Something about switching the topic

Switching a research topic is not so easy at first, but I'm taking great advantage from it now!! I can see how different branches of condensed matter physics are constructed in some parallel and similar way, also I can struggle to get some unifying understanding of the seemingly totally different branches!

During the undergraduate years I did multiferroics and now switched to topological insulators, and recently I found myself starting to get aware of their deep-lying connections!

1) Both are quite related to symmetry, no matter time-reversal symmetry or space inversion symmetry. Though seemingly both are broken in the former while preserved in the latter. But please take a look at the surface state of topological insulators, they are just residing on where the inversion symmetry of the system is most severely broken!

2) This point is also somehow related with the above. I just start to find symmetry analysis so important in constructing effective Hamiltonian to both. Group theory is actually a necessary tool (Well, maybe this is the best word for him) in condensed matter physics when tackling with the various fancy crystal structures!

3) Spin-orbital coupling is playing an essential role in both. In the former, it is determining the magnetic ground states by eliminating the degeneracy of d/f orbitals, also introducing spin-dependent hopping term (However, this energy scale is much smaller than the Hund rule effect?) While in the latter, SOC also determines which bands would be essential when accounting for the topological nature.


TO BE CONTINUED...

Symmetry requirements imposed on Physics Phenomena

Well it is a combination of a project for the lesson group theory and the seminar topic..

Symmetry of the crystal give many vectors the restriction.But it is a little like nonsense...

A sudden understanding of the conception of dilution

It just means some of the sites could be unoccupied! How silly I has been!
[tex]a^2[/tex]

Broken Symmetry in the world of solid

So fascinating the conception of broken symmetry!! Before this term I had thought that the broken symmetry only exist in the most elementary and theoretical physics such as the cosmology and the elementary particles which I thought interesting but not within my ability...(Maybe under the influence of A.Zee's Fearful Symmetry, illustrating the fascinating world of broken symmetry in the world of elementary particles..)

And so glad to find that in the world of condensed matter physics, the conception of broken symmetry also exist!! Not only exist, but even prevailing!!! Confirm me that I can also do some work concerning the broken symmetry someday!XDDD

Here are some notes from books concerning the broken symmetry in the world of condensed matter physics~

From P.W.Anderson's BASIC NOTIONS ON CONDENSED MATTER PHYSICS

Ch2.Broken Symmetry

• The laws that govern the particles with which we usually deal have a very high degree of symmetry: transitional, rotational, space-inversion and time-reversal. In the elementary fireball, it is supposed that the vacuum and all the particles with with their puzzling approximate symmetry of SU3, isospin, parity, etc, are the result of some cosmological phase transiton which originated from a state with some very high overall symmetry which is not manifest to us.
• In solid, the transitional symmetry is broken everywhere.

• The overwhelming majority of real physical systems with interactions between the particles tend to exhibit the phenomenon of a lowest-energy state's not having the full symmetry of space or of the Hamiltonian's describing their interactions.

• The Bose liquid break symmetry in a totally different way-for it could not be divided into independent parts, but rather have a position-dependent phase. Here the Gauge symmetry is broken.
• Broken time-reversal invariance,(ferromagnetic); Broken inversion symmetry(ferroelectric); superconductivity is also broken gauge symmetry.
• "Why"of the broken symmetry: under surprisingly general circumstances the lowest energy state of a system does not have the total symmetry group of its Hamiltonian, and so in the absence of thermal fluctuations th system assumes an unsymmetrical state.
• Remaining question for "what are the consequences": (i)discreteness of phase transitions and the resulting failure of continuation, disjointness of physical phases; (ii)development of collective excitations/fluctuations; (iii)generalized rigidity; (iv) defect structures: dissipation and topological considerations.
• The first theorem of solid-state, it is impossible to change symmetry gradually.Second order phase transition could only occur between phases of different symmetry.
• Mott transition in V2O3, which can be seen as a free-electron liquid-to-gas transition.
• Order parameter: an additional variable necessary to specify the microscopic state in the lower symmetry state.
  

Understanding of critical slowing down

Have been wondering why when approaching the critical point the time revolution would slow down...

From one of the Japanese textbook, near the critical point, when the difference between whether to have a symmetry is infinitsemal, fluctuations are relatively amplified. After reading this I wondered whether it is according to the fluctuations are turned something macroscopic, so the change must take some time? However this explanation is something not so pleasing....T_T

Today I started to read the chapter concerning phase transition in Landau (So great are his theoretical series!!!!!!) and found a pleasing-to-me explanation though it is a phenomenological explanation, that is when approaching the critical point, the free energy becomes flat with respect to the order parameter, thus the "restoring force", driving the system to its corresponding equilibrium state would be rather weak, which in sum would enlarge the time revolution scale.

