# Brillouin Zone Sampling of Chain with N Sites and Modified Periodic Boundary Conditions

Brillouin Zone Sampling of Chain with N Sites and Modified Periodic Boundary Conditions

This Demonstration shows the use of a complex exponential phase factor in the periodic boundary conditions of the Wannier tight-binding (TB) Hamiltonian of a chain with sites, TB hopping parameter , and on-site energy parameter . This requires modified periodic boundary conditions (MPBCs). The phase is given by , where is the number of sites, , and is the lattice parameter of the chain. The use of MPBCs allows sampling other points inside the first Brillouin zone (BZ) with a small number of sites, which otherwise could only be reached by choosing a system with a higher number of sites. For instance, the points sampled with four sites can also be obtained with a two-site chain and switching the phase between the allowed values and . When , the MPBCs are simply periodic boundary conditions, while for the MPBCs are anti-periodic boundary conditions (APBCs). This trick of sweeping is particularly important and useful in the field of strongly correlated electron systems, where one has to perform exact diagonalizations of huge matrices (e.g., Hubbard-like matrices) for investigating the BZ with the aid of finite-sized systems having a very small number of sites.

N

t

ϵ

ϕ=2πκ/(Na)

N

κ=0,1,…,N-1

a

ϕ

0

π

ϕ=0

ϕ=π

ϕ