... | @@ -14,31 +14,52 @@ problems in arbitrary dimensions. Examples include: |
... | @@ -14,31 +14,52 @@ problems in arbitrary dimensions. Examples include: |
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+ Finite scattering region attached to 1D leads
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+ Finite scattering region attached to 1D leads
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<img src="http://imgh.us/0D_1D.svg" width="50%"></img>
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+ 1D scattering region attached to 2D leads (+ higher dimensional
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+ 1D scattering region attached to 2D leads (+ higher dimensional
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generalizations)
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generalizations)
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<img src="http://imgh.us/1D_2D.svg" width="50%"></img>
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+ Finite scattering region embedded in 2D bulk (+ higher dimensional
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+ Finite scattering region embedded in 2D bulk (+ higher dimensional
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generalizations)
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generalizations)
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<img src="http://imgh.us/0D_2D_1.svg" width="50%"></img>
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**TODO**: put examples w/ illustrations here
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## `CompositeSystem` API
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## `CompositeSystem` API
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A `CompositeSystem` is essentially a collection of `System`s arranged in a
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A `CompositeSystem` is essentially a collection of `System`s arranged in a
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directed graph, where systems of higher symmetry are *attached* to systems
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directed graph, where systems of higher symmetry are *attached* to systems of a
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of a lower symmetry. The interface between two systems is specified by a
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lower symmetry. The interface between two systems is specified by a sequence of
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sequence of *terms* of the system with higher symmetry that join the
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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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fundamental domain of the system with lower symmetry to the system
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with higher symmetry. If all the terms associated with a particular
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with higher symmetry.
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group element (of the higher-symmetry system) are present in the
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interface, we say that the lower-symmetry system *fully interrupts*
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+ a sequence of *sites* in the system with lower symmetry that
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the higher-symmetry one.
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constitute the interface.
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The system with lower symmetry must contain a full fundamental domain
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The above information allows the self-energy matrix (defined over the interface
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of the system with higher symmetry, otherwise the `term` does not
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sites in the system with lower symmetry) to be defined.
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represent a well defined hopping into the fundamental domain of the
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lower symmetry.
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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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The following examples should serve to illustrate the point.
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**TODO**: put examples w/ illustrations here |
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### Finite scattering region attached to 1D leads
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##### Nearest neighbor hoppings
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##### Next-nearest neighbor hoppings
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### 1D scattering region attached to 2D leads
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### Finite scattering region embedded in 2D bulk
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Note that we *must* choose the unit cell of the system with higher symmetry to
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be large enough so that the finite scattering region fits inside it. Otherwise
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it is impossible to describe the interface using the scheme we have outlined
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above. |