Mermin–Wagner theorem

In quantum field theory and statistical mechanics, the Mermin–Wagner theorem (also known as Mermin–Wagner–Hohenberg theorem or Coleman theorem) states that continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions d ≤ 2. Intuitively, this means that long-range fluctuations can be created with little energy cost and since they increase the entropy they are favored.

This is because if such a spontaneous symmetry breaking occurred, then the corresponding Goldstone bosons, being massless, would have an infrared divergent correlation function.

The absence of spontaneous symmetry breaking in d ≤ 2 dimensional systems was rigorously proved by Sidney Coleman (1973) in quantum field theory and by David Mermin, Herbert Wagner and Pierre Hohenberg in statistical physics. That the theorem does not apply to discrete symmetries can be seen in the two-dimensional Ising model.

Introduction

Consider the free scalar field φ of mass m in two Euclidean dimensions. Its propagator is:

$G(x) = \left\langle \varphi (x)\varphi (0) \right\rangle = \int \frac{d^2 k}{(2\pi)^2} \frac{e^{ik \cdot x}}{k^2 + m^2}.$

For small m, G is a solution to Laplace's equation with a point source:

$\nabla^2 G = \delta(x).$

This is because the propagator is the reciprocal of ∇² in k space. To use Gauss's law, define the electric field analog to be E = ∇G. The divergence of the electric field is zero. In two dimensions, using a large Gaussian ring:

$E = {1\over 2\pi r}.$

So that the function G has a logarithmic divergence both at small and large r.

$G(r) = {1\over 2\pi} \log(r)$

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 A = Re^iθ has an expectation value but is free to slide in the θ direction, the angle θ will be random at large distances. This is the Mermin–Wagner theorem: there is no spontaneous breaking of a continuous symmetry in two dimensions.

Kosterlitz–Thouless transition

Main article: Kosterlitz–Thouless transition

Another example is the XY model. The Mermin–Wagner theorem prevents any spontaneous symmetry breaking of the model's continuous (internal) O(2) symmetry on a spatial lattice of dimension d ≤ 2, i.e. the (spin-)field's expectation value remains zero for any finite temperature (quantum phase transitions remain unaffected). However, the theorem does not prevent the existence of a phase transition in the sense of a diverging correlation length ξ. To this end, the model has two phases: a conventional disordered phase at high temperature with dominating exponential decay of the correlation function $G(r)\sim\exp(-r/\xi)$ for $r/\xi\gg1$, and a low-temperature phase with quasi-long-range order where G(r) decays according to some power law for "sufficiently large", but finite distance r (a ≪ r ≪ ξ with a the lattice spacing).

Heisenberg model

We will present an intuitive way to understand the mechanism that prevents symmetry breaking in low dimensions, through an application to the Heisenberg model, that is a system of n-component spins S_i of unit length |S_i| = 1, located at the sites of a d-dimensional square lattice, with nearest neighbor coupling J. Its Hamiltonian is

$H = - J\sum_{\left\langle {i,j} \right\rangle } \mathbf{S}_i \cdot \mathbf{S}_j.$

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 O(n) rotational symmetry of the system is spontaneously broken, or rather reduced to the O(n − 1) symmetry under rotations around this direction. We can parametrize the field in terms of independent fluctuations σ_α around this direction as follows:

$\mathbf{S} = \left(\sqrt{1 - \sum_\alpha \sigma_\alpha^2}, \left \{\sigma_\alpha \right\} \right), \qquad \alpha = 1, \cdots, n - 1.$

