Tight-Binding Chain¶

Motivations¶

chainTB package models a one-dimensional tight-binding dimer chain with real and/or complex-valued hoppings and onsite energies. Dimer chain defects can be introduced by locally modifying the on-site energies, the hoppings, or by changing the dimerization pattern i.e. switching intradimer with interdimer hopping. The package can reproduce Shockley [Sho39] and SSH [Su79] states. Note that these systems acquired recently a renewed interest considering complex onsites energies allowing to amplify the topologically protected midgap state [Sch13], [Pol15].

This package can then by useful to illustrate the concepts of:

Parity-Time symmetry and Parity-Time symmetry breaking.

Sublattice symmetry breaking (for an odd number of sites), resulting in a mode of zero energy.

Localized states introduced by dimerization defects in the dimerization pattern.

Topologically protected states:

Localized in one edge of dimer chain with alternating couplings and sublattice symmetry breaking, the Shokley state.

Localized at the dimerization defect in a dimer chain with with alternating couplings and sublattice symmetry breaking, the SSH state.

The chain is defined by:

a unit cell composed of two sites labeled A (blue) and B (red):

Note

The chain starts with a A site.

two hoppings (with non zeros real parts), the intradimer coupling $t_{ab}=t_a$ which links the A sites to the B sites, and the interdimer hopping $t_{ba} =t_b$ which links the B sites to the A sites.
two onsite energies $\epsilon_{a}$ for the A sites, and $\epsilon_{b}$ for the B sites.
For an even number of sites $n=2m$ the chain is composed of $m$ unit cells.

For $n=10$ :
For an odd number of sites $2n+1$ the chain is composed of $n$ unit cells plus an extra A site.

For $n=11$ :

The defects implemented:

change locally the onsite energies.

change locally the hoppings.

introduce a defect in the dimerization pattern.

hopping disorder.

onsite energy disorder.

chainTB can:

obtain the spectrum (eigenenergies of the tight-binding Hamiltonian) and the probability densities of the states of the system (absolute value squared eigenvectors of the Hamiltonian).

obtain sublattice polarization (sum of the probability densities associated to one sublattice).

select polarized states (revealing zero modes).

test robustness to disorder by implementing hopping disorder.

get the time evolution of the field (using the Crank-Nicolson method).

Installation¶

chainTB is available at https://github.com/cpoli/TB

To use chainTB, you need to install the programming language python and three additional packages:

python 3.x

numpy

scipy

matplotlib

See https://cpoli.github.io/python-doc.html for details, and the TB module https://github.com/cpoli/TB:

latticeTB

eigTB

plotTB

propTB

References¶

[Sho39]

W. Shockley, On the surface states associated with a periodic potential. Phys. Rev. 56, 317 (1939)

[Su79]

W.P. Su, J.R. Schrieffer, and A.J. Heeger, Solitons in conducting polymers. Phys. Rev. Lett. 42, 1698 (1979).

[Sch13]

Schomerus, H. Topologically protected midgap states in complex photonic lattices. Opt. Lett. 38, 1912–1914 (2013).

[Pol15]

C. Poli, M. Bellec, U.Kuhl, F. Mortessagne, and H. Schomerus, Selective enhancement of topologically induced interface states in a dielectric resonator chain. Nat. Commun. 6 6710, (2015).

Feedback¶

Please send comments or suggestions for improvement to cpoli83 at hotmail dot fr