Fiber bundles: Part-I

Well, since there are lots of discussions on algebraic topology, after the realization of topological insulators in condensed matter, an important concept I’ll try ellaborate here, which is the concept of fiber bundles.

I’ll try to control myself as less algebraic as possible. Let me first mention, what is going inside the quantum Hall market. There are two kind of businessmen. The businessman of the first kind (may be his name is Laughlin) will say, “It’s just a matter of gauge invariance”. The other businessmen (Look, they are four, perhaps their names are Thouless, Kohmoto, Nightingle, and Nijs. You can call them TKNN together in short.) will say, “The integer corresponds to the first Chern class of a $U(1)$ principal fiber bundle on a torus geometry.”

Today we’ll try to comprehend what the businessmen of the second kind wanted to mean.

So I shall first start with the theory of fiber bundles. The theory is a generalization of the direct product concept. Now what is a direct product?

Let me give you a task first. You have been given two circular plates: disk1 and disk2. Can you connect those plates by threads of equal lengths and make a solid cylinder? If you look at the picture of the ligament (a tissue connecting bones), you can easily see how it is possible. The fiber like structures inside the ligament is known as fiber bundles in biology.

Now you can write this job in a mathematical way as follows.
\bar{D}\times \mathbb R

Now you can make a hollow cylinder in the same way by threading two hollow discs, viz. ring1 and ring2:
S^1 \times \mathbb R

Now we can always find a projection \pi that maps disk1 to disk2 in the first example or ring1 to ring2 through the threads \mathbb R (lines, 1D in Euclidean real space) .  We would like to call disk1 or ring1 as the base space B and

the projected space (disk2 or ring1) as total space E. And from the ligament analogy, the threads will be also called as fibers F.

Thus we can write:  \phi: B\times F \to E.

We may find a projection \pi from E to B given

\pi: E \to B

Note that in our example E and B are differentiable manifolds and F is

a topological space. No need to worry about definitions of those terms. Just remember that all of them are subsets of Euclidean space, which is are most familiar  differentiable manifold). Now in addition to this, to form a fiber bundle we need a few more conditions. But I won’t go into that to avoid complication. But overall these topological spaces form a fiber bundle in presence

of a Lie group G operating on the fiber F from. And we often denote the fiber bundle as (E,\pi,B,F,G) or just F\to E\stackrel{\pi}{\to} B.

Now you can see that we can map coordinates from the base space to the total space or vice-versa through fibers.

In the ring example, if we construct a hollow cylinder, the fiber bundle will be trivial. However, if we keep a twist, i.e. form

a Moebius strip, then the fiber bundle will be non-trivial. The presence of twists gives a characteristic that decides the nontrivial

topological invariant, e.g. the Chern numbers (Will be discussed soon).

Ok, now don’t you see the fibers? ! Look at the clips below.

Now what about the quantum Hall effect? This will be discussed in Part-II .

5/2 Enigma: ‘Odd’ fraction with even denominator

The filling factor \nu=5/2 state bears the most enigmatic fraction in the family of fractional quantum Hall effect (FQHE). This is the one of peculiar states that cannot be described by Laughlin’s ansatz or the hierarchical constructions by Haldane and Haperin. The reason is the even denominator in \nu.

The off-diagonal component of the conductance tensor has the following expression

\sigma_{xy}=\nu \frac{e^2}{hc} .

Inverse of  \sigma_{xy} gives the Hall resistance R_H.

Willett, Eisenstein, English, and the trio (Tsui, Stormer, and Gossard) who  first discovered the \nu=1/3 FQHE, found this ‘odd’ fraction having the even denominator in 1987. Later in 1991, Moore and Read, by using conformal field theory (CFT), proposed  a new possible wavefunction

\Psi_{\rm{Pf}}=\rm{Pf}\big(\frac{1}{z_i-z_j}\big)\prod_{i<j}\exp\big(-\sum_i |z_i|^2 /(4l_B^2) \big)

where the Pf stands for the Pfaffian, square-root of the determinant of an anti-symmetric matrix or in other words anti-symmetrized sum over pairs:

{\rm{Pf}}\big(\frac{1}{z_i-z_j}\big)=\mathcal A \big(\frac{1}{z_1-z_2}\frac{1}{z_3-z_4}\cdots\big).

