Harper-Hofstadter Square Lattice

Purpose and gauge

This square lattice adds magnetic Peierls phases in Landau gauge. Horizontal hoppings are real; a vertical hopping in column $x$ carries

$$ -t\exp(2\pi ifx), $$

where $f$ is flux per plaquette in flux-quantum units.

Hofstadter plaquette phases

The single-particle dimension is $N_rN_c$. Dense and CSR builders are available.

from quantum_lattice_models import harper_hofstadter_square_lattice

H = harper_hofstadter_square_lattice(n_rows=8, n_cols=8, flux=0.25)

Builder	Parameter	Type	Default	Constraint
`harper_hofstadter_square_lattice`	`n_rows`	`int`	`4`	>= 1
`harper_hofstadter_square_lattice`	`n_cols`	`int`	`4`	>= 1
`harper_hofstadter_square_lattice`	`hopping`	`float`	`1.0`
`harper_hofstadter_square_lattice`	`flux`	`float`	`0.25`
`harper_hofstadter_square_lattice`	`onsite`	`float`	`0.0`
`harper_hofstadter_square_lattice`	`periodic_x`	`bool`	`False`
`harper_hofstadter_square_lattice`	`periodic_y`	`bool`	`False`
`harper_hofstadter_square_lattice_sparse`	`n_rows`	`int`	`8`	>= 1
`harper_hofstadter_square_lattice_sparse`	`n_cols`	`int`	`8`	>= 1
`harper_hofstadter_square_lattice_sparse`	`hopping`	`float`	`1.0`
`harper_hofstadter_square_lattice_sparse`	`flux`	`float`	`0.25`
`harper_hofstadter_square_lattice_sparse`	`onsite`	`float`	`0.0`
`harper_hofstadter_square_lattice_sparse`	`periodic_x`	`bool`	`False`
`harper_hofstadter_square_lattice_sparse`	`periodic_y`	`bool`	`False`

Representative vertical phases and dense/sparse equivalence are tested. A finite flux sweep is not the infinite-system Hofstadter butterfly.