PhyslibAlpha.CondensedMatter.TightBindingChain.Saturation

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

For an open tight-binding chain with $N$ sites and lattice spacing $a$, let $|\psi_{\text{max}}\rangle$ be the maximal current state and $X$ be the position operator. For any site index $m \in \{0, \dots, N-1\}$, the inner product of the $m$-th localized state $|m\rangle$ with the position fluctuation vector $(X - \langle X \rangle) |\psi_{\text{max}}\rangle$ is given by: $$\langle m | (X - \langle X \rangle) |\psi_{\text{max}}\rangle = a \left( m - \frac{N-1}{2} \right) \langle m | \psi_{\text{max}} \rangle$$ where $\langle X \rangle = \langle \psi_{\text{max}} | X | \psi_{\text{max}} \rangle$ is the expectation value of the position in the maximal current state.

Consider a tight-binding chain with $N$ sites and hopping parameter $t$. If $N \ge 2$ and $t \neq 0$, the centered Gram defect of the open Hamiltonian $H_{\text{open}}$ and the position observable $X$ in the maximal current vector state $\omega_{\text{max}}$ is zero if and only if $N = 2$ or $N = 3$. That is, $$\text{centeredGramDefect}(\omega_{\text{max}}, H_{\text{open}}, X) = 0 \iff N = 2 \text{ or } N = 3.$$

Consider an open tight-binding chain $T$ with $N \ge 2$ sites, hopping parameter $t \neq 0$, and lattice spacing $a$. Let $\omega_{\text{max}}$ be the maximal current vector state, $H_{\text{open}}$ be the open Hamiltonian observable, and $X$ be the position observable. The Robertson–Schrödinger uncertainty relation saturates, satisfying the equality $$\text{Cov}_{\omega_{\text{max}}}(H_{\text{open}}, X)^2 + \left( a t \cos \frac{\pi}{N + 1} \right)^2 = \text{Var}_{\omega_{\text{max}}}(H_{\text{open}}) \cdot \text{Var}_{\omega_{\text{max}}}(X)$$ if and only if the number of sites $N$ is 2 or 3.

For an open tight-binding chain $T$ with $N$ sites, lattice spacing $a$, and a non-zero hopping parameter $t \neq 0$, if the number of sites $N \ge 4$, then the Robertson–Schrödinger uncertainty relation for the energy (Hamiltonian $H_{\text{open}}$) and position ($X$) in the maximal current state $\omega_{\text{max}}$ is a strict inequality: $$\text{Cov}_{\omega_{\text{max}}}(H_{\text{open}}, X)^2 + \left( a t \cos \frac{\pi}{N + 1} \right)^2 < \text{Var}_{\omega_{\text{max}}}(H_{\text{open}}) \cdot \text{Var}_{\omega_{\text{max}}}(X)$$ where $\text{Cov}_{\omega_{\text{max}}}$ and $\text{Var}_{\omega_{\text{max}}}$ denote the covariance and variance in the state $\omega_{\text{max}}$, respectively.