Mermin-Wagner theorem

Prerequisites: Symmetry Breaking, Goldstone's theorem. Full path (13 pages)

• Why is it interesting?
• Intuitive
• Concrete
• Abstract

Why is it interesting?

A theorem due to Mermin and Wagner states that a continuous symmetry can only be spontaneously broken in a dimension larger than two. For a discrete symmetry this lower critical dimensionality is one. This is, in fact, well known since in quantum mechanics with finitely many degrees of freedom (corresponding to one-dimensional field theory) tunneling between degenerate classical minima allows for a unique symmetric ground state. […] The Mermin-Wagner theorem has been restated by Coleman in the framework of field theory. page 525 in Quantum Field Theory by Claude Itzykson, ‎Jean-Bernard Zuber

Intuitive

The Mermin-Wagner theorem explains, why the existence of phase transitions like Magnetization or Freezing depends on the dimension of the material undergoing the transition. Imagine water molecules close to the freezing point in the following configurations:

• a one-dimensional chain
• a two dimensional monolayer of water molecules
• a cube

The molecules are moving around because of their thermal energy but also interact with each other due to hydrogen bonding.

For water to freeze, the temperature has to be low enough such that the bonding between the molecules becomes strong enough to keep the molecules fixed at their place. This means that the number of water molecules that are neighbours determines how hard it is for a water molecule to move away from its position. We can imagine that there are more neighbours, the higher the dimension of the configuration is. Mermin-Wagner predicts that for certain problems there is a lower critical dimension below which there are not enough neighbors to build up a strong enough bond such that all the water molecules "freeze" at their position. This behaviour is also called cooperativity.

Water freezing is actually a very complex problem and the above explanation is only meant to simplify the visualization and should not be taken to be physically correct. But for simple models with a continuous symmetry, like the XY or Heisenberg model, it is possible to calculate this lower critical dimension to be 2.

Concrete

Statement. Mermin and Wagner proved in 1966 that the isotropic Heisenberg model $$H = -\sum_{i,j} J_{ij}\, \vec S_i\cdot\vec S_j - h\sum_i S^z_i$$ on a lattice in $d = 1$ or $d = 2$ has no spontaneous magnetisation at any temperature $T > 0$, provided the interactions are short ranged in the sense $\sum_j |J_{ij}|\,|\vec r_i - \vec r_j|^2 < \infty$. Precisely: $\lim_{h\to 0}\lim_{N\to\infty} \frac{1}{N}\sum_i\langle S^z_i\rangle = 0$, and the same holds for antiferromagnetic order. Hohenberg proved the analogous statement for superfluid and superconducting order. The theorem says nothing about $T = 0$, nor about discrete symmetries: the two-dimensional Ising model does order at low temperature.

Spin-wave argument. The physical reason is seen most easily in the classical XY model, spins of unit length in a plane at angle $\theta_i$, with $H = -J\sum_{\langle ij\rangle}\cos(\theta_i - \theta_j)$ and $J > 0$. Suppose the spins are nearly aligned at low $T$. Expanding the cosine and passing to the continuum on a hypercubic lattice with spacing $a$, $$H \approx E_0 + \frac{\rho_s}{2}\int d^dx\, (\nabla\theta)^2, \qquad \rho_s = J a^{2-d}.$$ This is a free field. Writing $\theta(x) = \int \frac{d^dk}{(2\pi)^d}\, e^{i k\cdot x}\,\theta_k$, equipartition gives each mode an average energy $\frac{1}{2}k_BT$, and so $$\langle \theta(x)^2 \rangle = \int_{|k| < \pi/a} \frac{d^dk}{(2\pi)^d}\, \frac{k_B T}{\rho_s k^2}.$$ For $d > 2$ this integral is finite. For $d \leq 2$ it diverges at small $k$. In a system of linear size $L$ the lower cutoff is $k \sim 1/L$, and in $d = 2$ $$\langle\theta^2\rangle \approx \frac{k_BT}{2\pi\rho_s}\,\ln\frac{L}{a}.$$ Since $\theta$ is Gaussian in this approximation, the magnetisation is $\langle\cos\theta\rangle = e^{-\langle\theta^2\rangle/2} \propto (a/L)^{k_BT/(4\pi\rho_s)}$, which goes to zero as $L \to \infty$. Long-wavelength spin waves cost arbitrarily little energy, and in two dimensions there are enough of them to destroy long-range order at any $T > 0$. These are exactly the Goldstone modes that the broken symmetry would require, so the assumption of order is inconsistent.

The correlation function in the same approximation is $$\langle \cos(\theta(x) - \theta(0))\rangle \sim \left(\frac{a}{|x|}\right)^{k_BT/(2\pi\rho_s)},$$ a power law instead of a nonzero constant. This quasi-long-range order survives up to the Berezinskii-Kosterlitz-Thouless transition, where unbound vortices make the correlations decay exponentially. So the two-dimensional XY model has a phase transition without any spontaneous symmetry breaking.

Rigorous proof. The proof uses the Bogoliubov inequality $$\frac{\beta}{2}\,\langle\{A, A^\dagger\}\rangle\,\langle[[C, H], C^\dagger]\rangle \geq |\langle [C, A]\rangle|^2,$$ valid for thermal expectation values at inverse temperature $\beta = 1/k_BT$. Choosing $C = S^+_{\vec k}$ and $A = S^-_{-\vec k}$ (Fourier components of the spin operators), the right side is proportional to $m^2$, with $m$ the magnetisation per site, the double commutator is bounded by a constant times $N(c\,k^2 + |h\,m|)$ with $c$ set by $\sum_j |J_{ij}|\,|\vec r_i - \vec r_j|^2$, and summing over $\vec k$ gives an upper bound on the magnetisation. In $d \leq 2$ the sum, which becomes $\int d^dk/(c\,k^2 + |h\,m|)$, diverges as $h \to 0$, which forces the magnetisation to zero. The same infrared integral as in the spin-wave argument appears.

Abstract

Take note that the Mermin-Wagner theorem is not so universal as it is presented most of the time:

In two dimensions, crystals provide another loophole in a well-known result, known as the Mermin–Wagner theorem. Hohenberg, Mermin, and Wagner, in a series of papers, proved in the 1960s that two-dimensional systems with a continuous symmetry cannot have a broken symmetry at finite temperature. At least, that is the English phrase everyone quotes when they discuss the theorem; they actually prove it for several particular systems, including superfluids, superconductors, magnets, and translational order in crystals. Indeed, crystals in two dimensions do not break the translational symmetry; at finite temperatures, the atoms wiggle enough so that the atoms do not sit in lock-step over infinite distances (their translational correlations decay slowly with distance). But the crystals do have a broken orientational symmetry: the crystal axes point in the same directions throughout space. (Mermin discusses this point in his paper on crystals.) The residual translational correlations (the local alignment into rows and columns of atoms) introduce long-range forces which force the crystalline axes to align, breaking the continuous rotational symmetry.http://pages.physics.cornell.edu/~sethna/StatMech/EntropyOrderParametersComplexity.pdf

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1. Calculus
2. Linear Algebra
3. Functional
4. Group Theory
5. Newton's Second Law
6. Vector Calculus
7. Newtonian Mechanics
8. Symmetry
9. Variational Calculus
10. Euler-Lagrange Equations
11. Lagrangian Formalism
12. Symmetry Breaking
13. Goldstone's theorem