User:ShaoTang

Collective modes and Broken Symmetry

1,What is symmetry in physics?

A symmetry transformation is a change in our point of view that does not change the result of possible experiments.In particular, a symmetry transformation that is infinitesimally close to being trivial can be represented by a linear unitary operator that is infinitesimally close to be trivial can be represented by a linear unitary operator that is infinitesimally close to the identity: $U=1+i\epsilon t$

with $\epsilon$ a real infintesimal.For this to be unitary and linear,t must be Hermitian and linear, so it is a candidate for an observable.Indeed, most(and perhaps all) of the observables of physics, such as angular momentum or momentum, arise in this way from symmetry transformations.

The set of symmetry transformations has certain properties that define it as a group.(From The Quantum Theory Of Fields Volume I,Steven Weinberg)

For a continuous symmetry,Neother's theorem states that there exists a corresponding conservation law.

There're several typical intrinsic symmetries in condensed matter systems. Examples:

Translation and Rotation symmetry(continuous),Parity symmetry(discrete)

$H=\sum _{i}{\frac {{\textbf {p}}_{i}^{2}}{2m}}+\sum _{i<j}V(\mid {\overrightarrow {{\textbf {r}}_{i}}}-{\overrightarrow {{\textbf {r}}_{j}}}\mid )$

This is a many particle hamiltonian which includes the information of their kinetic energy and pairwise interactions.Hamiltonian invariant under translation or rotation of all coordinates indicates the global Galilean invariance of the system(continuous).Addtionally,this hamiltonian also invariant under space inversion about any point which indicates the parity invariant.

Translation and Rotation symmetry(discrete)

$H={\frac {{\textbf {p}}^{2}}{2m}}+\sum _{\overrightarrow {\textbf {R}}}V({\overrightarrow {\textbf {r}}}-{\overrightarrow {\textbf {R}}})$

It can be used to describe the motion of electrons in a bravias lattace.The hamiltonian would present the point symmetry gained by the lattice.

Spin rotation symmetry(continuous)

Time reversal symmetry(discrete)

With the symmetry properties, we can obtain the conservation laws which would help us simplify the problems. What's more important, a conserved observable is related to some excitation.In the low temperature regimes, we would get some low energy excitations which dominates the gross properties of the system.Thus,when analyzing a certain condensed matter systems, we would first try to figure out its symmetry properties.

2,Symmetry breaking: Explicit symmetry breaking

Explicit symmetry breaking indicates a situation where the dynamical equations are not manifestly invariant under the symmetry group considered.

Spontaneous symmetry breaking

Spontaneous symmetry breaking where the laws are invariant but the system isn't because the background of the system, its vacuum, is non-invariant. Such a symmetry breaking is parameterized by an order parameter.(From wikipedia)

3,Why broken symmetry in low temperature?

We can build up the Free energy F, and then minimize it with respect to some field variable $\phi$. And then we can obtain several minimums, corresponds to several possible ground states. Basically, a system is dominated by its kinetic energy part while in high temperature and the potential part at low temperature . Thus, in order to minimize the potential energy, the system might go through a phase transition and break the original symmetry, there would come out some nonzero expectation value of operators which are the order parameters of the system. The quantity which would indicates that a phase transition has happened is the order parameters.

For example,the density wave of crystal, the magnetization of ferromagnet, the pair condensate of superconductor. The expressions are correspondingly $\sum _{\textbf {k}}\langle C_{\textbf {k}}^{\dagger }C_{\textbf {k+q}}\rangle$,$\sum _{\textbf {k}}\langle C_{{\textbf {k}}\uparrow }^{\dagger }C_{{\textbf {k}}\uparrow }-C_{{\textbf {k}}\downarrow }^{\dagger }C_{{\textbf {k}}\downarrow }\rangle$,$\langle C_{{\textbf {k}}\uparrow }C_{{\textbf {-k}}\downarrow }\rangle$.

We should be careful when we apply the fundamental ergodicity postulate of statistical mechanics, that phase space of the system under this situation, actually sperates to different parts which have large potential barriers between them. Thus we cannot simply take the average over all configurations. If there're two minimums for example, they are totally different macroscopical configurations, not merely microscopical ones. Thus, in this situation, if we take average including this two, basically we would get zero of our order parameters and nothing exists. Thus, we need to be refrain our calculations when system goes across the "barrier" into another different state.

Here, we could introduce the powerful $\phi ^{4}$ theory to study some interacting field theory. $Z=\int D\phi e^{-S[\phi ]},S[\phi ]=\int d^{d}x(k(\partial \phi )^{2}+r\phi ^{2}+g\phi ^{4})$ The $\phi ^{4}$ model is extremely useful. For example, close to the critical point, the Ising model can be described by the $\phi ^{4}$ action. More generally, the long-range behavior of classical statical systems with a single order parameter is described by the $\phi^4$ action. Within the context of statistical mechanics. $S[\phi ]$ is known as the Ginzburg-Landau free energy functional. And notice that this functional has $Z_{2}$ symmetry reagrding $\phi$.

Then we need to minimize our functional S which directly related to F. We obtain the following equation: $2k\partial _{x}^{2}\phi =2r\phi +4g\phi ^{3}$ For a uniform solution( it happens when there's no external field), we have $r\phi +2g\phi ^{3}=0$. Then wether we would get a real solution is based on the sign of r and g. Obviously, we can get a trivial solution that is $\phi =0$, another two is $\phi =\pm {\sqrt {\frac {-r}{2g}}}$

In the latter cases, our symmetry has already broken, and it happens when r and g has a different sign. This indicates that original solution $\phi =0$ may not be the stable one under some condition, $T<T_{c}$,etc.