這次的演講，老實說有點私心。因為我想要把這一篇論文的脈絡整理出來，"順便"分享給大家><，所以演講的內容就會有點多。加上這一篇文篇的命運很詭譎，所以也加深了演講或是寫作的不安。還有最重要的一點是，我的英文實在是很爛（這大家都知道吧，哈）。希望大家見諒。
順便分享一下我昨晚畫的圖，這是第三版了。第一版是晏榛畫的，因為太具有藝術性，不適合發表（大家都知道，我前老闆對這一點很要求，哈）。第二版，是你們演講看到的。不過因為把所有自旋加上去，太雜了，不好看（大家都知道，我前老闆對這一點很要求，再一次），所以我只好製作第三版（如圖）。希望可以滿足所有作者的口味。
真不愧是畫圖界中的木村拓栽!!
回覆刪除畫的真好!!
回覆刪除作者已經移除這則留言。
回覆刪除Cool!
回覆刪除這是用什麼樣的軟体繪製的呢？
CS illustrator !!
回覆刪除真不愧是畫圖神手
回覆刪除你那位前老闆看到這圖，應該是讚不絕口吧！畫圖可以達到這種境界，真是太厲害了⋯
回覆刪除有個小缺點 n_y +1 的 1都歪一邊了, 1 應該要跟晶格平面還有0.....n_y 這些東西平行, 還有黑線跟honeycomb lattice平面似乎也不平行.
回覆刪除謝謝小胖的提醒～
回覆刪除改進啦!!
我還是有個問題...
回覆刪除From Wen-Min's talk, I know the properties of the edge magnon of graphene are strongly correlated with its bulk properties. However, the calculation includes bulk + boundary is much harder then the boundary only! If we encounter a new material and want to know its edge properties, what should I do? For what kind of material, just considers the edge itself can give us the right answer?
It is my opinion...
回覆刪除The whole idea is can we have PURE 1D theory at the boundary or not? In our case, we know the answer is NO. But, maybe it will exist in some systems. In these system, if we only want to know physical phenomena at the boundary, we can totally ignore the bulk.
When we have a new material, we first will reduce the degree of freedom in this system to simplify the model, and do some prediction from it. If we can find a simple model at the boundary which only contains the boundary parameters, we can reduce the system dimension. It will be very useful. The whole theoretical concept is can we explain the system as simple as we can. However, I agree with nanodreamy. If we have a new system and we want to understand the boundary properties, it is too rough to only consider the boundary degree of freedom. One need further carefully check.
For you later question, I have no example so far...><. In my knowledge, for example, the mobility in the 2DEG, QHE, and graphene, the boundary properties will connect to the bulk degree of freedom. MAYBE, in some classical systems, it will exist.
One example, MAYBE, classical motion at the boundary ><.
回覆刪除那我好奇 你們是怎麼知道graphene用1D理論是錯的？
回覆刪除最簡單來看,直接比較就知道:
回覆刪除Ferromagnetic chain(or 2D bulk systems): number of magnons:1, when k->0 dispersion of magnons 是正比 k^2
antiferromagntic chain(or 2D bulk systems) : number of magnons:2, when k->0 dispersion of magnons 是正比 k
edge of 2D AFM(for zigzag) :number of edge magnons:1, k->0 dispersion of edge magnons 是正比 k.
->edge magnon 有 FM magnon的性質(number of magnon), 也有 AFM 的特性(k->0 dispersion of edge magnons ).
作者已經移除這則留言。
回覆刪除問一下小胖，那麼dispersion的圖是怎麼畫出來的？我們是怎麼知道那張dispersion就是那個材料真的dispersion圖？
回覆刪除算出公式就畫出來了 不同材料不同的系統都可以算
回覆刪除about little thief's questions....
回覆刪除It is my fault><. I don't explain it very well in the talk. Sorry!! Actually, there are two kinds of dispersions. One denotes the bulk states, which momenta are real and propagate as plane waves. The other represents the edge states, which momenta are imaginary and decay away from boundaries. Meanwhile, the dispersion of the bulk states is illustrates within the green shadow in the paper, and the red line is for the edge state. These dispersions of magnons are based on the spin-wave theory, which is started from a Neel state and find out possible excitations. The edge magnons are the main contribution of our previous paper and can be computed by the general Bloch theory.
