PhyslibAlpha.CondensedMatter.TightBindingChain.Cube

Given three open tight-binding chains $T_x, T_y$, and $T_z$ with $N_x, N_y$, and $N_z$ sites respectively, the Hilbert space of the cube is the finite-dimensional complex Hilbert space $\mathcal{H}$ whose basis is indexed by the Cartesian product of the sites of each chain, denoted as $\text{Fin } N_x \times \text{Fin } N_y \times \text{Fin } N_z$. This space represents the quantum states of a system defined on the 3D grid of sites $(n_x, n_y, n_z)$.

Given three open tight-binding chains $T_x$, $T_y$, and $T_z$, `CubeOperators` is the space of continuous linear operators acting on the Hilbert space of the cube $\mathcal{H}_{\text{cube}}$. Mathematically, it is the space of linear maps $\mathcal{L}(\mathcal{H}_{\text{cube}})$ from the Hilbert space to itself over the complex numbers $\mathbb{C}$, where $\mathcal{H}_{\text{cube}}$ is the finite-dimensional Hilbert space representing the 3D grid of sites.

Given an observable $a$ (a self-adjoint operator) acting on the Hilbert space $\mathcal{H}_x$ of a tight-binding chain $T_x$, `alongX a` defines a corresponding observable on the Hilbert space of the cube $\mathcal{H}_{\text{cube}} \cong \mathcal{H}_x \otimes \mathcal{H}_y \otimes \mathcal{H}_z$. This observable acts as $a$ on the $x$-axis and as the identity operator on the $y$ and $z$ axes, representing the tensor product operator $a \otimes I$.

Given an observable $b$ (a self-adjoint linear operator) acting on the Hilbert space $\mathcal{H}_y$ of a tight-binding chain $T_y$, `alongY b` is the corresponding observable on the cube Hilbert space $\mathcal{H}_{\text{cube}} \cong \mathcal{H}_x \otimes \mathcal{H}_y \otimes \mathcal{H}_z$. It represents the operator $I_x \otimes b \otimes I_z$, which acts as $b$ on the $y$-axis and as the identity operator on the $x$ and $z$ axes.

Given an observable $c$ acting on the Hilbert space $\mathcal{H}_z$ of a tight-binding chain along the $z$-axis, this definition constructs the corresponding observable on the 3D cube Hilbert space $\mathcal{H}_x \otimes \mathcal{H}_y \otimes \mathcal{H}_z$. Specifically, it represents the operator $I_x \otimes I_y \otimes c$, where $I_x$ and $I_y$ are the identity operators on the $x$ and $y$ axes, respectively.

In a tight-binding cube formed by three chains $T_x$, $T_y$, and $T_z$, let $a$ be an observable (a self-adjoint operator) acting on the Hilbert space $\mathcal{H}_x$ of the $x$-axis chain, and let $b$ be an observable acting on the Hilbert space $\mathcal{H}_y$ of the $y$-axis chain. Let $\text{alongX}(a)$ and $\text{alongY}(b)$ be the corresponding lifted observables acting on the cube's Hilbert space $\mathcal{H}_{\text{cube}} \cong \mathcal{H}_x \otimes \mathcal{H}_y \otimes \mathcal{H}_z$. Then the Lie bracket of these observables is zero: $$\llbracket \text{alongX}(a), \text{alongY}(b) \rrbracket = 0,$$ where the Lie bracket for self-adjoint operators is defined as $\llbracket A, B \rrbracket = -\frac{i}{2}(AB - BA)$.

Let $T_x, T_y,$ and $T_z$ be open tight-binding chains with corresponding Hilbert spaces $\mathcal{H}_x, \mathcal{H}_y,$ and $\mathcal{H}_z$. For any observable (self-adjoint operator) $a$ acting on $\mathcal{H}_x$ and any observable $c$ acting on $\mathcal{H}_z$, their representations on the cube Hilbert space $\mathcal{H}_x \otimes \mathcal{H}_y \otimes \mathcal{H}_z$, denoted by $\text{alongX}(a)$ and $\text{alongZ}(c)$, commute such that their Lie bracket is zero: $$\llbracket \text{alongX}(a), \text{alongZ}(c) \rrbracket = 0$$ where the Lie bracket is defined as $\llbracket \hat{A}, \hat{B} \rrbracket = -\frac{i}{2}(\hat{A}\hat{B} - \hat{B}\hat{A})$.

