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)

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

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)

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

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