To use all functions of this page, please activate cookies in your browser.

my.chemeurope.com

With an accout for my.chemeurope.com you can always see everything at a glance – and you can configure your own website and individual newsletter.

- My watch list
- My saved searches
- My saved topics
- My newsletter

## Mermin-Wagner theorem
In quantum field theory and statistical mechanics, the The absence of spontaneous symmetry breaking in one and two dimensional systems was rigorously proved by Coleman in quantum field theory and by Mermin, Wagner and Hohenberg in statistical physics.
## Additional recommended knowledge
## IntroductionConsider the free scalar field φ of mass For small m, G is a solution to Laplace's equation with a point source: This is because the propagator is the reciprocal of in k space. To use Gauss's law, define the electric field analog to be . The divergence of the electric field is zero. In two dimensions, using a large Gaussian ring: So that the function G has a logarithmic divergence both at small and large r. The interpretation of the divergence is that the field fluctuations cannot stay centered around a mean. If you start at a point where the field has the value 1, the divergence tells you that as you travel far away, the field is arbitrarily far from the starting value. This makes a two dimensional massless scalar field slightly tricky to define mathematically. If you define the field by a monte-carlo simulation, it doesn't stay put, it slides to infinitely large values with time. This happens in one dimension too, when the field is a one dimensional scalar field, a random walk in time. A random walk also moves arbitrarily far from its starting point, so that a one-dimensional or two-dimensional scalar does not have a well defined average value. If the field is an angle, θ, as it is in the mexican hat model where the complex field ## Kosterlitz-Thouless transition
Another example is the XY model. The Mermin-Wagner theorem prevents any spontaneous symmetry breaking of the XY U(1) symmetry. However, it ## Heisenberg modelWe will consider the Heisenberg model in The name of this model comes from its rotational symmetry. Let us consider the low temperature behavior of this system and assume that there exists a spontaneously broken, that is a phase where all spins point in the same direction, e.g. along the x-axis. Then the with and Taylor expand the resulting Hamiltonian. We have whence Ignoring the irrelevant constant term The field fluctuations σ To find if this hypothetical phase really exists we have to check if our assumption is self-consistent, that is if the expectation value of the magnetization, calculated in this framework, is finite as assumed. To this end we need to calculate the first order correction to the magnetization due to the fluctuations. This is the procedure followed in the derivation of the well-known Ginzburg criterion. The model is Gaussian to first order and so the momentum space correlation function is proportional to 1 / where The integral above is proportional to and so it is finite for We thus conclude that for our assumption that there exists a phase of spontaneous magnetization is incorrect for all
The result can also be extended to other geometries, such as Heisenberg films with an arbitrary number of layers, as well as to other lattice systems (Hubbard model, s-f model). (See ref. [4]) ## References1. P.C. Hohenberg: "Existence of Long-Range Order in One and Two Dimensions", Phys. Rev. 158, 383 (1967) 2. N.D. Mermin, H. Wagner: "Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models", Phys. Rev. Lett. 17, 1133–1136 (1966) 3. Sidney Coleman: "There are no Goldstone bosons in two dimensions", Commun. Math. Phys. 31, 259 (1973) 4. Axel Gelfert, Wolfgang Nolting: "The absence of finite-temperature phase transitions in low-dimensional many-body models: a survey and new results", J. Phys.: Condens. Matter 13, R505-R524 (2001) |
|||

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Mermin-Wagner_theorem". A list of authors is available in Wikipedia. |