PhyslibAlpha.CondensedMatter.TightBindingChain.MaxCurrentState

In an open tight-binding chain with $N$ sites, lattice constant $a$, and hopping parameter $t \geq 0$, the current eigenvalue $I_k$ associated with the quantum number $k$ satisfies $I_k \leq I_1$ for all $1 \leq k \leq N$. Here, the current eigenvalue is defined as $I_k = 2at \cos\left(\frac{k\pi}{N+1}\right)$.

For a tight-binding chain $T$ with $N$ sites, let $H_{\text{open}}$ be the Hamiltonian with open boundary conditions, $t$ be the hopping parameter, and $E_0$ be the on-site energy. For any quantum number $k \in \mathbb{N}$ and any site index $m \in \{0, \dots, N-1\}$, the matrix element of the Hamiltonian between the localized state $|m\rangle$ and the current eigenstate $|\psi_k\rangle$ is given by: $$\langle m | H_{\text{open}} | \psi_k \rangle = E_0 \langle m | \psi_k \rangle - 2 t \sin \theta \cdot i^{m+1} \cos((m+1) \theta)$$ where $\theta = \frac{k \pi}{N + 1}$ and $i$ is the imaginary unit.

For an open tight-binding chain $T$ with $N$ sites, the maximal current state $|\psi_{\text{max}}\rangle$ is a normalized vector in the Hilbert space defined as the current eigenstate corresponding to the quantum number $k=1$, scaled by a normalization factor $\sqrt{2/(N+1)}$. Expressed in the basis of localized states $\{|n\rangle\}_{n=0}^{N-1}$, the state is given by: $$|\psi_{\text{max}}\rangle = \sqrt{\frac{2}{N+1}} \sum_{n=0}^{N-1} i^n \sin\left(\frac{(n+1) \pi}{N+1}\right) |n\rangle$$ where $i$ is the imaginary unit.

For an open tight-binding chain $T$, the norm of the maximal current state $|\psi_{\text{max}}\rangle$ is equal to 1: $$\|\psi_{\text{max}}\| = 1$$

For an open tight-binding chain with $N$ sites, let $|\psi_{\text{max}}\rangle$ denote the maximal current state and $|m\rangle$ be the localized state at site $m$ for $0 \le m < N$. The amplitude of the maximal current state at site $m$, given by the complex inner product $\langle m | \psi_{\text{max}} \rangle$, is $$\langle m | \psi_{\text{max}} \rangle = \sqrt{\frac{2}{N+1}} i^m \sin\left(\frac{(m+1) \pi}{N+1}\right)$$ where $i$ is the imaginary unit.

For a tight-binding chain $T$ with $N$ sites, hopping parameter $t$, and on-site energy $E_0$, let $|m\rangle$ be the localized basis state at site $m \in \{0, \dots, N-1\}$. The matrix element of the open boundary Hamiltonian $H_{\text{open}}$ acting on the maximal current state $|\psi_{\text{max}}\rangle$, projected onto the site $m$, is given by: $$\langle m | H_{\text{open}} | \psi_{\text{max}} \rangle = E_0 \langle m | \psi_{\text{max}} \rangle - \sqrt{\frac{2}{N+1}} \left( 2t \sin\left(\frac{\pi}{N+1}\right) i^{m+1} \cos\left(\frac{(m+1) \pi}{N+1}\right) \right)$$ where $i$ is the imaginary unit and $\langle m | \psi_{\text{max}} \rangle$ is the component of the maximal current state at site $m$.

For an open tight-binding chain with $N$ sites, lattice constant $a$, and hopping parameter $t$, the maximal current state $|\psi_{\text{max}}\rangle$ is an eigenstate of the current operator $J$ with an eigenvalue corresponding to $k=1$. Specifically, $$J |\psi_{\text{max}}\rangle = 2 a t \cos\left(\frac{\pi}{N + 1}\right) |\psi_{\text{max}}\rangle$$ where $J = i[H, X]$ is the current operator and $2 a t \cos\left(\frac{\pi}{N + 1}\right)$ is the current eigenvalue for $k=1$.

For an open tight-binding chain $T$ with Hilbert space $\mathcal{H}$, the maximal current vector state $\omega_{\text{max}}$ is the state on the algebra of continuous linear operators $\mathcal{B}(\mathcal{H})$ defined by the unit vector $|\psi_{\text{max}}\rangle$. This state maps any operator $A \in \mathcal{B}(\mathcal{H})$ to its expectation value in the maximal current state, $\omega_{\text{max}}(A) = \langle \psi_{\text{max}} | A | \psi_{\text{max}} \rangle$.

For an open tight-binding chain $T$ with Hilbert space $\mathcal{H}$ and maximal current state $|\psi_{\text{max}}\rangle$, the expectation value of any observable (self-adjoint operator) $A$ in the state $\omega_{\text{max}}$ is given by the real part of the diagonal matrix element $\langle \psi_{\text{max}} | A | \psi_{\text{max}} \rangle$: $$\omega_{\text{max}}\langle A \rangle = \text{Re}(\langle \psi_{\text{max}} | A | \psi_{\text{max}} \rangle)$$ where $\omega_{\text{max}}\langle A \rangle$ denotes the expectation value of the observable $A$ in the maximal current state.

For an open tight-binding chain $T$, the expectation value of the open Hamiltonian observable $H_{\text{open}}$ in the maximal current state $\omega_{\text{max}}$ is equal to the onsite energy $E_0$, expressed as: $$\omega_{\text{max}}\langle H_{\text{open}} \rangle = E_0$$

For an open tight-binding chain $T$ with $N$ sites and lattice spacing $a$, the expectation value of the position observable $X$ in the maximal current state $\omega_{\text{max}}$ is given by: $$\omega_{\text{max}}\langle X \rangle = \frac{a(N - 1)}{2}$$ This indicates that the maximal current state is centered at the physical midpoint of the chain.

For an open tight-binding chain with $N$ sites, lattice spacing $a$, and hopping parameter $t$, let $H$ be the open Hamiltonian observable and $X$ be the position operator observable. The expectation value of the bracket $\llbracket H, X \rrbracket$ (defined as $-\frac{i}{2}(HX - XH)$) in the maximal current state $\omega_{\text{max}}$ is given by: $$\omega_{\text{max}} \langle \llbracket H, X \rrbracket \rangle = -at \cos \left( \frac{\pi}{N + 1} \right)$$

For an open tight-binding chain with $N$ sites, lattice spacing $a$, and hopping parameter $t$, the Robertson–Schrödinger uncertainty relation for the Hamiltonian $H$ and the position operator $X$ in the maximal current state $\omega_{\text{max}}$ is given by: $$\text{Cov}_{\omega_{\text{max}}}(H, X)^2 + \left( a t \cos \left( \frac{\pi}{N + 1} \right) \right)^2 \le \text{Var}_{\omega_{\text{max}}}(H) \cdot \text{Var}_{\omega_{\text{max}}}(X)$$ where $\text{Var}_{\omega_{\text{max}}}(\cdot)$ and $\text{Cov}_{\omega_{\text{max}}}(\cdot, \cdot)$ denote the variance and covariance of the observables in the state $\omega_{\text{max}}$, respectively.