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 the identity: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U=1+i\epsilon t}

with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)

Mathematically, for a system which has a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_2 } symmetry, the pitchfork bifurcation might involved.

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_\textbf{k}\langle C^\dagger_\textbf{k}C_\textbf{k+q}\rangle} ,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_\textbf{k}\langle C^\dagger_{\textbf{k}\uparrow} C_{\textbf{k}\uparrow}-C^\dagger_{\textbf{k}\downarrow} C_{\textbf{k}\downarrow}\rangle} ,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 refrain our calculations when system goes across the "barrier" into another different state.

Here, we could introduce the powerful Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi^4} theory to study some interacting field theory. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z=\int D\phi e^{-S[\phi]}, S[\phi]=\int d^dx(k(\partial\phi)^2+r\phi^2+g\phi^4) } The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi^4} model is extremely useful. For example, close to the critical point, the Ising model can be described by the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi^4} action. More generally, the long-range behavior of classical statical systems with a single order parameter is described by the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi^4} action. Within the context of statistical mechanics. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S[\phi]} is known as the Ginzburg-Landau free energy functional. And notice that this functional has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_2} symmetry reagrding Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} .

Then we need to minimize our functional S which directly related to F. We obtain the following equation: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2k\partial^2_x\phi=2r\phi+4g\phi^3} For a uniform solution( it happens when there's no external field), we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi=0} , another two is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi=0} may not be the stable one under some condition, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T<T_c} ,etc.

4 Collective phenomena

Minimum steps(for non-symmetry broken induced collective modes):

Write down the partition function as a path integral.

Introduce the auxiliary bosonic field.

Utilize the Hubbard–Stratonovich transformation, decouple the fields, evaluate the Matsubara frequency sum.

Take the factors of quadratic terms of fields, analytically continue it from imaginary time to real time retard Green's function, find out all the poles, We get the collective modes!!!

Symmetry broken case

(From Consented Matter Field Theory, Altland, Simons) The appearance of non-trivial ground states is just one manifestation of spontaneous symmetry breaking. Equally important, residual fluctuations around the ground state lead to the formation of soft modes (massless modes), i.e. field configurations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_\textbf{q}} whose action Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S[\phi]} vanishes in the limit of long wavelength, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{q}\rightarrow0} . Specifically, the soft modes formed on the top of a symmetry broken ground state are called Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{Goldstone modes}} . As a rule, the presence of soft modes in a continuum theory has important phenomenological consequences. To understand this point, notice that the general structure of a soft mode action is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S[\phi]=\sum_{\textbf{q},i}\phi_\textbf{q}(c^i_1|q_i|+c^i_2q^2_i)\phi_-\textbf{q}+ O(\phi^4,q^3)} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c^i_{1,2}} are coefficients. The absence of a constant contribution to the action(i.e a contribution that does not vanish in the limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{q}\rightarrow0} ) signals the existence of long-range power-law correlations in the system. As we will see shortly, the vanishing of the action in the long-wavelength limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{q}\rightarrow0} further implies that the contribution of the soft modes dominates practically all observable properties of the system.

What are the origin and nature of the soft Goldstone modes caused by the spontaneous breakdown of a symmetry? To address this point let us consider the action of a symmetry group element g on a symmetry broken ground state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_0} . By definition, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S[g\psi_0]=S[\psi_0]} still assumes its extremal value. Assuming that g is close to the group identity, we may express Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g=\exp(\sum_a\phi_a T_a)} , where the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {T_a}} are generators living in the Lie algebra of the group(differentiable) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_a} are some expansion coefficients. Expressing fluctuations around Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_0} in terms of the "coordinates" Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_a} , we conclude that the action Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S[\phi]=0} . However, if we promote the global transformation to one with a weakly fluctuating spatial profile, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\rightarrow g(\textbf{r}), \psi_0\rightarrow g(\textbf{r})\psi_0} , some price must be paid. That is, for a spatially fluctuating coordinate profile Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\phi_a(\textbf{r})}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S[\phi]\neq0} , where the energy cost depends inversely on the fluctuation rate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda } of the field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} . The expansion of S in terms of gradients of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi } is thus bound to lead to a soft mode action of the type as the previous equation.

In view of their physical significance, it is important to ask how many independent soft mode exist. The answer can be straightforwardly given on the basis of the geometric picture developed above. Suppose our symmetry group G has dimension r, i.e. its Lie algebra is spanned by r linearly independent generators Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_a} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=1,\ldots,r} . If the subgroup Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H\sub G } has dimension s<r, s of these generators can be chosen so as to leave the ground state invariant. On the other hand, the remaining p=r-s generators inevitably create Goldstone modes. In the language of group theory, these generators span the coset space G/H. For example, for the ferromagnet, H=O(2) is the one-dimensional subgroup of rotations around the quantization axis(the z-axis). Since the rotation group has dimension 3, there must be two independent Goldstone modes. These can be generated by the action of the rotation, or angular momentum generators Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_{x,y}} acting on the z-aligned ground state. The coset space O(3)/O(2) can be shown to be isomorphic to the 2-sphere, i.e. the sphere traced out by the spins as they fluctuate around the ground state.

Finally, the connection between the coordinates parameterizing the Goldstone modes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_i, i=1,\ldots,p} residual "massive modes" $k_{j},j=1,\ldots ,n-p$, and the original coordinates $\psi _{i},i=1,\ldots ,n$, of the problem, respectively, is usually nonlinear and sometimes not even very transparent. With problems more complex than the three prototypical examples mentioned above, it is usually profitable to first develop a good understanding of the geometry of the problem before specific coordinate systems are introduced.

Goldstone's theorem examines a generic continuous symmetry which is spontaneously broken; i.e., its currents are conserved, but the ground state (vacuum) is not invariant under the action of the corresponding charges. Then, necessarily, new massless (or light, if the symmetry is not exact) scalar particles appear in the spectrum of possible excitations. There is one scalar particle—called a Nambu–Goldstone boson—for each generator of the symmetry that is broken, i.e., that does not preserve the ground state. The Nambu–Goldstone mode is a long-wavelength fluctuation of the corresponding order parameter.(From wikipedia)

Basic concepts of BCS theory

Superconductivity involves an ordered state of conduction electrons in a metal, caused by the presence of a residual \textbf{attractive} interaction at the Fermi surface. At low temperatures, an attractive pairwise interaction can induce an instability of the electron gas towards the formation of bound pairs of time-reversed states ${\textbf {k}}_{\uparrow }$ and $-{\textbf {k}}_{\downarrow }$ in the vicinity of the Fermi surface.

From where does an attractive interaction between charged particles appear? In conventional (BCS) superconductors, attractive correlations between electrons are due to the exchange of lattice vibrations, or phonons: The motion of an electron through a metal causes a dynamic local distortion of the ionic crystal. Crucially, this process is governed by two totally different time scales. For an electron, it takes a time $\sim E_{F}^{-1}$ to traverse the immediate vicinity of a lattice ion and to trigger a distortion out of its equilibrium position into a configuration that both particles find energetically beneficial. There're some elementary excitations, the Bogoliubov quasiparticles which have a energy gap.