PhyslibAlpha.CondensedMatter.TightBindingChain.OpenBoundary

For a tight binding chain $T$ with $N$ sites and lattice spacing $a$, the position operator $X$ is the $\mathbb{C}$-linear operator on the Hilbert space $\mathcal{H}$ defined by: $$X = \sum_{n=0}^{N-1} (a \cdot n) |n\rangle\langle n|$$ where $n \in \{0, \dots, N-1\}$ is the site index, and $|n\rangle\langle n|$ is the outer product operator (projector) onto the localized basis state at site $n$. This operator measures the position of a particle relative to site $0$.

For a tight binding chain $T$ with $N$ sites and lattice spacing $a$, let $X$ be the position operator and $\{|n\rangle\}_{n=0}^{N-1}$ be the orthonormal basis of localized states. For any site index $n \in \{0, \dots, N-1\}$, the action of the position operator on the localized state $|n\rangle$ is given by: $$X |n\rangle = (a \cdot n) |n\rangle$$ This identity confirms that the localized states are the eigenstates of the position operator, with eigenvalues corresponding to the physical position of each site.

For any states $\psi$ and $\phi$ in the Hilbert space $\mathcal{H}$ of a tight binding chain $T$, the position operator $X$ is Hermitian, meaning it satisfies the identity: $$\langle X \psi, \phi \rangle = \langle \psi, X \phi \rangle$$ where $\langle \cdot, \cdot \rangle$ denotes the complex inner product on $\mathcal{H}$.

For a tight binding chain $T$ with $N$ sites and hopping parameter $t$, the open boundary Hamiltonian $H_{\text{open}}$ is a linear operator on the $N$-dimensional complex Hilbert space $\mathcal{H} \cong \mathbb{C}^N$. It is defined by modifying the periodic Hamiltonian $H$ to remove the hopping terms between the first site $|0\rangle$ and the last site $|N-1\rangle$: $$H_{\text{open}} = H + t (|0\rangle\langle N-1| + |N-1\rangle\langle 0|)$$ where $\{|n\rangle\}_{n=0}^{N-1}$ are the orthonormal localized basis states. This operator represents a 1D system where electrons can hop between adjacent sites but cannot hop from the last site back to the first.

For a tight binding chain $T$ with Hilbert space $\mathcal{H}$, the Hamiltonian with open boundary conditions $H_{\text{open}}$ is a Hermitian operator. Specifically, for any states $\psi, \phi \in \mathcal{H}$, the complex inner product satisfies: $$\langle H_{\text{open}} \psi, \phi \rangle = \langle \psi, H_{\text{open}} \phi \rangle$$

For a tight binding chain $T$ with $N$ sites, let $H_{\text{open}}$ be the Hamiltonian with open boundary conditions, $E_0$ the on-site energy, and $t$ the hopping parameter. Let $\{|n\rangle\}_{n=0}^{N-1}$ be the orthonormal basis of localized states. For any site indices $m, n \in \{0, \dots, N-1\}$, the matrix elements of the Hamiltonian are given by: $$\langle m | H_{\text{open}} | n \rangle = \begin{cases} E_0 & \text{if } m = n \\ -t & \text{if } m + 1 = n \text{ or } n + 1 = m \\ 0 & \text{otherwise} \end{cases}$$ This indicates that the particle can only hop between adjacent sites with amplitude $-t$, while the diagonal elements represent the on-site energy $E_0$.

Let $T$ be a tight binding chain with $N$ sites and lattice spacing $a$. Let $\mathcal{H}$ be its associated $N$-dimensional complex Hilbert space with an orthonormal basis of localized states $\{|n\rangle\}_{n=0}^{N-1}$. The position operator $X$ is defined as $X = \sum_{n=0}^{N-1} (a \cdot n) |n\rangle\langle n|$. For any linear operator $A$ on $\mathcal{H}$ and any site indices $m, n \in \{0, \dots, N-1\}$, the matrix elements of the commutator $[A, X] = AX - XA$ satisfy: $$\langle m | [A, X] | n \rangle = a(n - m) \langle m | A | n \rangle$$ where $\langle m | A | n \rangle$ denotes the inner product $\langle m, A n \rangle$.