Physlib.CondensedMatter.TightBindingChain.Basic

For any tight binding chain $T$, the number of sites $N$ in the chain is non-zero ($N \neq 0$).

The Hilbert space associated with a tight binding chain $T$ is the $N$-dimensional complex Hilbert space $\mathbb{C}^N$, where $N$ is the number of sites in the chain.

For a tight binding chain $T$ with $N$ sites, the `localizedState` is the orthonormal basis $\{|n\rangle\}_{n=0}^{N-1}$ of the associated Hilbert space $\mathcal{H} \cong \mathbb{C}^N$. Each basis vector $|n\rangle$ (mathematically the standard basis vector $e_n$) represents a quantum state where the particle is perfectly localized at the $n$-th site of the chain.

The notation $|n\rangle$ represents the localized state at the $n$-th site of the tight binding chain. It denotes the $n$-th vector in the orthonormal basis `localizedState` of the system's Hilbert space, where $n \in \{0, 1, \dots, N-1\}$.

The notation $\langle m | n \rangle$ denotes the complex inner product $\langle \psi_m, \psi_n \rangle_{\mathbb{C}}$ between two localized states in the tight binding chain, where $\psi_m$ and $\psi_n$ are the states corresponding to the electron being located at site $m$ and site $n$, respectively.

For a tight binding chain $T$ with $N$ sites, the set of localized states $\{|n\rangle\}_{n=0}^{N-1}$ is orthonormal in the associated Hilbert space $\mathcal{H} \cong \mathbb{C}^N$ over the complex numbers $\mathbb{C}$. That is, the inner product of any two localized states satisfies $\langle m | n \rangle = \delta_{mn}$, where $\delta_{mn}$ is the Kronecker delta.

For a tight binding chain $T$ with $N$ sites, let $|m\rangle$ and $|n\rangle$ be the localized states in the Hilbert space corresponding to site indices $m, n \in \{0, 1, \dots, N-1\}$. The inner product of these states satisfies $$\langle m | n \rangle = \begin{cases} 1 & \text{if } m = n \\ 0 & \text{if } m \neq n \end{cases}$$

For a tight binding chain $T$ with $N$ sites and associated Hilbert space $\mathcal{H} \cong \mathbb{C}^N$, let $|n\rangle$ and $|m\rangle$ be the orthonormal localized basis states corresponding to sites $n, m \in \{0, \dots, N-1\}$. The linear map `localizedComp` represents the outer product operator $|m\rangle\langle n| : \mathcal{H} \to \mathcal{H}$, which maps a state $|\psi\rangle \in \mathcal{H}$ to the vector $\langle n | \psi \rangle |m\rangle$, where $\langle n | \psi \rangle$ denotes the complex inner product of the state $|n\rangle$ and $|\psi\rangle$.

For site indices $n, m \in \{0, \dots, N-1\}$, the notation $|n\rangle\langle m|$ represents the outer product of the localized states $|n\rangle$ and $|m\rangle$ in the Hilbert space of the tight binding chain. This operator is a linear map that acts on a state $|\psi\rangle$ as $|n\rangle\langle m | \psi\rangle$, where $\langle m | \psi \rangle$ is the inner product of the state $|\psi\rangle$ with the localized state $|m\rangle$.

In a tight binding chain with $N$ sites, let $\{|n\rangle\}_{n=0}^{N-1}$ be the orthonormal basis of localized states in the Hilbert space $\mathcal{H} \cong \mathbb{C}^N$. For any site indices $m, n, p \in \{0, \dots, N-1\}$, the action of the outer product operator $|m\rangle\langle n|$ on a localized state $|p\rangle$ is given by $$(|m\rangle\langle n|) |p\rangle = \begin{cases} |m\rangle & \text{if } n = p \\ 0 & \text{if } n \neq p \end{cases}$$ which can be written using the Kronecker delta as $(|m\rangle\langle n|) |p\rangle = \delta_{np} |m\rangle$.

In the Hilbert space of a tight binding chain with $N$ sites, let $|m\rangle\langle n|$ be the outer product operator associated with the localized basis states for sites $m, n \in \{0, \dots, N-1\}$. For any quantum states $\psi$ and $\phi$ in the Hilbert space, the complex inner product satisfies $$\langle (|m\rangle\langle n|) \psi, \phi \rangle = \langle \psi, (|n\rangle\langle m|) \phi \rangle.$$ Consequently, the adjoint of the operator $|m\rangle\langle n|$ is $|n\rangle\langle m|$.

For a tight binding chain $T$ with $N$ sites, on-site energy $E_0$, and hopping parameter $t$, the Hamiltonian $H$ is a linear operator acting on the $N$-dimensional complex Hilbert space $\mathcal{H} \cong \mathbb{C}^N$. It is defined as: $$H = E_0 \sum_{n=0}^{N-1} |n\rangle\langle n| - t \sum_{n=0}^{N-1} (|n\rangle\langle n+1| + |n+1\rangle\langle n|)$$ where $\{|n\rangle\}_{n=0}^{N-1}$ is the orthonormal basis of localized states at each site. The periodic boundary conditions are implemented by treating the site indices $n$ modulo $N$, such that the state $|N\rangle$ is identified with $|0\rangle$.

