The Other Side of Moon

Scattered Thoughts

2011年11月14日发表评论

Conformal field theories in even space-time dimensions(e.g. 1+1, 3+1,…) have trace anomaly and one can define a central charge for the theory which effectively counts how many degrees of freedom there are in the theory. This is related to the stability of topologically-ordered phases in odd space-time dimensions, since their boundaries are described by CFT in even space-time dimensions. Therefore, we can understand the IQHE in 2+1 and 4+1 dimensions. Relation to chiral anomaly?

FQHE can be understood from parton(or slave particle) construction as the IQHE of partons. This approach has been developed by Wen to derive the effective field theories of various non-Abelian QHS. Recently discovered FQHE in flat-band lattice models can also be understood using this approach. An interesting question is what if the mean-field ansatz breaks the internal gauge symmetry(i.e. the mean-field ansatz has nontrivial internal gauge fluxes). The most extreme case would be that the gauge group is completely broken to its center, which is usually a discrete group(i.e. Z_n). Does the topological order remain the same? The Abelian case is studied by Lu & Ran. The non-Abelian case is still an open question.

A deep result due to Dijkgraaf and Witten states that the Chern-Simons actions for a compact gauge group G are in one–to–one correspondence with the elements of the cohomology group $H^4(BG, Z)$ of the classifying space BG with integer coeﬃcients Z. In particular, this classiﬁcation includes the case of ﬁnite groups H. The isomorphism $H^4(BH, Z)\simeq H^3(H, U(1))$ which is only valid for ﬁnite H then implies that the diﬀerent CS theories for a ﬁnite gauge group H correspond to the diﬀerent
elements $\omega \in H^3(H, U(1))$ , i.e. algebraic 3-cocycles ω taking values in U(1).

分类: Uncategorized

vim-latex for MacVim

2012年01月26日发表评论

See http://code.google.com/p/macvim/wiki/FAQ for instructions on how to install vim-latex(or known as latex-suite) on macvim.

However, on my Mac OS X Lion, there seems to be a problem with \lv using the default setting. When commanding \lv a pile of error messages jumped out. The fix I found is the following: go to ftplugin/latex-suite, open texrc, uncomment the line

TexLet g:Tex_TreatMacViewerAsUNIX = 1

and add here or to .vimrc the following:

TexLet g:Tex_ViewRule_pdf = ‘open -a Skim’

I use Skim as my pdf viewer. These two resolve the problem.

分类: Uncategorized

Computing Determinant in Fortran

2012年01月6日发表评论

It may seem that one should go to LAPACK and find a routine that does just this job. However, it turns out that LAPACK does not really have such a routine. So a little bit of detour is needed. If you want to deal with a general, complex matrix, not symmetric, not Hermitian, then the way to go is using Gauss elimination(GE). LAPACK provides routines to find the LU decomposition of a matrix, which is essentially GE and can be used to compute the determinant. Here is the code snippet:


 1 complex(kind=8) function det(N, mat)
 2     implicit none
 3     integer(kind=8), intent(in) :: N 
 4     complex(kind=8), intent(inout), dimension(:,:) :: mat
 5     integer(kind=8) :: i, info
 6     integer, allocatable :: ipiv(:)
 7     real(kind=8) :: sgn
 8 
 9     allocate(ipiv(N))
10 
11     ipiv = 0
12     
13     call zgetrf(N, N, mat, N, ipiv, info)
14     
15     det = ONE
16     do i = 1, N
17         det = det*mat(i, i)
18     end do
19     sgn = ONE
20     do i = 1, N
21         if(ipiv(i) /= i) then
22             sgn = -sgn
23         end if
24     end do
25     det = sgn*det   
26 end function det

分类: Uncategorized

Innocence

2011年12月2日发表评论

In our acquisition of knowledge of the Universe (whether mathematical or otherwise) that which renovates the quest is nothing more nor less than complete innocence. It is in this state of complete innocence that we receive everything from the moment of our birth. Although so often the object of our contempt and of our private fears, it is always in us. It alone can unite humility with boldness so as to allow us to penetrate to the heart of things, or allow things to enter us and taken possession of us.

This unique power is in no way a privilege given to “exceptional talents” – persons of incredible brain power (for example), who are better able to manipulate, with dexterity and ease, an enormous mass of data, ideas and specialized skills. Such gifts are undeniably valuable, and certainly worthy of envy from those who (like myself) were not so “endowed at birth, far beyond the ordinary”.

Yet it is not these gifts, nor the most determined ambition combined with irresistible will-power, that enables one to surmount the “invisible yet formidable boundaries” that encircle our universe. Only innocence can surmount them, which mere knowledge doesn’t even take into account, in those moments when we find ourselves able to listen to things, totally and intensely absorbed in child’s play.