This Pfaffian wavefunction has a close connection to the p+ip superconductors. Therefore understanding either of these two different phases may help us to understand the other one.





Falaco Soliton

I suddenly came across this video on youtube. The video shows a demonstration of Falaco solitons inside a pool. Solitons are by definition solitary waves which retain their shapes during their propagation with time. They arise as solutions of non-linear field equations. One such popular non-linear equation, studied extensively in physics, is the Sine-Gordon equation (see 10th Chapter of Lewis Ryder’s QFT book). Solitons are also considered as topological defects which may appear as kinks in 1D, vortices in 2D, and magnetic monopoles in 3D physical systems.



The video also provides a link to a paper, where the author claims for experimental evidences that show that physical systems can be non-Euclidean (i.e. not our conventional Cartesian space).

To know what Falaco solitons exactly are, I searched for the author (R. M. Kiehn)’s or his group’s webpage. I’m simply quoting from their page:

“The Falaco Soliton water vortices rotate in opposite directions relative to each other on the flat surface of the water. It has been experimentally determined but is not immediately evident from watching the video that the rotating pairs are dynamically connected by a (nearly) 1 dimensional “stringlike” feature that the discoverer (R.M. Kiehn) describes as a “wormhole”.”

Whoaaaah ! Wormhole !

Solitons in a pool may not be a new thing, since John Scott Russell first reported solitary waves during late 19th century in the Union Canal of Scotland.

However, I’ll strongly recommend to look at the page. It may contain the key to unwind the Bermuda Triangle mystery.

The other side

One big issue that we often encounter while doing theory is about the physical reality of a quantity. For example, a negative number. A quantity which has negative values is often discarded on the ground of physicality. Scalar quantities like temperature, mass, time, and energy were initially believed to be always positive. However, after Dirac introduced relativistic quantum mechanics, we find an interpretation of negative energy (energy of a ‘hole’, later identified as a positron by C. D.  Anderson in 1932, as the negative energy counter-part of an electron) . Similarly the concept of time reversal led us to go backward in time and hence can be negative. What about negative mass? But before we look for an answer, we must understand how a particle with negative should behave. If we recall Newton’s equation F=ma, then we realize that if we push a conventional positive mass particle,  it moves away from me and hence the acceleration is the direction of the force. For a negative mass particle the opposite thing will happen, i.e. it will move towards  me. Now Einstein’s general theory of relativity (GR) tells about the possibility of arising of negative mass through the wormhole. The idea came up from Einstein and Rozen in their 1935 Physical Review paper. I must confess that I am not very comfortable with GR. May be later time I’ll make a post on wormhole on this blog. If you are little disappointed by my confession, then you may take a look at (and of course, Wiki is always there).

Now the last quantity of our interest is the temperature. Can temperature be negative in absolute scale? You may tempt to say that laws of thermodynamics prevent us to go beyond zero Kelvin. But I’ll suggest you to take deep breath and think about what would happen if we could have gone  below the absolute zero. Can we imagine a different physics to come out?


3D Printer

I guess, the video below has become popular for the last few months. By printing we normally understand printing image on a paper. But we probably can generalize for higher dimension.  The video demonstrates, by scanning a wrench, the printer can really replicate another wrench and we can choose a different color for the wrench too.

The technology  apparently looks very simple. Use nylon powder as printing material and make 3D blocks by stacking 2D layers.

The next BBC video goes for a bigger challenge. It’s not a small wrench, it’s real, ride-worthy bike.

Isn’t it something truly amazing ?

Hello world!

Hi ! I am Himadri Barman, a PhD student from India. I am starting a science blog from blog from today. This blog is meant to discuss science topics in a free and friendly way. I’ll try to post recent developments and achievement in the field of science. I may become inclined to  physics issues since that’s my subject. Science provides intellectual pleasures and it’s a different fun-world.

To wander into the wonders of science and nature is my dream and to share the experiences is the purpose of this blog.

I must admit that I still bear student-sort of immaturity in my writing. Hope I’ll grow up or maybe not. Thanks.