そうか！という感じ。

And now also more understandable that why in the class the Brownian motion is mentioned. It is also the macroscopic effect of many microscopic actions and is in a slow time scale, and also where fluctuations take important place. In common they share the phenomenological coefficients and the random force(not exactly)? But the ways to deal with random process and fluctuations still seem difficult to me now....which makes the preference to experimentalist to theorist stronger and stronger...

website you can see the critical slowing down:
http://www.ibiblio.org/e-notes/Perc/rel.htm

2011/01/16

Booklist

books in my reading plan....in....these years.....

• Basic Notions on Condensed Matter Physics P.W.Anderson

Evaluated as a Bible-like book!! Though only read the first chapter of the broken symmetry when I first borrowed it from the library of Dept.Physics of UT(Like the library there so much!!!XDD), I find it full of profound and concise conceptions and summary, really insightful! One day I will buy one!!

• Quantum Field Theory of Many-Body Systems

By X.G.Wen
Wants to attend his lecture in THU but the time is during my exchange...
Said to be so profound and could explain the importance of condensed matter physics in the whole range of physics but to fully understand it may seem impossible for me....

Phase Transition Notes by a B3

Some summary from Statistical Mechanics II this term and also some understanding by myself. For this term as an exchange student, I take the Statistical Mechanics II without having taken the corresponding Statistical Mechanics I. It took a lot of effort but I think maybe now I can say that I understand the phase transition somehow:))

• What is a phase transition?

I think it would be looked as the transition of different macroscopic phases. It could be divided into different kinds, with respect to the number of orders of the thermodynamic functions(such as the Heat) to temperature.

First order phase transition has latent heat included, melting of ice, normal transition between gas and liquids are typical examples.
Second order has a discontinuous specific heat, ferromagnetics, ferroelectrics....
Third order, the Bose-Einstein condensation of ideal gas, derivative of the specific heat would have a jump at the critical point.

Phase below the critical point would be of higher order and less symmetry. So at the critical point there would be the breaking of symmetry-like the conception of broken symmetry, profound and trenchant, and physics-like XD

• Order parameter

Order parameter was first proposed by L.D.Landau.
Above the critical point the order parameter is zero and below the critical temperature it would show nonzero values, with the property ->0 as temperature->critical point. It is the characteristic of the extend of order of the phase under consideration, thus called the order parameter.

Each kind of order parameter has its own coupling external field. Usually the coupling term appear in the Hamiltonian. When the coupling external field is zero, the order parameter spontaneously emerges at the critical point. The coupling external field could change the behavior near the critical point, also the critical temperature could be altered.

At many situations the order parameters are used to build self-consistency equations, that is to say to calculate the order parameter in different ways,(say the m minimizing the free energy together with the fact that the order parameter is the microscopic average value of some microscopic quantity) and build an equation according to the consistency. Using this technique the critical temperature could be calculated.

Another important concept concerning the order parameter would be the susceptibility, derivative of the order parameter with respect to the external coupling field.

• Mean Field Theory and Phenomenological (misspelling maybe) Theory

The heart of mean field theory is to alter the many-body problem into 1 body problem, (By TA)representing the interactions with an average field value. Fluctuations are omitted.

In Landau's Phenomenological Theory, the thermodynamic potentials are analytic functions of the order parameter near the critical point, thus Taylor expansion is valid. In the absence of an external coupling field, where the plus and minus of the order parameter is equivalent, Free energy should be symmetry in both sign, thus only the even terms remain in the expansion.

Omitting all the fluctuations which would act an important part near the critical point MFT does not apply to the experimental results well so many modifications are made.Some only take into consideration the nearest neighbor, some to the second nearest neighbor with other simplifications... Where the reports in SM are difficult is the different models. In dimensions larger than 4 the MFT would be the accurate theory but unfortunately we live in a space with a dimension 3.

• Correlation Function and Correlation length

To get more exact solution the interactions between different sites should be taken into consideration.(Maybe interaction is always what make things complex..)

Correlation function of 2 sites (with respect to the microscopic quantity) is defined as the average of their product minus the product of their average respectively.(for when the correlation function is zero the two sites are irrelevant) Be careful that the correlation function is something characterize the equilibrium state.

Exploiting the partition function to get susceptibility across different site-for example the magnetization of site i with respect to the external magnetic field at site j-we could get the fluctuation-dissipation relation: connecting quantities characterize the equilibrium state with response to external field, in the linear response(R.Kubo, highly revered Japanese physicist).

Correlation length is the characteristic length in the e exponential of correlation function.