law for the n-point functions associated with the homogeneous dilatation of all distances are found for a classical Heisenberg ferromagnetic substance (Sec. 2); also expressions are found for the free energy of magnetic substances (Sees. 1 and 2) in a weak external field, from which the necessity of a phase transition in these two-dimensional systems follows.