... | ... | @@ -31,9 +31,8 @@ directed graph, where systems of higher symmetry are *attached* to systems of a |
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lower symmetry. The interface between two systems is specified by a sequence of
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two bits of information:
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+ a *term* of the system with higher symmetry that joins the
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fundamental domain of the system with lower symmetry to the system
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with higher symmetry.
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+ a *term* of the system with higher symmetry that joins a unit cell
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of the system with lower symmetry to the system with higher symmetry.
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+ a sequence of *sites* in the system with lower symmetry that
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constitute the interface.
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... | ... | @@ -45,29 +44,62 @@ The symmetry group of the system with lower symmetry must be a subgroup of |
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the symmetry group of the system with higher symmetry.
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The following examples should serve to illustrate the point.
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In the following illustrations, the circles correspond to a single unit cell
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of the system under consideration. Circles with dashed outline represent the
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image of the unit cell under the action of the symmetry of the system; they
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are shown for clarity. Full lines denote a set of *terms* associated with
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a symmetry group element.
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the symmetry group element that takes us from the fundamental unit cell
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to the unit cell joined by the line. Crosses on lines denote that the
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lines should be "cut" and joined instead to the system
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(of lower symmetry) to which we are attaching. Note that the "cuts"
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are also applied to all lines under the action of the symmetry group
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of the system with lower symmetry.
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### Finite scattering region attached to 1D leads
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This is the simplest and most common case. This is the only case that Kwant
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could treat in version 1.x.
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##### Nearest neighbor hoppings
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<img src="https://cdn.rawgit.com/wiki/jbweston/kwant/images/0D_1D.svg" width="50%"></img>
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We have a lead and a finite scattering region. We want to attach the lead
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by cutting the hopping between a unit cell and the cell on the left. As the
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finite scattering region has the trivial symmetry group, only one hopping
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of the infinite lead is cut. This produces 2 semi-infinite leads that are
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disconnected. The "cut" hopping is then attached to the finite scattering
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region. The figure below represents this schematically. The resultant system
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has two disconnected parts; we are interested in the outlined part, the other
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part is discarded.
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<img src="https://rawgit.com/wiki/jbweston/kwant/images/0D_1D.svg" width="50%"></img>
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##### Next-nearest neighbor hoppings
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<img src="https://cdn.rawgit.com/wiki/jbweston/kwant/images/0D_1D_nn.svg" width="50%"></img>
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We have a lead with next-nearest neighbor hoppings, in addition to nearest
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neighbor hoppings. This is essentially the same as the previous case.
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<img src="https://rawgit.com/wiki/jbweston/kwant/images/0D_1D_nn.svg" width="50%"></img>
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### 1D scattering region attached to 2D leads
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<img src="https://cdn.rawgit.com/wiki/jbweston/kwant/images/1D_2D.svg" width="50%"></img>
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<img src="https://cdn.rawgit.com/wiki/jbweston/kwant/images/1D_2D_nn.svg" width="50%"></img>
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Here we have a system that is translationally invariant in 2 dimensions that we
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want to attach to a system that is translationally invariant in 1 direction. We
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specify that we wish to cut the hopping to the left. Due to the (1D) translational
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invariance of the system to which we are attaching, this also cuts all the hoppings
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to the left for the cells above and below. We see that the resultant system
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is translationally invariant in the vertical direction
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<img src="https://rawgit.com/wiki/jbweston/kwant/images/1D_2D.svg" width="50%"></img>
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### Finite scattering region embedded in 2D bulk
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<img src="https://cdn.rawgit.com/wiki/jbweston/kwant/images/0D_2D.svg" width="50%"></img>
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<img src="https://cdn.rawgit.com/wiki/jbweston/kwant/images/0D_2D_weird_connection.svg" width="50%"></img> |
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Here we have a 2D translationally invariant system into which we wish to embed
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a finite scattering region. We accomplish this by cutting all the hoppings to
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a unit cell.
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<img src="https://rawgit.com/wiki/jbweston/kwant/images/0D_2D.svg" width="50%"></img>
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We could also have a weirder configuration, where we do not cut all the hoppings.
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This means that we do not cut the higher-dimensional system into disjoint parts.
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This would look like the following:
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<img src="https://rawgit.com/wiki/jbweston/kwant/images/0D_2D_weird_connection.svg" width="50%"></img> |