Mermin-Wagner theorem
Encyclopedia
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 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
.
φ of mass m in two Euclidean dimensions. Its propagator
is:
For small m, G is a solution to Laplace's equation with a point source:
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:
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 A = R ei θ 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.
. The Mermin–Wagner theorem prevents any spontaneous symmetry breaking of the model's U(1) symmetry. However, it does not prevent the existence of any phase transitions. In fact the model has two phases: a conventional disordered phase at high temperature, and a low-temperature phase with quasi-long-range order.
, that is a system of n-component spins Si of unit length |Si|=1, located at the sites of a d-dimensional square lattice, with nearest neighbor coupling J. Its Hamiltonian is
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 σα around this direction as follows:
with |σα|1, and Taylor expand the resulting Hamiltonian. We have
whence
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
The field fluctuations σα are called spin wave
s 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/k2. Thus the real space two-point correlation function for each of these modes is
where a is the lattice spacing. The average magnetization is
and the first order correction can now easily be calculated:
The integral above is proportional to
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 and is called 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).
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 two-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.
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
and statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
, 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
Spontaneous symmetry breaking
Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
occurred, then the corresponding Goldstone bosons, being massless, would have an infrared divergent correlation function
Correlation function
A correlation function is the correlation between random variables at two different points in space or time, usually as a function of the spatial or temporal distance between the points...
.
The absence of spontaneous symmetry breaking in d ≤ 2 dimensional systems was rigorously proved by in quantum field theory and by David Mermin
David Mermin
Nathaniel David Mermin is a solid-state physicist at Cornell University best known for the eponymous Mermin-Wagner theorem and his application of the term "Boojum" to superfluidity, and for the quote "Shut up and calculate!"Together with Neil W...
, Herbert Wagner
Herbert Wagner (physicist)
Herbert Wagner is a German theoretical physicist, who mainly works in statistical mechanics. He is a professor emeritus of Ludwig Maximilian University of Munich.-Biography:...
and Pierre Hohenberg
Pierre Hohenberg
Pierre C. Hohenberg is a French-American theoretical physicist, who works primarily on statistical mechanics....
in statistical physics. That the theorem does not apply to discrete symmetries can be seen in the two-dimensional Ising model
Ising model
The Ising model is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables called spins that can be in one of two states . The spins are arranged in a graph , and each spin interacts with its nearest neighbors...
.
Introduction
Consider the free scalar fieldFree field
In classical physics, a free field is a field whose equations of motion are given by linear partial differential equations. Such linear PDE's have a unique solution for a given initial condition....
φ of mass m in two Euclidean dimensions. Its propagator
Propagator
In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. Propagators are used to represent the contribution of virtual particles on the internal...
is:
For small m, G is a solution to Laplace's equation with a point source:
This is because the propagator is the reciprocal of ∇2 in k space. To use Gauss's law
Gauss's law
In physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. Gauss's law states that:...
, 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:
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
Goldstone boson
In particle and condensed matter physics, Goldstone bosons or Nambu–Goldstone bosons are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries...
where the complex field A = R ei θ 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
Another example is the XY modelXY model
The classical XY model is a model of statistical mechanics. It is the special case of the n-vector model for n=2.-Definition:...
. The Mermin–Wagner theorem prevents any spontaneous symmetry breaking of the model's U(1) symmetry. However, it does not prevent the existence of any phase transitions. In fact the model has two phases: a conventional disordered phase at high temperature, and a low-temperature phase with quasi-long-range order.
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 modelHeisenberg model
The Heisenberg model can refer to two models in statistical mechanics:*Heisenberg model , a classical nearest neighbour spin model*Heisenberg model , a model where the spins are treated quantum mechanically using Pauli matrices....
, that is a system of n-component spins Si of unit length |Si|=1, located at the sites of a d-dimensional square lattice, with nearest neighbor coupling J. Its Hamiltonian is
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 σα around this direction as follows:
with |σα|1, and Taylor expand the resulting Hamiltonian. We have
whence
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
The field fluctuations σα are called spin wave
Spin wave
Spin waves are propagating disturbances in the ordering of magnetic materials. These low-lying collective excitations occur in magnetic lattices with continuous symmetry. From the equivalent quasiparticle point of view, spin waves are known as magnons, which are boson modes of the spin lattice...
s 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
Magnetization
In classical electromagnetism, magnetization or magnetic polarization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material...
, 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
Ginzburg criterion
Mean field theory gives sensible results as long as we are able to neglect fluctuations in the system under consideration. The Ginzburg Criterion tells us quantitatively when mean field theory is valid...
.
The model is Gaussian to first order and so the momentum space correlation function is proportional to 1/k2. Thus the real space two-point correlation function for each of these modes is
where a is the lattice spacing. The average magnetization is
and the first order correction can now easily be calculated:
The integral above is proportional to
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 and is called 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 two-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.