Su–Schrieffer–Heeger model
In condensed matter physics, the Su–Schrieffer–Heeger (SSH) model or SSH chain is a one-dimensional lattice model that presents topological features.
[1] It was devised by Wu-Pei Su, John Robert Schrieffer, and Alan J. Heeger in 1979, to describe the increase of electrical conductivity of polyacetylene polymer chain when doped, based on the existence of solitonic defects.
[2][3] It is a quantum mechanical tight binding approach, that describes the hopping of spinless electrons in a chain with two alternating types of bonds.
For the finite chain, there exists an insulating phase, that is topologically non-trivial and allows for the existence of edge states that are localized at the boundaries.
[1] The model describes a half-filled one-dimensional lattice, with two sites per unit cell, A and B, which correspond to a single electron per unit cell.
In this configuration each electron can either hop inside the unit cell or hop to an adjacent cell through nearest neighbor sites.
If the gap lies at the Fermi level, then the system is considered to be an insulator.
The tight binding Hamiltonian in a chain with N sites can be written as[1] where h.c. denotes the Hermitian conjugate, v is the energy required to hop from a site A to B inside the unit cell, and w is the energy required to hop between unit cells.
The dispersion relation for the bulk can be obtained through a Fourier transform.
, we pass to k-space by doing which results in the following Hamiltonian where the eigenenergies are easily calculated as and the corresponding eigenstates are where The eigenenergies are symmetrical under swap of
By analyzing the energies, the problem is apparently symmetric about
Nevertheless, not all properties of the system are symmetrical, for example the eigenvectors are very different under swap of
It can be shown for example that the Berry connection integrated over the Brillouin zone
, produces different winding numbers:[1] showing that the two insulating phases,
The winding number remains undefined for the metallic case
This phenomenon is called a topological phase transition.
It is much harder to diagonalize the Hamiltonian analytically in the finite case due to the lack of translational symmetry.
[1] There exist two limiting cases for the finite chain, either
dimers and two unpaired sites at the edges of the chain.
By plotting the eigenstates of the finite chain as function of position, one can show that there are two distinct kinds of states.
For non-zero eigenenergies, the corresponding wavefunctions would be delocalized all along the chain while the zero energy eigenstates would portray localized amplitudes at the edge sites.
Even if the eigenenergies lie in the gap, the edge states are localized and correspond to an insulating phase.
The existence of edge states in one region and not in the other demonstrate the difference between insulating phases and it is this sharp transition at
[1] The bulk case allows to predict which insulating region would present edge states, depending on the value of the winding number in the bulk case.
in the bulk case, the corresponding finite chain would not.
This relation between winding numbers in the bulk and edge states in the finite chain is called the bulk-edge correspondence.
