Notes on gapped boundaries of Abelian topological phases

2014年08月15日 留下评论

This is going to be a collection of various results on Abelian topological phases, with the focus being gapped boundaries. Part of the motivation is to make some connections between the existing mathematical literature on quadratic forms and the physical applications.

First, recall that a topological phase (unitary modular tensor category) can have a gapped boundary if and only if it contains a maximal Lagrangian subalgebra. It is also known that all such topological phases must be the Drinfeld centers of unitary fusion categories, i.e. quantum doubles. For Abelian phases (i.e. pointed categories), it means all those with gapped boundaries are essentially discrete gauge theories with an Abelian gauge group G, possibly twisted by 3-cocycles in H^3(G, U(1)). If we represent them with Abelian Chern-Simons theory, their K matrices always take the following form (up to GL equivalence):

\displaystyle \mathbf{K}=\displaystyle \begin{pmatrix} 0 & \mathbf{A} \\ \mathbf{A}^{\mathrm{T}} & \mathbf{B} \end{pmatrix}

In particular, it implies that \det \mathbf{K} is a square.

We also note that time-reversal-invariant Abelian theories must have gapped boundaries. This can be shown following Levin and Stern.

First we define the time-reversal symmetry directly in Abelian Chern-Simons theory. Let us consider a general Abelian topological phase given by a \mathbf{K} matrix with dimension 2N. The time-reversal symmetry operation \mathcal{T} acts on the gauge fields as

a_I\rightarrow \mathbf{T}_{IJ}a_J.

Also, because t\rightarrow -t, time-reversal invariance implies that


For bosonic systems, we should have \mathcal{T}^2=1, which implies \mathbf{T}^2=1.

We now need to prove that there should be at least N null vectors in the theory which are mutually null. We can easily prove that \mathrm{Tr}\,(\mathbf{T})=0, and since \mathbf{T}^2=1, the eigenvalues of \mathbf{T} are \pm 1 and there must be equal number of 1 and -1. Let \mathbf{v}_i denote the eigenvectors of \mathbf{T} with +1 eigenvalue, i=1,\dots, N. We have

\mathbf{v}_i^{\mathrm{T}} \mathbf{K} \mathbf{v}_j=\mathbf{v}_i^{\mathrm{T}} \mathbf{T}^{\mathrm{T}} \mathbf{K} \mathbf{T}\mathbf{v}_j=-\mathbf{v}_i^{\mathrm{T}} \mathbf{K} \mathbf{v}_j.

Therefore \mathbf{v}_i^{\mathrm{T}} \mathbf{K} \mathbf{v}_j=0.

Levin has argued that \mathbf{v}_i can be chosen to be integer vectors. This is always possible since this eigenspace is spanned by the columns of 1 + \mathbf{T} , a matrix with integer entries. Therefore we have found a set of null vectors. QED.


Notes 08/15/13

2013年08月16日 留下评论


2013年07月13日 留下评论

I just found this python package developed by Vanderbilt group in Rutgers. It is incredibly useful. As self-categorizing as a member of the “topological community” in condensed matter, I often need to do some band structure calculations for tight-binding models on various lattices. These are elementary ones, but sometimes a lot of work — doing Fourier transformation, calculating Berry curvatures and mistakes can easily sneak in for careless people like me. But Pythtb just kills all tight-binding model caculations — maybe not all, but at least most of them — in one shot. With a few lines of python codes you can easily build up models even with complicated lattice geometry and hoppings, solve for the dispersions and plot them nicely. For those who are interested in topological stuff, it has a lot more than one could have thought: Berry phase, Berry curvature, Wannier center, finite lattice, edge states….I feel like I’m promoting it, but it is really a wonderful piece of work. I even think one should code a GUI for it and just “draw” models on the screen.


密码保护:Surface State of a 3D SPT

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2013年05月20日 1条评论

因朋友邀请写了一篇关于Majorana的科普文,比之前随便写的Race for Majorana稍微严整些。


Some useful stuff

2012年12月22日 留下评论

The online database of Vertex Operator Algebras and Modular Categories
A Catalogue of Lattices

This program can calculate almost all CFT properties of WZW models.


Numerically Computing Pfaffian (Mathematica)

2012年11月8日 1条评论

A simple method: given a real antisymmetric matrix, its Schur decomposition yields the canonical form. Then the Pfaffian can be easily read off.