diff --git a/docs/index.md b/docs/index.md index b439ad1..ceefb73 100644 --- a/docs/index.md +++ b/docs/index.md @@ -229,15 +229,24 @@ from math import pi, sin, cos, sqrt Source: Haldane, F. D. M. (1988). Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the" parity anomaly". Physical review letters, 61(18). We denote $\{\mathbf{a}_i\}$ are the vectors from a B site to its three nearest-neighbor A sites, and $\{\mathbf{b}_i\}$ are next-nearest-neighbor distance vectors, then we have + +$$ +{\mathbf{a}_1} = (0,a), +$$ +$$ +{\mathbf{a}_2} = (\sqrt 3 a/2, - a/2), +$$ +$$ +{\mathbf{a}_3} = ( - \sqrt 3 a/2, - a/2) +$$ +$$ +{\mathbf{b}_1} = {\mathbf{a}_2} - {\mathbf{a}_3} = (\sqrt 3 a,0), +$$ +$$ +{\mathbf{b}_2} = {\mathbf{a}_3} - {\mathbf{a}_1} = ( - \sqrt 3 a/2, - 3a/2), +$$ $$ -\begin{aligned} -{\mathbf{a}_1} &= (0,a),\\ -{\mathbf{a}_2} &= (\sqrt 3 a/2, - a/2),\\ -{\mathbf{a}_3} &= ( - \sqrt 3 a/2, - a/2)\\ -{\mathbf{b}_1} &= {\mathbf{a}_2} - {\mathbf{a}_3} = (\sqrt 3 a,0),\\ -{\mathbf{b}_2} &= {\mathbf{a}_3} - {\mathbf{a}_1} = ( - \sqrt 3 a/2, - 3a/2),\\ -{\mathbf{b}_3} &= {\mathbf{a}_1} - {\mathbf{a}_2} = ( - \sqrt 3 a/2,3a/2) -\end{aligned} +{\mathbf{b}_3} = {\mathbf{a}_1} - {\mathbf{a}_2} = ( - \sqrt 3 a/2,3a/2) $$ Then the Haldane model on a hexagonal lattice can be written as