# Clockwise and Anti-Clockwise next-nearest hopping on honeycomb lattice?

I have the difficulty of understanding, How we can distinguish that which next-nearest hopping on honeycomb lattice is clockwise or anticlockwise?

## 1 Answer

You cannot distinguish this, they are all symmetric. The hopping that is clockwise with respect to one cell is anticlockwise with respect to the neighboring cell. If you add this "with respect to", then orientation is meaningfull.

Well, any graph can be oriented. Just arbitrarily chose for each edge a positive direction and call the other one negative. What I'm saying is: you cannot chose an orientation which is in accord with the symmetry (translation, rotation, inversion) of the lattice.

What you maybe mean: the second-neighbour-hoppings can be given a direction. Since any of them is completely inside a cell, you can just call "clockwise" those which are clockwise with respect to their cell, which is well defined.
Or to put it differently, those 2-neighbor-hoppings form triangles, and there are two distinct types of triangles (inside a hexagon/in three hexagons or: pointing upwards/downwards). You can chose with respect to which type of the triangles you want to define the orientation, and any choice is a valid choice.

Since you cannot split the hexagons in two types (try! I mean two types such, that each bond belongs to two different types on either side), you cannot define an orientation for the nearest neighbours.