Let $T_x, T_y, T_z$ be three open tight-binding chains forming a 3D cube with the Hilbert space $\mathcal{H}_{\text{cube}} \cong \mathcal{H}_x \otimes \mathcal{H}_y \otimes \mathcal{H}_z$. For any observable $b$ acting on the Hilbert space $\mathcal{H}_y$ and any observable $c$ acting on the Hilbert space $\mathcal{H}_z$, let $b_y = I_x \otimes b \otimes I_z$ and $c_z = I_x \otimes I_y \otimes c$ be the corresponding observables acting on the cube. Then their Lie bracket vanishes: $$\llbracket b_y, c_z \rrbracket = 0$$ where the bracket is defined as $\llbracket A, B \rrbracket = -\frac{i}{2}(AB - BA)$.

Given three open tight-binding chains $T_x, T_y$, and $T_z$ with $N_x, N_y$, and $N_z$ sites respectively, the maximal current state of the cube $|\Psi_{\text{max}}\rangle$ is the product vector of the 1D maximal current states $|\psi_{\text{max}}\rangle$ of each individual chain. Specifically, $|\Psi_{\text{max}}\rangle = |\psi_{\text{max}, x}\rangle \otimes (|\psi_{\text{max}, y}\rangle \otimes |\psi_{\text{max}, z}\rangle)$, which resides in the cube Hilbert space $\mathcal{H} \cong \mathbb{C}^{N_x \times N_y \times N_z}$.

Given three open tight-binding chains $T_x$, $T_y$, and $T_z$, the norm of the maximal current state of the cube $|\Psi_{\text{max}}\rangle$ is equal to 1, i.e., $\||\Psi_{\text{max}}\rangle\| = 1$.

Given three open tight-binding chains $T_x, T_y$, and $T_z$, the maximal current vector state is the quantum state (unital positive linear map) on the algebra of cube operators $\mathcal{L}(\mathcal{H}_{\text{cube}})$ defined by the normalized maximal current state vector $|\Psi_{\text{max}}\rangle$. For any operator $A \in \mathcal{L}(\mathcal{H}_{\text{cube}})$, the state maps $A$ to the expectation value $\langle \Psi_{\text{max}}, A \Psi_{\text{max}} \rangle$.

Consider three open tight-binding chains $T_x$, $T_y$, and $T_z$. Let $\omega_{\text{cube}}$ be the maximal current vector state of the 3D cube formed by these chains, and let $\omega_x$ be the maximal current vector state of the individual chain $T_x$. For any observable $a$ acting on the Hilbert space of $T_x$, let $a_x$ be the corresponding observable acting along the $x$-axis of the cube (representing $a \otimes I_y \otimes I_z$). The variance of $a_x$ in the cube state $\omega_{\text{cube}}$ is equal to the variance of $a$ in the chain state $\omega_x$: $$\text{Var}_{\omega_{\text{cube}}}(a_x) = \text{Var}_{\omega_x}(a)$$

For three open tight-binding chains $T_x$, $T_y$, and $T_z$, let $\omega_{\text{cube}}$ be the maximal current vector state on the Hilbert space of the cube $\mathcal{H}_{\text{cube}}$, and let $\omega_x$ be the maximal current vector state on the Hilbert space $\mathcal{H}_x$ of the chain $T_x$. For any two observables $a, b$ acting on $\mathcal{H}_x$, let $a_x$ and $b_x$ be the corresponding observables acting along the $x$-axis of the cube (defined by the operators $a \otimes I_y \otimes I_z$ and $b \otimes I_y \otimes I_z$ respectively). The centered Gram defect of $a_x$ and $b_x$ with respect to the state $\omega_{\text{cube}}$ is equal to the centered Gram defect of $a$ and $b$ with respect to the state $\omega_x$: $$\text{centeredGramDefect}(\omega_{\text{cube}}, a_x, b_x) = \text{centeredGramDefect}(\omega_x, a, b)$$ where the centered Gram defect for a state $\omega$ and observables $u, v$ is defined as $\text{Var}_\omega(u) \cdot \text{Var}_\omega(v) - |\omega(\Delta u \Delta v)|^2$.

