... | ... | @@ -153,18 +153,21 @@ Formally we need to solve an infinite eigenvalue problem. In practice |
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we use the symmetry of the system to "fold" the problem into the fundamental
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domain. The finite eigenvalue problem to solve is then:
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[∑ ∑[Uₐ(kₐ)]ⁿ Hₐₙ ] u(k̲) = E(k̲) u(k̲)
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a nₐ
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where the first sum is over the symmetry generators, the second sum is over
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the cycle length of the generator. `Uₐ(kₐ)` is the `U(N)` representation
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of one of the symmetry group generators, `-π ≤ kₐ ≤ π` is the associated
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momentum, `k̲` is a vector of all the momenta. `Hₐₙ` is the Hamiltonian
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matrix linking the fundamental domain to the nth domain in the direction
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of the generator `a`. For any reasonable system all the `Hₐₙ` are 0 for `n`
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larger than a certain value (usually 1 or 2). `E(k̲)` is the energy
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[∑ Uₐ(k̲) Hₐ ] u(k̲) = E(k̲) u(k̲)
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a
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The sum is over all symmetry group elements, `a`, in the connection set, `k̲` is
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a vector of the momenta associated to each symmetry group generator. `Uₐ(k̲)` is
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a representation of `a` in `U(N)`. `Hₐ` is the Hamiltonian matrix linking the
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fundamental domain to the domain under the action of `a`. `E(k̲)` is the energy
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and `u(k̲)` is the wavefunction.
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For `a` that are pure translational symmetries the `Uₐ(k̲)` will just look like
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`exp[i k̲·R̲ₐ] ⊗ 𝟙ₙ` where `R̲ₐ` is the tuple of integers associated with `a` and
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the components of `k̲` are in `[0, 2π)`. If there is a rotational symmetry
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subgroup, then I guess the associated momenta will be quantized in units of
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`2π/m` for an m-fold rotational symmetry.
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We can see that the low-level format naturally provides the necessary
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ingredients to be able to set up such a system.
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... | ... | @@ -177,7 +180,6 @@ Write this. |
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easier to conform to the C system API.
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## `kwant.builder.System` Implemetation
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The low-level site families will correspond to the high-level site
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