Phase-space quantum distorted stability pattern for Aubry-André-Harper dynamics

TL;DR

This work develops an extended Weyl-Wigner phase-space framework for nonlinear Hamiltonians $H^{W}(q,p)=K(p)+V(q)$ and applies it to Aubry-André-Harper dynamics to quantify quantum distortions via Wigner currents. By mapping the AAH problem to a dimensionless Hamiltonian $\,\mathcal{H}_{AAH}(x,k)= w k + \cos(k) + a^2(w x + \cos(x))$, the authors derive analytic Wigner currents for Gaussian ensembles, unveiling how quantum fluctuations (parametrized by $\alpha$) modify phase-space flows and the equilibrium structure. Hyperbolic-equilibrium analysis, using the Jacobian of the quantum-corrected flow, yields leading-order stability criteria ${\rm Det}[j] \approx a^2(1-w^2-\alpha^2/6)$ and ${\rm Tr}[j] \approx (\alpha^4/3)(a^2-1) w \arcsin(w)$, showing that quantum effects can shift equilibria and alter stability while preserving hyperbolic behavior at small $\alpha$. The framework thus provides a robust, nonperturbative method to distinguish quantum fluctuations from nonlinear dynamics in quasiperiodic systems and can be extended to other quasicrystal models and experimental platforms.

Abstract

Instability features associated to topological quantum domains which emerge from the Weyl-Wigner (WW) quantum phase-space description of Gaussian ensembles driven by Aubry-André-Harper (AAH) Hamiltonians are investigated. Hyperbolic equilibrium and stability patterns are then identified and classified according to the associated (nonlinear) AAH Hamiltonian parameters. Besides providing the tools for quantifying the information content of AAH systems, the Wigner flow patterns here discussed suggest a systematic procedure for identifying the role of quantum fluctuations over equilibrium and stability, in a framework which can be straightforwardly extended to describe the evolution of similar/modified AAH systems.

Figures (5)

• Figure 1: (Color online) Classical portrait of AAH Hamiltonians. Phase-space trajectories and corresponding lattice designs are for $\hbox{Max}\{a^2-1,0\} < \vert\epsilon\vert < a^2+1$ corresponding to closed trajectories for $\epsilon > 0$ (black dashed lines) and for $\epsilon < 0$ (red thin lines), and for $0 < \vert\epsilon\vert < a^2-1$ corresponding to opened trajectories (blue thick lines), when they exist. All parameters are concerned with the original Harper pattern ($w=0$ (first row)). The limiar (opened-closed) value is given by $\vert\epsilon\vert = a^2 -1$. The plots are for $a^2=2$ (first column), with $\vert\epsilon\vert = 5/2,\, 2,\,3/2,\,\dots,\,0$, $a^2=1$ (second column), with $\vert\epsilon\vert = 5/2,\,2,\,3/2,\,\dots,\,0$ and $a^2=1/2$ (third column), with $\vert\epsilon\vert = 5/2,\, 2,\,3/2,\,\dots,\,0$. One also has $w=0$ (first row), $w=0.1$ (second row) and $w=0.4$ (third row), which have been chosen for comparative reasons.
• Figure 2: (Color online) First column: Features of the Wigner flow for the Gaussian ensemble, in the $x - k$ plane. At $\tau = 0$, Gaussian ensembles do not exhibit neither vortices nor stagnation points, in a kind of camouflage of the quantum distortions. The stationarity quantifier, $\hbox{\boldmath $\nabla$}_{\xi} \cdot \hbox{\boldmath $\mathcal{J}$}^{\alpha}$, is described according to the background color scheme. The results are for the increasing spreading characteristic of the Gaussian function, from $\alpha =1/2$ (first row), $1/4$ (second row) and $1/8$ (third row). Peaked Gaussian distributions ($\alpha \gtrsim 1$) localizes the quantum distortions which result into non-stationarity. The parameter $w=0.4$ was arbitrarily chosen. Second column: Liouvillian quantifier, $\omega^{-1}\hbox{\boldmath $\nabla$}_{\xi} \cdot \mathbf{w}$, depicted through the background color scheme, from darker regions, $\hbox{\boldmath $\nabla$}_{\xi} \cdot \mathbf{w} \sim 0$, to lighter regions, $\hbox{\boldmath $\nabla$}_{\xi} \cdot \mathbf{w} > 0$. In both columns, classical pattern is shown as a collection of black lines. The stationarity quantifier, $\hbox{\boldmath $\nabla$}_{\xi} \cdot \hbox{\boldmath $\mathcal{J}$}^{\alpha}$, is described according to the background color scheme, from which lighter regions correspond to non-vanishing local contributions to $\partial_t \mathcal{G}_{\alpha}(x,\,k)$.
• Figure 3: (Color online) Region plot scheme for the phase-space evolution of quantum critical points corresponding to quasi-stable (blue regions) and unstable (red regions) equilibrium points in terms of the Gaussian spreading $\alpha$. Results are for the Wigner flow with the equilibrium point (flux) surrounding envelop described by $\vert \vert\mathbf{w}\vert < 0.07$. For peaked Gaussians, $\alpha \gtrsim 3$, local effects compensate each other when sliced views of the Wigner flux for fixed $\alpha$ are considered, i.e. either when two vortices of opposite winding numbers match each other or when saddle points mutually annihilates one each other. The spreading behavior of the Gaussian ensemble, from red bubble (unstable) islands to the blue (stable) envelop, corresponding to decreasing values of $\alpha$, diffusively recovers the classical-like pattern for which the quantum imprint is just to the small displacement of the ( quasi-)stable equilibrium point. The results are for $w=4$ and $a=1.2$ (first row), $a=1$ (second row) and $a=0.8$ (third row). The portraits are the same for different angle views (columns).
• Figure 4: (Color online) First column: Classical periodic (dashed lines) and quantum quasiperiodic (solid lines) dynamics, $x(\tau)$ (red lines) and $k(\tau)$, for typical spreading Gaussian ensembles, with $\alpha = 1/2$, $a = 1/2,\,1$ and $0.8$ and $w=0.4$. Second column: Corresponding phase-space ( quasi-)stable and quasiperiodic trajectories for classical (dashed lines) and quantum (solid lines) patterns. The color scheme describes the quantum quasi-stable evolution from $\tau = 0$ (blue tone) to $\tau \gg 0$ (red tone). .
• Figure 5: (Color online) Hyperbolic equilibrium and stability for the AAH system, obtained from Eqs. (\ref{['imWA4CCD4mm']})-(\ref{['imWABCCD4mm']}) as function of the $\alpha(w)$ and the anisotropy parameter $a$.