A Lean 4 formalization accompanying the paper's rigorous results is released alongside the numerical code in the author's GitHub repository.
Abstract
We construct quantum Turing patterns in Lindblad lattice dynamics and establish a rigorous theory of their nonlinear order and quantum fluctuations. For an explicit family of completely positive lattice generators with finite-range couplings, the first-moment equations undergo a supercritical instability at a nonzero wave number and admit analytic site- and bond-centered commensurate stripe branches. These branches are locally asymptotically stable in their reflection-fixed period-cell spaces, and projected coherent states exhibit extensive Bragg order on every bounded time interval in the semiclassical limit. We prove $O(N^{-1/2})$ convergence of microscopic covariances to a nonautonomous Gaussian Lyapunov flow, transferring strict partial-transpose uncertainty violations to sufficiently large $N$. In the homogeneous Gaussian sector, a single dimensionless ratio controls both the Turing stability determinant and the logarithmic negativity of opposite momenta, relating wavelength selection directly to quantum entanglement. Differential transport shifts the strongest opposite-momentum correlations from the infrared to the selected Turing scale. Numerical continuation and two-dimensional simulations display stripe, spot, and labyrinth morphologies whose Fourier modes and fluctuation spectra concentrate at the same selected wave numbers.
Problem
Whether Turing pattern selection at a nonzero wave number can emerge from quantum Markov (Lindblad) lattice dynamics. A further question is how the selected wavelength shows up in quantum fluctuations and entanglement.
Approach
The author builds an explicit family of completely positive Lindblad lattice generators with finite-range couplings and differential transport. The first-moment equations are analyzed by spectral design and Lyapunov–Schmidt reduction to obtain supercritical commensurate stripe branches. Semiclassical coherent-state estimates give covariance convergence to a Gaussian Lyapunov flow. A Lean 4 formalization is provided with the numerical code; the paper text does not say which results it covers.
Figure 2. Finite wave-number Turing point. (a) The critical eigenvalue crosses zero at k_{*}=\pi/6 while the homogeneous mode remains stable; the shaded interval is the unstable band at \lambda=0.4 . (b,c) The computed determinant agrees with the exact factorization and has a quadratic zero at k_{*} . (d) The critical eigenvalue crosses with slope 1/4 .
Results
The paper proves analytic, locally stable site- and bond-centered stripe branches with extensive Bragg order, and O(N^{-1/2}) covariance convergence that carries partial-transpose uncertainty violations to large N. A single dimensionless ratio governs both the Turing stability determinant and the opposite-momentum logarithmic negativity. Two-dimensional simulations show stripe, spot, and labyrinth morphologies whose Fourier power and fluctuation spectra concentrate at the selected wave numbers.
Figure 1. Quantum Turing morphologies and selected wave numbers in one Lindblad family. The top row shows the first-moment fields, the middle row their two-dimensional Fourier power, and the bottom row the selected spectra together with the linear unstable bands. The stripe selects the pair \pm k_{*} , while the isotropic spot and labyrinth regimes concentrate power on the selected wave-number sheFigure 4. Momentum-resolved Gaussian entanglement and phase locking. (a) Longitudinal opposite-momentum spectra for the nonlinear stripe, the homogeneous differential-transport system, and a scalar-transport control; the detail panel resolves the neighborhood of k_{*} . (b) Critical-pair logarithmic negativity across the homogeneous threshold and along the continued patterned branch. (c) Phase-loc
