# Search results for query: *

1. ### A The hermicity of a k.p matrix?

Thanks very much, the reference is this one : https://aip.scitation.org/doi/10.1063/1.4890585
2. ### A The hermicity of a k.p matrix?

I am trying to use the k.p method to study quantum well band structure. One example Hamiltonian look like this [J. Appl. Phys., 116, 033709(2014)] where ##{{\hat k}_ \pm } = {{\hat k}_x} \pm i{{\hat k}_y}## and the matrix elements are function of ##{{\hat k}_i}## and if quantum well is...
3. ### A The k direction in a k.p model ?

Thanks very much, I will look into it.
4. ### A The k direction in a k.p model ?

Thanks That is what is said in the graph, which is taken from a different article. But if I only look at the Hamitonian and the basis functions, how can I tell?
5. ### A The k direction in a k.p model ?

I am trying to do some calculation based on a k.p model of GaN proposed by S. Chuang [Phys. Rev. B, 54, 2491]. It is a 8 by 8 Kane model with basis functions: The 8 by 8 Hamiltonian contain first order of k is where ##{k_ \pm } = {k_x} \pm i{k_y}## the reciprocal space and high symmetry...
6. ### A Why is the nearest hopping kept real in Haldane model?

I am leaning the Haldane model : https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.61.2015 Haldane imaged threading magnetic flux though a graphene sheet, and the net flux of a unit cell is zero. He argued that since the loop integral ##\exp [ie/\hbar \oint {A \cdot dr} ]## along a path...
7. ### A Can I change topology of the physical system smoothly?

Maybe the question is not so clear, I will use an example. A physical system is represented by k-dependent Hamiltonian operators ##\hat H(k)##, where ##k## is a point in the torus. Each linear operator ##\hat H(k)## has a set of discrete eigen-functions such that ##\hat H(k)\left| {{u_n}(k,r)}...
8. ### A Can I change topology of the physical system smoothly?

Is this possible? let's say there are two principal bundles ##{P_1}## and ##{P_2}## over ##M##, and ##{A_1}## and ##{A_2}## are corresponding gauge fields. Is it possible to smoothly change ##{A_1}## into ##{A_2}##?
9. ### A Integration along a loop in the base space of U(1) bundles

Thanks very much for the explanation.
10. ### A Integration along a loop in the base space of U(1) bundles

I mean should the Chern class be a even dimensional form like 2-form, 4-form..., and the circle is one dimensional. or you mean the 0-form is well defined? May be I am confused about the definition of Chern number. Does the U(1) bundle on a cirlce have a Chern number?
11. ### A Can I change topology of the physical system smoothly?

I am encountering this kind of problem in physics. The problem is like this: Some quantity ##A## is identified as a potential field of a ##U(1)## bundle on a space ##M## (usually a torus), because it transforms like this ##{A_j}(p) = {A_i}(p) + id\Lambda (p)## in the intersection between...
12. ### A Integration along a loop in the base space of U(1) bundles

I have to ask this again, should the Chern class be only defined on a bundle whose base space is even dimensional? And are the Chern classes defined on a ##U(1)## bundle on a circle?
13. ### A Can I find a smooth vector field on the patches of a torus?

Thanks very much, you said "it follows easily that an oriented ##U(1)## bundle over the cylinder is also trivial", I have two questions 1, what about other bundles such as ##U(N)##? The original text says for the second graph a vector bundle on patch A must be trivial, what is the reason for...
14. ### A Integration along a loop in the base space of U(1) bundles

Thanks again, I got a new question https://www.physicsforums.com/threa...ector-field-on-the-patches-of-a-torus.927959/ please take a look, this topic is driving me crazy.
15. ### A Integration along a loop in the base space of U(1) bundles

Hi Lavinia, Thanks very much, that is very helpful!
16. ### A Can I find a smooth vector field on the patches of a torus?

