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Thanks very much, the reference is this one : https://aip.scitation.org/doi/10.1063/1.4890585

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...

Thanks very much, I will look into it.

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?

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...

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...

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)}...

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}##?

Thanks very much for the explanation.

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?

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...

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?

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...

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.

Hi Lavinia, Thanks very much, that is very helpful!

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...

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 ?

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...

I got it, thanks very much!

Thanks very much. I think it makes great sense, and I will also learn a little bit more about the casualty.

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...

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

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...

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}} =...

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...

Thanks very much.

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)##.

Thanks very much.

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...

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...

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}...

Yes, that is generally possible right?

Thanks very much.

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 ?

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##?

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?

Hi guys, Thanks for the help, I think it really is a ring not 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...

So this is not a torus? I though the upper side and bottom side are identified as the same edge.

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.

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...

THanks very much!

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?

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...

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?

thanks very much

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...

Thanks very much, I got the point.

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...

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...