Peierls instability. The Su-Schrieffer-Heeger (SSH) model is a tight-binding model that describes a single spin-less electron on a two site unit cell 1D lattice. The two sites in a unit cell are labelled as $A$ and $B$. From here we will set the lattice constant, $2a=1$.

![Visual representation of the SSH Model[]{data-label="sshfig"}](fig1.pdf)

We will work with one electron per unit cell that corresponds to a half-filled lattice because we are considering spin-less electrons. Thus the only degree of freedom the electrons have is that they can hop around from one site to the other - let us call the hopping potentials as $v$ (for hopping within the unit cell) and $w$ (for hopping connecting neighbouring unit cells), and write the tight binding Hamiltonian as $$\begin{aligned}
H & = & v\sum_{n=1}^{N} (\ket{n,B}\bra{n,A}+\text{h.c.}) \nonumber \\
&& +w\sum_{n=1}^{N}(\ket{n+1,A}\bra{n,B}+\text{h.c})\end{aligned}$$ Here ‘h.c.’ denotes the hermitian conjugate of the term before it. The basis we have used to define the Hamiltonian can also be written as $\ket{n,A(\text{or }B)}=\ket{n}\otimes\ket{A(\text{or }B)}$. This representation tells us that the full Hilbert space is made of two parts - $\mathcal{H}_{\text{external}}\otimes\mathcal{H}_{\text{internal}}$. The internal Hilbert space is due to the two sub-lattice sites ($A$ and $B$) in the unit cell and the external 