For a tight binding chain $T$ with associated Hilbert space $\mathcal{H}$, the Hamiltonian operator $H$ is Hermitian (self-adjoint). That is, for any two quantum states $\psi, \phi \in \mathcal{H}$, the complex inner product satisfies: $$\langle H\psi, \phi \rangle = \langle \psi, H\phi \rangle$$

For a tight binding chain $T$ with $N$ sites, on-site energy $E_0$, and hopping parameter $t$, let $\{|n\rangle\}_{n=0}^{N-1}$ be the orthonormal basis of localized states in the Hilbert space. For any site index $n \in \{0, \dots, N-1\}$, the action of the Hamiltonian $H$ on the localized state $|n\rangle$ is given by $$H |n\rangle = E_0 |n\rangle - t (|n+1\rangle + |n-1\rangle)$$ where the site indices $n+1$ and $n-1$ are taken modulo $N$ to account for periodic boundary conditions.

For a tight binding chain $T$ with $N$ sites and on-site energy $E_0$, let $\{|n\rangle\}_{n=0}^{N-1}$ be the orthonormal basis of localized states in the Hilbert space. If the number of sites $N$ is greater than 1, then for any site index $n \in \{0, \dots, N-1\}$, the expectation value of the Hamiltonian $H$ in the localized state $|n\rangle$ is equal to the on-site energy $E_0$: $$\langle n | H | n \rangle = E_0$$

For a tight binding chain with lattice spacing $a$, the Brillouin zone is defined as the half-open interval $[-\pi/a, \pi/a)$. This is the set of values for which wave functions are uniquely defined within the model.

For a tight binding chain with $N$ sites and lattice spacing $a$, the set of quantized wavenumbers $\mathcal{K}$ associated with the energy eigenstates is defined as: \[ \mathcal{K} = \left\{ \frac{2\pi}{aN} \left( n - \left\lfloor \frac{N}{2} \right\rfloor \right) \mid n \in \{0, 1, \dots, N-1\} \right\} \] where $n$ is an index ranging over the sites of the chain. These wavenumbers are defined such that they lie within the Brillouin zone of the system.

For a tight binding chain with $N$ sites and lattice spacing $a$, the set of quantized wavenumbers $\mathcal{K}$, defined as \[ \mathcal{K} = \left\{ \frac{2\pi}{aN} \left( n - \left\lfloor \frac{N}{2} \right\rfloor \right) \mid n \in \{0, 1, \dots, N-1\} \right\}, \] is a subset of the Brillouin zone $[-\pi/a, \pi/a)$.

For a tight binding chain with $N$ sites and lattice spacing $a$, let $\mathcal{K}$ be the set of quantized wavenumbers. For any natural number $n$ and any quantized wavenumber $k \in \mathcal{K}$, the complex exponential satisfies: \[ e^{i k n N a} = 1 \] where $i$ is the imaginary unit.

For a tight binding chain with $N$ sites and lattice spacing $a$, let $k$ be a quantized wavenumber and $n \in \{0, 1, \dots, N-1\}$ be a site index. The following identity holds: \[ e^{i k (n - 1 \pmod N) a} = e^{i k n a} \cdot e^{-i k a} \] where $i$ is the imaginary unit and the subtraction $n - 1$ is performed modulo $N$ to account for periodic boundary conditions.

For a tight binding chain with $N$ sites and lattice spacing $a$, let $k$ be a quantized wavenumber and $n \in \{0, 1, \dots, N-1\}$ be a site index. The following identity holds: \[ e^{i k (n + 1 \pmod N) a} = e^{i k n a} \cdot e^{i k a} \] where $i$ is the imaginary unit and the addition $n + 1$ is performed modulo $N$ to account for periodic boundary conditions.

For a tight binding chain $T$ with lattice spacing $a$ and $N$ sites, the energy eigenstate associated with a quantized wavenumber $k$ is defined as the vector in the Hilbert space given by the linear combination of localized states $\{|n\rangle\}_{n=0}^{N-1}$: \[ \sum_{n=0}^{N-1} e^{ikna} |n\rangle \] where $i$ is the imaginary unit and $n$ ranges over the indices of the lattice sites.

For a tight binding chain $T$ with $N$ sites and lattice spacing $a$, let $|k\rangle$ be the energy eigenstate associated with a quantized wavenumber $k \in \mathcal{K}$. For any two distinct quantized wavenumbers $k_1, k_2 \in \mathcal{K}$ such that $k_1 \neq k_2$, the corresponding energy eigenstates are orthogonal: \[ \langle k_1 | k_2 \rangle = 0 \] where $\langle \cdot | \cdot \rangle$ denotes the inner product in the $N$-dimensional complex Hilbert space $\mathbb{C}^N$. This orthogonality arises because distinct wavenumbers $k_1 \neq k_2$ give rise to different $N$-th roots of unity $\exp(i(k_2 - k_1)a)$, and the sum of all $N$-th roots of unity vanishes.

For a tight binding chain $T$ with lattice spacing $a$, on-site energy $E_0$, and hopping parameter $t$, the energy eigenvalue associated with a quantized wavenumber $k$ is given by: \[ E(k) = E_0 - 2t \cos(ka) \] This formula describes the dispersion relation for an electron in a one-dimensional lattice under the tight-binding approximation.

For a tight binding chain $T$, let $H$ be the Hamiltonian operator, $|k\rangle$ be the energy eigenstate associated with a quantized wavenumber $k$, and $E(k)$ be the corresponding energy eigenvalue. Then the energy eigenstates satisfy the time-independent Schrödinger equation: \[ H |k\rangle = E(k) |k\rangle \] where $H |k\rangle$ represents the action of the Hamiltonian on the state $|k\rangle$.