I am looks at problems that use the line integrals ##\frac{i}{{2\pi }}\oint_C A ## over a closed loop to evaluate the Chern number ##\frac{i}{{2\pi }}\int_T F ## of a U(1) bundle on a torus . I am looking at two literatures, in the first one the torus is divided like this then the Chern number...
17. ### A Integration along a loop in the base space of U(1) bundles

Hi Lavia: Thanks very much, but what character characterize the circle bundle on a circle, I only know little about the Chern class, which seems to be only defined on a even denominational base space ?
18. ### A Integration along a loop in the base space of U(1) bundles

Let ##P## be a ##U(1)## principal bundle over base space ##M##. In physics there are phenomenons related to a loop integration in ##M##, such as the Berry's phase ##\gamma = \oint_C A ## where ##C(t)## is a loop in ##M##, and ##A## is the gauge potential (pull back of connection one-form of...
19. ### A A question about linear response and conductivity

I got it, thanks very much!
20. ### A A question about linear response and conductivity

Thanks very much. I think it makes great sense, and I will also learn a little bit more about the casualty.
21. ### A A question about linear response and conductivity

Thanks very much for your help. I can vaguely feel adding a small ## \epsilon ## here has something to do with the causality, which is the part I tried to skip when I was deriving the conductivity. But, like you said, we are studying physics, adding something non-physical can always make...
22. ### A A question about linear response and conductivity

I don't have a book, I am looking at files online, here are links of two files http://www.damtp.cam.ac.uk/user/tong/kintheory/four.pdf http://phy.ntnu.edu.tw/~changmc/Teach/SM/ch03_.pdf
23. ### A A question about linear response and conductivity

I mean we need to deal with a integral like ##\int_0^\infty {dt{e^{i\omega t}}} {e^{i({E_m} - {E_n})t}} = \int_0^\infty {dt{e^{i({E_m} - {E_n} + \omega )t}} = } \left. {\frac{{{e^{i({E_m} - {E_n} + \omega )t}}}}{{i({E_m} - {E_n} + \omega )}}} \right|_0^\infty ## if ##\omega ## have no...
24. ### A A question about linear response and conductivity

I am trying to derive the DC electrical conductivity using the pertubation theory in Interaction picture and linear response theory. If working in a energy eigen basis and using the density matrix, the Fourier transform of the susceptibility can be written as ##\chi {(\omega )_{ij}} =...
25. ### A About Chern number of U(1) principal bundle on a torus

You are right the limitation on "g" lead to a quantization of the monopole value. I honestly cannot explain well about this issue, which is very deep and is not my focus of research. One of the naive motivations is that the electric fields have source ##\nabla \cdot \vec E = \rho ##, where...
26. ### A About Chern number of U(1) principal bundle on a torus

Thanks very much.
27. ### A About Chern number of U(1) principal bundle on a torus

This is the same thing as I wrote in my previous post, right? if ##{g_{NM}} = \exp (i\phi (p))##, then ##{A_N} = {A_M} + id\phi (p)##.
28. ### A About Chern number of U(1) principal bundle on a torus

Thanks very much.
29. ### A About Chern number of U(1) principal bundle on a torus

Can you help me to check this example? It is one of the reasons I post this thread. It is the magnetic monopole problem. If there is a magnetic monopole, it should produce a magnetic field ##\vec B = g\vec r/{r^3}## so that ##\int_S {\vec B \cdot d\vec S} = 4\pi g##, where ##4\pi g## is a...
30. ### A About Chern number of U(1) principal bundle on a torus

Thanks for your explanation, that is very helpful. I think the mistake in my example is that I tried to build a globally defined gauge potential (one-form) on the base space first, then I used it to define the connection one-form on the bundle. This is not valid at all. One should always start...
31. ### A About Chern number of U(1) principal bundle on a torus

Sorry I was trying to say a every where non-zero form. So if I go back to my original post, and let ##A## be such a globally defined one-form on the torus. Since the bundle is not necessarily trivial. On certain charts the gauge potential has the form ##iA + id{\Lambda _1} + id{\Lambda _2}...
32. ### A About Chern number of U(1) principal bundle on a torus

