Quantum Backflow in Tight-Binding Systems

• Quantum backflow in tight-binding systems is a non-classical effect where a quantum particle with only positive momentum exhibits a negative probability current on a discrete lattice.
• The phenomenon is rigorously analyzed using tight-binding Hamiltonians, plane-wave eigenbases, and optimization via rank-2 eigenvalue problems over positive-momentum superpositions.
• Varying lattice parameters, such as bias and system size, reveals tunable backflow bounds that can exceed continuum limits, offering promising avenues for experimental verification.

Quantum backflow in tight-binding systems refers to the non-classical phenomenon wherein a quantum particle, represented as a superposition of positive-momentum states on a discrete lattice, exhibits a probability current flowing against its nominal momentum direction. This effect persists in tight-binding models with complex (time-reversal-breaking) hopping amplitudes, under both open and periodic boundary conditions, and can be rigorously quantified both instantaneously and as an integrated effect over time, offering a discrete counterpart to continuum backflow phenomena (Arvizu et al., 21 Nov 2025).

1. Tight-Binding Model Hamiltonians

Quantum backflow in lattices is studied in the context of tight-binding Hamiltonians defined on one-dimensional discrete chains. For the infinite (open) chain, site indices $j\in\mathbb{Z}$ label localized orbital basis states $|j\rangle$. The Hamiltonian is given by

$$H = -\tau[(1 + i\epsilon) S + (1 - i\epsilon) S^\dagger],$$

where $\tau > 0$ is the hopping strength, $\epsilon \in \mathbb{R}$ parameterizes the complex bias, $S|j\rangle = |j+1\rangle$, and $S^\dagger|j\rangle = |j-1\rangle$. The anti-Hermitian hopping breaks time-reversal symmetry, while $H$ remains Hermitian. For a periodic chain of length $N$, the same $H$ applies with $|j+N\rangle=|j\rangle$ and $S^N = \openone$.

2. Plane-Wave Eigenbasis and Momentum Structure

The tight-binding Hamiltonian's eigenfunctions are plane waves. For the open chain,

$\psi_k(j) = \frac{1}{\sqrt{2\pi}} e^{ikj},$

while for the periodic ring,

$$\psi_n(j) = \frac{1}{\sqrt{N}} e^{i \frac{2\pi n}{N} j},$$

with energy dispersion

$$E_k = -2\tau \sqrt{1+\epsilon^2} \cos(k + \xi), \quad \xi = \arctan \epsilon.$$

The discrete momentum operator derived via Heisenberg's equation defines "positive-momentum" modes as those with $\sin(k + \xi) > 0$, corresponding to $k \in [-\xi,\,\pi - \xi]$ (mod $2\pi$).

3. Lattice Current and the Backflow Criterion

Probability current in the lattice, from site $j-1$ to $j$, is

$$J(j,t) = \frac{\tau}{i\hbar} \left[ \Psi^*(j-1)\Psi(j) - \Psi(j-1)\Psi^*(j) + i\epsilon \left( \Psi^*(j-1)\Psi(j) + \Psi(j-1)\Psi^*(j) \right) \right].$$

The continuity equation for probability density directly follows. Quantum backflow is defined by the appearance of $J(j,t) < 0$ in a state composed solely of positive-momentum eigenmodes. This negative current, without negative momentum components, constitutes the discrete analogue of the quantum backflow phenomenon previously established in the continuum.

4. Optimization of Instantaneous Backflow

The maximal instantaneous backflow is obtained via extremal superpositions of positive-momentum eigenstates. For the periodic chain, considering two such modes:

$$\Psi(j,t) = \frac{1}{\sqrt{N}} \left[ \cos \frac{\theta}{2} e^{i(k_{m_1} j - E_{m_1} t)/\hbar} + \sin \frac{\theta}{2} e^{i(k_{m_2} j - E_{m_2} t)/\hbar + \gamma} \right],$$

minimization over the phase difference $\gamma$ and mixing angle $\theta$ yields the most negative $J$. In the general case for the infinite chain, the optimization reduces to a rank-2 Fredholm eigenvalue problem over the momentum distribution $\phi(k)$. The extremal eigenvalues are:

$$\lambda_\pm = \frac{(2 \pm \pi)\tau \sqrt{1+\epsilon^2}}{2\hbar\pi}.$$

Thus, the most negative current at a given site and time is $J_{\min} = \lambda_-$, and the most positive is $J_{\max} = \lambda_+$. Explicitly, at $\epsilon = 0$ and $\tau/\hbar=1$, $J_{\min} \approx -0.1817$, $J_{\max} \approx +0.7273$.

