static void MakeSpinOrbital()
{
// First, we assign an orbital index, say `5`. Note that we use 0-indexing,
// so this is the 6th orbital.
var orbitalIdx = 5;
// Second, we assign a spin index, say `Spin.u` for spin up or `Spin.d` for spin down.
var spin = Spin.d;
// The spin-orbital (5, ↓) is then
var spinOrbital0 = new SpinOrbital(orbitalIdx, spin);
// A tuple `(int, Spin)` is also implicitly recognized as a spin-orbital.
(int, Spin)tuple = (orbitalIdx, spin);
// We explicitly specify the type of `spinOrbital1` to demonstrate
// the implicit cast to `SpinOrbital`.
SpinOrbital spinOrbital1 = tuple;
Assert.True(spinOrbital1 == spinOrbital0);
}
static void LadderOperator()
{
// Let us use the spin orbital created in the previous snippet.
var spinOrbitalInteger = new SpinOrbital(5, Spin.d).ToInt();
// We specify either a creation or annihilation operator using
// the enumerable type `RaisingLowering.u` or `RaisingLowering.d`
// respectively;
var creationEnum = RaisingLowering.u;
// The type representing a creation operator is then initialized
// as follows. Here, we index these operators with integers.
// Hence we initialize the generic ladder operator with an
// integer index type.
var ladderOperator0 = new LadderOperator <int>(creationEnum, spinOrbitalInteger);
// An alternate constructor for a LadderOperator instead uses
// a tuple.
var ladderOperator1 = new LadderOperator <int>((creationEnum, spinOrbitalInteger));
Assert.Equal(ladderOperator0, ladderOperator1);
}
static void SpinOrbitalToInt()
{
// Let us use the spin orbital created in the previous snippet.
var spinOrbital = new SpinOrbital(5, Spin.d);
// Let us set the total number of orbitals to be say, `7`.
var nOrbitals = 7;
// This converts a spin-orbital index to a unique integer, in this case `12`,
// using the formula `g(j,σ)`.
var integerIndexHalfUp = spinOrbital.ToInt(IndexConvention.HalfUp, nOrbitals);
// This converts a spin-orbital index to a unique integer, in this case `11`,
// using the formula `h(j,σ)`.
var integerIndexUpDown = spinOrbital.ToInt(IndexConvention.UpDown);
// The default conversion uses the formula `h(j,σ)`, in this case `11`.
var integerIndexDefault = spinOrbital.ToInt();
Assert.Equal(11, integerIndexDefault);
Assert.Equal(12, integerIndexHalfUp);
}
static void CreateHubbardHamiltonianTest()
{
//////////////////////////////////////////////////////////////////////////
// Introduction //////////////////////////////////////////////////////////
//////////////////////////////////////////////////////////////////////////
// In this example, we will create a representation of a
// 1D Hubbard Hamiltonian using the quantum chemistry library.
// This representation allows one to easily simulate Hamiltonian
// time-evolution and obtain the cost of simulation.
// The 1D Hubbard model has `n` sites. Let `i` be the site index,
// `s` = 1,0 be the spin index, where 0 is up and 1 is down, `t` be the
// hopping coefficient, `u` the repulsion coefficient, and aᵢₛ the fermionic
// annihilation operator on the fermion indexed by `(i,s)`. The Hamiltonian
// of this model is
//
// H ≔ - t/2 Σᵢ (a†ᵢₛ aᵢ₊₁ₛ + a†ᵢ₊₁ₛ aᵢₛ) + u Σᵢ a†ᵢ₀ a†ᵢ₁ aᵢ₁ aᵢ₀
//
// Note that we use closed boundary conditions.
// Contents:
// - Spin-orbital representation
// - Hamiltonian term representation
// - Hamiltonian representation
// - Building the Hubbard Hamiltonian
// - Building the Hubbard Hamiltonian through orbital integrals
// - Jordan–Wigner representation
#region Spin-orbital representation
// In the vocabulary of chemistry, the site index is also called the
// orbital index. Thus a complete description of a fermion index is
// as spin-orbital. Let us build an example.
// First, we assign an orbital index, say `5`.
var orbitalIdx = 5;
// Second, we assign a spin index. In addition to
// assigning them up and down integer indices, a good memonic is to
// label then with a `Spin` enumeration type, say `u` for up and `d`
// for down.
var spin = Spin.d;
// A spin-orbital index is then
var spinOrbital0 = new SpinOrbital(orbitalIdx, spin);
// We may also map the composite spin-orbital index into a single integer `x`
// using the default formula `x = 2 * orbitalIdx + spin`;
var spinOrbital0Int = spinOrbital0.ToInt();
// Other indexing schemes are possible. For example, we may use the formula
// `x = orbitalIdx + nOrbitals * spin`
var spinOrbital0HalfUpInt = spinOrbital0.ToInt(IndexConvention.HalfUp, 6);
// Let us print these spin-orbitals to verify they contain the
// expected information.
Console.WriteLine($"Spin-orbital representation:");
Console.WriteLine($"spinOrbital0: (Orbital, Spin) index: ({spinOrbital0.Orbital},{spinOrbital0.Spin}).");
Console.WriteLine($"spinOrbital0Int: (2 * Orbital + Spin) index: ({spinOrbital0Int}).");
Console.WriteLine($"spinOrbital0HalfUpInt: (Orbital + nOrbitals * Spin) index: ({spinOrbital0HalfUpInt}).");
Console.WriteLine($"");
#endregion
#region Hamiltonian term representation
// Each term in the Hamiltonian is then labelled by an ordered sequence of
// spin-orbtal indices, and a coefficient. By default, normal-ordering is
// assumed, meaning that all creation operators are left of all annihilation
// operators. By default, the number of creation and annihilation operators
// are also assumed to be equal as chemistry Hamiltonian often conserve
// particle number.
// Let us represent the Hermitian fermion term 0.5 (a†ᵢₛ aᵢ₊₁ₛ + a†ᵢ₊₁ₛ aᵢₛ), where `i` is 5 and `s`
// is spin up. As mentioned, we require a coefficient
var coefficient = 0.5;
// We also require a sequence of spin-orbital indices.
var spinOrbitalIndices = new[] { (5, Spin.u), (6, Spin.u) };