Hamiltonian¶

QAssemble uses a Python dict-based input to define the Hamiltonian, split into a one-body (OneBody) and a two-body (TwoBody) part.

Hamiltonian = {
    'OneBody': {
        'Hopping': {
            ((0,0),(1,0)): {
                1.0: [[0,0,0],[-1,0,0],[0,-1,0]],
            },
        },
        'Onsite': {
            0: {(0,0): 0.0, (1,0): 0.0}
        }
    },
    'TwoBody': {
        'Local': {
            'Parameter': 'SlaterKanamori',
            'option': {
                (0,(0)): {'l': 0, 'U': 2.0, 'Up': 0.0},
                (1,(0)): {'l': 0, 'U': 2.0, 'Up': 0.0}
            }
        },
        'NonLocal': {
            ((0,0),(1,0)): {
                0.20: [[0,0,0],[-1,0,0],[0,-1,0]],
            },
        }
    }
}

OneBody¶

The one-body part describes the non-interacting Hamiltonian \(H_0\):

\[H_0 = \sum_{ij} t_{ij} c_i^\dagger c_j + \sum_i \epsilon_i c_i^\dagger c_i\]

Hopping¶

Defines the hopping amplitudes \(t_{ij}\) between orbital pairs.

'Hopping': {
    ((0,0),(1,0)): {        # orbital pair: (sublattice 0, orbital 0) → (sublattice 1, orbital 0)
        1.0: [[0,0,0],[-1,0,0],[0,-1,0]],   # amplitude: list of lattice vectors (R)
    },
},

Key ((i, orb_i), (j, orb_j)) — source and target (sublattice, orbital) pair
Sub-key — hopping amplitude \(t\)
Value — list of lattice vectors \(\mathbf{R}\) connecting the two sites

Onsite¶

Defines on-site energies \(\epsilon_i\) for each orbital.

'Onsite': {
    0: {(0,0): 0.0, (1,0): 0.0}   # spin index: {(sublattice, orbital): energy}
},

TwoBody¶

The two-body part defines Coulomb interaction terms.

Local¶

Specifies the on-site (local) interaction using a predefined parameterization.

'Local': {
    'Parameter': 'SlaterKanamori',   # interaction type
    'option': {
        (0,(0)): {'l': 0, 'U': 2.0, 'Up': 0.0},   # (sublattice, (orbital,)): params
        (1,(0)): {'l': 0, 'U': 2.0, 'Up': 0.0}
    }
},

Supported parameterizations:

Parameter	Description
`SlaterKanamori`	Slater-Kanamori interaction (\(U\), \(U'\), \(J\))
`Slater`	Full Slater integrals
`Kanamori`	Simplified Kanamori form

NonLocal¶

Defines inter-site (non-local) density-density interactions.

'NonLocal': {
    ((0,0),(1,0)): {          # orbital pair
        0.20: [[0,0,0],[-1,0,0],[0,-1,0]],   # amplitude: list of lattice vectors
    },
},

Supported non-local interaction types:

Type	Description
`Ohno`	Ohno interpolation \(V(r) = U / \sqrt{1 + (r/a)^2}\)
`Ohno-Yukawa`	Screened Ohno form
`JTH`	J-threading (JTH)