Mean-field Hamiltonian in k-space, 8-sites cell

This part of the library focuses on solving the system Hamiltonian $$\begin{aligned} H = &-t_0\sum_{<ij>}(c_i^\dagger c_j + H.c.)+J^A_x\sum_{i\in A}(c_{i}^\dagger c_{i+x} + H.c.) +J^A_y\sum_{i\in A}(c_{i}^\dagger c_{i+y} + H.c.)\\ &+J^B_x\sum_{i\in B}(c_{i}^\dagger c_{i+x} + H.c.) +J^B_y\sum_{i\in B}(c_{i}^\dagger c_{i+y} + H.c.) +V_1\sum_{<ij>}n_in_j+V_2\sum_{<<ij>>}n_in_j\\ &+V_3\sum_{<ij>_3}n_in_j+V_4\sum_{<ij>_4}n_in_j \end{aligned}$$ self-consistently with a restricted Hartree fock ansatz on the 8-site unit cell.

class checkerboard_eight_sites[source]

checkerboard_eight_sites()

Class defining the checkerboard lattice model.

Methods

Method	Description
__init__	Initialize lattice related parameters
fixed_point_method	Fixed-point search routine
get_hamiltonian	Define the mean-field Hamiltonian, dependent on mean-field coefficients given by the correlation matrix
diagH	Diagonalize the Hamiltonian for each $k$-vector in the Brilluoin zone. Obtain new k-space eigen-vectors and eigen-values (bands)
get_correlation_tensors	Calculate correlation matrix
get_FE	Calculate free energy
check_convergence	Stop the fixed-point search i) after itermax iterations or ii) after itermin iterations and if iia) the largest mean-field parameter does not converge but oscillates between $\pm n$ for consecutive_oscillations times, indicating a metallic phase (discard run) or iib) if both the free energy gradient and the correlation matrix gradient converged below convergence_thresh_cFE or the inverse ratios between the largest mean-field parameter and all others converged below convergence_thresh_mfops

Attributes

Attribute	type	Description
cellBravaisvecs	int nparray	Bravais vectors of the unit cell
iterations	int	Counts iterations
itermin	int	Minimun number of iterations
itermax	int	Maximun number of iterations
ncs	int	Minimum number of neighboring cells (+1) needed to achieve full lattice tiling
nsites	int	Number of lattice unit cell sites
site_coos	int nparray	Sites coordinates in the lattice unit cell
Vsarray	float nparray	Interaction amplitudes, nV terms including $V_0$
xiQAH_loop_mask	int nparray	Coordinates of the square plaquette over which $\xi_{\textrm{QAH}}$ is calculated

class mean_field_analysis[source]

mean_field_analysis(itermin=20, itermax=700, kgridsize=10, Vgrid=5, beta=100, corrm='comp', rydberg=False) :: checkerboard_eight_sites

Perform the mean-field analysis.

Methods

Method	Description
__init__	Initialize Hamiltonian related parameters, then call checkerboard_eight_sites.init()
prepare_run	Update the set of $V_i$, define Hamiltonian accordingly, initialize random correlation matrix, set and initialize other run parameters like k-space eigen-vectors and eigen-values (bands)