PhyslibAlpha.CondensedMatter.TightBindingChain.MandelstamTammMaxCurrent

For an open tight-binding chain $T$, the Mandelstam–Tamm ratio evaluated in the maximal current state $\omega_{\text{max}}$ is equal to $1/C_{\text{Nava}}^2$. Mathematically, $$\frac{\langle J \rangle_{\omega_{\text{max}}}^2}{4 \text{Var}_{\omega_{\text{max}}}(H) \cdot \text{Var}_{\omega_{\text{max}}}(X)} = \frac{1}{C_{\text{Nava}}^2}$$ where $\langle J \rangle_{\omega_{\text{max}}}$ is the expectation value of the current operator $J$, $\text{Var}_{\omega_{\text{max}}}(H)$ and $\text{Var}_{\omega_{\text{max}}}(X)$ are the variances of the Hamiltonian $H$ and the position operator $X$ in the state $\omega_{\text{max}}$, and $C_{\text{Nava}}$ is the uncertainty constant of the chain.

For an open tight-binding chain with $N \ge 2$ sites and a non-zero hopping parameter $t \neq 0$, the Mandelstam–Tamm ratio (representing the saturation of the Mandelstam–Tamm and quantum Cramér–Rao bounds) in the maximal current state $\omega_{\text{max}}$, defined as $$\frac{\langle J \rangle_{\omega_{\text{max}}}^2}{4 \text{Var}_{\omega_{\text{max}}}(H) \cdot \text{Var}_{\omega_{\text{max}}}(X)}$$ is equal to $1$ if and only if $N = 2$ or $N = 3$. Here, $\langle J \rangle$ is the expectation value of the current, and $\text{Var}(H)$ and $\text{Var}(X)$ are the variances of the Hamiltonian and position observables, respectively.

For a tight-binding chain $T$ with $N$ sites and a non-zero hopping parameter $t \neq 0$, if the number of sites $N$ is at least 4, then the Mandelstam–Tamm ratio of the maximal current state $\omega_{\text{max}}$ is strictly less than 1: $$\frac{\langle J \rangle_{\omega_{\text{max}}}^2}{4 \text{Var}_{\omega_{\text{max}}}(H) \cdot \text{Var}_{\omega_{\text{max}}}(X)} < 1.$$ Here, $\langle J \rangle_{\omega_{\text{max}}}$ is the expectation value of the current operator, and $\text{Var}_{\omega_{\text{max}}}(H)$ and $\text{Var}_{\omega_{\text{max}}}(X)$ are the variances of the Hamiltonian and position operators, respectively.

Consider a tight-binding cube constructed from three open chains $T_x$, $T_y$, and $T_z$. Let $\omega_{\text{max}}$ be the maximal current vector state of the cube. Let $J_x$, $H_x$, and $X_x$ be the current, Hamiltonian, and position observables of the chain $T_x$, respectively, embedded to act along the $x$-axis of the cube. The ratio of the squared expectation value of the current to the product of the variances of the Hamiltonian and position (the Mandelstam–Tamm ratio) along the $x$-axis is given by: $$\frac{\langle J_x \rangle_{\omega_{\text{max}}}^2}{4 \text{Var}_{\omega_{\text{max}}}(H_x) \text{Var}_{\omega_{\text{max}}}(X_x)} = \frac{1}{C_{\text{Nava}}^2}$$ where $C_{\text{Nava}}$ is the uncertainty constant of the tight-binding chain $T_x$.

Consider a 3D tight-binding cube formed by the combination of three open chains $T_x, T_y$, and $T_z$. Let $\omega_{\text{max,cube}}$ be the maximal current vector state of this cube. Let $J_y$, $H_y$, and $X_y$ denote the current operator, the open Hamiltonian, and the position operator of the chain $T_y$, respectively, each acting along the $y$-axis of the cube (i.e., as $I_x \otimes A_y \otimes I_z$ on the cube's Hilbert space). Then the Mandelstam–Tamm ratio for these observables in the state $\omega_{\text{max,cube}}$ satisfies: $$\frac{\langle J_y \rangle_{\omega_{\text{max,cube}}}^2}{4 \text{Var}_{\omega_{\text{max,cube}}}(H_y) \cdot \text{Var}_{\omega_{\text{max,cube}}}(X_y)} = \frac{1}{C_{\text{Nava}, y}^2}$$ where $C_{\text{Nava}, y}$ is the uncertainty constant associated with the open chain $T_y$.

Let $T_x, T_y$, and $T_z$ be three open tight-binding chains forming a 3D tight-binding cube. In the maximal current vector state $\omega$ of the cube, the Mandelstam–Tamm ratio along the $z$-axis—defined by the expectation value of the $z$-axis current observable $J_z$ and the variances of the $z$-axis Hamiltonian $H_z$ and position operator $X_z$—is given by: $$\frac{\omega\langle J_z \rangle^2}{4 \cdot \text{Var}_{\omega}(H_z) \cdot \text{Var}_{\omega}(X_z)} = \frac{1}{C_{\text{Nava}, z}^2}$$ where $C_{\text{Nava}, z}$ is the uncertainty constant $C_{\text{Nava}}$ associated with the chain $T_z$.