Let $T_x, T_y$, and $T_z$ be three open tight-binding chains. Let $\omega_{\text{cube, max}}$ be the maximal current state on the Hilbert space of the cube $\mathcal{H}_x \otimes \mathcal{H}_y \otimes \mathcal{H}_z$, and let $\omega_{y, \text{max}}$ be the maximal current state on the Hilbert space $\mathcal{H}_y$ of the chain $T_y$. For any observable $b'$ acting on the chain $T_y$, let $b'_y$ denote the corresponding observable acting along the $y$-axis of the cube (representing $I_x \otimes b' \otimes I_z$). Then the variance of $b'_y$ in the state $\omega_{\text{cube, max}}$ is equal to the variance of $b'$ in the state $\omega_{y, \text{max}}$: $$\text{Var}_{\omega_{\text{cube, max}}}(b'_y) = \text{Var}_{\omega_{y, \text{max}}}(b')$$

Consider three open tight-binding chains $T_x, T_y$, and $T_z$. Let $\Omega_{\text{max}}$ be the maximal current vector state of the tight-binding cube, and let $\omega_{\text{max}, y}$ be the maximal current vector state of the $y$-axis chain $T_y$. For any two observables $b'$ and $c'$ (self-adjoint linear operators) acting on the Hilbert space of $T_y$, let $B = I_x \otimes b' \otimes I_z$ and $C = I_x \otimes c' \otimes I_z$ be the corresponding lifted observables acting on the cube's Hilbert space. Then the centered Gram defect of $B$ and $C$ with respect to $\Omega_{\text{max}}$ is equal to the centered Gram defect of $b'$ and $c'$ with respect to $\omega_{\text{max}, y}$: $$\text{GramDefect}_{\Omega_{\text{max}}}(B, C) = \text{GramDefect}_{\omega_{\text{max}, y}}(b', c')$$ where the centered Gram defect for a state $\omega$ and observables $a, b$ is defined as the remainder in the Robertson–Schrödinger uncertainty relation: $\text{Var}_\omega(a) \cdot \text{Var}_\omega(b) - |\omega((a - \omega\langle a \rangle 1)(b - \omega\langle b \rangle 1))|^2$.

For three open tight-binding chains $T_x, T_y$, and $T_z$, let $\Omega$ be the maximal current vector state of the 3D cube and $\omega_z$ be the maximal current vector state of the $z$-axis chain $T_z$. For any observable $b''$ acting on the Hilbert space of $T_z$, let $b''_z$ be the corresponding observable acting along the $z$-axis of the cube (defined as $I_x \otimes I_y \otimes b''$). Then the variance of $b''_z$ in the state $\Omega$ is equal to the variance of $b''$ in the state $\omega_z$: $$\text{Var}_{\Omega}(b''_z) = \text{Var}_{\omega_z}(b'')$$

Let $T_x, T_y$, and $T_z$ be three open tight-binding chains. Let $\omega_{\text{cube}}$ be the maximal current vector state of the tight-binding cube, and let $\omega_z$ be the maximal current vector state of the chain $T_z$. For any two observables $b, c$ acting on the Hilbert space of the $z$-axis chain, let $B$ and $C$ be the corresponding observables acting on the cube's Hilbert space along the $z$-axis (representing $I_x \otimes I_y \otimes b$ and $I_x \otimes I_y \otimes c$ respectively). The centered Gram defect of $B$ and $C$ in the state $\omega_{\text{cube}}$ is equal to the centered Gram defect of $b$ and $c$ in the state $\omega_z$: $$\text{centeredGramDefect}(\omega_{\text{cube}}, B, C) = \text{centeredGramDefect}(\omega_z, b, c)$$ where the centered Gram defect for two observables $x, y$ in a state $\omega$ is defined as $\text{Var}_\omega(x) \cdot \text{Var}_\omega(y) - |\omega((x - \omega(x)1)(y - \omega(y)1))|^2$.