The second question is more subtle. In 1D, there is an exact solution for an antiferromagnetic Heisenbergy chain. You may wonder how good is the spin-wave theory approach comparing to exact solution in 1D. The answer is that the dispersion in spin-wave theory is the lowest boundary of the exact one. However, the spectral weights in experiments are quite sharp in the lowest boundary. Thus, the spin-wave approach is quite useful due to its simplicity. In 2D system, we do not have exact solutions, even without the boundary. We can not claim it is the "exact dispersion of magnons" in a honeycomb lattice. However, it is rather remarkable the dispersion of edge magnons is lower than the bulk states. It may inspire experimental scientists to verify the beautiful results.
表面波是很有趣的物理現象。舉例來說，高中學的水波就是表面波。但是，淺水波的波速卻與水深有關，那不是與文敏所講的邊磁子相似嗎？
回覆刪除就某個角度來看，是這樣沒錯。在彈性物質中，有兩類不同的彈性波：dilatational (primary) wave (P wave) and rotational (secondary) wave (S wave)，前者的速度較後者快。而在彈性介質的表面，會有表面波的生成，稱為Rayleigh wave (R wave)，速度又更慢一些。這是為何在地震時，三種不同的波以P、S、R的順序到達。經過一些理論分析後，可算出表面波的波速僅與材料特性有關，所以「表面—體」的關係確實存在。而且，表面波的波速一定比較慢，這與邊磁子的特性相同。
但是一般而言，表面波可以被等效的低維系統所描述，塊材僅提供相關參數而已。但是反鐵磁的邊磁子無法由等效的低維系統所描述，所以很特別。這表示「表面—體」不僅有關，而且無法切割。就這個角度來看，邊磁子的現象並非一般的表面波。
這好像很好玩，我好想聽喔 嗚嗚 T_T
回覆刪除是蠻有趣的，小胖的碩士論文，晏榛與文敏接力完成。關於「表面–體」的不可切割性，最近很紅，也就是江湖上所謂的拓樸絕緣體，其表面上必有低能量的表面波。
回覆刪除反過來說，如果系統具有「表面–體」的不可切割性，而且又有低能量的表面波，那是不是就找到了拓樸絕緣體呢？非也非也。不過當初小胖得到低能量的邊磁子時，我們其實是蠻希望，反鐵磁的基態具有特殊的拓樸特性。
如果從非均向(Ising anisotropy)的自旋作用出發，可得到基態為一絕緣體。但很可惜地，此時的邊磁子具有能隙，所以Ising-like反鐵磁的基態只是平凡無奇的絕緣體，含恨啊⋯
同意老師所說的,要有有趣的拓樸,或更正確說,穩定的拓樸態,需要系統體本身有能隙.有了這前提,沒能隙的邊界態才有可穩定存在,或是由所謂的拓樸場論來描述.像topological insulator或是 QHE.
回覆刪除但即使是不穩定的邊界態,仍有可能被特殊定義的拓樸數來描述,像是普通有 zigzag edge 的 graphene, Andreev bound state of d-wave superconductor,此時依然可以定義一些拓樸量,像Berry phase來描述(詳見 Hatsugai 的paper ). 有一件事我不太確定,但是就我所知的例子,這種情形往往跟細微的邊界結構有關(誰叫你 bulk沒有gap). 跟一般所謂之拓樸絕緣體相比,用處跟有趣度就有差了.雖然如此,能夠找到這種拓樸量,似乎至少能發PRL,但不能拿諾貝爾獎....
不過一個spin liquid的系統有沒可能出現類似拓樸絕緣體,我就不太清楚了
說到這個,就想到兩個星期前在交大 Miniworkshop on Mesoscopic and Spin Physics,我把edge magnon 跟 Furusaki 提了一下,然後問他有沒可能也用一般的 topological field theory來描述這個系統. 其時我只是想說,有沒比較系統的方法找拓樸量.就像 Fermion系統跟規範場偶和,積掉費米場後, Chern-Simon term前面係數就是拓樸量這樣. 而現在考慮的edge magnon 是 boson field,而且又是個excitation,跟我知道 fermion的例子不同. 我只是想說,如果可行,就當個練習題看看,然後要問他有沒有參考的ref.
回覆刪除結果他說,他不知道,但沒有 bulk 能隙讓他覺得應該不可能.