Mermin-Wagner theorem

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In quantum field theory and statistical mechanics (at a nonzero temperature), the Mermin-Wagner theorem (also known as Mermin-Wagner-Hohenberg theorem or Coleman theorem) states that continuous symmetries cannot be spontaneously broken in two dimensional theories. This is because if such a spontaneous symmetry breaking occurred, then the corresponding Goldstone bosons, being massless, would have an infrared divergent correlation function. Intuitively, this means that long-range fluctuations can be created with little energy cost and since they increase the entropy they are favored.

The absence of spontaneous symmetry breaking in two-dimensional systems was rigorously proved by Coleman in quantum field theory and by Mermin, Wagner and Hohenberg in statistical physics.

1 Introduction
2 Kosterlitz-Thouless transition
3 Heisenberg model
4 References

[edit] Introduction

We begin by investigating the free field model in which however we can see the basic idea that underlies our further discussion. Suppose we have a free scalar field $φ$ of mass m in d Euclidean dimensions and calculate its propagator

$\left\langle \phi (x)\phi (0) \right\rangle = \frac{1} {{\rm Z}}\int {D\phi } e^{-S[\phi ]} \phi (x)\phi (0) = \int {\frac{{d^d k}} {{(2\pi )^d }}\frac{{e^{ik \cdot x} }} {{k^2 + m^2 }}}$

For d=2, we have $\langle \phi(x) \phi(0) \rangle = \frac{1}{2\pi}K_0(mr)$ where K₀ is the modified Bessel function of order 0.

If $m = 0$ , we have shift symmetry and this integral becomes

$\int {\frac{{d^d k}} {{(2\pi )^d }}\frac{{e^{ik \cdot x} }} {{k^2 }}}$

which is infrared divergent for $d\leqslant 2$ .

When d=2,

$\langle \phi(x)\phi(0) \rangle = - \frac{1}{2\pi}\ln x + C$

where C is a constant to be determined.

If we are describing a Wick rotated quantum field theory, we need to satisfy reflection positivity. In particular,

$\left\langle \phi\left(\frac{x}{2}\right)\phi\left(-\frac{x}{2}\right) \right\rangle = - \frac{1}{2\pi}\ln x + C \geqslant 0$

for all positive x. But this is not possible for any finite C, which means that C has to be infinite.

If we are describing a statistical mechanics model, we need to satisfy positivity.

For all field theories, products like $\langle \phi(0)\phi(0) \rangle$ need to be regularized. Here, we will use a simple smearing. Define $\phi[f]\equiv \int d^2x f(x)\phi(x)$ with f being a smooth test function with compact support of radius R. The mean square fluctuation $\left\langle \phi[f]^2 \right\rangle$ goes as about $- α R 2 ln R + β R 2 + γ C R 2$ , where α, β and γ are positive numbers which depend upon the precise form of f. Positivity tells us that $\left\langle \phi[f]^2 \right\rangle \geqslant 0$ for all R but this is only possible if C is infinite.

If C is infinite, $\left\langle \phi[f]^2 \right\rangle$ is also infinite, which means that $\langle \phi[f] \rangle$ does not make sense. We note that this divergence is caused by the fact that the field is massless. However, it is important to note that this does not mean such a model does not make sense because this model is perfectly consistent. What it does mean is φ is not a good field to work with, although the difference φ(y)-φ(x) does not suffer from any IR divergence problems (To understand this, note that φ(x) requires knowing the absolute value of φ at x but φ(y)-φ(x) only requires a knowledge of the difference in the value of φ between two finitely separated points) and is perfectly valid to work with.

What people (i.e. string theorists) often work with is a related field $e i q φ$ where q is any real number (if φ is really a periodic field, then q can only take on discrete values). This family of redefined fields (one for each q) does not suffer from IR divergences and is better behaved as a result. Instead of the shift symmetry $\phi \rightarrow \phi + \theta$ , we now have a rotation $e^{iq\phi} \rightarrow e^{iq\theta}e^{iq\phi}$ . The somewhat surprising result is

$\langle e^{iq\phi}\rangle = 0$

This is not what we would expect if we have spontaneous symmetry breaking.