5. Total Backflow and Bracken–Melloy-Type Bounds

Total backflow quantifies the maximum probability transported against momentum over a finite time window. The backwards crossing probability is:

$\Delta P = -\int_{-T/2}^{T/2} J(1, t)\,dt,$

maximized over all positive-momentum initial states. For the infinite chain, this leads to a time-windowed integral eigenvalue problem for a kernel $K_T(k, k')$. The asymptotic bound, for large $\nu = \tau T/\hbar$, approaches

$$\lim_{\nu \to \infty} \lambda_p(\nu) \approx 0.07647 \sqrt{1+\epsilon^2},$$

approximately twice the corresponding continuum Bracken–Melloy value $c_{\rm BM} \approx 0.03845$.

For the periodic chain, the problem becomes a finite matrix eigenvalue computation. For large $N$, results converge to those of the infinite chain. Notably, for small $N$, the total backflow bound can exceed the corresponding continuum-ring limit ($c_{\rm ring}^{\rm(cont)} \approx 0.1168$), peaking at $N = 5$ with $\max\lambda_p \approx 0.13135$, about 12% above the continuum value.

6. Dependence on System Parameters

The amplitude and bounds of quantum backflow exhibit characteristic dependences on the lattice parameters:

• Bias ($\epsilon$): All backflow bounds scale as $\sqrt{1+\epsilon^2}$; thus, increasing the bias enhances both instantaneous and total backflow, and in periodic chains also amplifies oscillations in $J(t)$ away from the optimal point.
• Lattice size ($N$): Instantaneous backflow bounds in periodic chains converge to the infinite-chain limit as $N$ increases. For total backflow in rings, the discrete system can exceed the continuum bound at small $N$ and converges toward it from above as $N$ increases, with the maximum excess decaying approximately as $N^{-1.7}$ and the optimal $\nu$ scaling as $N^2$.
• Boundary conditions: The qualitative backflow phenomenon persists under both open and periodic boundary conditions; the spectral features shift from continuous $k$ in the infinite case to discrete $k_n = 2\pi n / N$ on a ring, with convergence as $N\to\infty$.

7. Numerical and Analytical Results

Direct computation and analytical formulas yield:

• Instantaneous current extremal values $J_{\min}$ and $J_{\max}$ agree with predictions for both chain types.
• As bias $\epsilon$ increases, the extrema of current deepen and rise proportionally to $\sqrt{1+\epsilon^2}$.
• In the infinite chain, the backflow eigenvalue $\lambda_p(\nu)$ as a function of $\nu$ peaks at a small value, then asymptotes, encapsulating the time-integrated backflow capacity. For periodic chains, $\lambda_p(\nu)$ for $N=10,\,100,\,1000$ demonstrates convergence to the continuous-ring bound; small $N$ chains temporarily exceed this asymptote by up to 12%.
• Key plots, such as the decay of $\max\lambda_p - c_{\rm ring}^{\rm(cont)}$ as $N^{-1.7}$ and the scaling of optimal $\nu$ as $N^2$, confirm the analytic scaling relations.

In summary, the tight-binding lattice provides a platform for robust quantum backflow effects analogous to continuum models, with analytic accessibility and tunable parameters (lattice bias, boundary conditions, finite size) that can be tuned to optimize or exceed known continuum backflow bounds. This identifies discrete lattices as promising candidates for experimental observation and detailed study of quantum backflow in condensed-matter or quantum simulation contexts (Arvizu et al., 21 Nov 2025).