PhyslibAlpha.CondensedMatter.TightBindingChain.SpeedLimit

For an open tight-binding chain with $N$ sites, lattice constant $a$, and hopping parameter $t$, the maximum current $J_{\text{max}}$ is defined as: $$J_{\text{max}} = 2 |at| \cos \left( \frac{\pi}{N + 1} \right)$$

For an open tight-binding chain with $N$ sites, lattice constant $a$, and hopping parameter $t$, let $J_k = 2 a t \cos\left(\frac{k \pi}{N + 1}\right)$ be the current eigenvalue for $1 \leq k \leq N$. Then the absolute value of $J_k$ is less than or equal to the maximum current $J_{\text{max}} = 2 |at| \cos \left( \frac{\pi}{N + 1} \right)$, that is: $$|J_k| \le J_{\text{max}}$$

For an open tight-binding chain $T$ with $N$ sites, the $k$-th normalized current basis vector (where $k \in \{0, \dots, N-1\}$) is defined as: $$\sqrt{\frac{2}{N+1}} |\psi_{k+1}\rangle$$ where $|\psi_{k+1}\rangle$ is the unnormalized current eigenstate associated with the quantum number $k+1$. These vectors form an orthonormal basis for the Hilbert space of the chain.

For an open tight-binding chain $T$ with $N$ sites, the $k$-th normalized current basis vector $|\psi'_k\rangle$ (where $k \in \{0, \dots, N-1\}$) has unit norm, i.e., $$\||\psi'_k\rangle\| = 1$$

For an open tight-binding chain $T$ with $N$ sites and a non-zero hopping parameter $t \neq 0$, the set of normalized current eigenstates $\{|\psi'_k\rangle\}_{k=0}^{N-1}$ forms an orthonormal system in the Hilbert space $\mathbb{C}^N$. That is, the complex inner product of any two such vectors satisfies $\langle \psi'_k | \psi'_l \rangle = \delta_{kl}$, where $\delta_{kl}$ is the Kronecker delta.

For an open tight-binding chain $T$ with $N$ sites and a non-zero hopping parameter $t \neq 0$, `currentBasis` is the orthonormal basis $\{|\psi'_k\rangle\}_{k=0}^{N-1}$ of the $N$-dimensional complex Hilbert space $\mathcal{H} \cong \mathbb{C}^N$. This basis consists of the normalized current eigenstates $|\psi'_k\rangle$ (where $k \in \{0, \dots, N-1\}$).

For an open tight-binding chain $T$ with $N$ sites and a non-zero hopping parameter $t \neq 0$, let $\{|\psi'_k\rangle\}_{k=0}^{N-1}$ be the orthonormal basis of current eigenstates. For each $k \in \{0, \dots, N-1\}$, the current operator $J$ acts on the $k$-th basis vector $|\psi'_k\rangle$ according to the eigenvalue equation: $$J |\psi'_k\rangle = \lambda_{k+1} |\psi'_k\rangle$$ where $\lambda_{k+1} = 2 a t \cos\left(\frac{(k+1) \pi}{N + 1}\right)$ is the current eigenvalue associated with the quantum number $k+1$.

For an open tight-binding chain with $N$ sites and a non-zero hopping parameter $t \neq 0$, let $J$ be the current operator and $\{|\psi'_k\rangle\}_{k=1}^N$ be the orthonormal basis of current eigenstates with corresponding eigenvalues $\lambda_k = 2at \cos\left(\frac{k\pi}{N+1}\right)$. For any vector $\psi$ in the Hilbert space, the real part of the expectation value of the current is given by the sum of the current eigenvalues weighted by the squared magnitudes of the projections of $\psi$ onto the respective basis vectors: $$\text{Re} \langle \psi, J \psi \rangle = \sum_{k=1}^N \lambda_k |\langle \psi'_k, \psi \rangle|^2$$ where $\langle \cdot, \cdot \rangle$ denotes the complex inner product.

For an open tight-binding chain with a non-zero hopping parameter $t$, let $\mathcal{H}$ be its Hilbert space and $J$ be the current operator. For any vector $\psi \in \mathcal{H}$, the absolute value of the real part of the inner product $\langle \psi, J \psi \rangle$ is bounded by the maximum current $J_{\text{max}}$ scaled by the squared norm of $\psi$: $$|\text{Re} \langle \psi, J \psi \rangle| \leq J_{\text{max}} \|\psi\|^2$$ where $J_{\text{max}} = 2 |at| \cos \left( \frac{\pi}{N + 1} \right)$ is the maximum possible current for the chain.

For an open tight-binding chain with $N$ sites, lattice constant $a$, and non-zero hopping parameter $t \neq 0$, let $J$ be the current operator. For any state $\omega$ of the system (defined as a unital positive linear map on the algebra of operators), the absolute value of the expectation value of the current $\omega\langle J \rangle$ is bounded by the maximum current $J_{\text{max}}$: $$|\omega\langle J \rangle| \leq J_{\text{max}}$$ where the "speed limit" is given by $J_{\text{max}} = 2 |at| \cos \left( \frac{\pi}{N + 1} \right)$.

For an open tight-binding chain with $N$ sites, lattice constant $a$, and hopping parameter $t$, let $J$ be the current observable and $\omega_{\text{max}}$ be the maximal current vector state. The expectation value of the current in this state is: $$\omega_{\text{max}}\langle J \rangle = 2 a t \cos \left( \frac{\pi}{N + 1} \right)$$

For an open tight-binding chain with $N$ sites, lattice constant $a$, and hopping parameter $t$, let $J$ be the current observable and $\omega_{\text{max}}$ be the maximal current vector state. The absolute value of the expectation value of the current in this state is equal to the maximum current $J_{\text{max}}$: $$|\omega_{\text{max}}\langle J \rangle| = J_{\text{max}}$$ where the maximum current (the "speed limit") is defined as $J_{\text{max}} = 2 |at| \cos \left( \frac{\pi}{N + 1} \right)$.