Consider a 3D tight-binding cube formed by three open chains $T_x, T_y$, and $T_z$ with $N_x, N_y$, and $N_z$ sites respectively. Let $\omega_{\text{cube}}$ be the maximal current state of the cube. Let $H_x$ be the Hamiltonian of the $x$-axis chain acting on the cube (i.e., $H_{\text{open}} \otimes I \otimes I$) and $X_x$ be the position operator of the $x$-axis chain acting on the cube (i.e., $X \otimes I \otimes I$). The following relationship holds: $$\frac{\sqrt{\text{Var}_{\omega_{\text{cube}}}(H_x) \cdot \text{Var}_{\omega_{\text{cube}}}(X_x)}}{\left| a_x t_x \cos\left( \frac{\pi}{N_x + 1} \right) \right|} = C_{\text{Nava}}(T_x)$$ where $a_x$ is the lattice spacing and $t_x$ is the hopping parameter of the chain $T_x$, and $C_{\text{Nava}}(T_x)$ is the uncertainty constant inherent to the standalone $x$-axis chain.

Consider a tight-binding cube formed by three open chains $T_x, T_y$, and $T_z$. Let $\omega_{\text{cube}}$ be the maximal current vector state of the cube. Let $H_{y, \text{cube}}$ and $X_{y, \text{cube}}$ denote the open Hamiltonian and the position operator of the $y$-axis chain acting on the cube's Hilbert space (represented by `alongY Ty.openHamiltonianObservable` and `alongY Ty.positionObservable`). The following relationship holds: $$\frac{\sqrt{\text{Var}_{\omega_{\text{cube}}}(H_{y, \text{cube}}) \cdot \text{Var}_{\omega_{\text{cube}}}(X_{y, \text{cube}})}}{\left| a_y t_y \cos\left( \frac{\pi}{N_y + 1} \right) \right|} = C_{\text{Nava}}(T_y)$$ where $a_y$ is the lattice spacing, $t_y$ is the hopping parameter, $N_y$ is the number of sites of the chain $T_y$, and $C_{\text{Nava}}(T_y)$ is the uncertainty constant of the $y$-axis chain.

For a system of three open tight-binding chains $T_x, T_y$, and $T_z$ forming a 3D cube, let $\omega_{\text{cube}}$ be the maximal current state of the cube. Let $H_z$ and $X_z$ be the observables corresponding to the open Hamiltonian and the position operator of the $z$-axis chain acting on the cube's Hilbert space. Then the uncertainty relation for the $z$-axis satisfies: $$\frac{\sqrt{\text{Var}_{\omega_{\text{cube}}}(H_z) \cdot \text{Var}_{\omega_{\text{cube}}}(X_z)}}{\left| a_z t_z \cos\left( \frac{\pi}{N_z+1} \right) \right|} = C_{\text{Nava}, z}$$ where $a_z$ is the lattice spacing, $t_z$ is the hopping parameter, $N_z$ is the number of sites, and $C_{\text{Nava}, z}$ is the uncertainty constant of the independent $z$-axis chain.

Consider three open tight-binding chains $T_x, T_y,$ and $T_z$ forming a cubic lattice. Let $N_x$ be the number of sites and $t$ be the hopping parameter for the chain along the $x$-axis. Suppose $t \neq 0$ and $N_x \geq 2$. Let $\omega$ be the maximal current vector state of the cube, and let $H_x$ and $X_x$ be the Hamiltonian and position observables acting along the $x$-axis, respectively. The centered Gram defect between $H_x$ and $X_x$ in state $\omega$ vanishes (meaning the Robertson–Schrödinger uncertainty relation becomes an equality) if and only if $N_x = 2$ or $N_x = 3$.