Another point worth emphasizing is that in the ordered phase, we still have massless "Goldstone bosons" (or quasiparticles without an energy gap) despite the absence of any spontaneous symmetry breaking. We will encounter two other examples later; the XY model and the Heisenberg model in the ordered phase. The Goldstone bosons in both cases are the spin waves.

Yet another point worth emphasizing is that we were only able to deduce that C is infinite because the total volume (or rather area) of space is infinite so that we can take R to infinity. If space is compact, this argument does not apply any longer and the Mermin-Wagner theorem is no longer applicable and spontaneous symmetry breaking of continuous symmetries is possible. This is why open and closed strings can be localized in space as the spatial cross section of the worldsheet is compact.

[edit] 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 XY U(1) symmetry. However, it does not prevent the existence of an ordered phase.

[edit] Heisenberg model

We will consider the Heisenberg model in d dimensions, that is a system of n-component spins ${\mathbf{S}}_i$ of unit length $|{\mathbf{S}}_i | = 1$ , located at the sites of a d-dimensional square lattice, with nearest neighbor coupling J. Thus the Hamiltonian is

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

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

${\mathbf{S}} = \left( {\sqrt {1 - \sum\limits_\alpha {\sigma _\alpha ^2 } } ,\{ \sigma _\alpha \} } \right), \,\,\, \alpha = 1,2,\dots ,n - 1$

with $|\sigma _\alpha | \ll 1$ and Taylor expand the resulting Hamiltonian. We have

${\mathbf{S}}_i \cdot {\mathbf{S}}_j = \sqrt {\left( {1 - \sum\limits_\alpha {\sigma ^2 _{i\alpha } } } \right)\left( {1 - \sum\limits_\alpha {\sigma ^2 _{j\alpha } } } \right)} + \sum\limits_\alpha {\sigma _{i\alpha } \sigma _{j\alpha } } = 1 - \tfrac{1}{2} \sum\limits_\alpha \left({{\sigma ^2 _{i\alpha }} + {\sigma ^2 _{j\alpha } } }\right) + \sum\limits_\alpha {\sigma _{i\alpha } \sigma _{j\alpha } } + \mathcal{O}(\sigma ^4 ) = 1 - \tfrac{1} {2}{\sum\limits_\alpha {(\sigma _{i\alpha } } - \sigma _{j\alpha } )^2 } + \ldots$

whence

$H = H_0 + \tfrac{1} {2}J\sum\limits_{\left\langle {i,j} \right\rangle } {\sum\limits_\alpha {(\sigma _{i\alpha } } - \sigma _{j\alpha } )^2 } + \ldots$

Ignoring the irrelevant constant term $H 0 = - J N d$ and passing to the continuum limit, given that we are interested in the low temperature phase where long-wavelength fluctuations dominate, we get

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

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 $1 / k 2$ . Thus the real space two-point correlation function for each of these modes is

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

where a is the lattice spacing. The average magnetization is $\left\langle {S_1} \right\rangle =1-\tfrac{1}{2}\sum\limits_\alpha \left\langle {{\sigma _\alpha ^2 }} \right\rangle + \ldots$ and the first order correction can now easily be calculated

$\sum\limits_\alpha \left\langle {{\sigma _\alpha ^2 (0) }} \right\rangle = (n-1) \frac{1} {{\beta J}}\int\limits_{}^{1/a} {\frac{{d^d k}} {{(2\pi )^d }}\frac{1} {{k^2 }}}$

The integral above is proportional to

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

and so it is finite for d>2, but appears to be logarithmically divergent for $d \leqslant 2$ . 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 $d \leqslant 2$ 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 and is called Mermin-Wagner-Hohenberg theorem:

There is no phase with spontaneous breaking of a continuous symmetry for T>0, in $d \leqslant 2$ 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). (See ref. [4])