Predefined Lattices - ALF — Algorithms for Lattice Fermions

Reference for the Bravais lattices available in ALF. These are implemented in Prog/Predefined_Latt_mod.F90 and selected via the Lattice_type string in the &VAR_lattice namelist.

Parameters¶

&VAR_lattice
L1 = 6
L2 = 6
Lattice_type = "Square"
Model = ""
/

L1, L2 — lattice dimensions (number of unit cells in each direction)
Lattice_type — string selecting the lattice (see table below)
Model — optional model variant string (used by some Hamiltonians)

Available Lattices¶

`Lattice_type`	Orbitals per cell	Coord. vectors	Primitive vectors	Notes
`"Square"`	1	1 (1D) or 2 (2D)	$\mathbf{a}_1 = (1,0)$ , $\mathbf{a}_2 = (0,1)$	Set `L2=1` for a 1D chain
`"N_leg_ladder"`	`L2`	1	$\mathbf{a}_1 = (1,0)$	`L2` orbitals per unit cell, `L1` rungs; 1D geometry
`"Bilayer_Square"`	2	2	$\mathbf{a}_1 = (1,0)$ , $\mathbf{a}_2 = (0,1)$	Two layers with 3D orbital positions
`"Triangular"`	1	3	$\mathbf{a}_1 = (1,0)$ , $\mathbf{a}_2 = (1/2, \sqrt{3}/2)$	2D only (`L1,L2 > 1`)
`"Honeycomb"`	2	3	$\mathbf{a}_1 = (1,0)$ , $\mathbf{a}_2 = (1/2, \sqrt{3}/2)$	A/B sublattice
`"Bilayer_Honeycomb"`	4	3	$\mathbf{a}_1 = (1,0)$ , $\mathbf{a}_2 = (1/2, \sqrt{3}/2)$	Two honeycomb layers with 3D orbital positions
`"Pi_Flux"`	2	4	$\mathbf{a}_1 = (1,1)$ , $\mathbf{a}_2 = (1,-1)$	2D only (`L1,L2 > 1`)
`"Kagome"`	3	4	$\mathbf{a}_1 = (1,0)$ , $\mathbf{a}_2 = (1/2, \sqrt{3}/2)$	2D only (`L1,L2 > 1`)

The total number of orbitals is Ndim = L1 * L2 * Norb where Norb is the number of orbitals per unit cell.

Key Data Structures¶

After calling Predefined_Latt, the following are available:

Variable	Type	Description
`Latt`	`Type(Lattice)`	Bravais lattice: unit cell count `N`, neighbor list `nnlist`, distance map `imj`
`Latt_unit`	`Type(Unit_cell)`	Unit cell: `Norb`, coordination number `N_coord`, orbital positions `Orb_pos_p`
`List(I,1)`	Integer array	Unit cell index of site `I`
`List(I,2)`	Integer array	Orbital index of site `I`
`Invlist(uc,orb)`	Integer array	Site index of orbital `orb` in unit cell `uc`
`Ndim`	Integer	Total number of orbitals (`Latt%N * Latt_unit%Norb`)

Defining a Custom Lattice¶

If the predefined lattices don’t fit your needs, you can define a custom lattice directly in your Hamiltonian’s Ham_Set subroutine using the lattice library in Libraries/Modules/lattices_v3_mod.F90:

Create a Type(Unit_cell) specifying Norb, N_coord, orbital positions, and coordination vectors
Call Make_Lattice(Latt, Latt_unit, L1, L2) to build the full lattice
Construct the List/Invlist mapping arrays

See the existing Hamiltonians for examples of this pattern.