— Alexander Grothendieck

分类: Uncategorized

RG=GR

2011年11月15日发表评论

今天听黑哥们Philip Phillips的报告，印象比较深的就是他介绍AdS/CFT的时候说RG=GR……黑哥们笑话的质量还是相当不赖的。

分类: Uncategorized

周作人《希腊女诗人》一文中所译萨福诗句

2011年07月2日发表评论

周作人《希腊女诗人》一文中所译萨福诗句：

其一：凉风嗫嚅，过棠棣枝间，睡意自流，自颤叶而下。

其二：月落星沉，良夜已半，光阴自逝，而吾今独卧。

其三：满月已升，女伴绕神坛而立，或作雅舞，践弱草之芳华。

其四：甘棠色赤于枝头，为采者所忘，
——非敢忘也，但不能及耳。

其五：如山上水仙，为牧人所践，花萎于地。

其六：爱摇吾心，如山风降于栎树。

（见钟叔河编辑《周作人文类编》第八卷《希腊之余光》，湖南文艺出版社1998年版）

分类: Uncategorized

Recent arXiv Digest

2011年05月28日发表评论

Topological Field Theory for p-wave Superconductors
T.H. Hansson, A. Karlhede, Masatoshi Sato

Functional renormalization group approach to correlated fermion systems
Walter Metzner, Manfred Salmhofer, Carsten Honerkamp, Volker Meden, Kurt Schoenhammer

The effects of interaction on quantum spin Hall insulators
Dung-Hai Lee

Observation of topologically protected bound states in a one dimensional photonic system
Takuya Kitagawa, Matthew A. Broome, Alessandro Fedrizzi, Mark S. Rudner, Erez Berg, Ivan Kassal, Alán Aspuru-Guzik, Eugene Demler, Andrew G. White

The Fate of the Photon in Topological Matter: Superconductivity, Confinement and the Vortex Quantum Hall Effect
M.C. Diamantini, C.A. Trugenberger

分类: Uncategorized

数列极限题之后续

2011年01月26日发表评论

在之前一篇post里我讨论了一个数列极限问题，题目如下：设有数列 $a_n, b_n$ 满足

$\displaystyle a_{n+1}=\frac{a_n+b_n}{2}, b_{n+1}=\sqrt{a_n b_n}$ .

研究 $a_n/b_n$ 的极限。题目本身不难，很容易证明极限存在并且为1。比较值得注意的是这个数列收敛速度非常快（可以估计出数列是以指数的指数这样的速度收敛的）。这个题目看起来像一个普通的高等数学练习，但实际上背景很不简单：历史上最早研究这个问题的是Gauss。可以想象，能引起了Gauss兴趣的问题必定不是泛泛。Gauss问的是：给定初始值 $a_0=a, b_0=b$ , 极限 $\lim_{n\rightarrow \infty}a_n$ 是多少？他特别考虑了 $a=1, b=\sqrt{2}$ 的情形。记 $\lim_{n\rightarrow \infty}a_n=M(a, b)$ . Gauss证明了下面的结论：

$\displaystyle \frac{\pi}{2M(1,\sqrt{2})}=\int_0^1\frac{\mathrm{d} x}{\sqrt{1-x^4}}$

事实上，有如下更一般的结论：

$\displaystyle \frac{\pi}{2M(1,y)}=K(\sqrt{y^2-1})$

也即 $M(a, b)$ 可以用椭圆积分来表示！ $K(k)$ 是所谓的第一类完全椭圆积分：

$\displaystyle K(k)= \int_0^{\pi/2}\frac{\mathrm{d} \theta}{\sqrt{1-k^2\sin^2\theta}}=\int_0^1 \frac{\mathrm{d} u}{\sqrt{(1-u^2)(1-k^2u^2)}}$

分类: Uncategorized

arXiv Digest 12/17/10

2010年12月17日发表评论

The landscape of the Hubbard model
Authors: Subir Sachdev

Electron-Electron Interactions in Graphene: Current Status and Perspectives
Authors: Valeri N. Kotov, Bruno Uchoa, Vitor M. Pereira, A. H. Castro Neto, F. Guinea

Fractional quantum Hall effect in a U(1)xSU(2) gauge field
Authors: Rebecca N. Palmer, Jiannis K. Pachos

Localization of preformed Cooper-pairs in disordered superconductors
Authors: B. Sacepe, T. Dubouchet, C. Chapelier, M. Sanquer, M. Ovadia, D. Shahar, M. Feigel’man, L. Ioffe

Anomalous Hall response of topological insulators
Authors: Dimitrie Culcer, S. Das Sarma

分类: arXiv Digest

arXiv Digest 12/01/10

2010年12月2日发表评论

Non-Abelian s-wave superconductivity in spin-orbit coupled systems: Bulk phases and quantum phase transitions
Sumanta Tewari, Tudor D. Stanescu, Jay D. Sau, S. Das Sarma

Evidence for electron-electron interaction in topological insulator thin films
Jian Wang, Ashley M. DaSilva, Cui-Zu Chang, Ke He, J. K. Jain, Nitin Samarth, Xu-Cun Ma, Qi-Kun Xue, Moses H. W. Chan

Majorana fermion exchange in quasi-one-dimensional networks
David J Clarke, Jay D. Sau, Sumanta Tewari

Fractionalization via $\mathbb{Z}_{2}$ Gauge Fields at a Cold Atom Quantum Hall Transition
Yafis Barlas, Kun Yang