Yes, that is generally possible right?
33. ### A About Chern number of U(1) principal bundle on a torus

Thanks very much.
34. ### A About Chern number of U(1) principal bundle on a torus

So, can I say that I can always have a globally defined one-form ##A## for an arbitrary manifold, as long as in the overlapping area of different charts it transform according to the transition function? Its exterior derivative ##dA## should be also globally defined? right ?
35. ### A About Chern number of U(1) principal bundle on a torus

Thanks, I am really confused about something here. Can I say that ##dθ## is not exact is because the circle cannot be parameterized completely by ##θ##?
36. ### A About Chern number of U(1) principal bundle on a torus

Thanks very much. You said "Chern class is zero" do you really mean Chern number ? So what is the reason ##\theta ## is not exact? Just because it cannot be globally defined? Also, can you explain more about how two U(1) bundles on the same base space are different?
37. ### A Stokes' theorem on a torus?

Hi guys, Thanks for the help, I think it really is a ring not torus.
38. ### A About Chern number of U(1) principal bundle on a torus

The U(1) bundle on a torus is a important math setup for a lot of physics problems. Since I am awkward on this subject and many of the physics material doesn't give a good introduction. I like to put some of my understanding here and please help me to check whether they are right or wrong. 1. A...
39. ### A Stokes' theorem on a torus?

So this is not a torus? I though the upper side and bottom side are identified as the same edge.
40. ### A Stokes' theorem on a torus?

Sorry for the confusion, the horizontal axis is parameterized with angle, both C1 and C2 are closed loops because the left and right sides are identified.
41. ### A Stokes' theorem on a torus?

I am now looking at a physics problem that should be a use of stokes' theorem on a torus. The picture (b) here is a torus that the upper and bottom sides are identified as the same, so are the left and right sides. ##A## is a 1-form and ##F = dA## is the corresponding curvature. As is shown in...
42. ### A Is there a natural paring between homology and cohomology?

THanks very much!
43. ### A Is there a natural paring between homology and cohomology?

Is it defined this way? If the cochain is defined as ##C_n^ * = Hom({C_n},R)##, and let ##\alpha \in Hom({C_n},R)## and ##\beta \in {C_n}##, so I can define a product ## < \alpha ,\beta > = \alpha (\beta ) \in R## is this right?
44. ### A Is there a natural paring between homology and cohomology?

I am looking at the definition of the characteristic numbers from the wikipedia https://en.wikipedia.org/wiki/Characteristic_class#Characteristic_numbers "one can pair a product of characteristic classes of total degree n with the fundamental class" I am not sure how is this paring defined here...
45. ### A Very basic question about cohomology.

Thanks very much, I need time to understand all this. But before that, given a space, is there a general procedure to find the co-chain group or cohomology group?
46. ### A Very basic question about cohomology.

thanks very much
47. ### A Very basic question about cohomology.

I am self leaning some basic cohomology theory and I managed to go through from the definition to the universal coefficient theorem. But I don't think I get the main point of this theory, I like to ask this questions: Is such an abstract theory practical? I would say that homology is...
48. ### A A question about split short exact sequence

Thanks very much, I got the point.
49. ### A A question about split short exact sequence

My bad, I was in a hurry this afternoon, there are big typos in 2 and 3 I was trying to say, by doing the following 3 things, can I find a subset ##B'## in ##B## that is isomorphic to C? 1. 0 is in ##B'## such that ##g(0) = 0## 2. For each generator ##{c_i} \in C##, take a ##{b_i} \in B## in...
50. ### A A question about split short exact sequence

For the isomorphism, I mean I can choose a subset ##B'## of ##B## in this way 1. 0 is in ##B'## such that ##g(0) = 0## 2. For each generator ##{c_i} \in C##, find a ##{b_i} \in B## such that ##g({b_i}) \in {c_i}## 3. Take ## - {c_i}## in ##B'## then if ##g## is a homomorphism and surjective...