Do we really need 11 dimensions? How about ,... let's say....... just one?
If you are a string theorist, and do not find stuff with less than 11 dimensions fun, you may discontinue reading at this point. (How you found your way to reading this blog-post would be the bigger curiosity, in that case)
My mentors have decided for me to pursue a bottom-up approach and so we start with only one dimension(bummer? not really!). The plan is we add dimensions as we go along. However, I am unsure about reaching the lofty goals of 11 dimensions in any imaginable eventuality, a modest goal of three dimensions would be my best guess.
Appropos, I should add, Clemens, Nathan and Shahnawaz are my mentors for GSoC 2019. I am sure you can figure out who they are just from their names, if you are curios about it.
No one has ever complained one dimension to be boring! In fact, Luttinger liquids only exist at one dimension, beat that!
(drum-rolls!)This post is about the much loved(or hated)-debated over-scrutinized SSH model!
For now I shall direct you to the following awesome lectures for its theory. I may write-up all about it later myself.
https://phyx.readthedocs.io/en/latest/TI/Lecture%20notes/1.html
It can degenerate into a single orbital cosine wave disperion when the inter and intra hopping energies are the same! (a humble feature!)
Dispersion relations tell you the eigen-values of infintely repeated unit cells. So a system having 10 unit cells with 20 atoms with open(hard-wall) boundary conditions would give sampled points on the dispersion relations, as shown on the figure on the left.
Periodic Boundary condition would make the eigen-values form pairs because of an additional two-fold symmetry imposed.
SSH model starts to get fun when the inter and intra hopping energies differ in value! Let's start with the larger intra-hopping! The model becomes gapped for the condition.
Insulators are more interesting than conductors!(because harder the problem, the more we like it) However it is still a trivial insulator(topologically). Going in the other direction, having larger inter-hopping is the most fun a SSH model has to offer.
Well, don't they look exactly the same? From the eigen-energies with Periodic Boundary Condition and disperion relations --yes. However, funny things appear in their eigen-functions! And if we plot the eigen-energie with hard-wall Boundary Conditions, we see eigen-energies right at the middle of the gap, the famous edge states, appearing at the boundary between a topologically non-trivial state to trivial state! Plotting the eigen-function clarifies their names, the show up close to the edge!!
The QuTip functions used to produce these results would be available publicly soon. I hope, I made my point! One dimension is anything but boring!
My mentors have decided for me to pursue a bottom-up approach and so we start with only one dimension(bummer? not really!). The plan is we add dimensions as we go along. However, I am unsure about reaching the lofty goals of 11 dimensions in any imaginable eventuality, a modest goal of three dimensions would be my best guess.
Appropos, I should add, Clemens, Nathan and Shahnawaz are my mentors for GSoC 2019. I am sure you can figure out who they are just from their names, if you are curios about it.
No one has ever complained one dimension to be boring! In fact, Luttinger liquids only exist at one dimension, beat that!
(drum-rolls!)This post is about the much loved(or hated)-debated over-scrutinized SSH model!
For now I shall direct you to the following awesome lectures for its theory. I may write-up all about it later myself.
https://phyx.readthedocs.io/en/latest/TI/Lecture%20notes/1.html
It can degenerate into a single orbital cosine wave disperion when the inter and intra hopping energies are the same! (a humble feature!)
Dispersion relations tell you the eigen-values of infintely repeated unit cells. So a system having 10 unit cells with 20 atoms with open(hard-wall) boundary conditions would give sampled points on the dispersion relations, as shown on the figure on the left.
Periodic Boundary condition would make the eigen-values form pairs because of an additional two-fold symmetry imposed.
SSH model starts to get fun when the inter and intra hopping energies differ in value! Let's start with the larger intra-hopping! The model becomes gapped for the condition.
Insulators are more interesting than conductors!(because harder the problem, the more we like it) However it is still a trivial insulator(topologically). Going in the other direction, having larger inter-hopping is the most fun a SSH model has to offer.
Well, don't they look exactly the same? From the eigen-energies with Periodic Boundary Condition and disperion relations --yes. However, funny things appear in their eigen-functions! And if we plot the eigen-energie with hard-wall Boundary Conditions, we see eigen-energies right at the middle of the gap, the famous edge states, appearing at the boundary between a topologically non-trivial state to trivial state! Plotting the eigen-function clarifies their names, the show up close to the edge!!
The QuTip functions used to produce these results would be available publicly soon. I hope, I made my point! One dimension is anything but boring!
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