Consider a tight-binding cube formed by three chains $T_x, T_y$, and $T_z$. Let $N_y$ be the number of sites and $t_y$ be the hopping parameter of the $y$-axis chain $T_y$. Let $\omega_{\text{cube}}$ be the maximal current vector state of the cube. Let $H_y$ and $X_y$ be the open Hamiltonian and the position operator acting along the $y$-axis of the cube, respectively. For $t_y \neq 0$ and $N_y \geq 2$, the centered Gram defect of $H_y$ and $X_y$ with respect to the state $\omega_{\text{cube}}$ is zero if and only if $N_y = 2$ or $N_y = 3$: $$\text{centeredGramDefect}_{\omega_{\text{cube}}}(H_y, X_y) = 0 \iff N_y = 2 \text{ or } N_y = 3$$

Let $T_x$, $T_y$, and $T_z$ be three open tight-binding chains forming a 3D cube. Let $N_z$ be the number of sites and $t_z$ be the hopping parameter of the $z$-axis chain. Suppose $t_z \neq 0$ and $N_z \geq 2$. Let $\omega$ be the maximal current vector state of the cube, and let $H_z$ and $X_z$ be the open Hamiltonian and position observables acting along the $z$-axis, respectively. Then, the centered Gram defect of $H_z$ and $X_z$ in the state $\omega$ is zero if and only if $N_z$ is 2 or 3: $$\text{Var}_\omega(H_z) \cdot \text{Var}_\omega(X_z) - |\omega(\Delta H_z \Delta X_z)|^2 = 0 \iff N_z = 2 \text{ or } N_z = 3$$ where $\Delta H_z = H_z - \omega(H_z)I$ and $\Delta X_z = X_z - \omega(X_z)I$ are the centered observables.

Consider a three-dimensional open tight-binding cube formed by three chains $T_x, T_y$, and $T_z$ with non-zero hopping parameters $t_x, t_y, t_z$ and at least 4 sites along each axis ($N_x, N_y, N_z \geq 4$). Let $\omega$ be the maximal current state of the cube. For each axis $i \in \{x, y, z\}$, let $H_i$ be the open Hamiltonian observable and $X_i$ be the position observable acting along that axis. Then the centered Gram defect $\mathcal{G}$ for each pair of observables is strictly positive: $$\mathcal{G}(\omega, H_x, X_x) > 0 \quad \text{and} \quad \mathcal{G}(\omega, H_y, X_y) > 0 \quad \text{and} \quad \mathcal{G}(\omega, H_z, X_z) > 0$$ where the centered Gram defect of two observables $a, b$ in state $\omega$ is defined as $\text{Var}_\omega(a) \cdot \text{Var}_\omega(b) - |\omega(\Delta a \Delta b)|^2$, representing the "slack" in the Robertson–Schrödinger uncertainty relation.

For a tight-binding chain $T$ with associated Hilbert space $\mathcal{H} \cong \mathbb{C}^N$, the **conjugate pair** is a function mapping the index set $\{0, 1\}$ to the space of observables (self-adjoint operators) in $\mathcal{L}(\mathcal{H})$. This pair consists of the open Hamiltonian $H$ and the position operator $X$ of the chain.

This function maps a pair of indices $(j, k) \in \{0, 1\} \times \{0, 1, 2\}$ to an observable (a self-adjoint operator) acting on the Hilbert space of the tight-binding cube $\mathcal{H}_{\text{cube}}$. The index $k$ specifies the axis of the cube ($x, y,$ or $z$), and the index $j$ selects the corresponding member of the conjugate pair $(H, X)$—the Hamiltonian and the position operator—of that axis. Each resulting observable acts on its specific axis while acting as the identity operator on the other two axes.

Given a quantum state $\omega$ on the algebra of operators of a tight-binding cube (formed by three chains $T_x, T_y$, and $T_z$) and an axis index $k \in \{0, 1, 2\}$, this definition calculates the expectation value $\omega\langle \llbracket H_k, X_k \rrbracket \rangle$ of the Lie bracket of the Hamiltonian $H_k$ and position operator $X_k$ corresponding to the $k$-th axis. The bracket is defined as $\llbracket A, B \rrbracket = -\frac{i}{2}(AB - BA)$, which ensures that the commutator of two self-adjoint operators results in another self-adjoint operator (an observable).