with |σ_α| ≪ 1, and Taylor expand the resulting Hamiltonian. We have

$\begin{align} \mathbf{S}_i \cdot \mathbf{S}_j &= \sqrt{\left(1 - \sum_\alpha \sigma^2_{i\alpha} \right)\left(1 - \sum_\alpha \sigma^2_{j\alpha } \right)} + \sum_\alpha \sigma_{i\alpha} \sigma_{j\alpha}\\ &= 1 - \tfrac{1}{2} \sum_\alpha \left(\sigma^2_{i\alpha} + \sigma^2_{j\alpha}\right) + \sum_\alpha \sigma _{i\alpha} \sigma _{j\alpha} + \mathcal{O}\left (\sigma ^4 \right )\\ &= 1 - \tfrac{1}{2} \sum_\alpha \left (\sigma _{i\alpha} - \sigma _{j\alpha } \right )^2 + \ldots \end{align}$

whence

$H = H_0 + \tfrac{1}{2} J\sum_{\left\langle i,j \right\rangle} \sum_\alpha \left (\sigma_{i\alpha}- \sigma_{j\alpha} \right )^2 + \cdots$

Ignoring the irrelevant constant term H₀ = −JNd and passing to the continuum limit, given that we are interested in the low temperature phase where long-wavelength fluctuations dominate, we get

$H = \tfrac{1}{2}J \int {d^d x\sum_\alpha {(\nabla \sigma _\alpha )^2 } } + \ldots.$

The field fluctuations σ_α are called spin waves and can be recognized as Goldstone bosons. Indeed, they are n-1 in number and they have zero mass since there is no mass term in the Hamiltonian.

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 k⁻². Thus the real space two-point correlation function for each of these modes is

$\left\langle \sigma_\alpha (r)\sigma_\alpha (0) \right\rangle = \frac{1}{\beta J} \int^{\frac{1}{a}} \frac{d^d k}{(2\pi)^d} \frac{e^{i\mathbf{k} \cdot \mathbf{r}}}{k^2}$

where a is the lattice spacing. The average magnetization is

$\left\langle S_1 \right\rangle =1-\tfrac{1}{2}\sum_\alpha\left\langle \sigma_\alpha^2 \right\rangle + \ldots$

and the first order correction can now easily be calculated:

$\sum_\alpha \left\langle \sigma_\alpha ^2 (0) \right\rangle = (n-1)\frac{1}{\beta J} \int^{\frac{1}{a}}\frac{d^d k}{(2\pi)^d} \frac{1}{k^2}.$

The integral above is proportional to

$\int^{\frac{1}{a}} k^{d-3} dk$

and so it is finite for d > 2, but appears to be logarithmically divergent for d ≤ 2. However, this is really an artifact of the linear approximation. In a more careful treatment, the average magnetization is zero.

We thus conclude that for d ≤ 2 our assumption that there exists a phase of spontaneous magnetization is incorrect for all T > 0, because the fluctuations are strong enough to destroy the spontaneous symmetry breaking. This is a general result:

Mermin–Wagner–Hohenberg Theorem. There is no phase with spontaneous breaking of a continuous symmetry for T > 0, in d ≤ 2 dimensions.

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).

Generalizations

Much stronger results than absence of magnetization can actually be proved, and the setting can be substantially more general. In particular:

1. The Hamiltonian can be invariant under the action of an arbitrary compact, connected Lie group G.
2. Long-range interactions can be allowed (provided that they decay fast enough; necessary and sufficient conditions are known).

In this general setting, Mermin–Wagner theorem admits the following strong form (stated here in an informal way):

All (infinite-volume) Gibbs states associated to this Hamiltonian are invariant under the action of G.

When the assumption that the Lie group be compact is dropped, a similar result holds, but with the conclusion that infinite-volume Gibbs states do not exist.

Finally, there are other important applications of these ideas and methods, most notably to the proof that there cannot be non-translation invariant Gibbs states in 2-dimensional systems. A typical such example would be the absence of crystalline states in a system of hard disks (with possibly additional attractive interactions).

It has been proved however that interactions of hard-core type can lead in general to violations of Mermin–Wagner theorem.

References

13. Mermin-Wagner Theorem at Scholarpedia, curated by Herbert Wagner.