Model lattice systems

Hubbard model (momentum-space)

hubbard_k {
    -- options,
}

Returns:

a system object.

hubbard_k creates a system object for the Hubbard model:

\[H = -t \sum_{\langle r,r' \rangle,\sigma} c^\dagger_{r,\sigma} c_{r',\sigma} + U \sum_r n_{r,\uparrow} n_{r,\downarrow}\]

using a single-particle basis of Bloch functions, \(\psi_k\):

\[\psi_k(r) = e^{ik.r} \sum_i \phi_i(r)\]

where \(\phi_i(r)\) is a single-particle basis function centred on site \(i\) in real space.

Options

sys

type: system object produced by a previous call.

Optional.

If provided, a previously created system object is updated with the new settings supplied, otherwise a new system object is created.

electrons

type: integer.

Required.

Number of electrons in the unit cell.

lattice

type: \(N\ N\)-dimensional vectors of floats.

Required.

Unit cell on which periodic boundary conditions are placed. See below.

ms

type: integer.

Required.

Set the spin polarisation of the system in units of electron spin (i.e. a single electron can take values 1 or -1).

sym

type: integer or string.

Optional. Default: aufbau.

Set the symmetry (i.e. crystal momentum) of the system if a reference determinant is not provided. This can be set to:

• An integer specifying the index of a specific wavevector; see the output produced by creating a system for possible values and their corresponding wavevectors.
• aufbau. Uses the symmetry of a determinant selected using the Aufbau principle.
• tot_sym. Uses the totally symmetric representation, whatever its index may be.

U

type: float.

Optional. Default: 1.

Specifies the \(U\) parameter in the Hamiltonian.

t

type: float.

Optional. Default: 1.

Specifies the \(t\) parameter in the Hamiltonian.

twist

type: \(N\)-dimensional vector

Optional. Default: 0 in each dimension.

Apply a twist to the wavevector grid. The twist is an ndim-dimensional vector in terms of the reciprocal lattice vectors. The twist angle is chosen to be within the primitive reciprocal cell, and hence the components should be between -0.5 and +0.5.

verbose

type: boolean.

Optional. Default: true.

Print out the single-particle basis set.

Hubbard model (real-space)

hubbard_real {
    -- options,
}

Returns:

a system object.

hubbard_real creates a system object for the Hubbard model:

\[H = -t \sum_{\langle r,r' \rangle,\sigma} c^\dagger_{r,\sigma} c_{r',\sigma} + U \sum_r n_{r,\uparrow} n_{r,\downarrow}\]

using a single-particle basis of functions in real-space.

Options

sys

type: system object produced by a previous call.

Optional.

If provided, a previously created system object is updated with the new settings supplied, otherwise a new system object is created.

electrons

type: integer.

Required.

Number of electrons in the unit cell.

lattice

type: \(N\ N\)-dimensional vectors of floats.

Required.

Unit cell on which periodic boundary conditions are placed. See below.

ms

type: integer.

Required.

Set the spin polarisation of the system in units of electron spin.

U

type: float.

Optional. Default: 1.

Specifies the \(U\) parameter in the Hamiltonian.

t

type: float.

Optional. Default: 1.

Specifies the \(t\) parameter in the Hamiltonian.

finite

type: boolean.

Optional. Default: false.

If false then periodic boundary conditions are applied to the unit cell, otherwise the system specified by the lattice is treated as an isolated set of sites.

verbose

type: boolean.

Optional. Default: true.

Print out the single-particle basis set.

Heisenberg model

heisenberg {
    -- options,
}

Returns:

a system object.

heisenberg creates a system object for the Heisenberg model, which models a set of spin 1/2 particles on a lattice:

\[\hat{H} = -J \sum_{\langle i,j \rangle} \hat{\boldsymbol{S}}_i \cdot \hat{\boldsymbol{S}}_j - h_z \sum_i \hat{S}_{iz} - h_z' \sum_i \hat{S}_{iz}^{\xi},\]

where \(h_z\) and \(h_z'\) denote the magnetic field strength and staggered magnetic field strength, respectively, and \(\xi\) is equal to +1 for sites on sublattice 1 and is equal to -1 for sites on sublattice 2.

Options

sys

type: system object produced by a previous call.

Optional.

If provided, a previously created system object is updated with the new settings supplied, otherwise a new system object is created.

lattice

type: \(N\ N\)-dimensional vectors of floats.

Required.

Unit cell on which periodic boundary conditions are placed. See below.

Warning

For efficiency reasons it is assumed that the smallest dimension lattice vector is greater than 2 if periodic boundary conditions are used.

ms

type: integer.

Required.

Set the spin polarisation of the system in units of 1/2.

J

type: float.

Optional. Default: 1.

Set the coupling constant for the Heisenberg model.

magnetic_field

type: float.

Optional. Default: 0.

staggered_magnetic_field

type: float.

Optional. Default: 0.

Note

Specifying non-zero values for both magnetic_field and staggered_magnetic_field is not currently possible.

finite

type: boolean.

Optional. Default: false.

If false then periodic boundary conditions are applied to the unit cell, otherwise the system specified by the lattice is treated as an isolated set of sites.

triangular

type: boolean.

Optional. Default: false.

If true, then a triangular lattice of sites on which the spins reside is used, requiring a 2D lattice. The default is to use a \(N\)-dimensional cubic arrangement of sites.

verbose

type: boolean.

Optional. Default: true.

Print out the single-particle basis set.

Chung-Landau model

chung_landau {
    -- options,
}

Returns:

a system object.

chung_landau creates a system object for the system of spinless fermions proposed by Chung and Landau:

\[H = -t \sum_{\langle r,r' \rangle} c^\dagger_{r} c_{r'} + U \sum_{\langle r,r' \rangle} n_{r} n_{r'}\]

using a single-particle basis of functions in real-space.

Options

sys

type: system object produced by a previous call.

Optional.

If provided, a previously created system object is updated with the new settings supplied, otherwise a new system object is created.

electrons

type: integer.

Required.

Number of fermions in the unit cell.

lattice

type: \(N\ N\)-dimensional vectors of floats.

Required.

Unit cell on which periodic boundary conditions are placed. See below.

U

type: float.

Optional. Default: 1.

Specifies the \(U\) parameter in the Hamiltonian.

t

type: float.

Optional. Default: 1.

Specifies the \(t\) parameter in the Hamiltonian.

finite

type: boolean.

Optional. Default: false.

If false then periodic boundary conditions are applied to the unit cell, otherwise the system specified by the lattice is treated as an isolated set of sites.

verbose

type: boolean.

Optional. Default: true.

Print out the single-particle basis set.

Specifying the lattice

The lattice is specified as a table of vectors. Sites (on which a spin or electron resides) are at unit locations on the grid. The unit cell (or, if periodic boundary conditions are not used, the geometry of the ‘flake’ essentially cut out of the infinite lattice) are given in this basis. The lattice variable hence requires \(N\) vectors, each of dimension \(N\). This is specified in lua by a nested table. For example:

lattice = { { 10 } }

sets a 1D system, with the unit cell containing 10 sites;

lattice = { { 2, 0 }, { 0, 2 } }

sets a 2D system, with the unit cell containing 4 sites; and

lattice = { { 3, 3 }, { 3, -3 } }

sets a 2D system, with the (square) unit cell containing 18 sites and rotated by \(45^\circ\) relative to the primitive lattice.

HANDE supports 1-, 2- and 3-dimensional lattices. Lattice vectors must be orthogonal.