For three open tight-binding chains $T_x, T_y,$ and $T_z$ forming a cube, and for each index $j \in \{0, 1\}$ representing either the Hamiltonian ($j=0$) or the position operator ($j=1$), the $j$-th member of the conjugate pair for the $x$-axis of the cube is equal to the $j$-th conjugate pair of the chain $T_x$ acting along the $x$-axis of the cube. Mathematically, for $j \in \{0, 1\}$: $$\text{cubePairs}(j, 0) = \text{alongX}(T_x.\text{conjugatePair}(j))$$ where the axis index $0$ denotes the $x$-axis, and $\text{alongX}$ lifts an operator from the $x$-chain's Hilbert space to the cube's Hilbert space by acting as the identity on the $y$ and $z$ axes.

For three open tight-binding chains $T_x, T_y$, and $T_z$, let $\text{cubePairs}(j, k)$ be the $j$-th observable in the conjugate pair (where $j=0$ denotes the Hamiltonian and $j=1$ denotes the position operator) for the $k$-th axis of the cube. For $j \in \{0, 1\}$, the $j$-th conjugate operator associated with the $y$-axis ($k=1$) is the $j$-th conjugate operator of the chain $T_y$ acting along the $y$-direction of the cube: $$\text{cubePairs}(j, 1) = \text{alongY}(\text{conjugatePair}_{T_y}(j))$$ where $\text{alongY}$ is the operator that acts as its argument on the $y$-axis Hilbert space and as the identity on the $x$ and $z$ axes.

For a tight-binding cube composed of three chains $T_x, T_y$, and $T_z$, let $\text{cubePairs}(j, k)$ be the operator on the cube's Hilbert space corresponding to the $j$-th conjugate variable (where $j=0$ is the Hamiltonian and $j=1$ is the position operator) along the $k$-th axis (where $k=2$ denotes the $z$-axis). For any $j \in \{0, 1\}$, it holds that $$\text{cubePairs}(j, 2) = \text{alongZ}(\text{conjugatePair}_{T_z}(j))$$ where $\text{conjugatePair}_{T_z}(j)$ is the $j$-th conjugate operator of the individual chain $T_z$, and $\text{alongZ}$ is the operation that embeds an operator $c$ acting on the $z$-axis Hilbert space into the full cube Hilbert space $\mathcal{H}_x \otimes \mathcal{H}_y \otimes \mathcal{H}_z$ as the tensor product $I_x \otimes I_y \otimes c$.

Let $T_x, T_y$, and $T_z$ be three open tight-binding chains. Let $p$ and $q$ be indices in $\{0, 1\} \times \{0, 1, 2\}$ that specify an observable type (Hamiltonian $H$ or position $X$) and an axis of the cube ($x, y$, or $z$). If $p$ and $q$ refer to different axes (i.e., $p_2 \neq q_2$), then the Lie bracket of the corresponding observables on the cube's Hilbert space is zero: $$\llbracket \text{cubePairs}(p), \text{cubePairs}(q) \rrbracket = 0$$ This implies that observables associated with different spatial axes commute.

For any quantum state $\omega$ on the operator algebra of a tight-binding cube (composed of three chains $T_x, T_y, T_z$), the $6 \times 6$ bracket matrix $\Omega$ of the conjugate pairs of observables $(H_k, X_k)$ for each axis $k \in \{x, y, z\}$ is block diagonal. Specifically, the matrix consists of three $2 \times 2$ antisymmetric blocks: $$\Omega = \text{diag}\left( \begin{pmatrix} 0 & \omega\langle \llbracket H_x, X_x \rrbracket \rangle \\ -\omega\langle \llbracket H_x, X_x \rrbracket \rangle & 0 \end{pmatrix}, \begin{pmatrix} 0 & \omega\langle \llbracket H_y, X_y \rrbracket \rangle \\ -\omega\langle \llbracket H_y, X_y \rrbracket \rangle & 0 \end{pmatrix}, \begin{pmatrix} 0 & \omega\langle \llbracket H_z, X_z \rrbracket \rangle \\ -\omega\langle \llbracket H_z, X_z \rrbracket \rangle & 0 \end{pmatrix} \right)$$ where $\llbracket A, B \rrbracket = -\frac{i}{2}(AB - BA)$ is the Lie bracket, and $\omega\langle \llbracket H_k, X_k \rrbracket \rangle$ is the expectation value of the bracket of the Hamiltonian and position operator along axis $k$.

Consider a quantum system on a tight-binding cube formed by three chains $T_x, T_y$, and $T_z$. For any quantum state $\omega$, let $H_k$ and $X_k$ be the Hamiltonian and position observables corresponding to the $k$-th axis for $k \in \{0, 1, 2\}$. Let $\Sigma$ be the $6 \times 6$ covariance matrix of these six observables in the state $\omega$. Then the determinant of the covariance matrix is bounded below by the squared product of the expectation values of the brackets $\llbracket H_k, X_k \rrbracket = -\frac{i}{2}(H_k X_k - X_k H_k)$: $$\left( \prod_{k=0}^2 \omega\langle \llbracket H_k, X_k \rrbracket \rangle \right)^2 \le \det \Sigma$$

Consider three open tight-binding chains $T_x, T_y$, and $T_z$. Let $\Omega_{\text{max}}$ be the maximal current vector state of the tight-binding cube, and let $\omega_{\text{max}}^x$ be the maximal current vector state of the chain $T_x$. For any two observables $a$ and $b$ acting on the Hilbert space of $T_x$, let $a_X$ and $b_X$ be the corresponding observables acting along the $x$-axis of the cube (i.e., acting as $a$ and $b$ on the $x$-axis and as the identity on the $y$ and $z$ axes). Then the covariance of $a_X$ and $b_X$ in the state $\Omega_{\text{max}}$ is equal to the covariance of $a$ and $b$ in the state $\omega_{\text{max}}^x$: $$\text{Cov}_{\Omega_{\text{max}}}(a_X, b_X) = \text{Cov}_{\omega_{\text{max}}^x}(a, b)$$

Consider a system of three open tight-binding chains $T_x, T_y$, and $T_z$ forming a 3D cube. Let $\Omega_{\text{max}}$ be the maximal current vector state of the cube, and let $\omega_y$ be the maximal current vector state of the individual chain $T_y$. For any two observables $b'$ and $c'$ acting on the Hilbert space of the chain $T_y$, let $b'^{(y)}$ and $c'^{(y)}$ be the corresponding observables acting along the $y$-axis of the cube (defined as $I_x \otimes b' \otimes I_z$ and $I_x \otimes c' \otimes I_z$). Then the covariance of these $y$-axis observables in the cube state $\Omega_{\text{max}}$ is equal to their covariance in the chain state $\omega_y$: $$\text{Cov}_{\Omega_{\text{max}}}(b'^{(y)}, c'^{(y)}) = \text{Cov}_{\omega_y}(b', c')$$

Let $T_x, T_y$, and $T_z$ be three open tight-binding chains. Let $\Omega_{\text{cube}}$ be the maximal current vector state of the 3D cube formed by these chains. For any two observables $b''$ and $c''$ acting on the Hilbert space of the $z$-axis chain $T_z$, let $b''_{\text{alongZ}}$ and $c''_{\text{alongZ}}$ denote their corresponding lifted observables on the cube's Hilbert space (representing $I_x \otimes I_y \otimes b''$ and $I_x \otimes I_y \otimes c''$ respectively). Then the covariance of these observables in the cube state $\Omega_{\text{cube}}$ is equal to the covariance of the original observables in the 1D maximal current vector state $\omega_{z}$ of the $z$-axis chain: $$\text{Cov}_{\Omega_{\text{cube}}}(b''_{\text{alongZ}}, c''_{\text{alongZ}}) = \text{Cov}_{\omega_{z}}(b'', c'')$$